%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP246^3 : 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 : n002.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 09:10:37 AM UTC 2026
% Result : Theorem 62.42s 14.85s
% Output : Refutation 62.42s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 5
% Syntax : Number of formulae : 26 ( 19 unt; 0 typ; 0 def)
% Number of atoms : 69 ( 30 equ; 0 cnn)
% Maximal formula atoms : 3 ( 2 avg)
% Number of connectives : 304 ( 17 ~; 13 |; 0 &; 274 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Number of types : 85 ( 84 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1538 (1535 usr; 71 con; 0-8 aty)
% Number of variables : 10 ( 0 ^; 10 !; 0 ?; 10 :)
% 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,
num: $tType ).
thf(type_def_11,type,
product_prod_int_int: $tType ).
thf(type_def_12,type,
product_prod_nat_nat: $tType ).
thf(type_def_13,type,
complex: $tType ).
thf(type_def_14,type,
option_num: $tType ).
thf(type_def_15,type,
product_prod_num_num: $tType ).
thf(type_def_16,type,
code_integer: $tType ).
thf(type_def_17,type,
produc6271795597528267376eger_o: $tType ).
thf(type_def_18,type,
produc8923325533196201883nteger: $tType ).
thf(type_def_19,type,
set_Extended_enat: $tType ).
thf(type_def_20,type,
extended_enat: $tType ).
thf(type_def_21,type,
set_fi4554929511873752355omplex: $tType ).
thf(type_def_22,type,
filter6041513312241820739omplex: $tType ).
thf(type_def_23,type,
set_fi7789364187291644575l_real: $tType ).
thf(type_def_24,type,
filter2146258269922977983l_real: $tType ).
thf(type_def_25,type,
set_real: $tType ).
thf(type_def_26,type,
set_set_nat: $tType ).
thf(type_def_27,type,
set_nat: $tType ).
thf(type_def_28,type,
filter_real: $tType ).
thf(type_def_29,type,
filter_nat: $tType ).
thf(type_def_30,type,
filter1242075044329608583at_nat: $tType ).
thf(type_def_31,type,
set_Pr5085853215250843933omplex: $tType ).
thf(type_def_32,type,
set_Pr6218003697084177305l_real: $tType ).
thf(type_def_33,type,
set_o: $tType ).
thf(type_def_34,type,
set_complex: $tType ).
thf(type_def_35,type,
set_int: $tType ).
thf(type_def_36,type,
set_list_nat: $tType ).
thf(type_def_37,type,
set_Product_unit: $tType ).
thf(type_def_38,type,
set_char: $tType ).
thf(type_def_39,type,
char: $tType ).
thf(type_def_40,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(type_def_41,type,
list_nat: $tType ).
thf(type_def_42,type,
literal: $tType ).
thf(type_def_43,type,
list_o: $tType ).
thf(type_def_44,type,
list_int: $tType ).
thf(type_def_45,type,
option_nat: $tType ).
thf(type_def_46,type,
vEBT_VEBT: $tType ).
thf(type_def_47,type,
set_Pr958786334691620121nt_int: $tType ).
thf(type_def_48,type,
set_Code_integer: $tType ).
thf(type_def_49,type,
set_rat: $tType ).
thf(type_def_50,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(type_def_51,type,
list_VEBT_VEBT: $tType ).
thf(type_def_52,type,
list_complex: $tType ).
thf(type_def_53,type,
list_Extended_enat: $tType ).
thf(type_def_54,type,
list_list_nat: $tType ).
thf(type_def_55,type,
list_real: $tType ).
thf(type_def_56,type,
set_VEBT_VEBT: $tType ).
thf(type_def_57,type,
list_num: $tType ).
thf(type_def_58,type,
list_P4002435161011370285od_o_o: $tType ).
thf(type_def_59,type,
product_prod_o_o: $tType ).
thf(type_def_60,type,
list_P3795440434834930179_o_int: $tType ).
thf(type_def_61,type,
product_prod_o_int: $tType ).
thf(type_def_62,type,
list_P6285523579766656935_o_nat: $tType ).
thf(type_def_63,type,
product_prod_o_nat: $tType ).
thf(type_def_64,type,
list_P7495141550334521929T_VEBT: $tType ).
thf(type_def_65,type,
produc2504756804600209347T_VEBT: $tType ).
thf(type_def_66,type,
list_P1726324292696863441at_num: $tType ).
thf(type_def_67,type,
product_prod_nat_num: $tType ).
thf(type_def_68,type,
list_P3744719386663036955um_num: $tType ).
thf(type_def_69,type,
list_P3126845725202233233VEBT_o: $tType ).
thf(type_def_70,type,
produc334124729049499915VEBT_o: $tType ).
thf(type_def_71,type,
list_P4547456442757143711BT_int: $tType ).
thf(type_def_72,type,
produc4894624898956917775BT_int: $tType ).
thf(type_def_73,type,
list_P7037539587688870467BT_nat: $tType ).
thf(type_def_74,type,
produc9072475918466114483BT_nat: $tType ).
thf(type_def_75,type,
list_P7413028617227757229T_VEBT: $tType ).
thf(type_def_76,type,
produc8243902056947475879T_VEBT: $tType ).
thf(type_def_77,type,
list_P7333126701944960589_nat_o: $tType ).
thf(type_def_78,type,
list_P5647936690300460905T_VEBT: $tType ).
thf(type_def_79,type,
option4927543243414619207at_nat: $tType ).
thf(type_def_80,type,
set_num: $tType ).
thf(type_def_81,type,
produc7248412053542808358at_nat: $tType ).
thf(type_def_82,type,
produc4471711990508489141at_nat: $tType ).
thf(type_def_83,type,
produc2963631642982155120at_num: $tType ).
thf(type_def_84,type,
produc3368934014287244435at_num: $tType ).
thf(type_def_85,type,
produc859450856879609959at_nat: $tType ).
thf(type_def_86,type,
produc7272778201969148633d_enat: $tType ).
thf(type_def_87,type,
produc4411394909380815293omplex: $tType ).
thf(type_def_88,type,
produc2422161461964618553l_real: $tType ).
thf(type_def_89,type,
product_unit: $tType ).
thf(func_def_0,type,
produc3368934014287244435at_num: $tType ).
thf(func_def_1,type,
produc4471711990508489141at_nat: $tType ).
thf(func_def_2,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(func_def_3,type,
produc859450856879609959at_nat: $tType ).
thf(func_def_4,type,
set_fi4554929511873752355omplex: $tType ).
thf(func_def_5,type,
list_P7413028617227757229T_VEBT: $tType ).
thf(func_def_6,type,
produc2963631642982155120at_num: $tType ).
thf(func_def_7,type,
produc7248412053542808358at_nat: $tType ).
thf(func_def_8,type,
set_fi7789364187291644575l_real: $tType ).
thf(func_def_9,type,
filter6041513312241820739omplex: $tType ).
thf(func_def_10,type,
list_P7037539587688870467BT_nat: $tType ).
thf(func_def_11,type,
list_P4547456442757143711BT_int: $tType ).
thf(func_def_12,type,
list_P5647936690300460905T_VEBT: $tType ).
thf(func_def_13,type,
produc8243902056947475879T_VEBT: $tType ).
thf(func_def_14,type,
set_Pr5085853215250843933omplex: $tType ).
thf(func_def_15,type,
produc8923325533196201883nteger: $tType ).
thf(func_def_16,type,
produc7272778201969148633d_enat: $tType ).
thf(func_def_17,type,
list_P3126845725202233233VEBT_o: $tType ).
thf(func_def_18,type,
list_P7495141550334521929T_VEBT: $tType ).
thf(func_def_19,type,
filter2146258269922977983l_real: $tType ).
thf(func_def_20,type,
option4927543243414619207at_nat: $tType ).
thf(func_def_21,type,
filter1242075044329608583at_nat: $tType ).
thf(func_def_22,type,
set_Pr6218003697084177305l_real: $tType ).
thf(func_def_23,type,
list_P3744719386663036955um_num: $tType ).
thf(func_def_24,type,
list_P1726324292696863441at_num: $tType ).
thf(func_def_25,type,
produc9072475918466114483BT_nat: $tType ).
thf(func_def_26,type,
produc4894624898956917775BT_int: $tType ).
thf(func_def_27,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(func_def_28,type,
set_Pr958786334691620121nt_int: $tType ).
thf(func_def_29,type,
produc4411394909380815293omplex: $tType ).
thf(func_def_30,type,
list_P7333126701944960589_nat_o: $tType ).
thf(func_def_31,type,
list_P6285523579766656935_o_nat: $tType ).
thf(func_def_32,type,
list_P3795440434834930179_o_int: $tType ).
thf(func_def_33,type,
produc334124729049499915VEBT_o: $tType ).
thf(func_def_34,type,
produc2504756804600209347T_VEBT: $tType ).
thf(func_def_35,type,
produc6271795597528267376eger_o: $tType ).
thf(func_def_36,type,
produc2422161461964618553l_real: $tType ).
thf(func_def_37,type,
product_prod_num_num: $tType ).
thf(func_def_38,type,
product_prod_nat_num: $tType ).
thf(func_def_39,type,
product_prod_nat_nat: $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,
list_list_nat: $tType ).
thf(func_def_43,type,
list_VEBT_VEBT: $tType ).
thf(func_def_44,type,
set_list_nat: $tType ).
thf(func_def_45,type,
product_prod_o_nat: $tType ).
thf(func_def_46,type,
product_prod_o_int: $tType ).
thf(func_def_47,type,
set_VEBT_VEBT: $tType ).
thf(func_def_48,type,
set_set_nat: $tType ).
thf(func_def_49,type,
set_Code_integer: $tType ).
thf(func_def_50,type,
list_Extended_enat: $tType ).
thf(func_def_51,type,
set_Product_unit: $tType ).
thf(func_def_52,type,
set_Extended_enat: $tType ).
thf(func_def_53,type,
list_complex: $tType ).
thf(func_def_54,type,
product_prod_o_o: $tType ).
thf(func_def_55,type,
set_complex: $tType ).
thf(func_def_56,type,
filter_real: $tType ).
thf(func_def_57,type,
option_num: $tType ).
thf(func_def_58,type,
option_nat: $tType ).
thf(func_def_59,type,
filter_nat: $tType ).
thf(func_def_60,type,
set_char: $tType ).
thf(func_def_61,type,
list_real: $tType ).
thf(func_def_62,type,
set_real: $tType ).
thf(func_def_63,type,
list_num: $tType ).
thf(func_def_64,type,
list_nat: $tType ).
thf(func_def_65,type,
list_int: $tType ).
thf(func_def_66,type,
vEBT_VEBT: $tType ).
thf(func_def_67,type,
set_rat: $tType ).
thf(func_def_68,type,
set_num: $tType ).
thf(func_def_69,type,
set_nat: $tType ).
thf(func_def_70,type,
set_int: $tType ).
thf(func_def_71,type,
code_integer: $tType ).
thf(func_def_72,type,
product_unit: $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,
literal: $tType ).
thf(func_def_77,type,
set_o: $tType ).
thf(func_def_78,type,
char: $tType ).
thf(func_def_79,type,
real: $tType ).
thf(func_def_80,type,
rat: $tType ).
thf(func_def_81,type,
num: $tType ).
thf(func_def_82,type,
nat: $tType ).
thf(func_def_83,type,
int: $tType ).
thf(func_def_84,type,
archim2889992004027027881ng_rat: rat > int ).
thf(func_def_85,type,
archim7802044766580827645g_real: real > int ).
thf(func_def_86,type,
archim3151403230148437115or_rat: rat > int ).
thf(func_def_87,type,
archim6058952711729229775r_real: real > int ).
thf(func_def_88,type,
archimedean_frac_rat: rat > rat ).
thf(func_def_89,type,
archim2898591450579166408c_real: real > real ).
thf(func_def_90,type,
archim7778729529865785530nd_rat: rat > int ).
thf(func_def_91,type,
archim8280529875227126926d_real: real > int ).
thf(func_def_92,type,
bNF_re1962705104956426057at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > $o ).
thf(func_def_93,type,
bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
thf(func_def_94,type,
bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
thf(func_def_95,type,
bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
thf(func_def_96,type,
bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
thf(func_def_97,type,
bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
thf(func_def_98,type,
bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
thf(func_def_99,type,
bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(func_def_100,type,
bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
thf(func_def_101,type,
bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_102,type,
bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
thf(func_def_103,type,
bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_104,type,
bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(func_def_105,type,
bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_106,type,
bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
thf(func_def_107,type,
bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
thf(func_def_108,type,
bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
thf(func_def_109,type,
bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
thf(func_def_110,type,
bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
thf(func_def_111,type,
bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
thf(func_def_112,type,
bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
thf(func_def_113,type,
bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
thf(func_def_114,type,
bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
thf(func_def_115,type,
bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
thf(func_def_116,type,
bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
thf(func_def_117,type,
bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
thf(func_def_118,type,
bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
thf(func_def_119,type,
bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
thf(func_def_120,type,
bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
thf(func_def_121,type,
bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
thf(func_def_122,type,
bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(func_def_123,type,
bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).
thf(func_def_124,type,
bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
thf(func_def_125,type,
binomial: nat > nat > nat ).
thf(func_def_126,type,
gbinomial_complex: complex > nat > complex ).
thf(func_def_127,type,
gbinomial_int: int > nat > int ).
thf(func_def_128,type,
gbinomial_nat: nat > nat > nat ).
thf(func_def_129,type,
gbinomial_rat: rat > nat > rat ).
thf(func_def_130,type,
gbinomial_real: real > nat > real ).
thf(func_def_131,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_132,type,
bit_and_not_num: num > num > option_num ).
thf(func_def_133,type,
bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_134,type,
bit_concat_bit: nat > int > int > int ).
thf(func_def_135,type,
bit_or_not_num_neg: num > num > num ).
thf(func_def_136,type,
bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_137,type,
bit_ri7632146776885996613nteger: code_integer > code_integer ).
thf(func_def_138,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(func_def_139,type,
bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
thf(func_def_140,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(func_def_141,type,
bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).
thf(func_def_142,type,
bit_se725231765392027082nd_int: int > int > int ).
thf(func_def_143,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(func_def_144,type,
bit_se8568078237143864401it_int: nat > int > int ).
thf(func_def_145,type,
bit_se8570568707652914677it_nat: nat > nat > nat ).
thf(func_def_146,type,
bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
thf(func_def_147,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(func_def_148,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(func_def_149,type,
bit_se2119862282449309892nteger: nat > code_integer ).
thf(func_def_150,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(func_def_151,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(func_def_152,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(func_def_153,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(func_def_154,type,
bit_se7788150548672797655nteger: nat > code_integer > code_integer ).
thf(func_def_155,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(func_def_156,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(func_def_157,type,
bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
thf(func_def_158,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(func_def_159,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(func_def_160,type,
bit_se1745604003318907178nteger: nat > code_integer > code_integer ).
thf(func_def_161,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(func_def_162,type,
bit_se2925701944663578781it_nat: nat > nat > nat ).
thf(func_def_163,type,
bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
thf(func_def_164,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(func_def_165,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(func_def_166,type,
bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).
thf(func_def_167,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(func_def_168,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(func_def_169,type,
bit_se9216721137139052372nteger: code_integer > nat > $o ).
thf(func_def_170,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(func_def_171,type,
bit_se1148574629649215175it_nat: nat > nat > $o ).
thf(func_def_172,type,
bit_take_bit_num: nat > num > option_num ).
thf(func_def_173,type,
bit_un1837492267222099188nd_num: num > num > option_num ).
thf(func_def_174,type,
bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_175,type,
bit_un6178654185764691216or_num: num > num > option_num ).
thf(func_def_176,type,
bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_177,type,
bit_un7362597486090784418nd_num: num > num > option_num ).
thf(func_def_178,type,
bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_179,type,
bit_un2480387367778600638or_num: num > num > option_num ).
thf(func_def_180,type,
bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_181,type,
code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
thf(func_def_182,type,
code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_183,type,
code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_184,type,
code_int_of_integer: code_integer > int ).
thf(func_def_185,type,
code_integer_of_int: int > code_integer ).
thf(func_def_186,type,
code_integer_of_num: num > code_integer ).
thf(func_def_187,type,
code_nat_of_integer: code_integer > nat ).
thf(func_def_188,type,
code_num_of_integer: code_integer > num ).
thf(func_def_189,type,
comple2295165028678016749d_enat: set_Extended_enat > extended_enat ).
thf(func_def_190,type,
comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
thf(func_def_191,type,
comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
thf(func_def_192,type,
comple4887499456419720421f_real: set_real > real ).
thf(func_def_193,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(func_def_194,type,
comple4398354569131411667d_enat: set_Extended_enat > extended_enat ).
thf(func_def_195,type,
comple1385675409528146559p_real: set_real > real ).
thf(func_def_196,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(func_def_197,type,
arg: complex > real ).
thf(func_def_198,type,
cis: real > complex ).
thf(func_def_199,type,
cnj: complex > complex ).
thf(func_def_200,type,
complex2: real > real > complex ).
thf(func_def_201,type,
im: complex > real ).
thf(func_def_202,type,
re: complex > real ).
thf(func_def_203,type,
csqrt: complex > complex ).
thf(func_def_204,type,
imaginary_unit: complex ).
thf(func_def_205,type,
rcis: real > real > complex ).
thf(func_def_206,type,
differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
thf(func_def_207,type,
has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
thf(func_def_208,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(func_def_209,type,
has_ve631408500373753343e_real: ( real > real ) > real > filter_real > $o ).
thf(func_def_210,type,
adjust_div: product_prod_int_int > int ).
thf(func_def_211,type,
adjust_mod: int > int > int ).
thf(func_def_212,type,
divmod_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_213,type,
eucl_rel_int: int > int > product_prod_int_int > $o ).
thf(func_def_214,type,
unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).
thf(func_def_215,type,
unique6319869463603278526ux_int: product_prod_int_int > $o ).
thf(func_def_216,type,
unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
thf(func_def_217,type,
unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
thf(func_def_218,type,
unique5052692396658037445od_int: num > num > product_prod_int_int ).
thf(func_def_219,type,
unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
thf(func_def_220,type,
unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_221,type,
unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
thf(func_def_222,type,
unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_223,type,
euclid4774559944035922753ze_int: int > nat ).
thf(func_def_224,type,
euclid4777050414544973029ze_nat: nat > nat ).
thf(func_def_225,type,
euclid3395696857347342551nt_int: int > int ).
thf(func_def_226,type,
euclid3398187327856392827nt_nat: nat > nat ).
thf(func_def_227,type,
extended_eSuc: extended_enat > extended_enat ).
thf(func_def_228,type,
extended_enat2: nat > extended_enat ).
thf(func_def_229,type,
extended_case_enat_o: ( nat > $o ) > $o > extended_enat > $o ).
thf(func_def_230,type,
extend3600170679010898289d_enat: ( nat > extended_enat ) > extended_enat > extended_enat > extended_enat ).
thf(func_def_231,type,
extend5688581933313929465d_enat: extended_enat ).
thf(func_def_232,type,
extended_the_enat: extended_enat > nat ).
thf(func_def_233,type,
comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
thf(func_def_234,type,
comm_s2602460028002588243omplex: complex > nat > complex ).
thf(func_def_235,type,
comm_s4660882817536571857er_int: int > nat > int ).
thf(func_def_236,type,
comm_s4663373288045622133er_nat: nat > nat > nat ).
thf(func_def_237,type,
comm_s4028243227959126397er_rat: rat > nat > rat ).
thf(func_def_238,type,
comm_s7457072308508201937r_real: real > nat > real ).
thf(func_def_239,type,
semiri3624122377584611663nteger: nat > code_integer ).
thf(func_def_240,type,
semiri5044797733671781792omplex: nat > complex ).
thf(func_def_241,type,
semiri1406184849735516958ct_int: nat > int ).
thf(func_def_242,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(func_def_243,type,
semiri773545260158071498ct_rat: nat > rat ).
thf(func_def_244,type,
semiri2265585572941072030t_real: nat > real ).
thf(func_def_245,type,
invers8013647133539491842omplex: complex > complex ).
thf(func_def_246,type,
inverse_inverse_rat: rat > rat ).
thf(func_def_247,type,
inverse_inverse_real: real > real ).
thf(func_def_248,type,
at_bot_real: filter_real ).
thf(func_def_249,type,
at_top_nat: filter_nat ).
thf(func_def_250,type,
at_top_real: filter_real ).
thf(func_def_251,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(func_def_252,type,
eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).
thf(func_def_253,type,
eventually_real: ( real > $o ) > filter_real > $o ).
thf(func_def_254,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(func_def_255,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(func_def_256,type,
filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
thf(func_def_257,type,
filtermap_real_real: ( real > real ) > filter_real > filter_real ).
thf(func_def_258,type,
princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
thf(func_def_259,type,
princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
thf(func_def_260,type,
prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).
thf(func_def_261,type,
finite_card_o: set_o > nat ).
thf(func_def_262,type,
finite_card_complex: set_complex > nat ).
thf(func_def_263,type,
finite_card_int: set_int > nat ).
thf(func_def_264,type,
finite_card_list_nat: set_list_nat > nat ).
thf(func_def_265,type,
finite_card_nat: set_nat > nat ).
thf(func_def_266,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(func_def_267,type,
finite_card_char: set_char > nat ).
thf(func_def_268,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(func_def_269,type,
finite4001608067531595151d_enat: set_Extended_enat > $o ).
thf(func_def_270,type,
finite_finite_int: set_int > $o ).
thf(func_def_271,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_272,type,
bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
thf(func_def_273,type,
bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
thf(func_def_274,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(func_def_275,type,
comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_276,type,
comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_277,type,
comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
thf(func_def_278,type,
comp_int_real_real: ( int > real ) > ( real > int ) > real > real ).
thf(func_def_279,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(func_def_280,type,
comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
thf(func_def_281,type,
id_o: $o > $o ).
thf(func_def_282,type,
id_nat: nat > nat ).
thf(func_def_283,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_284,type,
inj_on_nat_char: ( nat > char ) > set_nat > $o ).
thf(func_def_285,type,
inj_on_real_real: ( real > real ) > set_real > $o ).
thf(func_def_286,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_287,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_288,type,
map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).
thf(func_def_289,type,
map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).
thf(func_def_290,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_291,type,
map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
thf(func_def_292,type,
map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).
thf(func_def_293,type,
map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
thf(func_def_294,type,
map_fu1532550112467129777l_real: ( real > nat > rat ) > ( ( ( nat > rat ) > nat > rat ) > real > real ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > real > real > real ).
thf(func_def_295,type,
map_fu7146612038024189824t_real: ( real > nat > rat ) > ( ( nat > rat ) > real ) > ( ( nat > rat ) > nat > rat ) > real > real ).
thf(func_def_296,type,
map_fu1856342031159181835at_o_o: ( real > nat > rat ) > ( $o > $o ) > ( ( nat > rat ) > $o ) > real > $o ).
thf(func_def_297,type,
strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_298,type,
the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
thf(func_def_299,type,
fun_pair_leq: set_Pr8693737435421807431at_nat ).
thf(func_def_300,type,
fun_pair_less: set_Pr8693737435421807431at_nat ).
thf(func_def_301,type,
gcd_Gcd_nat: set_nat > nat ).
thf(func_def_302,type,
bezw: nat > nat > product_prod_int_int ).
thf(func_def_303,type,
bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_304,type,
gcd_gcd_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_305,type,
gcd_gcd_int: int > int > int ).
thf(func_def_306,type,
gcd_gcd_nat: nat > nat > nat ).
thf(func_def_307,type,
gcd_lcm_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_308,type,
gcd_lcm_int: int > int > int ).
thf(func_def_309,type,
gcd_lcm_nat: nat > nat > nat ).
thf(func_def_310,type,
gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_311,type,
abs_abs_Code_integer: code_integer > code_integer ).
thf(func_def_312,type,
abs_abs_complex: complex > complex ).
thf(func_def_313,type,
abs_abs_int: int > int ).
thf(func_def_314,type,
abs_abs_rat: rat > rat ).
thf(func_def_315,type,
abs_abs_real: real > real ).
thf(func_def_316,type,
minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).
thf(func_def_317,type,
minus_2020553357622893040enat_o: ( extended_enat > $o ) > ( extended_enat > $o ) > extended_enat > $o ).
thf(func_def_318,type,
minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_319,type,
minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
thf(func_def_320,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_321,type,
minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
thf(func_def_322,type,
minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
thf(func_def_323,type,
minus_minus_complex: complex > complex > complex ).
thf(func_def_324,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_325,type,
minus_minus_int: int > int > int ).
thf(func_def_326,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_327,type,
minus_minus_rat: rat > rat > rat ).
thf(func_def_328,type,
minus_minus_real: real > real > real ).
thf(func_def_329,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(func_def_330,type,
minus_925952699566721837d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).
thf(func_def_331,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(func_def_332,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(func_def_333,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_334,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(func_def_335,type,
one_one_Code_integer: code_integer ).
thf(func_def_336,type,
one_one_complex: complex ).
thf(func_def_337,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(func_def_338,type,
one_one_int: int ).
thf(func_def_339,type,
one_one_nat: nat ).
thf(func_def_340,type,
one_one_rat: rat ).
thf(func_def_341,type,
one_one_real: real ).
thf(func_def_342,type,
plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
thf(func_def_343,type,
plus_plus_complex: complex > complex > complex ).
thf(func_def_344,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_345,type,
plus_plus_int: int > int > int ).
thf(func_def_346,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_347,type,
plus_plus_num: num > num > num ).
thf(func_def_348,type,
plus_plus_rat: rat > rat > rat ).
thf(func_def_349,type,
plus_plus_real: real > real > real ).
thf(func_def_350,type,
plus_plus_literal: literal > literal > literal ).
thf(func_def_351,type,
sgn_sgn_Code_integer: code_integer > code_integer ).
thf(func_def_352,type,
sgn_sgn_complex: complex > complex ).
thf(func_def_353,type,
sgn_sgn_int: int > int ).
thf(func_def_354,type,
sgn_sgn_rat: rat > rat ).
thf(func_def_355,type,
sgn_sgn_real: real > real ).
thf(func_def_356,type,
times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
thf(func_def_357,type,
times_times_complex: complex > complex > complex ).
thf(func_def_358,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_359,type,
times_times_int: int > int > int ).
thf(func_def_360,type,
times_times_nat: nat > nat > nat ).
thf(func_def_361,type,
times_times_num: num > num > num ).
thf(func_def_362,type,
times_times_rat: rat > rat > rat ).
thf(func_def_363,type,
times_times_real: real > real > real ).
thf(func_def_364,type,
uminus1351360451143612070nteger: code_integer > code_integer ).
thf(func_def_365,type,
uminus1482373934393186551omplex: complex > complex ).
thf(func_def_366,type,
uminus_uminus_int: int > int ).
thf(func_def_367,type,
uminus_uminus_rat: rat > rat ).
thf(func_def_368,type,
uminus_uminus_real: real > real ).
thf(func_def_369,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(func_def_370,type,
zero_z3403309356797280102nteger: code_integer ).
thf(func_def_371,type,
zero_zero_complex: complex ).
thf(func_def_372,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(func_def_373,type,
zero_zero_int: int ).
thf(func_def_374,type,
zero_zero_nat: nat ).
thf(func_def_375,type,
zero_zero_rat: rat ).
thf(func_def_376,type,
zero_zero_real: real ).
thf(func_def_377,type,
zero_zero_literal: literal ).
thf(func_def_378,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(func_def_379,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(func_def_380,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_381,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(func_def_382,type,
groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).
thf(func_def_383,type,
groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).
thf(func_def_384,type,
groups766887009212190081x_real: ( complex > real ) > set_complex > real ).
thf(func_def_385,type,
groups4622424608036095791omplex: ( extended_enat > complex ) > set_Extended_enat > complex ).
thf(func_def_386,type,
groups2878480467620962989at_int: ( extended_enat > int ) > set_Extended_enat > int ).
thf(func_def_387,type,
groups2880970938130013265at_nat: ( extended_enat > nat ) > set_Extended_enat > nat ).
thf(func_def_388,type,
groups2245840878043517529at_rat: ( extended_enat > rat ) > set_Extended_enat > rat ).
thf(func_def_389,type,
groups97031904164794029t_real: ( extended_enat > real ) > set_Extended_enat > real ).
thf(func_def_390,type,
groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).
thf(func_def_391,type,
groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
thf(func_def_392,type,
groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).
thf(func_def_393,type,
groups2316167850115554303t_real: ( int > real ) > set_int > real ).
thf(func_def_394,type,
groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).
thf(func_def_395,type,
groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
thf(func_def_396,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_397,type,
groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).
thf(func_def_398,type,
groups129246275422532515t_real: ( nat > real ) > set_nat > real ).
thf(func_def_399,type,
groups713298508707869441omplex: ( real > complex ) > set_real > complex ).
thf(func_def_400,type,
groups4694064378042380927al_int: ( real > int ) > set_real > int ).
thf(func_def_401,type,
groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).
thf(func_def_402,type,
groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).
thf(func_def_403,type,
groups1681761925125756287l_real: ( real > real ) > set_real > real ).
thf(func_def_404,type,
groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
thf(func_def_405,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(func_def_406,type,
the_int: ( int > $o ) > int ).
thf(func_def_407,type,
the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).
thf(func_def_408,type,
the_real: ( real > $o ) > real ).
thf(func_def_409,type,
if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).
thf(func_def_410,type,
if_nat_int_int: $o > ( nat > int > int ) > ( nat > int > int ) > nat > int > int ).
thf(func_def_411,type,
if_nat_nat_nat: $o > ( nat > nat > nat ) > ( nat > nat > nat ) > nat > nat > nat ).
thf(func_def_412,type,
if_nat_rat: $o > ( nat > rat ) > ( nat > rat ) > nat > rat ).
thf(func_def_413,type,
if_Code_integer: $o > code_integer > code_integer > code_integer ).
thf(func_def_414,type,
if_complex: $o > complex > complex > complex ).
thf(func_def_415,type,
if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
thf(func_def_416,type,
if_int: $o > int > int > int ).
thf(func_def_417,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(func_def_418,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(func_def_419,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_420,type,
if_num: $o > num > num > num ).
thf(func_def_421,type,
if_option_nat: $o > option_nat > option_nat > option_nat ).
thf(func_def_422,type,
if_option_num: $o > option_num > option_num > option_num ).
thf(func_def_423,type,
if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
thf(func_def_424,type,
if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_425,type,
if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
thf(func_def_426,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_427,type,
if_rat: $o > rat > rat > rat ).
thf(func_def_428,type,
if_real: $o > real > real > real ).
thf(func_def_429,type,
if_set_int: $o > set_int > set_int > set_int ).
thf(func_def_430,type,
if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_431,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(func_def_432,type,
abs_Integ: product_prod_nat_nat > int ).
thf(func_def_433,type,
rep_Integ: int > product_prod_nat_nat ).
thf(func_def_434,type,
int_ge_less_than: int > set_Pr958786334691620121nt_int ).
thf(func_def_435,type,
int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
thf(func_def_436,type,
intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_437,type,
nat2: int > nat ).
thf(func_def_438,type,
pcr_int: product_prod_nat_nat > int > $o ).
thf(func_def_439,type,
power_int_real: real > int > real ).
thf(func_def_440,type,
ring_11222124179247155820nteger: set_Code_integer ).
thf(func_def_441,type,
ring_1_Ints_complex: set_complex ).
thf(func_def_442,type,
ring_1_Ints_int: set_int ).
thf(func_def_443,type,
ring_1_Ints_rat: set_rat ).
thf(func_def_444,type,
ring_1_Ints_real: set_real ).
thf(func_def_445,type,
ring_18347121197199848620nteger: int > code_integer ).
thf(func_def_446,type,
ring_17405671764205052669omplex: int > complex ).
thf(func_def_447,type,
ring_1_of_int_int: int > int ).
thf(func_def_448,type,
ring_1_of_int_rat: int > rat ).
thf(func_def_449,type,
ring_1_of_int_real: int > real ).
thf(func_def_450,type,
inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_451,type,
inf_inf_nat: nat > nat > nat ).
thf(func_def_452,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_453,type,
sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_454,type,
sup_sup_nat: nat > nat > nat ).
thf(func_def_455,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_456,type,
sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_457,type,
lattic921264341876707157d_enat: set_Extended_enat > extended_enat ).
thf(func_def_458,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(func_def_459,type,
quotie3684837364556693515t_real: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > real ) > ( real > nat > rat ) > ( ( nat > rat ) > real > $o ) > $o ).
thf(func_def_460,type,
bfun_nat_real: ( nat > real ) > filter_nat > $o ).
thf(func_def_461,type,
append_int: list_int > list_int > list_int ).
thf(func_def_462,type,
append_nat: list_nat > list_nat > list_nat ).
thf(func_def_463,type,
drop_nat: nat > list_nat > list_nat ).
thf(func_def_464,type,
last_nat: list_nat > nat ).
thf(func_def_465,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(func_def_466,type,
cons_int: int > list_int > list_int ).
thf(func_def_467,type,
cons_nat: nat > list_nat > list_nat ).
thf(func_def_468,type,
nil_int: list_int ).
thf(func_def_469,type,
nil_nat: list_nat ).
thf(func_def_470,type,
hd_nat: list_nat > nat ).
thf(func_def_471,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(func_def_472,type,
map_VE8901447254227204932T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > list_VEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_473,type,
set_o2: list_o > set_o ).
thf(func_def_474,type,
set_complex2: list_complex > set_complex ).
thf(func_def_475,type,
set_Extended_enat2: list_Extended_enat > set_Extended_enat ).
thf(func_def_476,type,
set_int2: list_int > set_int ).
thf(func_def_477,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(func_def_478,type,
set_nat2: list_nat > set_nat ).
thf(func_def_479,type,
set_real2: list_real > set_real ).
thf(func_def_480,type,
set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
thf(func_def_481,type,
size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
thf(func_def_482,type,
tl_nat: list_nat > list_nat ).
thf(func_def_483,type,
list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_484,type,
nth_o: list_o > nat > $o ).
thf(func_def_485,type,
nth_complex: list_complex > nat > complex ).
thf(func_def_486,type,
nth_Extended_enat: list_Extended_enat > nat > extended_enat ).
thf(func_def_487,type,
nth_int: list_int > nat > int ).
thf(func_def_488,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(func_def_489,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_490,type,
nth_num: list_num > nat > num ).
thf(func_def_491,type,
nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
thf(func_def_492,type,
nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
thf(func_def_493,type,
nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).
thf(func_def_494,type,
nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
thf(func_def_495,type,
nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).
thf(func_def_496,type,
nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).
thf(func_def_497,type,
nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
thf(func_def_498,type,
nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
thf(func_def_499,type,
nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
thf(func_def_500,type,
nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
thf(func_def_501,type,
nth_real: list_real > nat > real ).
thf(func_def_502,type,
nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_503,type,
product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
thf(func_def_504,type,
product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
thf(func_def_505,type,
product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
thf(func_def_506,type,
product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
thf(func_def_507,type,
product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
thf(func_def_508,type,
product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).
thf(func_def_509,type,
produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
thf(func_def_510,type,
product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
thf(func_def_511,type,
product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
thf(func_def_512,type,
produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
thf(func_def_513,type,
produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
thf(func_def_514,type,
produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
thf(func_def_515,type,
replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_516,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(func_def_517,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(func_def_518,type,
take_nat: nat > list_nat > list_nat ).
thf(func_def_519,type,
take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_520,type,
upt: nat > nat > list_nat ).
thf(func_def_521,type,
upto: int > int > list_int ).
thf(func_def_522,type,
upto_aux: int > int > list_int > list_int ).
thf(func_def_523,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_524,type,
suc: nat > nat ).
thf(func_def_525,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_526,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(func_def_527,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_528,type,
case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
thf(func_def_529,type,
pred: nat > nat ).
thf(func_def_530,type,
semiri4939895301339042750nteger: nat > code_integer ).
thf(func_def_531,type,
semiri8010041392384452111omplex: nat > complex ).
thf(func_def_532,type,
semiri4216267220026989637d_enat: nat > extended_enat ).
thf(func_def_533,type,
semiri1314217659103216013at_int: nat > int ).
thf(func_def_534,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(func_def_535,type,
semiri681578069525770553at_rat: nat > rat ).
thf(func_def_536,type,
semiri5074537144036343181t_real: nat > real ).
thf(func_def_537,type,
size_size_list_o: list_o > nat ).
thf(func_def_538,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(func_def_539,type,
size_s3941691890525107288d_enat: list_Extended_enat > nat ).
thf(func_def_540,type,
size_size_list_int: list_int > nat ).
thf(func_def_541,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(func_def_542,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_543,type,
size_size_list_num: list_num > nat ).
thf(func_def_544,type,
size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
thf(func_def_545,type,
size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
thf(func_def_546,type,
size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).
thf(func_def_547,type,
size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
thf(func_def_548,type,
size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
thf(func_def_549,type,
size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
thf(func_def_550,type,
size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
thf(func_def_551,type,
size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
thf(func_def_552,type,
size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
thf(func_def_553,type,
size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
thf(func_def_554,type,
size_size_list_real: list_real > nat ).
thf(func_def_555,type,
size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
thf(func_def_556,type,
size_size_num: num > nat ).
thf(func_def_557,type,
size_size_VEBT_VEBT: vEBT_VEBT > nat ).
thf(func_def_558,type,
nat_list_decode: nat > list_nat ).
thf(func_def_559,type,
nat_list_decode_rel: nat > nat > $o ).
thf(func_def_560,type,
nat_list_encode: list_nat > nat ).
thf(func_def_561,type,
nat_list_encode_rel: list_nat > list_nat > $o ).
thf(func_def_562,type,
nat_prod_decode: nat > product_prod_nat_nat ).
thf(func_def_563,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(func_def_564,type,
nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_565,type,
nat_prod_encode: product_prod_nat_nat > nat ).
thf(func_def_566,type,
nat_set_decode: nat > set_nat ).
thf(func_def_567,type,
nat_set_encode: set_nat > nat ).
thf(func_def_568,type,
nat_triangle: nat > nat ).
thf(func_def_569,type,
root: nat > real > real ).
thf(func_def_570,type,
sqrt: real > real ).
thf(func_def_571,type,
bitM: num > num ).
thf(func_def_572,type,
inc: num > num ).
thf(func_def_573,type,
neg_nu8804712462038260780nteger: code_integer > code_integer ).
thf(func_def_574,type,
neg_nu7009210354673126013omplex: complex > complex ).
thf(func_def_575,type,
neg_numeral_dbl_int: int > int ).
thf(func_def_576,type,
neg_numeral_dbl_rat: rat > rat ).
thf(func_def_577,type,
neg_numeral_dbl_real: real > real ).
thf(func_def_578,type,
neg_nu7757733837767384882nteger: code_integer > code_integer ).
thf(func_def_579,type,
neg_nu6511756317524482435omplex: complex > complex ).
thf(func_def_580,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(func_def_581,type,
neg_nu3179335615603231917ec_rat: rat > rat ).
thf(func_def_582,type,
neg_nu6075765906172075777c_real: real > real ).
thf(func_def_583,type,
neg_nu5831290666863070958nteger: code_integer > code_integer ).
thf(func_def_584,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(func_def_585,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(func_def_586,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(func_def_587,type,
neg_nu8295874005876285629c_real: real > real ).
thf(func_def_588,type,
neg_numeral_sub_int: num > num > int ).
thf(func_def_589,type,
bit0: num > num ).
thf(func_def_590,type,
bit1: num > num ).
thf(func_def_591,type,
one: num ).
thf(func_def_592,type,
case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
thf(func_def_593,type,
size_num: num > nat ).
thf(func_def_594,type,
num_of_nat: nat > num ).
thf(func_def_595,type,
numera6620942414471956472nteger: num > code_integer ).
thf(func_def_596,type,
numera6690914467698888265omplex: num > complex ).
thf(func_def_597,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(func_def_598,type,
numeral_numeral_int: num > int ).
thf(func_def_599,type,
numeral_numeral_nat: num > nat ).
thf(func_def_600,type,
numeral_numeral_rat: num > rat ).
thf(func_def_601,type,
numeral_numeral_real: num > real ).
thf(func_def_602,type,
pow: num > num > num ).
thf(func_def_603,type,
pred_numeral: num > nat ).
thf(func_def_604,type,
sqr: num > num ).
thf(func_def_605,type,
none_nat: option_nat ).
thf(func_def_606,type,
none_num: option_num ).
thf(func_def_607,type,
none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
thf(func_def_608,type,
some_nat: nat > option_nat ).
thf(func_def_609,type,
some_num: num > option_num ).
thf(func_def_610,type,
some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
thf(func_def_611,type,
case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
thf(func_def_612,type,
case_option_int_num: int > ( num > int ) > option_num > int ).
thf(func_def_613,type,
case_option_num_num: num > ( num > num ) > option_num > num ).
thf(func_def_614,type,
case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
thf(func_def_615,type,
map_option_num_num: ( num > num ) > option_num > option_num ).
thf(func_def_616,type,
the_nat: option_nat > nat ).
thf(func_def_617,type,
bot_bot_nat_o: nat > $o ).
thf(func_def_618,type,
bot_bo4199563552545308370d_enat: extended_enat ).
thf(func_def_619,type,
bot_bot_nat: nat ).
thf(func_def_620,type,
bot_bo7653980558646680370d_enat: set_Extended_enat ).
thf(func_def_621,type,
bot_bot_set_int: set_int ).
thf(func_def_622,type,
bot_bot_set_nat: set_nat ).
thf(func_def_623,type,
bot_bot_set_num: set_num ).
thf(func_def_624,type,
bot_bot_set_rat: set_rat ).
thf(func_def_625,type,
bot_bot_set_real: set_real ).
thf(func_def_626,type,
bot_bot_set_set_nat: set_set_nat ).
thf(func_def_627,type,
ord_Le1955565732374568822d_enat: ( extended_enat > $o ) > extended_enat ).
thf(func_def_628,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(func_def_629,type,
ord_Least_real: ( real > $o ) > real ).
thf(func_def_630,type,
ord_less_complex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(func_def_631,type,
ord_le8499522857272258027enat_o: ( extended_enat > $o ) > ( extended_enat > $o ) > $o ).
thf(func_def_632,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_633,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_634,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_635,type,
ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
thf(func_def_636,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(func_def_637,type,
ord_less_int: int > int > $o ).
thf(func_def_638,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_639,type,
ord_less_num: num > num > $o ).
thf(func_def_640,type,
ord_less_rat: rat > rat > $o ).
thf(func_def_641,type,
ord_less_real: real > real > $o ).
thf(func_def_642,type,
ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).
thf(func_def_643,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(func_def_644,type,
ord_le2529575680413868914d_enat: set_Extended_enat > set_Extended_enat > $o ).
thf(func_def_645,type,
ord_less_set_int: set_int > set_int > $o ).
thf(func_def_646,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_647,type,
ord_less_set_num: set_num > set_num > $o ).
thf(func_def_648,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(func_def_649,type,
ord_less_set_real: set_real > set_real > $o ).
thf(func_def_650,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_651,type,
ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
thf(func_def_652,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(func_def_653,type,
ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
thf(func_def_654,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_655,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_656,type,
ord_less_eq_num: num > num > $o ).
thf(func_def_657,type,
ord_less_eq_rat: rat > rat > $o ).
thf(func_def_658,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_659,type,
ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).
thf(func_def_660,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(func_def_661,type,
ord_le7203529160286727270d_enat: set_Extended_enat > set_Extended_enat > $o ).
thf(func_def_662,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(func_def_663,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_664,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(func_def_665,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(func_def_666,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_667,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_668,type,
ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_669,type,
ord_max_int: int > int > int ).
thf(func_def_670,type,
ord_max_nat: nat > nat > nat ).
thf(func_def_671,type,
ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_672,type,
ord_min_nat: nat > nat > nat ).
thf(func_def_673,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(func_def_674,type,
order_9091379641038594480t_real: ( nat > real ) > $o ).
thf(func_def_675,type,
order_4130057895858720880d_enat: ( extended_enat > extended_enat ) > $o ).
thf(func_def_676,type,
order_mono_nat_nat: ( nat > nat ) > $o ).
thf(func_def_677,type,
order_mono_nat_real: ( nat > real ) > $o ).
thf(func_def_678,type,
order_5726023648592871131at_nat: ( nat > nat ) > $o ).
thf(func_def_679,type,
order_7092887310737990675l_real: ( real > real ) > $o ).
thf(func_def_680,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(func_def_681,type,
top_to3028658606643905974d_enat: extended_enat ).
thf(func_def_682,type,
top_top_set_o: set_o ).
thf(func_def_683,type,
top_top_set_nat: set_nat ).
thf(func_def_684,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(func_def_685,type,
top_top_set_real: set_real ).
thf(func_def_686,type,
top_top_set_char: set_char ).
thf(func_def_687,type,
power_8256067586552552935nteger: code_integer > nat > code_integer ).
thf(func_def_688,type,
power_power_complex: complex > nat > complex ).
thf(func_def_689,type,
power_power_int: int > nat > int ).
thf(func_def_690,type,
power_power_nat: nat > nat > nat ).
thf(func_def_691,type,
power_power_rat: rat > nat > rat ).
thf(func_def_692,type,
power_power_real: real > nat > real ).
thf(func_def_693,type,
produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).
thf(func_def_694,type,
produc851828971589881931at_num: ( nat > num > num ) > produc2963631642982155120at_num > produc3368934014287244435at_num ).
thf(func_def_695,type,
product_Pair_o_o: $o > $o > product_prod_o_o ).
thf(func_def_696,type,
product_Pair_o_int: $o > int > product_prod_o_int ).
thf(func_def_697,type,
product_Pair_o_nat: $o > nat > product_prod_o_nat ).
thf(func_def_698,type,
produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
thf(func_def_699,type,
produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
thf(func_def_700,type,
produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_701,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(func_def_702,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_703,type,
product_Pair_nat_num: nat > num > product_prod_nat_num ).
thf(func_def_704,type,
produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
thf(func_def_705,type,
produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num ).
thf(func_def_706,type,
product_Pair_num_num: num > num > product_prod_num_num ).
thf(func_def_707,type,
produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
thf(func_def_708,type,
produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
thf(func_def_709,type,
produc581526299967858633d_enat: vEBT_VEBT > extended_enat > produc7272778201969148633d_enat ).
thf(func_def_710,type,
produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
thf(func_def_711,type,
produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
thf(func_def_712,type,
produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
thf(func_def_713,type,
produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
thf(func_def_714,type,
produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_715,type,
produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
thf(func_def_716,type,
produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
thf(func_def_717,type,
produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
thf(func_def_718,type,
produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
thf(func_def_719,type,
produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_720,type,
produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
thf(func_def_721,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(func_def_722,type,
produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).
thf(func_def_723,type,
produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_724,type,
produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_725,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_726,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_727,type,
produc2761476792215241774st_nat: ( nat > nat > list_nat ) > product_prod_nat_nat > list_nat ).
thf(func_def_728,type,
produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
thf(func_def_729,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_730,type,
produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
thf(func_def_731,type,
produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
thf(func_def_732,type,
produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).
thf(func_def_733,type,
product_fst_int_int: product_prod_int_int > int ).
thf(func_def_734,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_735,type,
produc6174133586879617921nteger: produc8923325533196201883nteger > code_integer ).
thf(func_def_736,type,
product_snd_int_int: product_prod_int_int > int ).
thf(func_def_737,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_738,type,
abs_Rat: product_prod_int_int > rat ).
thf(func_def_739,type,
fract: int > int > rat ).
thf(func_def_740,type,
frct: product_prod_int_int > rat ).
thf(func_def_741,type,
rep_Rat: rat > product_prod_int_int ).
thf(func_def_742,type,
field_5140801741446780682s_real: set_real ).
thf(func_def_743,type,
field_7254667332652039916t_real: rat > real ).
thf(func_def_744,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(func_def_745,type,
pcr_rat: product_prod_int_int > rat > $o ).
thf(func_def_746,type,
positive: rat > $o ).
thf(func_def_747,type,
quotient_of: rat > product_prod_int_int ).
thf(func_def_748,type,
ratrel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_749,type,
real2: ( nat > rat ) > real ).
thf(func_def_750,type,
cauchy: ( nat > rat ) > $o ).
thf(func_def_751,type,
cr_real: ( nat > rat ) > real > $o ).
thf(func_def_752,type,
pcr_real: ( nat > rat ) > real > $o ).
thf(func_def_753,type,
positive2: real > $o ).
thf(func_def_754,type,
realrel: ( nat > rat ) > ( nat > rat ) > $o ).
thf(func_def_755,type,
rep_real: real > nat > rat ).
thf(func_def_756,type,
vanishes: ( nat > rat ) > $o ).
thf(func_def_757,type,
real_V2521375963428798218omplex: set_complex ).
thf(func_def_758,type,
real_V5970128139526366754l_real: ( real > real ) > $o ).
thf(func_def_759,type,
real_V3694042436643373181omplex: complex > complex > real ).
thf(func_def_760,type,
real_V975177566351809787t_real: real > real > real ).
thf(func_def_761,type,
real_V1022390504157884413omplex: complex > real ).
thf(func_def_762,type,
real_V7735802525324610683m_real: real > real ).
thf(func_def_763,type,
real_V4546457046886955230omplex: real > complex ).
thf(func_def_764,type,
real_V1803761363581548252l_real: real > real ).
thf(func_def_765,type,
real_V2046097035970521341omplex: real > complex > complex ).
thf(func_def_766,type,
real_V1485227260804924795R_real: real > real > real ).
thf(func_def_767,type,
transp_nat_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).
thf(func_def_768,type,
algebr932160517623751201me_int: int > int > $o ).
thf(func_def_769,type,
algebr934650988132801477me_nat: nat > nat > $o ).
thf(func_def_770,type,
divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
thf(func_def_771,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(func_def_772,type,
divide_divide_int: int > int > int ).
thf(func_def_773,type,
divide_divide_nat: nat > nat > nat ).
thf(func_def_774,type,
divide_divide_rat: rat > rat > rat ).
thf(func_def_775,type,
divide_divide_real: real > real > real ).
thf(func_def_776,type,
dvd_dvd_Code_integer: code_integer > code_integer > $o ).
thf(func_def_777,type,
dvd_dvd_complex: complex > complex > $o ).
thf(func_def_778,type,
dvd_dvd_int: int > int > $o ).
thf(func_def_779,type,
dvd_dvd_nat: nat > nat > $o ).
thf(func_def_780,type,
dvd_dvd_rat: rat > rat > $o ).
thf(func_def_781,type,
dvd_dvd_real: real > real > $o ).
thf(func_def_782,type,
modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
thf(func_def_783,type,
modulo_modulo_int: int > int > int ).
thf(func_def_784,type,
modulo_modulo_nat: nat > nat > nat ).
thf(func_def_785,type,
zero_n356916108424825756nteger: $o > code_integer ).
thf(func_def_786,type,
zero_n1201886186963655149omplex: $o > complex ).
thf(func_def_787,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(func_def_788,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(func_def_789,type,
zero_n2052037380579107095ol_rat: $o > rat ).
thf(func_def_790,type,
zero_n3304061248610475627l_real: $o > real ).
thf(func_def_791,type,
suminf_complex: ( nat > complex ) > complex ).
thf(func_def_792,type,
suminf_real: ( nat > real ) > real ).
thf(func_def_793,type,
summable_complex: ( nat > complex ) > $o ).
thf(func_def_794,type,
summable_real: ( nat > real ) > $o ).
thf(func_def_795,type,
sums_real: ( nat > real ) > real > $o ).
thf(func_def_796,type,
collect_o: ( $o > $o ) > set_o ).
thf(func_def_797,type,
collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
thf(func_def_798,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(func_def_799,type,
collec4429806609662206161d_enat: ( extended_enat > $o ) > set_Extended_enat ).
thf(func_def_800,type,
collect_int: ( int > $o ) > set_int ).
thf(func_def_801,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(func_def_802,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_803,type,
collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
thf(func_def_804,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(func_def_805,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(func_def_806,type,
collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
thf(func_def_807,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_808,type,
collect_VEBT_VEBT: ( vEBT_VEBT > $o ) > set_VEBT_VEBT ).
thf(func_def_809,type,
image_80655429650038917d_enat: ( extended_enat > extended_enat ) > set_Extended_enat > set_Extended_enat ).
thf(func_def_810,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(func_def_811,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(func_def_812,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_813,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(func_def_814,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(func_def_815,type,
image_nat_char: ( nat > char ) > set_nat > set_char ).
thf(func_def_816,type,
image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
thf(func_def_817,type,
image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
thf(func_def_818,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(func_def_819,type,
image_char_nat: ( char > nat ) > set_char > set_nat ).
thf(func_def_820,type,
insert_int: int > set_int > set_int ).
thf(func_def_821,type,
insert_nat: nat > set_nat > set_nat ).
thf(func_def_822,type,
insert_real: real > set_real > set_real ).
thf(func_def_823,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_824,type,
set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
thf(func_def_825,type,
set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
thf(func_def_826,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(func_def_827,type,
set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
thf(func_def_828,type,
set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
thf(func_def_829,type,
set_or5403411693681687835d_enat: extended_enat > extended_enat > set_Extended_enat ).
thf(func_def_830,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(func_def_831,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(func_def_832,type,
set_or7049704709247886629st_num: num > num > set_num ).
thf(func_def_833,type,
set_or633870826150836451st_rat: rat > rat > set_rat ).
thf(func_def_834,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(func_def_835,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(func_def_836,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(func_def_837,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(func_def_838,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(func_def_839,type,
set_ord_atLeast_real: real > set_real ).
thf(func_def_840,type,
set_ord_atMost_nat: nat > set_nat ).
thf(func_def_841,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(func_def_842,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(func_def_843,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(func_def_844,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(func_def_845,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(func_def_846,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(func_def_847,type,
set_or5849166863359141190n_real: real > set_real ).
thf(func_def_848,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(func_def_849,type,
set_or5984915006950818249n_real: real > set_real ).
thf(func_def_850,type,
abort_real: literal > ( product_unit > real ) > real ).
thf(func_def_851,type,
literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(func_def_852,type,
ascii_of: char > char ).
thf(func_def_853,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(func_def_854,type,
comm_s629917340098488124ar_nat: char > nat ).
thf(func_def_855,type,
integer_of_char: char > code_integer ).
thf(func_def_856,type,
unique3096191561947761185of_nat: nat > char ).
thf(func_def_857,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(func_def_858,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(func_def_859,type,
topolo6980174941875973593q_real: ( nat > real ) > $o ).
thf(func_def_860,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(func_def_861,type,
topolo7531315842566124627t_real: ( nat > real ) > $o ).
thf(func_def_862,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(func_def_863,type,
topolo6517432010174082258omplex: ( nat > complex ) > $o ).
thf(func_def_864,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(func_def_865,type,
topolo896644834953643431omplex: filter6041513312241820739omplex ).
thf(func_def_866,type,
topolo1511823702728130853y_real: filter2146258269922977983l_real ).
thf(func_def_867,type,
arccos: real > real ).
thf(func_def_868,type,
arcosh_real: real > real ).
thf(func_def_869,type,
arcsin: real > real ).
thf(func_def_870,type,
arctan: real > real ).
thf(func_def_871,type,
arsinh_real: real > real ).
thf(func_def_872,type,
artanh_real: real > real ).
thf(func_def_873,type,
cos_complex: complex > complex ).
thf(func_def_874,type,
cos_real: real > real ).
thf(func_def_875,type,
cos_coeff: nat > real ).
thf(func_def_876,type,
cosh_complex: complex > complex ).
thf(func_def_877,type,
cosh_real: real > real ).
thf(func_def_878,type,
cot_complex: complex > complex ).
thf(func_def_879,type,
cot_real: real > real ).
thf(func_def_880,type,
exp_complex: complex > complex ).
thf(func_def_881,type,
exp_real: real > real ).
thf(func_def_882,type,
ln_ln_real: real > real ).
thf(func_def_883,type,
log: real > real > real ).
thf(func_def_884,type,
pi: real ).
thf(func_def_885,type,
powr_real: real > real > real ).
thf(func_def_886,type,
powr_real2: real > real > real ).
thf(func_def_887,type,
sin_complex: complex > complex ).
thf(func_def_888,type,
sin_real: real > real ).
thf(func_def_889,type,
sin_coeff: nat > real ).
thf(func_def_890,type,
sinh_complex: complex > complex ).
thf(func_def_891,type,
sinh_real: real > real ).
thf(func_def_892,type,
tan_complex: complex > complex ).
thf(func_def_893,type,
tan_real: real > real ).
thf(func_def_894,type,
tanh_complex: complex > complex ).
thf(func_def_895,type,
tanh_real: real > real ).
thf(func_def_896,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_897,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_898,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(func_def_899,type,
vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_900,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(func_def_901,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(func_def_902,type,
vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
thf(func_def_903,type,
vEBT_V312737461966249ad_rel: produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ).
thf(func_def_904,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(func_def_905,type,
vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
thf(func_def_906,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(func_def_907,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(func_def_908,type,
vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_909,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(func_def_910,type,
vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_911,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(func_def_912,type,
vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_913,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(func_def_914,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_915,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(func_def_916,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(func_def_917,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_918,type,
vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_919,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(func_def_920,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(func_def_921,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_922,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(func_def_923,type,
vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_924,type,
vEBT_VEBT_add: option_nat > option_nat > option_nat ).
thf(func_def_925,type,
vEBT_VEBT_greater: option_nat > option_nat > $o ).
thf(func_def_926,type,
vEBT_VEBT_less: option_nat > option_nat > $o ).
thf(func_def_927,type,
vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
thf(func_def_928,type,
vEBT_VEBT_max_in_set: set_nat > nat > $o ).
thf(func_def_929,type,
vEBT_VEBT_min_in_set: set_nat > nat > $o ).
thf(func_def_930,type,
vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
thf(func_def_931,type,
vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
thf(func_def_932,type,
vEBT_VEBT_power: option_nat > option_nat > option_nat ).
thf(func_def_933,type,
vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
thf(func_def_934,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_935,type,
vEBT_vebt_mint: vEBT_VEBT > option_nat ).
thf(func_def_936,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_937,type,
vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
thf(func_def_938,type,
vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
thf(func_def_939,type,
vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_940,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(func_def_941,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
thf(func_def_942,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(func_def_943,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_944,type,
accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
thf(func_def_945,type,
accp_P6183159247885693666d_enat: ( produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ) > produc7272778201969148633d_enat > $o ).
thf(func_def_946,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(func_def_947,type,
accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
thf(func_def_948,type,
less_than: set_Pr1261947904930325089at_nat ).
thf(func_def_949,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(func_def_950,type,
fChoice_real: ( real > $o ) > real ).
thf(func_def_951,type,
member_o: $o > set_o > $o ).
thf(func_def_952,type,
member_Code_integer: code_integer > set_Code_integer > $o ).
thf(func_def_953,type,
member_complex: complex > set_complex > $o ).
thf(func_def_954,type,
member_Extended_enat: extended_enat > set_Extended_enat > $o ).
thf(func_def_955,type,
member_int: int > set_int > $o ).
thf(func_def_956,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(func_def_957,type,
member_nat: nat > set_nat > $o ).
thf(func_def_958,type,
member_num: num > set_num > $o ).
thf(func_def_959,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_960,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(func_def_961,type,
member_rat: rat > set_rat > $o ).
thf(func_def_962,type,
member_real: real > set_real > $o ).
thf(func_def_963,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_964,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_965,type,
deg: nat ).
thf(func_def_966,type,
m: nat ).
thf(func_def_967,type,
ma: nat ).
thf(func_def_968,type,
mi: nat ).
thf(func_def_969,type,
miny: nat ).
thf(func_def_970,type,
na: nat ).
thf(func_def_971,type,
res: nat ).
thf(func_def_972,type,
sc: nat ).
thf(func_def_973,type,
summary: vEBT_VEBT ).
thf(func_def_974,type,
treeList: list_VEBT_VEBT ).
thf(func_def_975,type,
xa: nat ).
thf(func_def_976,type,
za: nat ).
thf(func_def_995,type,
sP0: int > ( int > $o ) > $o ).
thf(func_def_996,type,
sP1: nat > nat > nat > $o ).
thf(func_def_997,type,
sP2: nat > nat > nat > $o ).
thf(func_def_998,type,
sP3: nat > nat > nat > $o ).
thf(func_def_999,type,
sP4: int > int > int > $o ).
thf(func_def_1000,type,
sP5: int > int > int > $o ).
thf(func_def_1001,type,
sP6: int > int > int > $o ).
thf(func_def_1002,type,
sP7: vEBT_VEBT > nat > $o ).
thf(func_def_1003,type,
sP8: rat > rat > rat > $o ).
thf(func_def_1004,type,
sP9: rat > rat > rat > $o ).
thf(func_def_1005,type,
sP10: rat > rat > rat > $o ).
thf(func_def_1006,type,
sP11: vEBT_VEBT > option_nat > $o ).
thf(func_def_1007,type,
sP12: ( int > $o ) > $o ).
thf(func_def_1008,type,
sP13: nat > $o > vEBT_VEBT > $o ).
thf(func_def_1009,type,
sP14: $o > vEBT_VEBT > $o ).
thf(func_def_1010,type,
sP15: vEBT_VEBT > nat > $o ).
thf(func_def_1011,type,
sP16: real > ( real > real > $o ) > real > $o ).
thf(func_def_1012,type,
sK17: nat > nat > nat > nat ).
thf(func_def_1013,type,
sK18: ( nat > rat ) > nat ).
thf(func_def_1014,type,
sK19: ( nat > rat ) > nat ).
thf(func_def_1015,type,
sK20: ( real > real ) > real ).
thf(func_def_1016,type,
sK21: ( real > real ) > real ).
thf(func_def_1017,type,
sK22: rat > rat ).
thf(func_def_1018,type,
sK23: real > int ).
thf(func_def_1019,type,
sK24: ( rat > real ) > rat ).
thf(func_def_1020,type,
sK25: ( rat > real ) > rat ).
thf(func_def_1021,type,
sK26: int > int ).
thf(func_def_1022,type,
sK27: real > ( real > $o ) > real ).
thf(func_def_1023,type,
sK28: real > ( real > $o ) > real ).
thf(func_def_1024,type,
sK29: real > ( real > $o ) > real ).
thf(func_def_1025,type,
sK30: real > ( real > $o ) > real ).
thf(func_def_1026,type,
sK31: ( nat > $o ) > nat ).
thf(func_def_1027,type,
sK32: ( nat > $o ) > nat ).
thf(func_def_1028,type,
sK33: real > nat > real ).
thf(func_def_1029,type,
sK34: ( nat > rat ) > nat ).
thf(func_def_1030,type,
sK35: ( nat > $o ) > nat ).
thf(func_def_1031,type,
sK36: nat > nat > nat ).
thf(func_def_1032,type,
sK37: real > nat ).
thf(func_def_1033,type,
sK38: int > nat ).
thf(func_def_1034,type,
sK39: int > nat ).
thf(func_def_1035,type,
sK40: nat > nat > nat > nat ).
thf(func_def_1036,type,
sK41: ( rat > rat ) > rat ).
thf(func_def_1037,type,
sK42: ( rat > rat ) > rat ).
thf(func_def_1038,type,
sK43: real > real > real > real ).
thf(func_def_1039,type,
sK44: ( nat > nat ) > nat ).
thf(func_def_1040,type,
sK45: real > real ).
thf(func_def_1041,type,
sK46: nat > nat > nat ).
thf(func_def_1042,type,
sK47: rat > rat ).
thf(func_def_1043,type,
sK48: real ).
thf(func_def_1044,type,
sK49: ( real > rat ) > real ).
thf(func_def_1045,type,
sK50: ( real > rat ) > real ).
thf(func_def_1046,type,
sK51: int > int ).
thf(func_def_1047,type,
sK52: nat > nat ).
thf(func_def_1048,type,
sK53: ( rat > int ) > rat ).
thf(func_def_1049,type,
sK54: ( rat > int ) > rat ).
thf(func_def_1050,type,
sK55: nat > nat ).
thf(func_def_1051,type,
sK56: ( nat > rat ) > nat ).
thf(func_def_1052,type,
sK57: ( nat > rat ) > nat ).
thf(func_def_1053,type,
sK58: ( int > rat ) > int ).
thf(func_def_1054,type,
sK59: ( int > rat ) > int ).
thf(func_def_1055,type,
sK60: rat > rat ).
thf(func_def_1056,type,
sK61: nat > nat > nat ).
thf(func_def_1057,type,
sK62: nat > nat > nat ).
thf(func_def_1058,type,
sK63: nat > nat > nat ).
thf(func_def_1059,type,
sK64: nat > nat > nat ).
thf(func_def_1060,type,
sK65: nat > nat > nat ).
thf(func_def_1061,type,
sK66: real > real > real ).
thf(func_def_1062,type,
sK67: real > real > real ).
thf(func_def_1063,type,
sK68: ( real > $o ) > nat ).
thf(func_def_1064,type,
sK69: ( real > $o ) > real ).
thf(func_def_1065,type,
sK70: ( real > $o ) > real ).
thf(func_def_1066,type,
sK71: ( rat > rat ) > rat ).
thf(func_def_1067,type,
sK72: ( rat > rat ) > rat ).
thf(func_def_1068,type,
sK73: int > ( int > $o ) > int ).
thf(func_def_1069,type,
sK74: int > ( int > $o ) > int ).
thf(func_def_1070,type,
sK75: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1071,type,
sK76: ( int > $o ) > int ).
thf(func_def_1072,type,
sK77: real > real ).
thf(func_def_1073,type,
sK78: ( rat > int ) > rat ).
thf(func_def_1074,type,
sK79: ( rat > int ) > rat ).
thf(func_def_1075,type,
sK80: ( rat > int ) > rat ).
thf(func_def_1076,type,
sK81: ( rat > int ) > rat ).
thf(func_def_1077,type,
sK82: ( real > real ) > real ).
thf(func_def_1078,type,
sK83: ( real > real ) > real ).
thf(func_def_1079,type,
sK84: ( int > $o ) > int ).
thf(func_def_1080,type,
sK85: ( int > $o ) > int > int ).
thf(func_def_1081,type,
sK86: ( int > $o ) > int > int ).
thf(func_def_1082,type,
sK87: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1083,type,
sK88: nat > nat > nat ).
thf(func_def_1084,type,
sK89: rat > rat > rat ).
thf(func_def_1085,type,
sK90: rat > rat > rat > rat ).
thf(func_def_1086,type,
sK91: nat > nat > nat ).
thf(func_def_1087,type,
sK92: int > int ).
thf(func_def_1088,type,
sK93: ( rat > rat ) > rat ).
thf(func_def_1089,type,
sK94: ( rat > rat ) > rat ).
thf(func_def_1090,type,
sK95: ( int > real ) > int ).
thf(func_def_1091,type,
sK96: ( int > real ) > int ).
thf(func_def_1092,type,
sK97: ( rat > rat ) > rat ).
thf(func_def_1093,type,
sK98: ( rat > rat ) > rat ).
thf(func_def_1094,type,
sK99: ( nat > $o ) > nat ).
thf(func_def_1095,type,
sK100: int > nat ).
thf(func_def_1096,type,
sK101: int > nat ).
thf(func_def_1097,type,
sK102: ( int > $o ) > int > int ).
thf(func_def_1098,type,
sK103: ( real > rat ) > real ).
thf(func_def_1099,type,
sK104: ( real > rat ) > real ).
thf(func_def_1100,type,
sK105: int > nat ).
thf(func_def_1101,type,
sK106: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1102,type,
sK107: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1103,type,
sK108: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1104,type,
sK109: int > ( int > $o ) > int ).
thf(func_def_1105,type,
sK110: ( int > $o ) > int > int ).
thf(func_def_1106,type,
sK111: real > real ).
thf(func_def_1107,type,
sK112: real > real ).
thf(func_def_1108,type,
sK113: real > real > real > real ).
thf(func_def_1109,type,
sK114: ( nat > $o ) > nat ).
thf(func_def_1110,type,
sK115: nat > nat > nat ).
thf(func_def_1111,type,
sK116: real > real ).
thf(func_def_1112,type,
sK117: ( rat > nat ) > rat ).
thf(func_def_1113,type,
sK118: ( rat > nat ) > rat ).
thf(func_def_1114,type,
sK119: int > int > int ).
thf(func_def_1115,type,
sK120: ( nat > real ) > nat ).
thf(func_def_1116,type,
sK121: ( real > real ) > real ).
thf(func_def_1117,type,
sK122: ( real > real ) > real ).
thf(func_def_1118,type,
sK123: nat > nat ).
thf(func_def_1119,type,
sK124: ( nat > nat ) > nat ).
thf(func_def_1120,type,
sK125: nat > nat ).
thf(func_def_1121,type,
sK126: real > real > real ).
thf(func_def_1122,type,
sK127: ( nat > nat ) > nat ).
thf(func_def_1123,type,
sK128: ( rat > real ) > rat ).
thf(func_def_1124,type,
sK129: ( rat > real ) > rat ).
thf(func_def_1125,type,
sK130: int > int ).
thf(func_def_1126,type,
sK131: int > int ).
thf(func_def_1127,type,
sK132: ( int > $o ) > nat ).
thf(func_def_1128,type,
sK133: ( int > $o ) > nat ).
thf(func_def_1129,type,
sK134: ( real > nat ) > real ).
thf(func_def_1130,type,
sK135: ( real > nat ) > real ).
thf(func_def_1131,type,
sK136: nat > nat ).
thf(func_def_1132,type,
sK137: int > int ).
thf(func_def_1133,type,
sK138: real > int ).
thf(func_def_1134,type,
sK139: ( rat > rat ) > rat ).
thf(func_def_1135,type,
sK140: ( rat > rat ) > rat ).
thf(func_def_1136,type,
sK141: real > real ).
thf(func_def_1137,type,
sK142: int > int > int ).
thf(func_def_1138,type,
sK143: ( real > $o ) > nat ).
thf(func_def_1139,type,
sK144: ( real > $o ) > real ).
thf(func_def_1140,type,
sK145: ( real > $o ) > real ).
thf(func_def_1141,type,
sK146: rat > rat ).
thf(func_def_1142,type,
sK147: ( int > real ) > int ).
thf(func_def_1143,type,
sK148: ( int > real ) > int ).
thf(func_def_1144,type,
sK149: rat > rat ).
thf(func_def_1145,type,
sK150: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1146,type,
sK151: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1147,type,
sK152: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1148,type,
sK153: int > int > int ).
thf(func_def_1149,type,
sK154: ( nat > int ) > nat ).
thf(func_def_1150,type,
sK155: ( nat > nat > $o ) > nat ).
thf(func_def_1151,type,
sK156: ( nat > nat > $o ) > nat ).
thf(func_def_1152,type,
sK157: ( nat > nat > $o ) > nat ).
thf(func_def_1153,type,
sK158: ( nat > nat > $o ) > nat ).
thf(func_def_1154,type,
sK159: nat > nat > nat > nat ).
thf(func_def_1155,type,
sK160: nat > nat > nat > nat ).
thf(func_def_1156,type,
sK161: nat > ( nat > $o ) > nat ).
thf(func_def_1157,type,
sK162: ( nat > rat ) > nat ).
thf(func_def_1158,type,
sK163: ( int > int > $o ) > int ).
thf(func_def_1159,type,
sK164: ( int > int > $o ) > int ).
thf(func_def_1160,type,
sK165: ( int > int > $o ) > int ).
thf(func_def_1161,type,
sK166: ( int > int > $o ) > int ).
thf(func_def_1162,type,
sK167: ( int > int > $o ) > int ).
thf(func_def_1163,type,
sK168: real > real > nat ).
thf(func_def_1164,type,
sK169: rat > rat ).
thf(func_def_1165,type,
sK170: real > real > real ).
thf(func_def_1166,type,
sK171: real > int ).
thf(func_def_1167,type,
sK172: nat > nat ).
thf(func_def_1168,type,
sK173: nat > nat ).
thf(func_def_1169,type,
sK174: int > ( int > $o ) > int ).
thf(func_def_1170,type,
sK175: real > real > nat ).
thf(func_def_1171,type,
sK176: nat > nat ).
thf(func_def_1172,type,
sK177: ( rat > rat ) > rat ).
thf(func_def_1173,type,
sK178: ( rat > rat ) > rat ).
thf(func_def_1174,type,
sK179: ( nat > nat > $o ) > nat ).
thf(func_def_1175,type,
sK180: ( nat > nat > $o ) > nat ).
thf(func_def_1176,type,
sK181: ( nat > nat > $o ) > nat ).
thf(func_def_1177,type,
sK182: ( nat > rat ) > nat ).
thf(func_def_1178,type,
sK183: ( nat > rat ) > nat ).
thf(func_def_1179,type,
sK184: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1180,type,
sK185: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1181,type,
sK186: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1182,type,
sK187: ( nat > $o ) > nat ).
thf(func_def_1183,type,
sK188: vEBT_VEBT > nat > nat ).
thf(func_def_1184,type,
sK189: vEBT_VEBT > nat > nat ).
thf(func_def_1185,type,
sK190: vEBT_VEBT > nat > nat ).
thf(func_def_1186,type,
sK191: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1187,type,
sK192: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1188,type,
sK193: nat > vEBT_VEBT > $o ).
thf(func_def_1189,type,
sK194: nat > vEBT_VEBT > $o ).
thf(func_def_1190,type,
sK195: ( nat > rat ) > nat ).
thf(func_def_1191,type,
sK196: ( nat > rat ) > nat ).
thf(func_def_1192,type,
sK197: nat > nat > nat ).
thf(func_def_1193,type,
sK198: int > nat ).
thf(func_def_1194,type,
sK199: int > nat ).
thf(func_def_1195,type,
sK200: rat > rat ).
thf(func_def_1196,type,
sK201: int > int ).
thf(func_def_1197,type,
sK202: int > int > int ).
thf(func_def_1198,type,
sK203: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1199,type,
sK204: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1200,type,
sK205: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1201,type,
sK206: nat > nat > nat ).
thf(func_def_1202,type,
sK207: ( nat > nat ) > nat ).
thf(func_def_1203,type,
sK208: ( nat > nat ) > nat ).
thf(func_def_1204,type,
sK209: int > ( int > $o ) > int ).
thf(func_def_1205,type,
sK210: nat > nat ).
thf(func_def_1206,type,
sK211: real > real > nat ).
thf(func_def_1207,type,
sK212: rat > rat ).
thf(func_def_1208,type,
sK213: ( real > real ) > real ).
thf(func_def_1209,type,
sK214: ( real > real ) > real ).
thf(func_def_1210,type,
sK215: ( nat > nat > $o ) > nat ).
thf(func_def_1211,type,
sK216: ( nat > nat > $o ) > nat ).
thf(func_def_1212,type,
sK217: ( nat > nat > $o ) > nat ).
thf(func_def_1213,type,
sK218: ( nat > nat > $o ) > nat ).
thf(func_def_1214,type,
sK219: ( nat > nat > $o ) > nat ).
thf(func_def_1215,type,
sK220: ( rat > int ) > rat ).
thf(func_def_1216,type,
sK221: ( rat > int ) > rat ).
thf(func_def_1217,type,
sK222: rat > rat ).
thf(func_def_1218,type,
sK223: nat > nat > nat > nat ).
thf(func_def_1219,type,
sK224: nat > nat > nat > nat ).
thf(func_def_1220,type,
sK225: ( nat > nat > $o ) > nat ).
thf(func_def_1221,type,
sK226: ( nat > nat > $o ) > nat ).
thf(func_def_1222,type,
sK227: ( nat > nat > $o ) > nat ).
thf(func_def_1223,type,
sK228: ( nat > nat > $o ) > nat ).
thf(func_def_1224,type,
sK229: rat > rat > rat ).
thf(func_def_1225,type,
sK230: ( nat > real ) > nat ).
thf(func_def_1226,type,
sK231: ( nat > real ) > nat ).
thf(func_def_1227,type,
sK232: ( real > int ) > real ).
thf(func_def_1228,type,
sK233: ( real > int ) > real ).
thf(func_def_1229,type,
sK234: int > nat ).
thf(func_def_1230,type,
sK235: list_VEBT_VEBT > list_VEBT_VEBT > nat ).
thf(func_def_1231,type,
sK236: real > real ).
thf(func_def_1232,type,
sK237: real > int ).
thf(func_def_1233,type,
sK238: ( real > int ) > real ).
thf(func_def_1234,type,
sK239: ( real > int ) > real ).
thf(func_def_1235,type,
sK240: nat > nat > nat ).
thf(func_def_1236,type,
sK241: ( rat > int ) > rat ).
thf(func_def_1237,type,
sK242: ( rat > int ) > rat ).
thf(func_def_1238,type,
sK243: ( rat > rat ) > rat ).
thf(func_def_1239,type,
sK244: ( rat > rat ) > rat ).
thf(func_def_1240,type,
sK245: nat > nat > nat ).
thf(func_def_1241,type,
sK246: ( nat > real ) > nat ).
thf(func_def_1242,type,
sK247: ( real > real ) > real ).
thf(func_def_1243,type,
sK248: ( real > real ) > real ).
thf(func_def_1244,type,
sK249: int > nat ).
thf(func_def_1245,type,
sK250: real > real > real ).
thf(func_def_1246,type,
sK251: code_integer > code_integer ).
thf(func_def_1247,type,
sK252: ( nat > $o ) > nat > nat ).
thf(func_def_1248,type,
sK253: ( nat > $o ) > nat ).
thf(func_def_1249,type,
sK254: nat > nat > nat > nat ).
thf(func_def_1250,type,
sK255: nat > nat > nat ).
thf(func_def_1251,type,
sK256: nat > nat > nat ).
thf(func_def_1252,type,
sK257: nat > nat > nat ).
thf(func_def_1253,type,
sK258: real > nat ).
thf(func_def_1254,type,
sK259: real > real ).
thf(func_def_1255,type,
sK260: rat > rat > rat ).
thf(func_def_1256,type,
sK261: nat > nat ).
thf(func_def_1257,type,
sK262: real > nat ).
thf(func_def_1258,type,
sK263: real > real ).
thf(func_def_1259,type,
sK264: real > int ).
thf(func_def_1260,type,
sK265: real > real ).
thf(func_def_1261,type,
sK266: ( nat > $o ) > nat ).
thf(func_def_1262,type,
sK267: nat > ( nat > $o ) > nat ).
thf(func_def_1263,type,
sK268: nat > nat ).
thf(func_def_1264,type,
sK269: ( int > rat ) > int ).
thf(func_def_1265,type,
sK270: ( int > rat ) > int ).
thf(func_def_1266,type,
sK271: ( nat > rat ) > nat ).
thf(func_def_1267,type,
sK272: ( nat > rat ) > nat ).
thf(func_def_1268,type,
sK273: int > ( int > $o ) > int ).
thf(func_def_1269,type,
sK274: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1270,type,
sK275: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1271,type,
sK276: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1272,type,
sK277: rat > rat > rat > rat ).
thf(func_def_1273,type,
sK278: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1274,type,
sK279: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1275,type,
sK280: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1276,type,
sK281: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1277,type,
sK282: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1278,type,
sK283: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1279,type,
sK284: ( rat > int ) > rat ).
thf(func_def_1280,type,
sK285: ( rat > int ) > rat ).
thf(func_def_1281,type,
sK286: rat > rat > rat ).
thf(func_def_1282,type,
sK287: ( rat > nat ) > rat ).
thf(func_def_1283,type,
sK288: ( rat > nat ) > rat ).
thf(func_def_1284,type,
sK289: int > int > int ).
thf(func_def_1285,type,
sK290: real > real ).
thf(func_def_1286,type,
sK291: nat > nat ).
thf(func_def_1287,type,
sK292: real > real ).
thf(func_def_1288,type,
sK293: nat ).
thf(func_def_1289,type,
sK294: int > int > int > int ).
thf(func_def_1290,type,
sK295: ( real > rat ) > real ).
thf(func_def_1291,type,
sK296: ( real > rat ) > real ).
thf(func_def_1292,type,
sK297: ( nat > $o ) > nat ).
thf(func_def_1293,type,
sK298: nat > nat ).
thf(func_def_1294,type,
sK299: ( real > nat ) > real ).
thf(func_def_1295,type,
sK300: ( real > nat ) > real ).
thf(func_def_1296,type,
sK301: real > real > real ).
thf(func_def_1297,type,
sK302: int > int > int ).
thf(func_def_1298,type,
sK303: nat > nat > nat ).
thf(func_def_1299,type,
sK304: nat > real > real ).
thf(func_def_1300,type,
sK305: real > real ).
thf(func_def_1301,type,
sK306: ( int > rat ) > int ).
thf(func_def_1302,type,
sK307: ( int > rat ) > int ).
thf(func_def_1303,type,
sK308: real > real > real ).
thf(func_def_1304,type,
sK309: ( nat > $o ) > nat > nat > nat ).
thf(func_def_1305,type,
sK310: rat > rat > rat ).
thf(func_def_1306,type,
sK311: int > int ).
thf(func_def_1307,type,
sK312: nat > vEBT_VEBT > nat ).
thf(func_def_1308,type,
sK313: real > real > real > real ).
thf(func_def_1309,type,
sK314: vEBT_VEBT > $o ).
thf(func_def_1310,type,
sK315: vEBT_VEBT > $o ).
thf(func_def_1311,type,
sK316: int > int > int ).
thf(func_def_1312,type,
sK317: nat > nat ).
thf(func_def_1313,type,
sK318: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1314,type,
sK319: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1315,type,
sK320: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1316,type,
sK321: ( real > real > $o ) > real ).
thf(func_def_1317,type,
sK322: ( rat > rat ) > rat ).
thf(func_def_1318,type,
sK323: ( rat > rat ) > rat ).
thf(func_def_1319,type,
sK324: ( rat > real ) > rat ).
thf(func_def_1320,type,
sK325: ( rat > real ) > rat ).
thf(func_def_1321,type,
sK326: ( nat > rat ) > nat ).
thf(func_def_1322,type,
sK327: ( nat > rat ) > nat ).
thf(func_def_1323,type,
sK328: ( nat > rat ) > nat ).
thf(func_def_1324,type,
sK329: real > nat ).
thf(func_def_1325,type,
sK330: real > real ).
thf(func_def_1326,type,
sK331: ( rat > real ) > rat ).
thf(func_def_1327,type,
sK332: ( rat > real ) > rat ).
thf(func_def_1328,type,
sK333: ( rat > nat ) > rat ).
thf(func_def_1329,type,
sK334: ( rat > nat ) > rat ).
thf(func_def_1330,type,
sK335: ( rat > rat ) > rat ).
thf(func_def_1331,type,
sK336: ( rat > rat ) > rat ).
thf(func_def_1332,type,
sK337: int > int ).
thf(func_def_1333,type,
sK338: ( rat > nat ) > rat ).
thf(func_def_1334,type,
sK339: ( rat > nat ) > rat ).
thf(func_def_1335,type,
sK340: int > int ).
thf(func_def_1336,type,
sK341: ( rat > nat ) > rat ).
thf(func_def_1337,type,
sK342: ( rat > nat ) > rat ).
thf(func_def_1338,type,
sK343: rat > rat ).
thf(func_def_1339,type,
sK344: ( nat > nat ) > nat ).
thf(func_def_1340,type,
sK345: ( nat > nat ) > nat ).
thf(func_def_1341,type,
sK346: real > real ).
thf(func_def_1342,type,
sK347: ( nat > $o ) > nat ).
thf(func_def_1343,type,
sK348: nat > real > real ).
thf(func_def_1344,type,
sK349: vEBT_VEBT > option_nat > $o ).
thf(func_def_1345,type,
sK350: vEBT_VEBT > option_nat > $o ).
thf(func_def_1346,type,
sK351: vEBT_VEBT > nat ).
thf(func_def_1347,type,
sK352: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1348,type,
sK353: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1349,type,
sK354: option_nat > vEBT_VEBT > nat ).
thf(func_def_1350,type,
sK355: option_nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1351,type,
sK356: option_nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1352,type,
sK357: option_nat > vEBT_VEBT > nat ).
thf(func_def_1353,type,
sK358: option_nat > vEBT_VEBT > nat ).
thf(func_def_1354,type,
sK359: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1355,type,
sK360: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1356,type,
sK361: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1357,type,
sK362: nat > nat ).
thf(func_def_1358,type,
sK363: rat > rat ).
thf(func_def_1359,type,
sK364: ( nat > int ) > nat ).
thf(func_def_1360,type,
sK365: ( real > $o ) > real > real ).
thf(func_def_1361,type,
sK366: ( real > $o ) > real > real ).
thf(func_def_1362,type,
sK367: ( real > $o ) > real > real ).
thf(func_def_1363,type,
sK368: ( real > $o ) > real > real ).
thf(func_def_1364,type,
sK369: real > real > real > real ).
thf(func_def_1365,type,
sK370: nat > nat > nat ).
thf(func_def_1366,type,
sK371: nat > nat ).
thf(func_def_1367,type,
sK372: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1368,type,
sK373: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1369,type,
sK374: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1370,type,
sK375: ( real > rat ) > real ).
thf(func_def_1371,type,
sK376: ( real > rat ) > real ).
thf(func_def_1372,type,
sK377: nat > nat > nat ).
thf(func_def_1373,type,
sK378: nat > nat > ( nat > $o ) > nat ).
thf(func_def_1374,type,
sK379: ( rat > nat ) > rat ).
thf(func_def_1375,type,
sK380: ( rat > nat ) > rat ).
thf(func_def_1376,type,
sK381: int > int ).
thf(func_def_1377,type,
sK382: real > real > nat ).
thf(func_def_1378,type,
sK383: int > ( int > $o ) > int ).
thf(func_def_1379,type,
sK384: ( nat > nat > $o ) > nat ).
thf(func_def_1380,type,
sK385: ( nat > nat > $o ) > nat ).
thf(func_def_1381,type,
sK386: ( nat > nat > $o ) > nat ).
thf(func_def_1382,type,
sK387: ( nat > nat > $o ) > nat ).
thf(func_def_1383,type,
sK388: ( nat > nat > $o ) > nat ).
thf(func_def_1384,type,
sK389: real > nat ).
thf(func_def_1385,type,
sK390: ( nat > rat ) > nat ).
thf(func_def_1386,type,
sK391: ( nat > rat ) > nat ).
thf(func_def_1387,type,
sK392: ( rat > int ) > rat ).
thf(func_def_1388,type,
sK393: ( rat > int ) > rat ).
thf(func_def_1389,type,
sK394: real > real > real ).
thf(func_def_1390,type,
sK395: rat > rat ).
thf(func_def_1391,type,
sK396: ( rat > nat ) > rat ).
thf(func_def_1392,type,
sK397: ( rat > nat ) > rat ).
thf(func_def_1393,type,
sK398: nat > nat > nat ).
thf(func_def_1394,type,
sK399: code_integer > code_integer > code_integer > code_integer ).
thf(func_def_1395,type,
sK400: ( nat > rat ) > nat ).
thf(func_def_1396,type,
sK401: ( nat > $o ) > nat > nat ).
thf(func_def_1397,type,
sK402: nat > nat ).
thf(func_def_1398,type,
sK403: num > num ).
thf(func_def_1399,type,
sK404: num > num ).
thf(func_def_1400,type,
sK405: int > int ).
thf(func_def_1401,type,
sK406: ( list_VEBT_VEBT > $o ) > list_VEBT_VEBT ).
thf(func_def_1402,type,
sK407: nat > nat > nat > nat ).
thf(func_def_1403,type,
sK408: nat > nat > nat > nat ).
thf(func_def_1404,type,
sK409: real > real > real ).
thf(func_def_1405,type,
sK410: int > int ).
thf(func_def_1406,type,
sK411: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1407,type,
sK412: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1408,type,
sK413: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1409,type,
sK414: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1410,type,
sK415: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1411,type,
sK416: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1412,type,
sK417: ( nat > nat ) > nat ).
thf(func_def_1413,type,
sK418: int > nat ).
thf(func_def_1414,type,
sK419: int > nat ).
thf(func_def_1415,type,
sK420: ( real > real > $o ) > real ).
thf(func_def_1416,type,
sK421: ( real > real > $o ) > real ).
thf(func_def_1417,type,
sK422: ( real > real > $o ) > real ).
thf(func_def_1418,type,
sK423: ( real > real > $o ) > real ).
thf(func_def_1419,type,
sK424: ( real > real > $o ) > real ).
thf(func_def_1420,type,
sK425: ( nat > $o ) > nat ).
thf(func_def_1421,type,
sK426: ( nat > $o ) > nat ).
thf(func_def_1422,type,
sK427: ( int > real ) > int ).
thf(func_def_1423,type,
sK428: ( int > real ) > int ).
thf(func_def_1424,type,
sK429: nat > nat > nat ).
thf(func_def_1425,type,
sK430: rat > rat ).
thf(func_def_1426,type,
sK431: real > real ).
thf(func_def_1427,type,
sK432: nat > nat > nat ).
thf(func_def_1428,type,
sK433: nat > nat > nat ).
thf(func_def_1429,type,
sK434: nat > nat > nat ).
thf(func_def_1430,type,
sK435: ( nat > $o ) > nat ).
thf(func_def_1431,type,
sK436: ( rat > real ) > rat ).
thf(func_def_1432,type,
sK437: ( rat > real ) > rat ).
thf(func_def_1433,type,
sK438: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1434,type,
sK439: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1435,type,
sK440: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1436,type,
sK441: rat > rat ).
thf(func_def_1437,type,
sK442: int > int > int > int ).
thf(func_def_1438,type,
sK443: int > int > int > int ).
thf(func_def_1439,type,
sK444: int > int ).
thf(func_def_1440,type,
sK445: ( nat > $o ) > nat ).
thf(func_def_1441,type,
sK446: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1442,type,
sK447: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1443,type,
sK448: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1444,type,
sK449: ( real > real ) > real ).
thf(func_def_1445,type,
sK450: ( real > real ) > real ).
thf(func_def_1446,type,
sK451: ( nat > $o ) > nat > nat ).
thf(func_def_1447,type,
sK452: real > real > real ).
thf(func_def_1448,type,
sK453: int > nat ).
thf(func_def_1449,type,
sK454: ( rat > rat > $o ) > rat ).
thf(func_def_1450,type,
sK455: ( rat > rat > $o ) > rat ).
thf(func_def_1451,type,
sK456: ( rat > rat > $o ) > rat ).
thf(func_def_1452,type,
sK457: ( rat > rat > $o ) > rat ).
thf(func_def_1453,type,
sK458: real > real ).
thf(func_def_1454,type,
sK459: ( rat > rat ) > rat ).
thf(func_def_1455,type,
sK460: ( rat > rat ) > rat ).
thf(func_def_1456,type,
sK461: ( real > rat ) > real ).
thf(func_def_1457,type,
sK462: ( real > rat ) > real ).
thf(func_def_1458,type,
sK463: real > real ).
thf(func_def_1459,type,
sK464: ( int > int > $o ) > int ).
thf(func_def_1460,type,
sK465: ( int > int > $o ) > int ).
thf(func_def_1461,type,
sK466: ( int > int > $o ) > int ).
thf(func_def_1462,type,
sK467: ( int > int > $o ) > int ).
thf(func_def_1463,type,
sK468: ( rat > nat ) > rat ).
thf(func_def_1464,type,
sK469: ( rat > nat ) > rat ).
thf(func_def_1465,type,
sK470: ( nat > $o ) > nat ).
thf(func_def_1466,type,
sK471: ( nat > real ) > nat ).
thf(func_def_1467,type,
sK472: ( nat > real ) > nat ).
thf(func_def_1468,type,
sK473: ( nat > real ) > nat ).
thf(func_def_1469,type,
sK474: ( nat > real ) > nat ).
thf(func_def_1470,type,
sK475: real > real ).
thf(func_def_1471,type,
sK476: rat > rat > rat ).
thf(func_def_1472,type,
sK477: ( rat > rat ) > rat ).
thf(func_def_1473,type,
sK478: ( rat > rat ) > rat ).
thf(func_def_1474,type,
sK479: real > real > nat ).
thf(func_def_1475,type,
sK480: ( nat > $o ) > nat > nat ).
thf(func_def_1476,type,
sK481: ( nat > $o ) > nat > nat ).
thf(func_def_1477,type,
sK482: code_integer > code_integer ).
thf(func_def_1478,type,
sK483: rat > rat ).
thf(func_def_1479,type,
sK484: rat > rat ).
thf(func_def_1480,type,
sK485: code_integer > code_integer > code_integer ).
thf(func_def_1481,type,
sK486: real > real > real ).
thf(func_def_1482,type,
sK487: ( real > nat ) > real ).
thf(func_def_1483,type,
sK488: ( real > nat ) > real ).
thf(func_def_1484,type,
sK489: ( rat > nat ) > rat ).
thf(func_def_1485,type,
sK490: ( rat > nat ) > rat ).
thf(func_def_1486,type,
sK491: ( nat > int ) > nat ).
thf(func_def_1487,type,
sK492: nat > nat ).
thf(func_def_1488,type,
sK493: ( rat > real ) > rat ).
thf(func_def_1489,type,
sK494: ( rat > real ) > rat ).
thf(func_def_1490,type,
sK495: ( int > $o ) > int > int ).
thf(func_def_1491,type,
sK496: ( int > $o ) > int > int ).
thf(func_def_1492,type,
sK497: ( int > $o ) > int > int ).
thf(func_def_1493,type,
sK498: ( int > $o ) > int > int ).
thf(func_def_1494,type,
sK499: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1495,type,
sK500: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1496,type,
sK501: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1497,type,
sK502: ( nat > $o ) > nat > nat ).
thf(func_def_1498,type,
sK503: ( nat > $o ) > nat > nat ).
thf(func_def_1499,type,
sK504: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1500,type,
sK505: nat > vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_1501,type,
sK506: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1502,type,
sK507: nat > nat ).
thf(func_def_1503,type,
sK508: ( nat > $o ) > nat > nat ).
thf(func_def_1504,type,
sK509: real > real > real ).
thf(func_def_1505,type,
sK510: ( nat > int ) > nat ).
thf(func_def_1506,type,
sK511: code_integer > code_integer > code_integer ).
thf(func_def_1507,type,
sK512: int > int ).
thf(func_def_1508,type,
sK513: ( int > $o ) > int > int ).
thf(func_def_1509,type,
sK514: ( int > $o ) > int > int ).
thf(func_def_1510,type,
sK515: ( int > $o ) > int > int ).
thf(func_def_1511,type,
sK516: ( int > $o ) > int > int ).
thf(func_def_1512,type,
sK517: real > nat ).
thf(func_def_1513,type,
sK518: ( rat > real ) > rat ).
thf(func_def_1514,type,
sK519: ( rat > real ) > rat ).
thf(func_def_1515,type,
sK520: vEBT_VEBT > $o ).
thf(func_def_1516,type,
sK521: vEBT_VEBT > $o ).
thf(func_def_1517,type,
sK522: nat > nat ).
thf(func_def_1518,type,
sK523: rat > rat ).
thf(func_def_1519,type,
sK524: real > nat ).
thf(func_def_1520,type,
sK525: nat > nat ).
thf(func_def_1521,type,
sK526: nat > nat > nat ).
thf(func_def_1522,type,
sK527: rat > rat ).
thf(func_def_1523,type,
sK528: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1524,type,
sK529: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1525,type,
sK530: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1526,type,
sK531: ( rat > int ) > rat ).
thf(func_def_1527,type,
sK532: ( rat > int ) > rat ).
thf(func_def_1528,type,
sK533: real > real ).
thf(func_def_1529,type,
sK534: ( real > rat ) > real ).
thf(func_def_1530,type,
sK535: ( real > rat ) > real ).
thf(func_def_1531,type,
sK536: ( int > rat ) > int ).
thf(func_def_1532,type,
sK537: ( int > rat ) > int ).
thf(func_def_1533,type,
sK538: ( int > $o ) > int ).
thf(func_def_1534,type,
sK539: ( int > $o ) > int ).
thf(func_def_1535,type,
sK540: int > int > int ).
thf(func_def_1536,type,
sK541: nat > nat ).
thf(func_def_1537,type,
sK542: nat > nat ).
thf(func_def_1538,type,
sK543: int > int ).
thf(func_def_1539,type,
sK544: ( nat > nat > $o ) > nat ).
thf(func_def_1540,type,
sK545: ( nat > nat > $o ) > nat ).
thf(func_def_1541,type,
sK546: ( nat > nat > $o ) > nat ).
thf(func_def_1542,type,
sK547: ( nat > nat > $o ) > nat ).
thf(func_def_1543,type,
sK548: ( nat > nat > $o ) > nat ).
thf(func_def_1544,type,
sK549: ( rat > rat > $o ) > rat ).
thf(func_def_1545,type,
sK550: ( rat > rat > $o ) > rat ).
thf(func_def_1546,type,
sK551: ( rat > rat > $o ) > rat ).
thf(func_def_1547,type,
sK552: ( rat > rat > $o ) > rat ).
thf(func_def_1548,type,
sK553: ( rat > rat > $o ) > rat ).
thf(func_def_1549,type,
sK554: ( rat > real ) > rat ).
thf(func_def_1550,type,
sK555: ( rat > real ) > rat ).
thf(func_def_1551,type,
sK556: ( int > $o ) > int > int ).
thf(func_def_1552,type,
sK557: vEBT_VEBT > nat ).
thf(func_def_1553,type,
sK558: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1554,type,
sK559: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1555,type,
sK560: nat > $o > vEBT_VEBT > nat ).
thf(func_def_1556,type,
sK561: nat > $o > vEBT_VEBT > nat ).
thf(func_def_1557,type,
sK562: nat > $o > vEBT_VEBT > nat ).
thf(func_def_1558,type,
sK563: nat > $o > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1559,type,
sK564: nat > $o > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1560,type,
sK565: nat > vEBT_VEBT > $o > $o ).
thf(func_def_1561,type,
sK566: nat > vEBT_VEBT > $o > $o ).
thf(func_def_1562,type,
sK567: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1563,type,
sK568: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1564,type,
sK569: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1565,type,
sK570: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1566,type,
sK571: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1567,type,
sK572: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1568,type,
sK573: ( rat > rat ) > rat ).
thf(func_def_1569,type,
sK574: ( rat > rat ) > rat ).
thf(func_def_1570,type,
sK575: nat > nat > nat ).
thf(func_def_1571,type,
sK576: int > int > int > int ).
thf(func_def_1572,type,
sK577: int > int > int > int ).
thf(func_def_1573,type,
sK578: vEBT_VEBT > nat > nat ).
thf(func_def_1574,type,
sK579: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1575,type,
sK580: vEBT_VEBT > nat > nat ).
thf(func_def_1576,type,
sK581: vEBT_VEBT > nat > nat ).
thf(func_def_1577,type,
sK582: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1578,type,
sK583: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1579,type,
sK584: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1580,type,
sK585: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1581,type,
sK586: nat > vEBT_VEBT > $o ).
thf(func_def_1582,type,
sK587: nat > vEBT_VEBT > $o ).
thf(func_def_1583,type,
sK588: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1584,type,
sK589: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1585,type,
sK590: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1586,type,
sK591: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1587,type,
sK592: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1588,type,
sK593: vEBT_VEBT > nat ).
thf(func_def_1589,type,
sK594: ( real > int ) > real ).
thf(func_def_1590,type,
sK595: ( real > int ) > real ).
thf(func_def_1591,type,
sK596: ( nat > nat > $o ) > nat ).
thf(func_def_1592,type,
sK597: ( nat > nat > $o ) > nat ).
thf(func_def_1593,type,
sK598: ( nat > nat > $o ) > nat ).
thf(func_def_1594,type,
sK599: ( nat > nat > $o ) > nat ).
thf(func_def_1595,type,
sK600: real > ( real > real > $o ) > real > real ).
thf(func_def_1596,type,
sK601: real > real > ( real > real > $o ) > real > real ).
thf(func_def_1597,type,
sK602: real > real > ( real > real > $o ) > real > real ).
thf(func_def_1598,type,
sK603: ( real > real > $o ) > real ).
thf(func_def_1599,type,
sK604: ( real > real > $o ) > real ).
thf(func_def_1600,type,
sK605: ( real > real > $o ) > real ).
thf(func_def_1601,type,
sK606: nat > ( nat > $o ) > nat ).
thf(func_def_1602,type,
sK607: nat > ( nat > $o ) > nat ).
thf(func_def_1603,type,
sK608: nat > ( nat > $o ) > nat > nat ).
thf(func_def_1604,type,
sK609: real > nat ).
thf(func_def_1605,type,
sK610: real > nat ).
thf(func_def_1606,type,
sK611: real > nat ).
thf(func_def_1607,type,
sK612: nat > nat ).
thf(func_def_1608,type,
sK613: int > int > int ).
thf(func_def_1609,type,
sK614: real > nat ).
thf(func_def_1610,type,
sK615: real > nat ).
thf(func_def_1611,type,
sK616: real > int ).
thf(func_def_1612,type,
sK617: code_integer > ( code_integer > $o ) > code_integer ).
thf(func_def_1613,type,
sK618: code_integer > ( code_integer > $o ) > code_integer ).
thf(func_def_1614,type,
sK619: nat > ( nat > $o ) > nat > nat ).
thf(func_def_1615,type,
sK620: nat > ( nat > $o ) > nat > nat ).
thf(func_def_1616,type,
sK621: nat > nat > nat ).
thf(func_def_1617,type,
sK622: real > int ).
thf(func_def_1618,type,
sK623: real > int ).
thf(func_def_1619,type,
sK624: ( int > $o ) > int > int ).
thf(func_def_1620,type,
sK625: ( int > $o ) > int > int ).
thf(func_def_1621,type,
sK626: nat > nat > nat > nat ).
thf(func_def_1622,type,
sK627: nat > nat > nat > nat ).
thf(func_def_1623,type,
sK628: ( nat > $o ) > nat > nat ).
thf(func_def_1624,type,
sK629: ( nat > $o ) > nat > nat ).
thf(func_def_1625,type,
sK630: ( nat > $o ) > nat > nat ).
thf(func_def_1626,type,
sK631: real > nat ).
thf(func_def_1627,type,
sK632: real > nat ).
thf(func_def_1628,type,
sK633: real > nat ).
thf(func_def_1629,type,
sK634: nat > ( nat > $o ) > nat ).
thf(func_def_1630,type,
sK635: nat > ( nat > $o ) > nat ).
thf(func_def_1631,type,
sK636: nat > ( nat > $o ) > nat ).
thf(func_def_1632,type,
sK637: nat > nat > nat ).
thf(func_def_1633,type,
sK638: vEBT_VEBT > $o ).
thf(func_def_1634,type,
sK639: vEBT_VEBT > $o ).
thf(func_def_1635,type,
sK640: real > int ).
thf(func_def_1636,type,
ph641:
!>[X0: $tType] : X0 ).
thf(f46774,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f46773,f43977]) ).
thf(f43977,plain,
( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) )
!= ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) ),
inference(backward_demodulation,[status(thm)],[f30550,f28429]) ).
thf(f28429,plain,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ one_one_nat ) ),
inference(cnf_transformation,[status(thm)],[f336]) ).
thf(f336,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ one_one_nat ) ),
file('/export/starexec/sandbox2/tmp/tmp.5KBf3uDrBi/DTF2THF_21372.p',fact_335_Suc__1) ).
thf(f30550,plain,
( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
!= ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f22876]) ).
thf(f22876,plain,
( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
!= ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ),
inference(flattening,[status(thm)],[f10258]) ).
thf(f10258,negated_conjecture,
( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
!= ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f10257]) ).
thf(f10257,conjecture,
( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.5KBf3uDrBi/DTF2THF_21372.p',conj_0) ).
thf(f46773,plain,
( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) )
= ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) ),
inference(subsumption_resolution,[status(thm)],[f46765,f43979]) ).
thf(f43979,plain,
( ( ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f32345,f28429]) ).
thf(f32345,plain,
( ( ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f15945]) ).
thf(f15945,plain,
( ( ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f15944]) ).
thf(f15944,plain,
ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.5KBf3uDrBi/DTF2THF_21372.p',fact_2_aaaa) ).
thf(f46765,plain,
( ( ( ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) )
!= $true )
| ( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) )
= ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f46750]) ).
thf(f46750,plain,
( ( ( ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) )
!= $true )
| ( $true != $true )
| ( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) )
= ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) ) ),
inference(superposition,[status(thm)],[f29645,f43976]) ).
thf(f43976,plain,
( ( ord_less_eq_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( suc @ one_one_nat ) ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f30397,f28429]) ).
thf(f30397,plain,
( ( ord_less_eq_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f14471]) ).
thf(f14471,plain,
( ( ord_less_eq_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f14470]) ).
thf(f14470,plain,
ord_less_eq_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
ord_less_eq_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.5KBf3uDrBi/DTF2THF_21372.p',fact_0_bbbb) ).
thf(f29645,plain,
! [X0: nat,X1: nat] :
( ( ( ord_less_eq_nat @ X1 @ X0 )
!= $true )
| ( ( ord_less_eq_nat @ X0 @ X1 )
!= $true )
| ( X0 = X1 ) ),
inference(cnf_transformation,[status(thm)],[f22966]) ).
thf(f22966,plain,
! [X0: nat,X1: nat] :
( ( ( ord_less_eq_nat @ X0 @ X1 )
!= $true )
| ( X0 = X1 )
| ( ( ord_less_eq_nat @ X1 @ X0 )
!= $true ) ),
inference(flattening,[status(thm)],[f17277]) ).
thf(f17277,plain,
! [X0: nat,X1: nat] :
( ( X0 = X1 )
| ( ( ord_less_eq_nat @ X0 @ X1 )
!= $true )
| ( ( ord_less_eq_nat @ X1 @ X0 )
!= $true ) ),
inference(fool_elimination,[status(thm)],[f17276]) ).
thf(f17276,plain,
! [X0: nat,X1: nat] :
( ( X0 = X1 )
| ~ ( ord_less_eq_nat @ X0 @ X1 )
| ~ ( ord_less_eq_nat @ X1 @ X0 ) ),
inference(rectify,[status(thm)],[f50]) ).
thf(f50,axiom,
! [X6: nat,X11: nat] :
( ( X6 = X11 )
| ~ ( ord_less_eq_nat @ X6 @ X11 )
| ~ ( ord_less_eq_nat @ X11 @ X6 ) ),
file('/export/starexec/sandbox2/tmp/tmp.5KBf3uDrBi/DTF2THF_21372.p',fact_49_verit__la__disequality) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : ITP246^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.00/0.10 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n002.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.30 % CPULimit : 300
% 0.11/0.30 % WCLimit : 300
% 0.11/0.30 % DateTime : Thu Feb 26 21:45:18 EST 2026
% 0.11/0.30 % CPUTime :
% 0.11/0.30 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.39 ---- Original DTF file ---
% 0.17/0.39 thf(spec,logic,$$dhol).
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 % File : ITP246^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.17/0.39 % Domain : Interactive Theorem Proving
% 0.17/0.39 % Problem : Sledgehammer problem VEBT_Succ 00795_053196
% 0.17/0.39 % Version : [Des22] axioms.
% 0.17/0.39 % English :
% 0.17/0.39
% 0.17/0.39 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.17/0.39 % : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.17/0.39 % Source : [Des22]
% 0.17/0.39 % Names : 0070_VEBT_Succ_00795_053196 [Des22]
% 0.17/0.39
% 0.17/0.39 % Status : Theorem
% 0.17/0.39 % Rating : ? v9.2.1
% 0.17/0.39 % Syntax : Number of formulae : 11234 (6275 unt; 977 typ; 0 def)
% 0.17/0.39 % Number of atoms : 26809 (11987 equ; 0 cnn)
% 0.17/0.39 % Maximal formula atoms : 71 ( 2 avg)
% 0.17/0.39 % Number of connectives : 104965 (2569 ~; 512 |;1523 &;91781 @)
% 0.17/0.39 % ( 0 <=>;8580 =>; 0 <=; 0 <~>)
% 0.17/0.39 % Maximal formula depth : 39 ( 5 avg)
% 0.17/0.39 % Number of types : 85 ( 84 usr)
% 0.17/0.39 % Number of type conns : 3106 (3106 >; 0 *; 0 +; 0 <<)
% 0.17/0.39 % Number of symbols : 896 ( 893 usr; 69 con; 0-8 aty)
% 0.17/0.39 % Number of variables : 23057 (1889 ^;20505 !; 663 ?;23057 :)
% 0.17/0.39 % SPC : TH0_THM_EQU_NAR
% 0.17/0.39
% 0.17/0.39 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.17/0.39 % from the van Emde Boas Trees session in the Archive of Formal
% 0.17/0.39 % proofs -
% 0.17/0.39 % www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.17/0.39 % 2022-02-18 01:26:34.367
% 0.17/0.39 % Bugfixes : Added ()s around negated sides of equations.
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 % Could-be-implicit typings (84)
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J_J,type,
% 0.17/0.39 produc3368934014287244435at_num: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.17/0.39 produc4471711990508489141at_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.17/0.39 set_Pr8693737435421807431at_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.17/0.39 produc859450856879609959at_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_J,type,
% 0.17/0.39 set_fi4554929511873752355omplex: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.17/0.39 list_P7413028617227757229T_VEBT: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
% 0.17/0.39 produc2963631642982155120at_num: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.17/0.39 produc7248412053542808358at_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
% 0.17/0.39 set_fi7789364187291644575l_real: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.17/0.39 filter6041513312241820739omplex: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 0.17/0.39 list_P7037539587688870467BT_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 0.17/0.39 list_P4547456442757143711BT_int: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.17/0.39 list_P5647936690300460905T_VEBT: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.17/0.39 produc8243902056947475879T_VEBT: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.17/0.39 set_Pr5085853215250843933omplex: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.17/0.39 produc8923325533196201883nteger: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
% 0.17/0.39 produc7272778201969148633d_enat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 0.17/0.39 list_P3126845725202233233VEBT_o: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.17/0.39 list_P7495141550334521929T_VEBT: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.17/0.39 filter2146258269922977983l_real: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.17/0.39 option4927543243414619207at_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.17/0.39 filter1242075044329608583at_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.17/0.39 set_Pr6218003697084177305l_real: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
% 0.17/0.39 list_P3744719386663036955um_num: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
% 0.17/0.39 list_P1726324292696863441at_num: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.17/0.39 produc9072475918466114483BT_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 0.17/0.39 produc4894624898956917775BT_int: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.17/0.39 set_Pr1261947904930325089at_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.17/0.39 set_Pr958786334691620121nt_int: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.17/0.39 produc4411394909380815293omplex: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 0.17/0.39 list_P7333126701944960589_nat_o: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 0.17/0.39 list_P6285523579766656935_o_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 0.17/0.39 list_P3795440434834930179_o_int: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 0.17/0.39 produc334124729049499915VEBT_o: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.17/0.39 produc2504756804600209347T_VEBT: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.17/0.39 produc6271795597528267376eger_o: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.17/0.39 produc2422161461964618553l_real: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 0.17/0.39 product_prod_num_num: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 0.17/0.39 product_prod_nat_num: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.39 product_prod_nat_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.17/0.39 product_prod_int_int: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 0.17/0.39 list_P4002435161011370285od_o_o: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.17/0.39 list_list_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.17/0.39 list_VEBT_VEBT: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.17/0.39 set_list_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 0.17/0.39 product_prod_o_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 0.17/0.39 product_prod_o_int: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 0.17/0.39 set_VEBT_VEBT: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.17/0.39 set_set_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 0.17/0.39 set_Code_integer: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Extended____Nat__Oenat_J,type,
% 0.17/0.39 list_Extended_enat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
% 0.17/0.39 set_Product_unit: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
% 0.17/0.39 set_Extended_enat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.17/0.39 list_complex: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 0.17/0.39 product_prod_o_o: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.17/0.39 set_complex: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 0.17/0.39 filter_real: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
% 0.17/0.39 option_num: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
% 0.17/0.39 option_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
% 0.17/0.39 filter_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
% 0.17/0.39 set_char: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
% 0.17/0.39 list_real: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.17/0.39 set_real: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
% 0.17/0.39 list_num: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.39 list_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
% 0.17/0.39 list_int: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__VEBT____Definitions__OVEBT,type,
% 0.17/0.39 vEBT_VEBT: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
% 0.17/0.39 set_rat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
% 0.17/0.39 set_num: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.39 set_nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 0.17/0.39 set_int: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Code____Numeral__Ointeger,type,
% 0.17/0.39 code_integer: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Product____Type__Ounit,type,
% 0.17/0.39 product_unit: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Extended____Nat__Oenat,type,
% 0.17/0.39 extended_enat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__List__Olist_I_Eo_J,type,
% 0.17/0.39 list_o: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Complex__Ocomplex,type,
% 0.17/0.39 complex: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__String__Oliteral,type,
% 0.17/0.39 literal: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Set__Oset_I_Eo_J,type,
% 0.17/0.39 set_o: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__String__Ochar,type,
% 0.17/0.39 char: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Real__Oreal,type,
% 0.17/0.39 real: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Rat__Orat,type,
% 0.17/0.39 rat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Num__Onum,type,
% 0.17/0.39 num: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Nat__Onat,type,
% 0.17/0.39 nat: $tType ).
% 0.17/0.39
% 0.17/0.39 thf(ty_n_t__Int__Oint,type,
% 0.17/0.39 int: $tType ).
% 0.17/0.39
% 0.17/0.39 % Explicit typings (893)
% 0.17/0.39 thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
% 0.17/0.39 archim2889992004027027881ng_rat: rat > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
% 0.17/0.39 archim7802044766580827645g_real: real > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
% 0.17/0.39 archim3151403230148437115or_rat: rat > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
% 0.17/0.39 archim6058952711729229775r_real: real > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
% 0.17/0.39 archimedean_frac_rat: rat > rat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
% 0.17/0.39 archim2898591450579166408c_real: real > real ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
% 0.17/0.39 archim7778729529865785530nd_rat: rat > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
% 0.17/0.39 archim8280529875227126926d_real: real > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
% 0.17/0.39 bNF_re1962705104956426057at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
% 0.17/0.39 bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
% 0.17/0.39 bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.17/0.39 bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_Eo_J_001_062_It__Real__Oreal_M_Eo_J,type,
% 0.17/0.39 bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
% 0.17/0.39 bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
% 0.17/0.39 bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 0.17/0.39 bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.17/0.39 bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
% 0.17/0.39 bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 0.17/0.39 bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.17/0.39 bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.39 bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
% 0.17/0.39 bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.39 bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
% 0.17/0.39 bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
% 0.17/0.39 bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.17/0.39 bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 0.17/0.39 bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.17/0.39 bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 0.17/0.39 bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
% 0.17/0.39 bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 0.17/0.39 bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 0.17/0.39 bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.17/0.39 bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
% 0.17/0.39 bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.39 bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.17/0.39 bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.17/0.39 bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.17/0.39 bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 0.17/0.39 bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.39 bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.39 bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Binomial_Obinomial,type,
% 0.17/0.39 binomial: nat > nat > nat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
% 0.17/0.39 gbinomial_complex: complex > nat > complex ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
% 0.17/0.39 gbinomial_int: int > nat > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
% 0.17/0.39 gbinomial_nat: nat > nat > nat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
% 0.17/0.39 gbinomial_rat: rat > nat > rat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
% 0.17/0.39 gbinomial_real: real > nat > real ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 0.17/0.39 bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 0.17/0.39 bit_and_not_num: num > num > option_num ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 0.17/0.39 bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 0.17/0.39 bit_concat_bit: nat > int > int > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 0.17/0.39 bit_or_not_num_neg: num > num > num ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
% 0.17/0.39 bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
% 0.17/0.39 bit_ri7632146776885996613nteger: code_integer > code_integer ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
% 0.17/0.39 bit_ri7919022796975470100ot_int: int > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
% 0.17/0.39 bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
% 0.17/0.39 bit_ri631733984087533419it_int: nat > int > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
% 0.17/0.39 bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
% 0.17/0.39 bit_se725231765392027082nd_int: int > int > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
% 0.17/0.39 bit_se727722235901077358nd_nat: nat > nat > nat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
% 0.17/0.39 bit_se8568078237143864401it_int: nat > int > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
% 0.17/0.39 bit_se8570568707652914677it_nat: nat > nat > nat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
% 0.17/0.39 bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
% 0.17/0.39 bit_se2159334234014336723it_int: nat > int > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
% 0.17/0.39 bit_se2161824704523386999it_nat: nat > nat > nat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Code____Numeral__Ointeger,type,
% 0.17/0.39 bit_se2119862282449309892nteger: nat > code_integer ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
% 0.17/0.39 bit_se2000444600071755411sk_int: nat > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
% 0.17/0.39 bit_se2002935070580805687sk_nat: nat > nat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
% 0.17/0.39 bit_se1409905431419307370or_int: int > int > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
% 0.17/0.39 bit_se1412395901928357646or_nat: nat > nat > nat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
% 0.17/0.39 bit_se7788150548672797655nteger: nat > code_integer > code_integer ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
% 0.17/0.39 bit_se545348938243370406it_int: nat > int > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
% 0.17/0.39 bit_se547839408752420682it_nat: nat > nat > nat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
% 0.17/0.39 bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
% 0.17/0.39 bit_se7879613467334960850it_int: nat > int > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
% 0.17/0.39 bit_se7882103937844011126it_nat: nat > nat > nat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Ointeger,type,
% 0.17/0.39 bit_se1745604003318907178nteger: nat > code_integer > code_integer ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
% 0.17/0.39 bit_se2923211474154528505it_int: nat > int > int ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
% 0.17/0.39 bit_se2925701944663578781it_nat: nat > nat > nat ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
% 0.17/0.39 bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
% 0.17/0.39
% 0.17/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
% 0.17/0.40 bit_se4203085406695923979it_int: nat > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
% 0.17/0.40 bit_se4205575877204974255it_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
% 0.17/0.40 bit_se6526347334894502574or_int: int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
% 0.17/0.40 bit_se6528837805403552850or_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 bit_se9216721137139052372nteger: code_integer > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
% 0.17/0.40 bit_se1146084159140164899it_int: int > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
% 0.17/0.40 bit_se1148574629649215175it_nat: nat > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Otake__bit__num,type,
% 0.17/0.40 bit_take_bit_num: nat > num > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
% 0.17/0.40 bit_un1837492267222099188nd_num: num > num > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num__rel,type,
% 0.17/0.40 bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
% 0.17/0.40 bit_un6178654185764691216or_num: num > num > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num__rel,type,
% 0.17/0.40 bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
% 0.17/0.40 bit_un7362597486090784418nd_num: num > num > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
% 0.17/0.40 bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
% 0.17/0.40 bit_un2480387367778600638or_num: num > num > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
% 0.17/0.40 bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Code__Numeral_Obit__cut__integer,type,
% 0.17/0.40 code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Code__Numeral_Odivmod__abs,type,
% 0.17/0.40 code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Code__Numeral_Odivmod__integer,type,
% 0.17/0.40 code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
% 0.17/0.40 code_int_of_integer: code_integer > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
% 0.17/0.40 code_integer_of_int: int > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Code__Numeral_Ointeger__of__num,type,
% 0.17/0.40 code_integer_of_num: num > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Code__Numeral_Onat__of__integer,type,
% 0.17/0.40 code_nat_of_integer: code_integer > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Code__Numeral_Onum__of__integer,type,
% 0.17/0.40 code_num_of_integer: code_integer > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 comple2295165028678016749d_enat: set_Extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.17/0.40 comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.17/0.40 comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
% 0.17/0.40 comple4887499456419720421f_real: set_real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 comple7806235888213564991et_nat: set_set_nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 comple4398354569131411667d_enat: set_Extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
% 0.17/0.40 comple1385675409528146559p_real: set_real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 comple7399068483239264473et_nat: set_set_nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complex_OArg,type,
% 0.17/0.40 arg: complex > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complex_Ocis,type,
% 0.17/0.40 cis: real > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complex_Ocnj,type,
% 0.17/0.40 cnj: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complex_Ocomplex_OComplex,type,
% 0.17/0.40 complex2: real > real > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complex_Ocomplex_OIm,type,
% 0.17/0.40 im: complex > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complex_Ocomplex_ORe,type,
% 0.17/0.40 re: complex > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complex_Ocsqrt,type,
% 0.17/0.40 csqrt: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complex_Oimaginary__unit,type,
% 0.17/0.40 imaginary_unit: complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Complex_Orcis,type,
% 0.17/0.40 rcis: real > real > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
% 0.17/0.40 has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Deriv_Ohas__vector__derivative_001t__Real__Oreal,type,
% 0.17/0.40 has_ve631408500373753343e_real: ( real > real ) > real > filter_real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Oadjust__div,type,
% 0.17/0.40 adjust_div: product_prod_int_int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Oadjust__mod,type,
% 0.17/0.40 adjust_mod: int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Odivmod__nat,type,
% 0.17/0.40 divmod_nat: nat > nat > product_prod_nat_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Oeucl__rel__int,type,
% 0.17/0.40 eucl_rel_int: int > int > product_prod_int_int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
% 0.17/0.40 unique6319869463603278526ux_int: product_prod_int_int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
% 0.17/0.40 unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
% 0.17/0.40 unique5052692396658037445od_int: num > num > product_prod_int_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
% 0.17/0.40 unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
% 0.17/0.40 unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
% 0.17/0.40 unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Int__Oint,type,
% 0.17/0.40 euclid4774559944035922753ze_int: int > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Nat__Onat,type,
% 0.17/0.40 euclid4777050414544973029ze_nat: nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Int__Oint,type,
% 0.17/0.40 euclid3395696857347342551nt_int: int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Nat__Onat,type,
% 0.17/0.40 euclid3398187327856392827nt_nat: nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Extended__Nat_OeSuc,type,
% 0.17/0.40 extended_eSuc: extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Extended__Nat_Oenat,type,
% 0.17/0.40 extended_enat2: nat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001_Eo,type,
% 0.17/0.40 extended_case_enat_o: ( nat > $o ) > $o > extended_enat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 extend3600170679010898289d_enat: ( nat > extended_enat ) > extended_enat > extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 extend5688581933313929465d_enat: extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Extended__Nat_Othe__enat,type,
% 0.17/0.40 extended_the_enat: extended_enat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
% 0.17/0.40 comm_s2602460028002588243omplex: complex > nat > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
% 0.17/0.40 comm_s4660882817536571857er_int: int > nat > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
% 0.17/0.40 comm_s4663373288045622133er_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
% 0.17/0.40 comm_s4028243227959126397er_rat: rat > nat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
% 0.17/0.40 comm_s7457072308508201937r_real: real > nat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 semiri3624122377584611663nteger: nat > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
% 0.17/0.40 semiri5044797733671781792omplex: nat > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
% 0.17/0.40 semiri1406184849735516958ct_int: nat > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
% 0.17/0.40 semiri1408675320244567234ct_nat: nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
% 0.17/0.40 semiri773545260158071498ct_rat: nat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
% 0.17/0.40 semiri2265585572941072030t_real: nat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
% 0.17/0.40 invers8013647133539491842omplex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
% 0.17/0.40 inverse_inverse_rat: rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 0.17/0.40 inverse_inverse_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 0.17/0.40 at_bot_real: filter_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 0.17/0.40 at_top_nat: filter_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 0.17/0.40 at_top_real: filter_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 0.17/0.40 eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
% 0.17/0.40 eventually_real: ( real > $o ) > filter_real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.17/0.40 filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Ofiltermap_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 filtermap_real_real: ( real > real ) > filter_real > filter_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.17/0.40 princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.17/0.40 princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Finite__Set_Ocard_001_Eo,type,
% 0.17/0.40 finite_card_o: set_o > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
% 0.17/0.40 finite_card_complex: set_complex > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
% 0.17/0.40 finite_card_int: set_int > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.40 finite_card_list_nat: set_list_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 0.17/0.40 finite_card_nat: set_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
% 0.17/0.40 finite410649719033368117t_unit: set_Product_unit > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
% 0.17/0.40 finite_card_char: set_char > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
% 0.17/0.40 finite3207457112153483333omplex: set_complex > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Finite__Set_Ofinite_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 finite4001608067531595151d_enat: set_Extended_enat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
% 0.17/0.40 finite_finite_int: set_int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 0.17/0.40 finite_finite_nat: set_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.17/0.40 bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 0.17/0.40 bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
% 0.17/0.40 comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 comp_int_real_real: ( int > real ) > ( real > int ) > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.17/0.40 comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Oid_001_Eo,type,
% 0.17/0.40 id_o: $o > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
% 0.17/0.40 id_nat: nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
% 0.17/0.40 inj_on_nat_char: ( nat > char ) > set_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 inj_on_real_real: ( real > real ) > set_real > $o ).
% 0.17/0.40
% 0.17/0.40 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,
% 0.17/0.40 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 ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.17/0.40 map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 0.17/0.40 map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.17/0.40 map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 0.17/0.40 map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 0.17/0.40 map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 0.17/0.40 map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.17/0.40 map_fu1532550112467129777l_real: ( real > nat > rat ) > ( ( ( nat > rat ) > nat > rat ) > real > real ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > real > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
% 0.17/0.40 map_fu7146612038024189824t_real: ( real > nat > rat ) > ( ( nat > rat ) > real ) > ( ( nat > rat ) > nat > rat ) > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
% 0.17/0.40 map_fu1856342031159181835at_o_o: ( real > nat > rat ) > ( $o > $o ) > ( ( nat > rat ) > $o ) > real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun__Def_Opair__leq,type,
% 0.17/0.40 fun_pair_leq: set_Pr8693737435421807431at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Fun__Def_Opair__less,type,
% 0.17/0.40 fun_pair_less: set_Pr8693737435421807431at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
% 0.17/0.40 gcd_Gcd_nat: set_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_GCD_Obezw,type,
% 0.17/0.40 bezw: nat > nat > product_prod_int_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_GCD_Obezw__rel,type,
% 0.17/0.40 bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 gcd_gcd_Code_integer: code_integer > code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
% 0.17/0.40 gcd_gcd_int: int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
% 0.17/0.40 gcd_gcd_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_GCD_Ogcd__class_Olcm_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 gcd_lcm_Code_integer: code_integer > code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_GCD_Ogcd__class_Olcm_001t__Int__Oint,type,
% 0.17/0.40 gcd_lcm_int: int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_GCD_Ogcd__class_Olcm_001t__Nat__Onat,type,
% 0.17/0.40 gcd_lcm_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_GCD_Ogcd__nat__rel,type,
% 0.17/0.40 gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 abs_abs_Code_integer: code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
% 0.17/0.40 abs_abs_complex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
% 0.17/0.40 abs_abs_int: int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
% 0.17/0.40 abs_abs_rat: rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 0.17/0.40 abs_abs_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
% 0.17/0.40 minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Extended____Nat__Oenat_M_Eo_J,type,
% 0.17/0.40 minus_2020553357622893040enat_o: ( extended_enat > $o ) > ( extended_enat > $o ) > extended_enat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
% 0.17/0.40 minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
% 0.17/0.40 minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.17/0.40 minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
% 0.17/0.40 minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
% 0.17/0.40 minus_minus_complex: complex > complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 0.17/0.40 minus_minus_int: int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.17/0.40 minus_minus_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
% 0.17/0.40 minus_minus_rat: rat > rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 0.17/0.40 minus_minus_real: real > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.17/0.40 minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
% 0.17/0.40 minus_925952699566721837d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
% 0.17/0.40 minus_minus_set_int: set_int > set_int > set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.17/0.40 minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 minus_minus_set_nat: set_nat > set_nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.17/0.40 minus_minus_set_real: set_real > set_real > set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 one_one_Code_integer: code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
% 0.17/0.40 one_one_complex: complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 one_on7984719198319812577d_enat: extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
% 0.17/0.40 one_one_int: int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.17/0.40 one_one_nat: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
% 0.17/0.40 one_one_rat: rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 0.17/0.40 one_one_real: real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
% 0.17/0.40 plus_plus_complex: complex > complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 0.17/0.40 plus_plus_int: int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.17/0.40 plus_plus_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
% 0.17/0.40 plus_plus_num: num > num > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
% 0.17/0.40 plus_plus_rat: rat > rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 0.17/0.40 plus_plus_real: real > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Oplus__class_Oplus_001t__String__Oliteral,type,
% 0.17/0.40 plus_plus_literal: literal > literal > literal ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 sgn_sgn_Code_integer: code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
% 0.17/0.40 sgn_sgn_complex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
% 0.17/0.40 sgn_sgn_int: int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
% 0.17/0.40 sgn_sgn_rat: rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
% 0.17/0.40 sgn_sgn_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
% 0.17/0.40 times_times_complex: complex > complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
% 0.17/0.40 times_times_int: int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 0.17/0.40 times_times_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
% 0.17/0.40 times_times_num: num > num > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
% 0.17/0.40 times_times_rat: rat > rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
% 0.17/0.40 times_times_real: real > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 uminus1351360451143612070nteger: code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
% 0.17/0.40 uminus1482373934393186551omplex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 0.17/0.40 uminus_uminus_int: int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
% 0.17/0.40 uminus_uminus_rat: rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 0.17/0.40 uminus_uminus_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 uminus5710092332889474511et_nat: set_nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 zero_z3403309356797280102nteger: code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
% 0.17/0.40 zero_zero_complex: complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 zero_z5237406670263579293d_enat: extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 0.17/0.40 zero_zero_int: int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.17/0.40 zero_zero_nat: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
% 0.17/0.40 zero_zero_rat: rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.17/0.40 zero_zero_real: real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups_Ozero__class_Ozero_001t__String__Oliteral,type,
% 0.17/0.40 zero_zero_literal: literal ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.17/0.40 groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
% 0.17/0.40 groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.17/0.40 groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.17/0.40 groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Rat__Orat,type,
% 0.17/0.40 groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Real__Oreal,type,
% 0.17/0.40 groups766887009212190081x_real: ( complex > real ) > set_complex > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Complex__Ocomplex,type,
% 0.17/0.40 groups4622424608036095791omplex: ( extended_enat > complex ) > set_Extended_enat > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Int__Oint,type,
% 0.17/0.40 groups2878480467620962989at_int: ( extended_enat > int ) > set_Extended_enat > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
% 0.17/0.40 groups2880970938130013265at_nat: ( extended_enat > nat ) > set_Extended_enat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Rat__Orat,type,
% 0.17/0.40 groups2245840878043517529at_rat: ( extended_enat > rat ) > set_Extended_enat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
% 0.17/0.40 groups97031904164794029t_real: ( extended_enat > real ) > set_Extended_enat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Complex__Ocomplex,type,
% 0.17/0.40 groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
% 0.17/0.40 groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Rat__Orat,type,
% 0.17/0.40 groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Real__Oreal,type,
% 0.17/0.40 groups2316167850115554303t_real: ( int > real ) > set_int > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 0.17/0.40 groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
% 0.17/0.40 groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Rat__Orat,type,
% 0.17/0.40 groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.17/0.40 groups129246275422532515t_real: ( nat > real ) > set_nat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Complex__Ocomplex,type,
% 0.17/0.40 groups713298508707869441omplex: ( real > complex ) > set_real > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Int__Oint,type,
% 0.17/0.40 groups4694064378042380927al_int: ( real > int ) > set_real > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.17/0.40 groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Rat__Orat,type,
% 0.17/0.40 groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 groups1681761925125756287l_real: ( real > real ) > set_real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
% 0.17/0.40 groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
% 0.17/0.40 groups4561878855575611511st_nat: list_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_HOL_OThe_001t__Int__Oint,type,
% 0.17/0.40 the_int: ( int > $o ) > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.17/0.40 the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
% 0.17/0.40 the_real: ( real > $o ) > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.17/0.40 if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.17/0.40 if_nat_int_int: $o > ( nat > int > int ) > ( nat > int > int ) > nat > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.17/0.40 if_nat_nat_nat: $o > ( nat > nat > nat ) > ( nat > nat > nat ) > nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
% 0.17/0.40 if_nat_rat: $o > ( nat > rat ) > ( nat > rat ) > nat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 if_Code_integer: $o > code_integer > code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Complex__Ocomplex,type,
% 0.17/0.40 if_complex: $o > complex > complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Int__Oint,type,
% 0.17/0.40 if_int: $o > int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
% 0.17/0.40 if_list_int: $o > list_int > list_int > list_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.40 if_list_nat: $o > list_nat > list_nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Nat__Onat,type,
% 0.17/0.40 if_nat: $o > nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Num__Onum,type,
% 0.17/0.40 if_num: $o > num > num > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
% 0.17/0.40 if_option_nat: $o > option_nat > option_nat > option_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.17/0.40 if_option_num: $o > option_num > option_num > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.17/0.40 if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.17/0.40 if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.17/0.40 if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Rat__Orat,type,
% 0.17/0.40 if_rat: $o > rat > rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Real__Oreal,type,
% 0.17/0.40 if_real: $o > real > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
% 0.17/0.40 if_set_int: $o > set_int > set_int > set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
% 0.17/0.40 infini8530281810654367211te_nat: set_nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_OAbs__Integ,type,
% 0.17/0.40 abs_Integ: product_prod_nat_nat > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_ORep__Integ,type,
% 0.17/0.40 rep_Integ: int > product_prod_nat_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oint__ge__less__than,type,
% 0.17/0.40 int_ge_less_than: int > set_Pr958786334691620121nt_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oint__ge__less__than2,type,
% 0.17/0.40 int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Ointrel,type,
% 0.17/0.40 intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Onat,type,
% 0.17/0.40 nat2: int > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Opcr__int,type,
% 0.17/0.40 pcr_int: product_prod_nat_nat > int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
% 0.17/0.40 power_int_real: real > int > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 ring_11222124179247155820nteger: set_Code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
% 0.17/0.40 ring_1_Ints_complex: set_complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
% 0.17/0.40 ring_1_Ints_int: set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
% 0.17/0.40 ring_1_Ints_rat: set_rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 0.17/0.40 ring_1_Ints_real: set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 ring_18347121197199848620nteger: int > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
% 0.17/0.40 ring_17405671764205052669omplex: int > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 0.17/0.40 ring_1_of_int_int: int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
% 0.17/0.40 ring_1_of_int_rat: int > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 0.17/0.40 ring_1_of_int_real: int > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 0.17/0.40 inf_inf_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
% 0.17/0.40 semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 0.17/0.40 sup_sup_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.17/0.40 sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 lattic921264341876707157d_enat: set_Extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 0.17/0.40 lattic8265883725875713057ax_nat: set_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Lifting_OQuotient_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
% 0.17/0.40 quotie3684837364556693515t_real: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > real ) > ( real > nat > rat ) > ( ( nat > rat ) > real > $o ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.17/0.40 bfun_nat_real: ( nat > real ) > filter_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 0.17/0.40 append_int: list_int > list_int > list_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 0.17/0.40 append_nat: list_nat > list_nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Odrop_001t__Nat__Onat,type,
% 0.17/0.40 drop_nat: nat > list_nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olast_001t__Nat__Onat,type,
% 0.17/0.40 last_nat: list_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 0.17/0.40 linord2614967742042102400et_nat: set_nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 0.17/0.40 cons_int: int > list_int > list_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 0.17/0.40 cons_nat: nat > list_nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 0.17/0.40 nil_int: list_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 0.17/0.40 nil_nat: list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
% 0.17/0.40 hd_nat: list_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 map_VE8901447254227204932T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > list_VEBT_VEBT > list_VEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Oset_001_Eo,type,
% 0.17/0.40 set_o2: list_o > set_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 0.17/0.40 set_complex2: list_complex > set_complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Oset_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 set_Extended_enat2: list_Extended_enat > set_Extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 0.17/0.40 set_int2: list_int > set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.40 set_list_nat2: list_list_nat > set_list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 0.17/0.40 set_nat2: list_nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 0.17/0.40 set_real2: list_real > set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
% 0.17/0.40 tl_nat: list_nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001_Eo,type,
% 0.17/0.40 nth_o: list_o > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 0.17/0.40 nth_complex: list_complex > nat > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 nth_Extended_enat: list_Extended_enat > nat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Int__Oint,type,
% 0.17/0.40 nth_int: list_int > nat > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.40 nth_list_nat: list_list_nat > nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 0.17/0.40 nth_nat: list_nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Num__Onum,type,
% 0.17/0.40 nth_num: list_num > nat > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 0.17/0.40 nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 0.17/0.40 nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 0.17/0.40 nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.17/0.40 nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 0.17/0.40 nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 0.17/0.40 nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 0.17/0.40 nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 0.17/0.40 nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.17/0.40 nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.17/0.40 nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 0.17/0.40 nth_real: list_real > nat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 0.17/0.40 product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 0.17/0.40 product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
% 0.17/0.40 product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 0.17/0.40 product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
% 0.17/0.40 product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
% 0.17/0.40 product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 0.17/0.40 product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 0.17/0.40 produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 0.17/0.40 produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 0.17/0.40 sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 0.17/0.40 sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 0.17/0.40 take_nat: nat > list_nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oupt,type,
% 0.17/0.40 upt: nat > nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oupto,type,
% 0.17/0.40 upto: int > int > list_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oupto__aux,type,
% 0.17/0.40 upto_aux: int > int > list_int > list_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_List_Oupto__rel,type,
% 0.17/0.40 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_OSuc,type,
% 0.17/0.40 suc: nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 0.17/0.40 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 0.17/0.40 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.17/0.40 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Onat_Opred,type,
% 0.17/0.40 pred: nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 semiri4939895301339042750nteger: nat > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 0.17/0.40 semiri8010041392384452111omplex: nat > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 semiri4216267220026989637d_enat: nat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.17/0.40 semiri1314217659103216013at_int: nat > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.17/0.40 semiri1316708129612266289at_nat: nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 0.17/0.40 semiri681578069525770553at_rat: nat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 0.17/0.40 semiri5074537144036343181t_real: nat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 0.17/0.40 size_size_list_o: list_o > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.17/0.40 size_s3451745648224563538omplex: list_complex > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
% 0.17/0.40 size_s3941691890525107288d_enat: list_Extended_enat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 0.17/0.40 size_size_list_int: list_int > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.17/0.40 size_s3023201423986296836st_nat: list_list_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.40 size_size_list_nat: list_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 0.17/0.40 size_size_list_num: list_num > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 0.17/0.40 size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 0.17/0.40 size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 0.17/0.40 size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.17/0.40 size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 0.17/0.40 size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.17/0.40 size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 0.17/0.40 size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 0.17/0.40 size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 0.17/0.40 size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.17/0.40 size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 0.17/0.40 size_size_list_real: list_real > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.17/0.40 size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 0.17/0.40 size_size_num: num > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat__Bijection_Olist__decode,type,
% 0.17/0.40 nat_list_decode: nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
% 0.17/0.40 nat_list_decode_rel: nat > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 0.17/0.40 nat_list_encode: list_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 0.17/0.40 nat_list_encode_rel: list_nat > list_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat__Bijection_Oprod__decode,type,
% 0.17/0.40 nat_prod_decode: nat > product_prod_nat_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 0.17/0.40 nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.17/0.40 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 0.17/0.40 nat_prod_encode: product_prod_nat_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 0.17/0.40 nat_set_decode: nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.17/0.40 nat_set_encode: set_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Nat__Bijection_Otriangle,type,
% 0.17/0.40 nat_triangle: nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_NthRoot_Oroot,type,
% 0.17/0.40 root: nat > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_NthRoot_Osqrt,type,
% 0.17/0.40 sqrt: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_OBitM,type,
% 0.17/0.40 bitM: num > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oinc,type,
% 0.17/0.40 inc: num > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 0.17/0.40 neg_nu7009210354673126013omplex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 0.17/0.40 neg_numeral_dbl_int: int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 0.17/0.40 neg_numeral_dbl_rat: rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 0.17/0.40 neg_numeral_dbl_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 0.17/0.40 neg_nu6511756317524482435omplex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 0.17/0.40 neg_nu3811975205180677377ec_int: int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 0.17/0.40 neg_nu3179335615603231917ec_rat: rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 0.17/0.40 neg_nu6075765906172075777c_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 neg_nu5831290666863070958nteger: code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 0.17/0.40 neg_nu8557863876264182079omplex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
% 0.17/0.40 neg_nu5851722552734809277nc_int: int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 0.17/0.40 neg_nu5219082963157363817nc_rat: rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 0.17/0.40 neg_nu8295874005876285629c_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 0.17/0.40 neg_numeral_sub_int: num > num > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onum_OBit0,type,
% 0.17/0.40 bit0: num > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onum_OBit1,type,
% 0.17/0.40 bit1: num > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onum_OOne,type,
% 0.17/0.40 one: num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.17/0.40 case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onum_Osize__num,type,
% 0.17/0.40 size_num: num > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onum__of__nat,type,
% 0.17/0.40 num_of_nat: nat > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 numera6620942414471956472nteger: num > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 0.17/0.40 numera6690914467698888265omplex: num > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 numera1916890842035813515d_enat: num > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 0.17/0.40 numeral_numeral_int: num > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 0.17/0.40 numeral_numeral_nat: num > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 0.17/0.40 numeral_numeral_rat: num > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 0.17/0.40 numeral_numeral_real: num > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Opow,type,
% 0.17/0.40 pow: num > num > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Opred__numeral,type,
% 0.17/0.40 pred_numeral: num > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Num_Osqr,type,
% 0.17/0.40 sqr: num > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
% 0.17/0.40 none_nat: option_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 0.17/0.40 none_num: option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
% 0.17/0.40 some_nat: nat > option_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 0.17/0.40 some_num: num > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
% 0.17/0.40 case_option_int_num: int > ( num > int ) > option_num > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
% 0.17/0.40 case_option_num_num: num > ( num > num ) > option_num > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
% 0.17/0.40 case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
% 0.17/0.40 map_option_num_num: ( num > num ) > option_num > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
% 0.17/0.40 the_nat: option_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.17/0.40 bot_bot_nat_o: nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 bot_bo4199563552545308370d_enat: extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 0.17/0.40 bot_bot_nat: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
% 0.17/0.40 bot_bo7653980558646680370d_enat: set_Extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
% 0.17/0.40 bot_bot_set_int: set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 bot_bot_set_nat: set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
% 0.17/0.40 bot_bot_set_num: set_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
% 0.17/0.40 bot_bot_set_rat: set_rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.17/0.40 bot_bot_set_real: set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.17/0.40 bot_bot_set_set_nat: set_set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_OLeast_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 ord_Le1955565732374568822d_enat: ( extended_enat > $o ) > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
% 0.17/0.40 ord_Least_nat: ( nat > $o ) > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
% 0.17/0.40 ord_Least_real: ( real > $o ) > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Complex__Ocomplex_M_Eo_J,type,
% 0.17/0.40 ord_less_complex_o: ( complex > $o ) > ( complex > $o ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Extended____Nat__Oenat_M_Eo_J,type,
% 0.17/0.40 ord_le8499522857272258027enat_o: ( extended_enat > $o ) > ( extended_enat > $o ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
% 0.17/0.40 ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.17/0.40 ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
% 0.17/0.40 ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.17/0.40 ord_less_int: int > int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.17/0.40 ord_less_nat: nat > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
% 0.17/0.40 ord_less_num: num > num > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
% 0.17/0.40 ord_less_rat: rat > rat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.17/0.40 ord_less_real: real > real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 0.17/0.40 ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.17/0.40 ord_less_set_complex: set_complex > set_complex > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
% 0.17/0.40 ord_le2529575680413868914d_enat: set_Extended_enat > set_Extended_enat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
% 0.17/0.40 ord_less_set_int: set_int > set_int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 ord_less_set_nat: set_nat > set_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
% 0.17/0.40 ord_less_set_num: set_num > set_num > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
% 0.17/0.40 ord_less_set_rat: set_rat > set_rat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.17/0.40 ord_less_set_real: set_real > set_real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.17/0.40 ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
% 0.17/0.40 ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.17/0.40 ord_less_eq_int: int > int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.17/0.40 ord_less_eq_nat: nat > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
% 0.17/0.40 ord_less_eq_num: num > num > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
% 0.17/0.40 ord_less_eq_rat: rat > rat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.17/0.40 ord_less_eq_real: real > real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 0.17/0.40 ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.17/0.40 ord_le211207098394363844omplex: set_complex > set_complex > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
% 0.17/0.40 ord_le7203529160286727270d_enat: set_Extended_enat > set_Extended_enat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
% 0.17/0.40 ord_less_eq_set_int: set_int > set_int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
% 0.17/0.40 ord_less_eq_set_num: set_num > set_num > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
% 0.17/0.40 ord_less_eq_set_rat: set_rat > set_rat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.17/0.40 ord_less_eq_set_real: set_real > set_real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.17/0.40 ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
% 0.17/0.40 ord_max_int: int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
% 0.17/0.40 ord_max_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
% 0.17/0.40 ord_min_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
% 0.17/0.40 order_Greatest_nat: ( nat > $o ) > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.17/0.40 order_9091379641038594480t_real: ( nat > real ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oorder__class_Omono_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 order_4130057895858720880d_enat: ( extended_enat > extended_enat ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 order_mono_nat_nat: ( nat > nat ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.17/0.40 order_mono_nat_real: ( nat > real ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 order_5726023648592871131at_nat: ( nat > nat ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 order_7092887310737990675l_real: ( real > real ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
% 0.17/0.40 ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 top_to3028658606643905974d_enat: extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
% 0.17/0.40 top_top_set_o: set_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 top_top_set_nat: set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 0.17/0.40 top_to1996260823553986621t_unit: set_Product_unit ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.17/0.40 top_top_set_real: set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
% 0.17/0.40 top_top_set_char: set_char ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 power_8256067586552552935nteger: code_integer > nat > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
% 0.17/0.40 power_power_complex: complex > nat > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 0.17/0.40 power_power_int: int > nat > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 0.17/0.40 power_power_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
% 0.17/0.40 power_power_rat: rat > nat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 0.17/0.40 power_power_real: real > nat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.17/0.40 produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
% 0.17/0.40 produc851828971589881931at_num: ( nat > num > num ) > produc2963631642982155120at_num > produc3368934014287244435at_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
% 0.17/0.40 product_Pair_o_o: $o > $o > product_prod_o_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
% 0.17/0.40 product_Pair_o_int: $o > int > product_prod_o_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
% 0.17/0.40 product_Pair_o_nat: $o > nat > product_prod_o_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
% 0.17/0.40 produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
% 0.17/0.40 product_Pair_int_int: int > int > product_prod_int_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
% 0.17/0.40 product_Pair_nat_num: nat > num > product_prod_nat_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 0.17/0.40 produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
% 0.17/0.40 product_Pair_num_num: num > num > product_prod_num_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 0.17/0.40 produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 produc581526299967858633d_enat: vEBT_VEBT > extended_enat > produc7272778201969148633d_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 0.17/0.40 produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 0.17/0.40 produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 0.17/0.40 produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 0.17/0.40 produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 0.17/0.40 produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.17/0.40 produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.17/0.40 produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 0.17/0.40 produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 0.17/0.40 produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 0.17/0.40 produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.17/0.40 produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.17/0.40 produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.17/0.40 produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 0.17/0.40 produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.40 produc2761476792215241774st_nat: ( nat > nat > list_nat ) > product_prod_nat_nat > list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.17/0.40 produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 0.17/0.40 produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ofst_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 0.17/0.40 product_fst_int_int: product_prod_int_int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 product_fst_nat_nat: product_prod_nat_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 produc6174133586879617921nteger: produc8923325533196201883nteger > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 0.17/0.40 product_snd_int_int: product_prod_int_int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 product_snd_nat_nat: product_prod_nat_nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rat_OAbs__Rat,type,
% 0.17/0.40 abs_Rat: product_prod_int_int > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rat_OFract,type,
% 0.17/0.40 fract: int > int > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rat_OFrct,type,
% 0.17/0.40 frct: product_prod_int_int > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rat_ORep__Rat,type,
% 0.17/0.40 rep_Rat: rat > product_prod_int_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 0.17/0.40 field_5140801741446780682s_real: set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
% 0.17/0.40 field_7254667332652039916t_real: rat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rat_Onormalize,type,
% 0.17/0.40 normalize: product_prod_int_int > product_prod_int_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rat_Opcr__rat,type,
% 0.17/0.40 pcr_rat: product_prod_int_int > rat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rat_Opositive,type,
% 0.17/0.40 positive: rat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rat_Oquotient__of,type,
% 0.17/0.40 quotient_of: rat > product_prod_int_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rat_Oratrel,type,
% 0.17/0.40 ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real_OReal,type,
% 0.17/0.40 real2: ( nat > rat ) > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real_Ocauchy,type,
% 0.17/0.40 cauchy: ( nat > rat ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real_Ocr__real,type,
% 0.17/0.40 cr_real: ( nat > rat ) > real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real_Opcr__real,type,
% 0.17/0.40 pcr_real: ( nat > rat ) > real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real_Opositive,type,
% 0.17/0.40 positive2: real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real_Orealrel,type,
% 0.17/0.40 realrel: ( nat > rat ) > ( nat > rat ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real_Orep__real,type,
% 0.17/0.40 rep_real: real > nat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real_Ovanishes,type,
% 0.17/0.40 vanishes: ( nat > rat ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
% 0.17/0.40 real_V2521375963428798218omplex: set_complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 real_V5970128139526366754l_real: ( real > real ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 0.17/0.40 real_V3694042436643373181omplex: complex > complex > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 0.17/0.40 real_V975177566351809787t_real: real > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 0.17/0.40 real_V1022390504157884413omplex: complex > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 0.17/0.40 real_V7735802525324610683m_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 0.17/0.40 real_V4546457046886955230omplex: real > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
% 0.17/0.40 real_V1803761363581548252l_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
% 0.17/0.40 real_V2046097035970521341omplex: real > complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 0.17/0.40 real_V1485227260804924795R_real: real > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Relation_Otransp_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
% 0.17/0.40 transp_nat_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
% 0.17/0.40 algebr932160517623751201me_int: int > int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
% 0.17/0.40 algebr934650988132801477me_nat: nat > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
% 0.17/0.40 divide1717551699836669952omplex: complex > complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 0.17/0.40 divide_divide_int: int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 0.17/0.40 divide_divide_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
% 0.17/0.40 divide_divide_rat: rat > rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 0.17/0.40 divide_divide_real: real > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 dvd_dvd_Code_integer: code_integer > code_integer > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
% 0.17/0.40 dvd_dvd_complex: complex > complex > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
% 0.17/0.40 dvd_dvd_int: int > int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 0.17/0.40 dvd_dvd_nat: nat > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
% 0.17/0.40 dvd_dvd_rat: rat > rat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
% 0.17/0.40 dvd_dvd_real: real > real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
% 0.17/0.40 modulo_modulo_int: int > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
% 0.17/0.40 modulo_modulo_nat: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 zero_n356916108424825756nteger: $o > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
% 0.17/0.40 zero_n1201886186963655149omplex: $o > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
% 0.17/0.40 zero_n2684676970156552555ol_int: $o > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
% 0.17/0.40 zero_n2687167440665602831ol_nat: $o > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
% 0.17/0.40 zero_n2052037380579107095ol_rat: $o > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
% 0.17/0.40 zero_n3304061248610475627l_real: $o > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
% 0.17/0.40 suminf_complex: ( nat > complex ) > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
% 0.17/0.40 suminf_real: ( nat > real ) > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
% 0.17/0.40 summable_complex: ( nat > complex ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
% 0.17/0.40 summable_real: ( nat > real ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Series_Osums_001t__Real__Oreal,type,
% 0.17/0.40 sums_real: ( nat > real ) > real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001_Eo,type,
% 0.17/0.40 collect_o: ( $o > $o ) > set_o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
% 0.17/0.40 collect_complex: ( complex > $o ) > set_complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 collec4429806609662206161d_enat: ( extended_enat > $o ) > set_Extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__Int__Oint,type,
% 0.17/0.40 collect_int: ( int > $o ) > set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.40 collect_list_nat: ( list_nat > $o ) > set_list_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.17/0.40 collect_nat: ( nat > $o ) > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.17/0.40 collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.17/0.40 collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.17/0.40 collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.17/0.40 collect_real: ( real > $o ) > set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 collect_VEBT_VEBT: ( vEBT_VEBT > $o ) > set_VEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oimage_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 image_80655429650038917d_enat: ( extended_enat > extended_enat ) > set_Extended_enat > set_Extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
% 0.17/0.40 image_int_int: ( int > int ) > set_int > set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
% 0.17/0.40 image_nat_int: ( nat > int ) > set_nat > set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.17/0.40 image_nat_real: ( nat > real ) > set_nat > set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
% 0.17/0.40 image_nat_char: ( nat > char ) > set_nat > set_char ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.17/0.40 image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.17/0.40 image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 image_real_real: ( real > real ) > set_real > set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
% 0.17/0.40 image_char_nat: ( char > nat ) > set_char > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
% 0.17/0.40 insert_int: int > set_int > set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 0.17/0.40 insert_nat: nat > set_nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
% 0.17/0.40 insert_real: real > set_real > set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.17/0.40 vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
% 0.17/0.40 set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
% 0.17/0.40 set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
% 0.17/0.40 set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
% 0.17/0.40 set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
% 0.17/0.40 set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 set_or5403411693681687835d_enat: extended_enat > extended_enat > set_Extended_enat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
% 0.17/0.40 set_or1266510415728281911st_int: int > int > set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
% 0.17/0.40 set_or1269000886237332187st_nat: nat > nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
% 0.17/0.40 set_or7049704709247886629st_num: num > num > set_num ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
% 0.17/0.40 set_or633870826150836451st_rat: rat > rat > set_rat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
% 0.17/0.40 set_or1222579329274155063t_real: real > real > set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
% 0.17/0.40 set_or4662586982721622107an_int: int > int > set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
% 0.17/0.40 set_or4665077453230672383an_nat: nat > nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
% 0.17/0.40 set_ord_atLeast_nat: nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
% 0.17/0.40 set_ord_atLeast_real: real > set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 0.17/0.40 set_ord_atMost_nat: nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
% 0.17/0.40 set_or6656581121297822940st_int: int > int > set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
% 0.17/0.40 set_or6659071591806873216st_nat: nat > nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 0.17/0.40 set_or5832277885323065728an_int: int > int > set_int ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 0.17/0.40 set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 0.17/0.40 set_or1633881224788618240n_real: real > real > set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 0.17/0.40 set_or1210151606488870762an_nat: nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 0.17/0.40 set_or5849166863359141190n_real: real > set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 0.17/0.40 set_ord_lessThan_nat: nat > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 0.17/0.40 set_or5984915006950818249n_real: real > set_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_String_OCode_Oabort_001t__Real__Oreal,type,
% 0.17/0.40 abort_real: literal > ( product_unit > real ) > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_String_OLiteral,type,
% 0.17/0.40 literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_String_Oascii__of,type,
% 0.17/0.40 ascii_of: char > char ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_String_Ochar_OChar,type,
% 0.17/0.40 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 0.17/0.40 comm_s629917340098488124ar_nat: char > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_String_Ointeger__of__char,type,
% 0.17/0.40 integer_of_char: char > code_integer ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 0.17/0.40 unique3096191561947761185of_nat: nat > char ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.17/0.40 topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 0.17/0.40 topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 0.17/0.40 topolo2177554685111907308n_real: real > set_real > filter_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
% 0.17/0.40 topolo7531315842566124627t_real: ( nat > real ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 0.17/0.40 topolo2815343760600316023s_real: real > filter_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
% 0.17/0.40 topolo6517432010174082258omplex: ( nat > complex ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 0.17/0.40 topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 0.17/0.40 topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 0.17/0.40 topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Oarccos,type,
% 0.17/0.40 arccos: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 0.17/0.40 arcosh_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Oarcsin,type,
% 0.17/0.40 arcsin: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Oarctan,type,
% 0.17/0.40 arctan: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 0.17/0.40 arsinh_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 0.17/0.40 artanh_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 0.17/0.40 cos_complex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 0.17/0.40 cos_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Ocos__coeff,type,
% 0.17/0.40 cos_coeff: nat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
% 0.17/0.40 cosh_complex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 0.17/0.40 cosh_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
% 0.17/0.40 cot_complex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 0.17/0.40 cot_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 0.17/0.40 exp_complex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 0.17/0.40 exp_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 0.17/0.40 ln_ln_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Olog,type,
% 0.17/0.40 log: real > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Opi,type,
% 0.17/0.40 pi: real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 0.17/0.40 powr_real: real > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Opowr__real,type,
% 0.17/0.40 powr_real2: real > real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 0.17/0.40 sin_complex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 0.17/0.40 sin_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Osin__coeff,type,
% 0.17/0.40 sin_coeff: nat > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
% 0.17/0.40 sinh_complex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 0.17/0.40 sinh_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 0.17/0.40 tan_complex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 0.17/0.40 tan_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 0.17/0.40 tanh_complex: complex > complex ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 0.17/0.40 tanh_real: real > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 0.17/0.40 transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
% 0.17/0.40 transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 0.17/0.40 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 0.17/0.40 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 0.17/0.40 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 0.17/0.40 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
% 0.17/0.40 vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
% 0.17/0.40 vEBT_V312737461966249ad_rel: produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 0.17/0.40 vEBT_VEBT_high: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 0.17/0.40 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 0.17/0.40 vEBT_VEBT_low: nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 0.17/0.40 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 0.17/0.40 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 0.17/0.40 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 0.17/0.40 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 0.17/0.40 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 0.17/0.40 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 0.17/0.40 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 0.17/0.40 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 0.17/0.40 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 0.17/0.40 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 0.17/0.40 vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 0.17/0.40 vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 0.17/0.40 vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 0.17/0.40 vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 0.17/0.40 vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Member_Ovebt__member,type,
% 0.17/0.40 vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 0.17/0.40 vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 0.17/0.40 vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 0.17/0.40 vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 0.17/0.40 vEBT_VEBT_less: option_nat > option_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 0.17/0.40 vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 0.17/0.40 vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 0.17/0.40 vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 0.17/0.40 vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 0.17/0.40 vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 0.17/0.40 vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 0.17/0.40 vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 0.17/0.40 vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 0.17/0.40 vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 0.17/0.40 vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
% 0.17/0.40 vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Succ_Ovebt__succ,type,
% 0.17/0.40 vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
% 0.17/0.40 vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.40 accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 0.17/0.40 accp_nat: ( nat > nat > $o ) > nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.17/0.40 accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 0.17/0.40 accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
% 0.17/0.40 accp_P6183159247885693666d_enat: ( produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ) > produc7272778201969148633d_enat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.17/0.40 accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Wellfounded_Oless__than,type,
% 0.17/0.40 less_than: set_Pr1261947904930325089at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_Wellfounded_Opred__nat,type,
% 0.17/0.40 pred_nat: set_Pr1261947904930325089at_nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_fChoice_001t__Real__Oreal,type,
% 0.17/0.40 fChoice_real: ( real > $o ) > real ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001_Eo,type,
% 0.17/0.40 member_o: $o > set_o > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
% 0.17/0.40 member_Code_integer: code_integer > set_Code_integer > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 0.17/0.40 member_complex: complex > set_complex > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__Extended____Nat__Oenat,type,
% 0.17/0.40 member_Extended_enat: extended_enat > set_Extended_enat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__Int__Oint,type,
% 0.17/0.40 member_int: int > set_int > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 0.17/0.40 member_list_nat: list_nat > set_list_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__Nat__Onat,type,
% 0.17/0.40 member_nat: nat > set_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__Num__Onum,type,
% 0.17/0.40 member_num: num > set_num > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.17/0.40 member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.17/0.40 member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__Rat__Orat,type,
% 0.17/0.40 member_rat: rat > set_rat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__Real__Oreal,type,
% 0.17/0.40 member_real: real > set_real > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.17/0.40 member_set_nat: set_nat > set_set_nat > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 0.17/0.40 member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_deg____,type,
% 0.17/0.40 deg: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_m____,type,
% 0.17/0.40 m: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_ma____,type,
% 0.17/0.40 ma: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_mi____,type,
% 0.17/0.40 mi: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_miny____,type,
% 0.17/0.40 miny: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_na____,type,
% 0.17/0.40 na: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_res____,type,
% 0.17/0.40 res: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_sc____,type,
% 0.17/0.40 sc: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_summary____,type,
% 0.17/0.40 summary: vEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_treeList____,type,
% 0.17/0.40 treeList: list_VEBT_VEBT ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_xa____,type,
% 0.17/0.40 xa: nat ).
% 0.17/0.40
% 0.17/0.40 thf(sy_v_za____,type,
% 0.17/0.40 za: nat ).
% 0.17/0.40
% 0.17/0.40 % Relevant facts (10210)
% 0.17/0.40 thf(fact_0_bbbb,axiom,
% 0.17/0.40 ord_less_eq_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % bbbb
% 0.17/0.40 thf(fact_1__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
% 0.17/0.40 ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 0.17/0.40
% 0.17/0.40 % \<open>2 \<le> deg\<close>
% 0.17/0.40 thf(fact_2_aaaa,axiom,
% 0.17/0.40 ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % aaaa
% 0.17/0.40 thf(fact_3_max__in__set__def,axiom,
% 0.17/0.40 ( vEBT_VEBT_max_in_set
% 0.17/0.40 = ( ^ [Xs: set_nat,X: nat] :
% 0.17/0.40 ( ( member_nat @ X @ Xs )
% 0.17/0.40 & ! [Y: nat] :
% 0.17/0.40 ( ( member_nat @ Y @ Xs )
% 0.17/0.40 => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % max_in_set_def
% 0.17/0.40 thf(fact_4_min__in__set__def,axiom,
% 0.17/0.40 ( vEBT_VEBT_min_in_set
% 0.17/0.40 = ( ^ [Xs: set_nat,X: nat] :
% 0.17/0.40 ( ( member_nat @ X @ Xs )
% 0.17/0.40 & ! [Y: nat] :
% 0.17/0.40 ( ( member_nat @ Y @ Xs )
% 0.17/0.40 => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % min_in_set_def
% 0.17/0.40 thf(fact_5__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
% 0.17/0.40 ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = na ) ).
% 0.17/0.40
% 0.17/0.40 % \<open>deg div 2 = n\<close>
% 0.17/0.40 thf(fact_6_semiring__norm_I85_J,axiom,
% 0.17/0.40 ! [M: num] :
% 0.17/0.40 ( ( bit0 @ M )
% 0.17/0.40 != one ) ).
% 0.17/0.40
% 0.17/0.40 % semiring_norm(85)
% 0.17/0.40 thf(fact_7_semiring__norm_I83_J,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( one
% 0.17/0.40 != ( bit0 @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % semiring_norm(83)
% 0.17/0.40 thf(fact_8_numeral__Bit0__div__2,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = ( numeral_numeral_nat @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_Bit0_div_2
% 0.17/0.40 thf(fact_9_numeral__Bit0__div__2,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.40 = ( numeral_numeral_int @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_Bit0_div_2
% 0.17/0.40 thf(fact_10_high__def,axiom,
% 0.17/0.40 ( vEBT_VEBT_high
% 0.17/0.40 = ( ^ [X: nat,N2: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % high_def
% 0.17/0.40 thf(fact_11__092_060open_062high_Az_A_Ideg_Adiv_A2_J_A_060_Asc_092_060close_062,axiom,
% 0.17/0.40 ord_less_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ).
% 0.17/0.40
% 0.17/0.40 % \<open>high z (deg div 2) < sc\<close>
% 0.17/0.40 thf(fact_12_divide__numeral__1,axiom,
% 0.17/0.40 ! [A: complex] :
% 0.17/0.40 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.17/0.40 = A ) ).
% 0.17/0.40
% 0.17/0.40 % divide_numeral_1
% 0.17/0.40 thf(fact_13_divide__numeral__1,axiom,
% 0.17/0.40 ! [A: real] :
% 0.17/0.40 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 0.17/0.40 = A ) ).
% 0.17/0.40
% 0.17/0.40 % divide_numeral_1
% 0.17/0.40 thf(fact_14_divide__numeral__1,axiom,
% 0.17/0.40 ! [A: rat] :
% 0.17/0.40 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.17/0.40 = A ) ).
% 0.17/0.40
% 0.17/0.40 % divide_numeral_1
% 0.17/0.40 thf(fact_15_verit__eq__simplify_I8_J,axiom,
% 0.17/0.40 ! [X2: num,Y2: num] :
% 0.17/0.40 ( ( ( bit0 @ X2 )
% 0.17/0.40 = ( bit0 @ Y2 ) )
% 0.17/0.40 = ( X2 = Y2 ) ) ).
% 0.17/0.40
% 0.17/0.40 % verit_eq_simplify(8)
% 0.17/0.40 thf(fact_16_semiring__norm_I87_J,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ( bit0 @ M )
% 0.17/0.40 = ( bit0 @ N ) )
% 0.17/0.40 = ( M = N ) ) ).
% 0.17/0.40
% 0.17/0.40 % semiring_norm(87)
% 0.17/0.40 thf(fact_17_numeral__eq__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ( numera6690914467698888265omplex @ M )
% 0.17/0.40 = ( numera6690914467698888265omplex @ N ) )
% 0.17/0.40 = ( M = N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_eq_iff
% 0.17/0.40 thf(fact_18_numeral__eq__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ( numeral_numeral_real @ M )
% 0.17/0.40 = ( numeral_numeral_real @ N ) )
% 0.17/0.40 = ( M = N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_eq_iff
% 0.17/0.40 thf(fact_19_numeral__eq__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ( numeral_numeral_rat @ M )
% 0.17/0.40 = ( numeral_numeral_rat @ N ) )
% 0.17/0.40 = ( M = N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_eq_iff
% 0.17/0.40 thf(fact_20_numeral__eq__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ( numeral_numeral_nat @ M )
% 0.17/0.40 = ( numeral_numeral_nat @ N ) )
% 0.17/0.40 = ( M = N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_eq_iff
% 0.17/0.40 thf(fact_21_numeral__eq__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ( numeral_numeral_int @ M )
% 0.17/0.40 = ( numeral_numeral_int @ N ) )
% 0.17/0.40 = ( M = N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_eq_iff
% 0.17/0.40 thf(fact_22_numeral__le__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.17/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_le_iff
% 0.17/0.40 thf(fact_23_numeral__le__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.17/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_le_iff
% 0.17/0.40 thf(fact_24_numeral__le__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.17/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_le_iff
% 0.17/0.40 thf(fact_25_numeral__le__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.17/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_le_iff
% 0.17/0.40 thf(fact_26_numeral__less__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.17/0.40 = ( ord_less_num @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_less_iff
% 0.17/0.40 thf(fact_27_numeral__less__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.17/0.40 = ( ord_less_num @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_less_iff
% 0.17/0.40 thf(fact_28_numeral__less__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.17/0.40 = ( ord_less_num @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_less_iff
% 0.17/0.40 thf(fact_29_numeral__less__iff,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.17/0.40 = ( ord_less_num @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_less_iff
% 0.17/0.40 thf(fact_30__092_060open_062z_A_060_A2_A_094_Adeg_092_060close_062,axiom,
% 0.17/0.40 ord_less_nat @ za @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.17/0.40
% 0.17/0.40 % \<open>z < 2 ^ deg\<close>
% 0.17/0.40 thf(fact_31__C5_Ohyps_C_I10_J,axiom,
% 0.17/0.40 ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.17/0.40
% 0.17/0.40 % "5.hyps"(10)
% 0.17/0.40 thf(fact_32_verit__comp__simplify1_I3_J,axiom,
% 0.17/0.40 ! [B: real,A2: real] :
% 0.17/0.40 ( ( ~ ( ord_less_eq_real @ B @ A2 ) )
% 0.17/0.40 = ( ord_less_real @ A2 @ B ) ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(3)
% 0.17/0.40 thf(fact_33_verit__comp__simplify1_I3_J,axiom,
% 0.17/0.40 ! [B: rat,A2: rat] :
% 0.17/0.40 ( ( ~ ( ord_less_eq_rat @ B @ A2 ) )
% 0.17/0.40 = ( ord_less_rat @ A2 @ B ) ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(3)
% 0.17/0.40 thf(fact_34_verit__comp__simplify1_I3_J,axiom,
% 0.17/0.40 ! [B: num,A2: num] :
% 0.17/0.40 ( ( ~ ( ord_less_eq_num @ B @ A2 ) )
% 0.17/0.40 = ( ord_less_num @ A2 @ B ) ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(3)
% 0.17/0.40 thf(fact_35_verit__comp__simplify1_I3_J,axiom,
% 0.17/0.40 ! [B: nat,A2: nat] :
% 0.17/0.40 ( ( ~ ( ord_less_eq_nat @ B @ A2 ) )
% 0.17/0.40 = ( ord_less_nat @ A2 @ B ) ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(3)
% 0.17/0.40 thf(fact_36_verit__comp__simplify1_I3_J,axiom,
% 0.17/0.40 ! [B: int,A2: int] :
% 0.17/0.40 ( ( ~ ( ord_less_eq_int @ B @ A2 ) )
% 0.17/0.40 = ( ord_less_int @ A2 @ B ) ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(3)
% 0.17/0.40 thf(fact_37_verit__comp__simplify1_I2_J,axiom,
% 0.17/0.40 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(2)
% 0.17/0.40 thf(fact_38_verit__comp__simplify1_I2_J,axiom,
% 0.17/0.40 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(2)
% 0.17/0.40 thf(fact_39_verit__comp__simplify1_I2_J,axiom,
% 0.17/0.40 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(2)
% 0.17/0.40 thf(fact_40_verit__comp__simplify1_I2_J,axiom,
% 0.17/0.40 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(2)
% 0.17/0.40 thf(fact_41_verit__comp__simplify1_I2_J,axiom,
% 0.17/0.40 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(2)
% 0.17/0.40 thf(fact_42_verit__comp__simplify1_I1_J,axiom,
% 0.17/0.40 ! [A: real] :
% 0.17/0.40 ~ ( ord_less_real @ A @ A ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(1)
% 0.17/0.40 thf(fact_43_verit__comp__simplify1_I1_J,axiom,
% 0.17/0.40 ! [A: rat] :
% 0.17/0.40 ~ ( ord_less_rat @ A @ A ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(1)
% 0.17/0.40 thf(fact_44_verit__comp__simplify1_I1_J,axiom,
% 0.17/0.40 ! [A: num] :
% 0.17/0.40 ~ ( ord_less_num @ A @ A ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(1)
% 0.17/0.40 thf(fact_45_verit__comp__simplify1_I1_J,axiom,
% 0.17/0.40 ! [A: nat] :
% 0.17/0.40 ~ ( ord_less_nat @ A @ A ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(1)
% 0.17/0.40 thf(fact_46_verit__comp__simplify1_I1_J,axiom,
% 0.17/0.40 ! [A: int] :
% 0.17/0.40 ~ ( ord_less_int @ A @ A ) ).
% 0.17/0.40
% 0.17/0.40 % verit_comp_simplify1(1)
% 0.17/0.40 thf(fact_47_verit__la__disequality,axiom,
% 0.17/0.40 ! [A: rat,B2: rat] :
% 0.17/0.40 ( ( A = B2 )
% 0.17/0.40 | ~ ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.40 | ~ ( ord_less_eq_rat @ B2 @ A ) ) ).
% 0.17/0.40
% 0.17/0.40 % verit_la_disequality
% 0.17/0.40 thf(fact_48_verit__la__disequality,axiom,
% 0.17/0.40 ! [A: num,B2: num] :
% 0.17/0.40 ( ( A = B2 )
% 0.17/0.40 | ~ ( ord_less_eq_num @ A @ B2 )
% 0.17/0.40 | ~ ( ord_less_eq_num @ B2 @ A ) ) ).
% 0.17/0.40
% 0.17/0.40 % verit_la_disequality
% 0.17/0.40 thf(fact_49_verit__la__disequality,axiom,
% 0.17/0.40 ! [A: nat,B2: nat] :
% 0.17/0.40 ( ( A = B2 )
% 0.17/0.40 | ~ ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.40 | ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% 0.17/0.40
% 0.17/0.40 % verit_la_disequality
% 0.17/0.40 thf(fact_50_verit__la__disequality,axiom,
% 0.17/0.40 ! [A: int,B2: int] :
% 0.17/0.40 ( ( A = B2 )
% 0.17/0.40 | ~ ( ord_less_eq_int @ A @ B2 )
% 0.17/0.40 | ~ ( ord_less_eq_int @ B2 @ A ) ) ).
% 0.17/0.40
% 0.17/0.40 % verit_la_disequality
% 0.17/0.40 thf(fact_51_div__le__dividend,axiom,
% 0.17/0.40 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 0.17/0.40
% 0.17/0.40 % div_le_dividend
% 0.17/0.40 thf(fact_52_div__le__mono,axiom,
% 0.17/0.40 ! [M: nat,N: nat,K: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.40 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % div_le_mono
% 0.17/0.40 thf(fact_53_verit__eq__simplify_I10_J,axiom,
% 0.17/0.40 ! [X2: num] :
% 0.17/0.40 ( one
% 0.17/0.40 != ( bit0 @ X2 ) ) ).
% 0.17/0.40
% 0.17/0.40 % verit_eq_simplify(10)
% 0.17/0.40 thf(fact_54__092_060open_062high_Ares_A_Ideg_Adiv_A2_J_A_061_Asc_092_060close_062,axiom,
% 0.17/0.40 ( ( vEBT_VEBT_high @ res @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.40 = sc ) ).
% 0.17/0.40
% 0.17/0.40 % \<open>high res (deg div 2) = sc\<close>
% 0.17/0.40 thf(fact_55__092_060open_062x_A_060_Ares_092_060close_062,axiom,
% 0.17/0.40 ord_less_nat @ xa @ res ).
% 0.17/0.40
% 0.17/0.40 % \<open>x < res\<close>
% 0.17/0.40 thf(fact_56__092_060open_062sc_A_060_A2_A_094_Am_092_060close_062,axiom,
% 0.17/0.40 ord_less_nat @ sc @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).
% 0.17/0.40
% 0.17/0.40 % \<open>sc < 2 ^ m\<close>
% 0.17/0.40 thf(fact_57_power2__nat__le__imp__le,axiom,
% 0.17/0.40 ! [M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.17/0.40 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % power2_nat_le_imp_le
% 0.17/0.40 thf(fact_58_power2__nat__le__eq__le,axiom,
% 0.17/0.40 ! [M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.40 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % power2_nat_le_eq_le
% 0.17/0.40 thf(fact_59_self__le__ge2__pow,axiom,
% 0.17/0.40 ! [K: nat,M: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.17/0.40 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % self_le_ge2_pow
% 0.17/0.40 thf(fact_60_less__exp,axiom,
% 0.17/0.40 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % less_exp
% 0.17/0.40 thf(fact_61_high__bound__aux,axiom,
% 0.17/0.40 ! [Ma: nat,N: nat,M: nat] :
% 0.17/0.40 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.17/0.40 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % high_bound_aux
% 0.17/0.40 thf(fact_62_power__minus__is__div,axiom,
% 0.17/0.40 ! [B2: nat,A: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ B2 @ A )
% 0.17/0.40 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B2 ) )
% 0.17/0.40 = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_minus_is_div
% 0.17/0.40 thf(fact_63_pow__sum,axiom,
% 0.17/0.40 ! [A: nat,B2: nat] :
% 0.17/0.40 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.17/0.40 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ).
% 0.17/0.40
% 0.17/0.40 % pow_sum
% 0.17/0.40 thf(fact_64__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
% 0.17/0.40 ord_less_eq_nat @ one_one_nat @ na ).
% 0.17/0.40
% 0.17/0.40 % \<open>1 \<le> n\<close>
% 0.17/0.40 thf(fact_65__092_060open_062mi_A_092_060le_062_Ax_092_060close_062,axiom,
% 0.17/0.40 ord_less_eq_nat @ mi @ xa ).
% 0.17/0.40
% 0.17/0.40 % \<open>mi \<le> x\<close>
% 0.17/0.40 thf(fact_66__C5_Ohyps_C_I9_J,axiom,
% 0.17/0.40 ord_less_eq_nat @ mi @ ma ).
% 0.17/0.40
% 0.17/0.40 % "5.hyps"(9)
% 0.17/0.40 thf(fact_67__C5_Ohyps_C_I6_J,axiom,
% 0.17/0.40 ( deg
% 0.17/0.40 = ( plus_plus_nat @ na @ m ) ) ).
% 0.17/0.40
% 0.17/0.40 % "5.hyps"(6)
% 0.17/0.40 thf(fact_68__092_060open_062z_A_092_060noteq_062_Ami_092_060close_062,axiom,
% 0.17/0.40 za != mi ).
% 0.17/0.40
% 0.17/0.40 % \<open>z \<noteq> mi\<close>
% 0.17/0.40 thf(fact_69__092_060open_062mi_A_092_060noteq_062_Ama_092_060close_062,axiom,
% 0.17/0.40 mi != ma ).
% 0.17/0.40
% 0.17/0.40 % \<open>mi \<noteq> ma\<close>
% 0.17/0.40 thf(fact_70__092_060open_062res_A_092_060le_062_Ama_092_060close_062,axiom,
% 0.17/0.40 ord_less_eq_nat @ res @ ma ).
% 0.17/0.40
% 0.17/0.40 % \<open>res \<le> ma\<close>
% 0.17/0.40 thf(fact_71_mem__Collect__eq,axiom,
% 0.17/0.40 ! [A: extended_enat,P: extended_enat > $o] :
% 0.17/0.40 ( ( member_Extended_enat @ A @ ( collec4429806609662206161d_enat @ P ) )
% 0.17/0.40 = ( P @ A ) ) ).
% 0.17/0.40
% 0.17/0.40 % mem_Collect_eq
% 0.17/0.40 thf(fact_72_mem__Collect__eq,axiom,
% 0.17/0.40 ! [A: complex,P: complex > $o] :
% 0.17/0.40 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 0.17/0.40 = ( P @ A ) ) ).
% 0.17/0.40
% 0.17/0.40 % mem_Collect_eq
% 0.17/0.40 thf(fact_73_mem__Collect__eq,axiom,
% 0.17/0.40 ! [A: real,P: real > $o] :
% 0.17/0.40 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.17/0.40 = ( P @ A ) ) ).
% 0.17/0.40
% 0.17/0.40 % mem_Collect_eq
% 0.17/0.40 thf(fact_74_mem__Collect__eq,axiom,
% 0.17/0.40 ! [A: list_nat,P: list_nat > $o] :
% 0.17/0.40 ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 0.17/0.40 = ( P @ A ) ) ).
% 0.17/0.40
% 0.17/0.40 % mem_Collect_eq
% 0.17/0.40 thf(fact_75_mem__Collect__eq,axiom,
% 0.17/0.40 ! [A: nat,P: nat > $o] :
% 0.17/0.40 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 0.17/0.40 = ( P @ A ) ) ).
% 0.17/0.40
% 0.17/0.40 % mem_Collect_eq
% 0.17/0.40 thf(fact_76_mem__Collect__eq,axiom,
% 0.17/0.40 ! [A: int,P: int > $o] :
% 0.17/0.40 ( ( member_int @ A @ ( collect_int @ P ) )
% 0.17/0.40 = ( P @ A ) ) ).
% 0.17/0.40
% 0.17/0.40 % mem_Collect_eq
% 0.17/0.40 thf(fact_77_Collect__mem__eq,axiom,
% 0.17/0.40 ! [A3: set_Extended_enat] :
% 0.17/0.40 ( ( collec4429806609662206161d_enat
% 0.17/0.40 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A3 ) )
% 0.17/0.40 = A3 ) ).
% 0.17/0.40
% 0.17/0.40 % Collect_mem_eq
% 0.17/0.40 thf(fact_78_Collect__mem__eq,axiom,
% 0.17/0.40 ! [A3: set_complex] :
% 0.17/0.40 ( ( collect_complex
% 0.17/0.40 @ ^ [X: complex] : ( member_complex @ X @ A3 ) )
% 0.17/0.40 = A3 ) ).
% 0.17/0.40
% 0.17/0.40 % Collect_mem_eq
% 0.17/0.40 thf(fact_79_Collect__mem__eq,axiom,
% 0.17/0.40 ! [A3: set_real] :
% 0.17/0.40 ( ( collect_real
% 0.17/0.40 @ ^ [X: real] : ( member_real @ X @ A3 ) )
% 0.17/0.40 = A3 ) ).
% 0.17/0.40
% 0.17/0.40 % Collect_mem_eq
% 0.17/0.40 thf(fact_80_Collect__mem__eq,axiom,
% 0.17/0.40 ! [A3: set_list_nat] :
% 0.17/0.40 ( ( collect_list_nat
% 0.17/0.40 @ ^ [X: list_nat] : ( member_list_nat @ X @ A3 ) )
% 0.17/0.40 = A3 ) ).
% 0.17/0.40
% 0.17/0.40 % Collect_mem_eq
% 0.17/0.40 thf(fact_81_Collect__mem__eq,axiom,
% 0.17/0.40 ! [A3: set_nat] :
% 0.17/0.40 ( ( collect_nat
% 0.17/0.40 @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
% 0.17/0.40 = A3 ) ).
% 0.17/0.40
% 0.17/0.40 % Collect_mem_eq
% 0.17/0.40 thf(fact_82_Collect__mem__eq,axiom,
% 0.17/0.40 ! [A3: set_int] :
% 0.17/0.40 ( ( collect_int
% 0.17/0.40 @ ^ [X: int] : ( member_int @ X @ A3 ) )
% 0.17/0.40 = A3 ) ).
% 0.17/0.40
% 0.17/0.40 % Collect_mem_eq
% 0.17/0.40 thf(fact_83_Collect__cong,axiom,
% 0.17/0.40 ! [P: complex > $o,Q: complex > $o] :
% 0.17/0.40 ( ! [X3: complex] :
% 0.17/0.40 ( ( P @ X3 )
% 0.17/0.40 = ( Q @ X3 ) )
% 0.17/0.40 => ( ( collect_complex @ P )
% 0.17/0.40 = ( collect_complex @ Q ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % Collect_cong
% 0.17/0.40 thf(fact_84_Collect__cong,axiom,
% 0.17/0.40 ! [P: real > $o,Q: real > $o] :
% 0.17/0.40 ( ! [X3: real] :
% 0.17/0.40 ( ( P @ X3 )
% 0.17/0.40 = ( Q @ X3 ) )
% 0.17/0.40 => ( ( collect_real @ P )
% 0.17/0.40 = ( collect_real @ Q ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % Collect_cong
% 0.17/0.40 thf(fact_85_Collect__cong,axiom,
% 0.17/0.40 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.17/0.40 ( ! [X3: list_nat] :
% 0.17/0.40 ( ( P @ X3 )
% 0.17/0.40 = ( Q @ X3 ) )
% 0.17/0.40 => ( ( collect_list_nat @ P )
% 0.17/0.40 = ( collect_list_nat @ Q ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % Collect_cong
% 0.17/0.40 thf(fact_86_Collect__cong,axiom,
% 0.17/0.40 ! [P: nat > $o,Q: nat > $o] :
% 0.17/0.40 ( ! [X3: nat] :
% 0.17/0.40 ( ( P @ X3 )
% 0.17/0.40 = ( Q @ X3 ) )
% 0.17/0.40 => ( ( collect_nat @ P )
% 0.17/0.40 = ( collect_nat @ Q ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % Collect_cong
% 0.17/0.40 thf(fact_87_Collect__cong,axiom,
% 0.17/0.40 ! [P: int > $o,Q: int > $o] :
% 0.17/0.40 ( ! [X3: int] :
% 0.17/0.40 ( ( P @ X3 )
% 0.17/0.40 = ( Q @ X3 ) )
% 0.17/0.40 => ( ( collect_int @ P )
% 0.17/0.40 = ( collect_int @ Q ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % Collect_cong
% 0.17/0.40 thf(fact_88__092_060open_062mi_A_060_Ares_092_060close_062,axiom,
% 0.17/0.40 ord_less_nat @ mi @ res ).
% 0.17/0.40
% 0.17/0.40 % \<open>mi < res\<close>
% 0.17/0.40 thf(fact_89_add__numeral__left,axiom,
% 0.17/0.40 ! [V: num,W: num,Z: complex] :
% 0.17/0.40 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.17/0.40 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.17/0.40
% 0.17/0.40 % add_numeral_left
% 0.17/0.40 thf(fact_90_add__numeral__left,axiom,
% 0.17/0.40 ! [V: num,W: num,Z: real] :
% 0.17/0.40 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.17/0.40 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.17/0.40
% 0.17/0.40 % add_numeral_left
% 0.17/0.40 thf(fact_91_add__numeral__left,axiom,
% 0.17/0.40 ! [V: num,W: num,Z: rat] :
% 0.17/0.40 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.17/0.40 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.17/0.40
% 0.17/0.40 % add_numeral_left
% 0.17/0.40 thf(fact_92_add__numeral__left,axiom,
% 0.17/0.40 ! [V: num,W: num,Z: nat] :
% 0.17/0.40 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.17/0.40 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.17/0.40
% 0.17/0.40 % add_numeral_left
% 0.17/0.40 thf(fact_93_add__numeral__left,axiom,
% 0.17/0.40 ! [V: num,W: num,Z: int] :
% 0.17/0.40 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.17/0.40 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.17/0.40
% 0.17/0.40 % add_numeral_left
% 0.17/0.40 thf(fact_94_numeral__plus__numeral,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.17/0.40 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_plus_numeral
% 0.17/0.40 thf(fact_95_numeral__plus__numeral,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.17/0.40 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_plus_numeral
% 0.17/0.40 thf(fact_96_numeral__plus__numeral,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.17/0.40 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_plus_numeral
% 0.17/0.40 thf(fact_97_numeral__plus__numeral,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.17/0.40 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_plus_numeral
% 0.17/0.40 thf(fact_98_numeral__plus__numeral,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.17/0.40 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_plus_numeral
% 0.17/0.40 thf(fact_99_power__one,axiom,
% 0.17/0.40 ! [N: nat] :
% 0.17/0.40 ( ( power_power_rat @ one_one_rat @ N )
% 0.17/0.40 = one_one_rat ) ).
% 0.17/0.40
% 0.17/0.40 % power_one
% 0.17/0.40 thf(fact_100_power__one,axiom,
% 0.17/0.40 ! [N: nat] :
% 0.17/0.40 ( ( power_power_nat @ one_one_nat @ N )
% 0.17/0.40 = one_one_nat ) ).
% 0.17/0.40
% 0.17/0.40 % power_one
% 0.17/0.40 thf(fact_101_power__one,axiom,
% 0.17/0.40 ! [N: nat] :
% 0.17/0.40 ( ( power_power_real @ one_one_real @ N )
% 0.17/0.40 = one_one_real ) ).
% 0.17/0.40
% 0.17/0.40 % power_one
% 0.17/0.40 thf(fact_102_power__one,axiom,
% 0.17/0.40 ! [N: nat] :
% 0.17/0.40 ( ( power_power_int @ one_one_int @ N )
% 0.17/0.40 = one_one_int ) ).
% 0.17/0.40
% 0.17/0.40 % power_one
% 0.17/0.40 thf(fact_103_power__one,axiom,
% 0.17/0.40 ! [N: nat] :
% 0.17/0.40 ( ( power_power_complex @ one_one_complex @ N )
% 0.17/0.40 = one_one_complex ) ).
% 0.17/0.40
% 0.17/0.40 % power_one
% 0.17/0.40 thf(fact_104_power__one__right,axiom,
% 0.17/0.40 ! [A: nat] :
% 0.17/0.40 ( ( power_power_nat @ A @ one_one_nat )
% 0.17/0.40 = A ) ).
% 0.17/0.40
% 0.17/0.40 % power_one_right
% 0.17/0.40 thf(fact_105_power__one__right,axiom,
% 0.17/0.40 ! [A: real] :
% 0.17/0.40 ( ( power_power_real @ A @ one_one_nat )
% 0.17/0.40 = A ) ).
% 0.17/0.40
% 0.17/0.40 % power_one_right
% 0.17/0.40 thf(fact_106_power__one__right,axiom,
% 0.17/0.40 ! [A: int] :
% 0.17/0.40 ( ( power_power_int @ A @ one_one_nat )
% 0.17/0.40 = A ) ).
% 0.17/0.40
% 0.17/0.40 % power_one_right
% 0.17/0.40 thf(fact_107_power__one__right,axiom,
% 0.17/0.40 ! [A: complex] :
% 0.17/0.40 ( ( power_power_complex @ A @ one_one_nat )
% 0.17/0.40 = A ) ).
% 0.17/0.40
% 0.17/0.40 % power_one_right
% 0.17/0.40 thf(fact_108_semiring__norm_I78_J,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.17/0.40 = ( ord_less_num @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % semiring_norm(78)
% 0.17/0.40 thf(fact_109_semiring__norm_I71_J,axiom,
% 0.17/0.40 ! [M: num,N: num] :
% 0.17/0.40 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.17/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % semiring_norm(71)
% 0.17/0.40 thf(fact_110_semiring__norm_I75_J,axiom,
% 0.17/0.40 ! [M: num] :
% 0.17/0.40 ~ ( ord_less_num @ M @ one ) ).
% 0.17/0.40
% 0.17/0.40 % semiring_norm(75)
% 0.17/0.40 thf(fact_111_semiring__norm_I68_J,axiom,
% 0.17/0.40 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 0.17/0.40
% 0.17/0.40 % semiring_norm(68)
% 0.17/0.40 thf(fact_112_numeral__eq__one__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ( numera6690914467698888265omplex @ N )
% 0.17/0.40 = one_one_complex )
% 0.17/0.40 = ( N = one ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_eq_one_iff
% 0.17/0.40 thf(fact_113_numeral__eq__one__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ( numeral_numeral_real @ N )
% 0.17/0.40 = one_one_real )
% 0.17/0.40 = ( N = one ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_eq_one_iff
% 0.17/0.40 thf(fact_114_numeral__eq__one__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ( numeral_numeral_rat @ N )
% 0.17/0.40 = one_one_rat )
% 0.17/0.40 = ( N = one ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_eq_one_iff
% 0.17/0.40 thf(fact_115_numeral__eq__one__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ( numeral_numeral_nat @ N )
% 0.17/0.40 = one_one_nat )
% 0.17/0.40 = ( N = one ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_eq_one_iff
% 0.17/0.40 thf(fact_116_numeral__eq__one__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ( numeral_numeral_int @ N )
% 0.17/0.40 = one_one_int )
% 0.17/0.40 = ( N = one ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_eq_one_iff
% 0.17/0.40 thf(fact_117_one__eq__numeral__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( one_one_complex
% 0.17/0.40 = ( numera6690914467698888265omplex @ N ) )
% 0.17/0.40 = ( one = N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_eq_numeral_iff
% 0.17/0.40 thf(fact_118_one__eq__numeral__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( one_one_real
% 0.17/0.40 = ( numeral_numeral_real @ N ) )
% 0.17/0.40 = ( one = N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_eq_numeral_iff
% 0.17/0.40 thf(fact_119_one__eq__numeral__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( one_one_rat
% 0.17/0.40 = ( numeral_numeral_rat @ N ) )
% 0.17/0.40 = ( one = N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_eq_numeral_iff
% 0.17/0.40 thf(fact_120_one__eq__numeral__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( one_one_nat
% 0.17/0.40 = ( numeral_numeral_nat @ N ) )
% 0.17/0.40 = ( one = N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_eq_numeral_iff
% 0.17/0.40 thf(fact_121_one__eq__numeral__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( one_one_int
% 0.17/0.40 = ( numeral_numeral_int @ N ) )
% 0.17/0.40 = ( one = N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_eq_numeral_iff
% 0.17/0.40 thf(fact_122_power__inject__exp,axiom,
% 0.17/0.40 ! [A: real,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_real @ one_one_real @ A )
% 0.17/0.40 => ( ( ( power_power_real @ A @ M )
% 0.17/0.40 = ( power_power_real @ A @ N ) )
% 0.17/0.40 = ( M = N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_inject_exp
% 0.17/0.40 thf(fact_123_power__inject__exp,axiom,
% 0.17/0.40 ! [A: rat,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_rat @ one_one_rat @ A )
% 0.17/0.40 => ( ( ( power_power_rat @ A @ M )
% 0.17/0.40 = ( power_power_rat @ A @ N ) )
% 0.17/0.40 = ( M = N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_inject_exp
% 0.17/0.40 thf(fact_124_power__inject__exp,axiom,
% 0.17/0.40 ! [A: nat,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_nat @ one_one_nat @ A )
% 0.17/0.40 => ( ( ( power_power_nat @ A @ M )
% 0.17/0.40 = ( power_power_nat @ A @ N ) )
% 0.17/0.40 = ( M = N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_inject_exp
% 0.17/0.40 thf(fact_125_power__inject__exp,axiom,
% 0.17/0.40 ! [A: int,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_int @ one_one_int @ A )
% 0.17/0.40 => ( ( ( power_power_int @ A @ M )
% 0.17/0.40 = ( power_power_int @ A @ N ) )
% 0.17/0.40 = ( M = N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_inject_exp
% 0.17/0.40 thf(fact_126_semiring__norm_I76_J,axiom,
% 0.17/0.40 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % semiring_norm(76)
% 0.17/0.40 thf(fact_127_semiring__norm_I69_J,axiom,
% 0.17/0.40 ! [M: num] :
% 0.17/0.40 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 0.17/0.40
% 0.17/0.40 % semiring_norm(69)
% 0.17/0.40 thf(fact_128__C5_Ohyps_C_I5_J,axiom,
% 0.17/0.40 ( m
% 0.17/0.40 = ( suc @ na ) ) ).
% 0.17/0.40
% 0.17/0.40 % "5.hyps"(5)
% 0.17/0.40 thf(fact_129_numeral__plus__one,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 0.17/0.40 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_plus_one
% 0.17/0.40 thf(fact_130_numeral__plus__one,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.17/0.40 = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_plus_one
% 0.17/0.40 thf(fact_131_numeral__plus__one,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.17/0.40 = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_plus_one
% 0.17/0.40 thf(fact_132_numeral__plus__one,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.17/0.40 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_plus_one
% 0.17/0.40 thf(fact_133_numeral__plus__one,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.17/0.40 = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_plus_one
% 0.17/0.40 thf(fact_134_one__plus__numeral,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 0.17/0.40 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_plus_numeral
% 0.17/0.40 thf(fact_135_one__plus__numeral,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.17/0.40 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_plus_numeral
% 0.17/0.40 thf(fact_136_one__plus__numeral,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.17/0.40 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_plus_numeral
% 0.17/0.40 thf(fact_137_one__plus__numeral,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.17/0.40 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_plus_numeral
% 0.17/0.40 thf(fact_138_one__plus__numeral,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.17/0.40 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_plus_numeral
% 0.17/0.40 thf(fact_139_power__strict__increasing__iff,axiom,
% 0.17/0.40 ! [B2: real,X4: nat,Y3: nat] :
% 0.17/0.40 ( ( ord_less_real @ one_one_real @ B2 )
% 0.17/0.40 => ( ( ord_less_real @ ( power_power_real @ B2 @ X4 ) @ ( power_power_real @ B2 @ Y3 ) )
% 0.17/0.40 = ( ord_less_nat @ X4 @ Y3 ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_strict_increasing_iff
% 0.17/0.40 thf(fact_140_power__strict__increasing__iff,axiom,
% 0.17/0.40 ! [B2: rat,X4: nat,Y3: nat] :
% 0.17/0.40 ( ( ord_less_rat @ one_one_rat @ B2 )
% 0.17/0.40 => ( ( ord_less_rat @ ( power_power_rat @ B2 @ X4 ) @ ( power_power_rat @ B2 @ Y3 ) )
% 0.17/0.40 = ( ord_less_nat @ X4 @ Y3 ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_strict_increasing_iff
% 0.17/0.40 thf(fact_141_power__strict__increasing__iff,axiom,
% 0.17/0.40 ! [B2: nat,X4: nat,Y3: nat] :
% 0.17/0.40 ( ( ord_less_nat @ one_one_nat @ B2 )
% 0.17/0.40 => ( ( ord_less_nat @ ( power_power_nat @ B2 @ X4 ) @ ( power_power_nat @ B2 @ Y3 ) )
% 0.17/0.40 = ( ord_less_nat @ X4 @ Y3 ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_strict_increasing_iff
% 0.17/0.40 thf(fact_142_power__strict__increasing__iff,axiom,
% 0.17/0.40 ! [B2: int,X4: nat,Y3: nat] :
% 0.17/0.40 ( ( ord_less_int @ one_one_int @ B2 )
% 0.17/0.40 => ( ( ord_less_int @ ( power_power_int @ B2 @ X4 ) @ ( power_power_int @ B2 @ Y3 ) )
% 0.17/0.40 = ( ord_less_nat @ X4 @ Y3 ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_strict_increasing_iff
% 0.17/0.40 thf(fact_143_one__add__one,axiom,
% 0.17/0.40 ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 0.17/0.40 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_add_one
% 0.17/0.40 thf(fact_144_one__add__one,axiom,
% 0.17/0.40 ( ( plus_plus_real @ one_one_real @ one_one_real )
% 0.17/0.40 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_add_one
% 0.17/0.40 thf(fact_145_one__add__one,axiom,
% 0.17/0.40 ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 0.17/0.40 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_add_one
% 0.17/0.40 thf(fact_146_one__add__one,axiom,
% 0.17/0.40 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.17/0.40 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_add_one
% 0.17/0.40 thf(fact_147_one__add__one,axiom,
% 0.17/0.40 ( ( plus_plus_int @ one_one_int @ one_one_int )
% 0.17/0.40 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_add_one
% 0.17/0.40 thf(fact_148_power__increasing__iff,axiom,
% 0.17/0.40 ! [B2: real,X4: nat,Y3: nat] :
% 0.17/0.40 ( ( ord_less_real @ one_one_real @ B2 )
% 0.17/0.40 => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ X4 ) @ ( power_power_real @ B2 @ Y3 ) )
% 0.17/0.40 = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_increasing_iff
% 0.17/0.40 thf(fact_149_power__increasing__iff,axiom,
% 0.17/0.40 ! [B2: rat,X4: nat,Y3: nat] :
% 0.17/0.40 ( ( ord_less_rat @ one_one_rat @ B2 )
% 0.17/0.40 => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ X4 ) @ ( power_power_rat @ B2 @ Y3 ) )
% 0.17/0.40 = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_increasing_iff
% 0.17/0.40 thf(fact_150_power__increasing__iff,axiom,
% 0.17/0.40 ! [B2: nat,X4: nat,Y3: nat] :
% 0.17/0.40 ( ( ord_less_nat @ one_one_nat @ B2 )
% 0.17/0.40 => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X4 ) @ ( power_power_nat @ B2 @ Y3 ) )
% 0.17/0.40 = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_increasing_iff
% 0.17/0.40 thf(fact_151_power__increasing__iff,axiom,
% 0.17/0.40 ! [B2: int,X4: nat,Y3: nat] :
% 0.17/0.40 ( ( ord_less_int @ one_one_int @ B2 )
% 0.17/0.40 => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X4 ) @ ( power_power_int @ B2 @ Y3 ) )
% 0.17/0.40 = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_increasing_iff
% 0.17/0.40 thf(fact_152_add__self__div__2,axiom,
% 0.17/0.40 ! [M: nat] :
% 0.17/0.40 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = M ) ).
% 0.17/0.40
% 0.17/0.40 % add_self_div_2
% 0.17/0.40 thf(fact_153_numeral__le__one__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.17/0.40 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_le_one_iff
% 0.17/0.40 thf(fact_154_numeral__le__one__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.17/0.40 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_le_one_iff
% 0.17/0.40 thf(fact_155_numeral__le__one__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.17/0.40 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_le_one_iff
% 0.17/0.40 thf(fact_156_numeral__le__one__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.17/0.40 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_le_one_iff
% 0.17/0.40 thf(fact_157_one__less__numeral__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.17/0.40 = ( ord_less_num @ one @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_less_numeral_iff
% 0.17/0.40 thf(fact_158_one__less__numeral__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.17/0.40 = ( ord_less_num @ one @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_less_numeral_iff
% 0.17/0.40 thf(fact_159_one__less__numeral__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.17/0.40 = ( ord_less_num @ one @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_less_numeral_iff
% 0.17/0.40 thf(fact_160_one__less__numeral__iff,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.17/0.40 = ( ord_less_num @ one @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_less_numeral_iff
% 0.17/0.40 thf(fact_161_is__num__normalize_I1_J,axiom,
% 0.17/0.40 ! [A: real,B2: real,C: real] :
% 0.17/0.40 ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.17/0.40 = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % is_num_normalize(1)
% 0.17/0.40 thf(fact_162_is__num__normalize_I1_J,axiom,
% 0.17/0.40 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.40 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.17/0.40 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % is_num_normalize(1)
% 0.17/0.40 thf(fact_163_is__num__normalize_I1_J,axiom,
% 0.17/0.40 ! [A: int,B2: int,C: int] :
% 0.17/0.40 ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.17/0.40 = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % is_num_normalize(1)
% 0.17/0.40 thf(fact_164_one__plus__numeral__commute,axiom,
% 0.17/0.40 ! [X4: num] :
% 0.17/0.40 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X4 ) )
% 0.17/0.40 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X4 ) @ one_one_complex ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_plus_numeral_commute
% 0.17/0.40 thf(fact_165_one__plus__numeral__commute,axiom,
% 0.17/0.40 ! [X4: num] :
% 0.17/0.40 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X4 ) )
% 0.17/0.40 = ( plus_plus_real @ ( numeral_numeral_real @ X4 ) @ one_one_real ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_plus_numeral_commute
% 0.17/0.40 thf(fact_166_one__plus__numeral__commute,axiom,
% 0.17/0.40 ! [X4: num] :
% 0.17/0.40 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X4 ) )
% 0.17/0.40 = ( plus_plus_rat @ ( numeral_numeral_rat @ X4 ) @ one_one_rat ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_plus_numeral_commute
% 0.17/0.40 thf(fact_167_one__plus__numeral__commute,axiom,
% 0.17/0.40 ! [X4: num] :
% 0.17/0.40 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X4 ) )
% 0.17/0.40 = ( plus_plus_nat @ ( numeral_numeral_nat @ X4 ) @ one_one_nat ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_plus_numeral_commute
% 0.17/0.40 thf(fact_168_one__plus__numeral__commute,axiom,
% 0.17/0.40 ! [X4: num] :
% 0.17/0.40 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X4 ) )
% 0.17/0.40 = ( plus_plus_int @ ( numeral_numeral_int @ X4 ) @ one_one_int ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_plus_numeral_commute
% 0.17/0.40 thf(fact_169_le__num__One__iff,axiom,
% 0.17/0.40 ! [X4: num] :
% 0.17/0.40 ( ( ord_less_eq_num @ X4 @ one )
% 0.17/0.40 = ( X4 = one ) ) ).
% 0.17/0.40
% 0.17/0.40 % le_num_One_iff
% 0.17/0.40 thf(fact_170_le__numeral__extra_I4_J,axiom,
% 0.17/0.40 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.17/0.40
% 0.17/0.40 % le_numeral_extra(4)
% 0.17/0.40 thf(fact_171_le__numeral__extra_I4_J,axiom,
% 0.17/0.40 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 0.17/0.40
% 0.17/0.40 % le_numeral_extra(4)
% 0.17/0.40 thf(fact_172_le__numeral__extra_I4_J,axiom,
% 0.17/0.40 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.17/0.40
% 0.17/0.40 % le_numeral_extra(4)
% 0.17/0.40 thf(fact_173_le__numeral__extra_I4_J,axiom,
% 0.17/0.40 ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.17/0.40
% 0.17/0.40 % le_numeral_extra(4)
% 0.17/0.40 thf(fact_174_less__numeral__extra_I4_J,axiom,
% 0.17/0.40 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.17/0.40
% 0.17/0.40 % less_numeral_extra(4)
% 0.17/0.40 thf(fact_175_less__numeral__extra_I4_J,axiom,
% 0.17/0.40 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 0.17/0.40
% 0.17/0.40 % less_numeral_extra(4)
% 0.17/0.40 thf(fact_176_less__numeral__extra_I4_J,axiom,
% 0.17/0.40 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.17/0.40
% 0.17/0.40 % less_numeral_extra(4)
% 0.17/0.40 thf(fact_177_less__numeral__extra_I4_J,axiom,
% 0.17/0.40 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.17/0.40
% 0.17/0.40 % less_numeral_extra(4)
% 0.17/0.40 thf(fact_178_one__le__power,axiom,
% 0.17/0.40 ! [A: real,N: nat] :
% 0.17/0.40 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.17/0.40 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_le_power
% 0.17/0.40 thf(fact_179_one__le__power,axiom,
% 0.17/0.40 ! [A: rat,N: nat] :
% 0.17/0.40 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.17/0.40 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_le_power
% 0.17/0.40 thf(fact_180_one__le__power,axiom,
% 0.17/0.40 ! [A: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.17/0.40 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_le_power
% 0.17/0.40 thf(fact_181_one__le__power,axiom,
% 0.17/0.40 ! [A: int,N: nat] :
% 0.17/0.40 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.17/0.40 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_le_power
% 0.17/0.40 thf(fact_182_power__one__over,axiom,
% 0.17/0.40 ! [A: complex,N: nat] :
% 0.17/0.40 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 0.17/0.40 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_one_over
% 0.17/0.40 thf(fact_183_power__one__over,axiom,
% 0.17/0.40 ! [A: real,N: nat] :
% 0.17/0.40 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 0.17/0.40 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_one_over
% 0.17/0.40 thf(fact_184_power__one__over,axiom,
% 0.17/0.40 ! [A: rat,N: nat] :
% 0.17/0.40 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 0.17/0.40 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_one_over
% 0.17/0.40 thf(fact_185_nat__1__add__1,axiom,
% 0.17/0.40 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.17/0.40 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % nat_1_add_1
% 0.17/0.40 thf(fact_186_numeral__Bit0,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 0.17/0.40 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_Bit0
% 0.17/0.40 thf(fact_187_numeral__Bit0,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 0.17/0.40 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_Bit0
% 0.17/0.40 thf(fact_188_numeral__Bit0,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 0.17/0.40 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_Bit0
% 0.17/0.40 thf(fact_189_numeral__Bit0,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.17/0.40 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_Bit0
% 0.17/0.40 thf(fact_190_numeral__Bit0,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 0.17/0.40 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_Bit0
% 0.17/0.40 thf(fact_191_power__strict__increasing,axiom,
% 0.17/0.40 ! [N: nat,N3: nat,A: real] :
% 0.17/0.40 ( ( ord_less_nat @ N @ N3 )
% 0.17/0.40 => ( ( ord_less_real @ one_one_real @ A )
% 0.17/0.40 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_strict_increasing
% 0.17/0.40 thf(fact_192_power__strict__increasing,axiom,
% 0.17/0.40 ! [N: nat,N3: nat,A: rat] :
% 0.17/0.40 ( ( ord_less_nat @ N @ N3 )
% 0.17/0.40 => ( ( ord_less_rat @ one_one_rat @ A )
% 0.17/0.40 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_strict_increasing
% 0.17/0.40 thf(fact_193_power__strict__increasing,axiom,
% 0.17/0.40 ! [N: nat,N3: nat,A: nat] :
% 0.17/0.40 ( ( ord_less_nat @ N @ N3 )
% 0.17/0.40 => ( ( ord_less_nat @ one_one_nat @ A )
% 0.17/0.40 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_strict_increasing
% 0.17/0.40 thf(fact_194_power__strict__increasing,axiom,
% 0.17/0.40 ! [N: nat,N3: nat,A: int] :
% 0.17/0.40 ( ( ord_less_nat @ N @ N3 )
% 0.17/0.40 => ( ( ord_less_int @ one_one_int @ A )
% 0.17/0.40 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_strict_increasing
% 0.17/0.40 thf(fact_195_power__less__imp__less__exp,axiom,
% 0.17/0.40 ! [A: real,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_real @ one_one_real @ A )
% 0.17/0.40 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.17/0.40 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_less_imp_less_exp
% 0.17/0.40 thf(fact_196_power__less__imp__less__exp,axiom,
% 0.17/0.40 ! [A: rat,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_rat @ one_one_rat @ A )
% 0.17/0.40 => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.17/0.40 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_less_imp_less_exp
% 0.17/0.40 thf(fact_197_power__less__imp__less__exp,axiom,
% 0.17/0.40 ! [A: nat,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_nat @ one_one_nat @ A )
% 0.17/0.40 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.17/0.40 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_less_imp_less_exp
% 0.17/0.40 thf(fact_198_power__less__imp__less__exp,axiom,
% 0.17/0.40 ! [A: int,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_int @ one_one_int @ A )
% 0.17/0.40 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.17/0.40 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_less_imp_less_exp
% 0.17/0.40 thf(fact_199_power__increasing,axiom,
% 0.17/0.40 ! [N: nat,N3: nat,A: real] :
% 0.17/0.40 ( ( ord_less_eq_nat @ N @ N3 )
% 0.17/0.40 => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.17/0.40 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_increasing
% 0.17/0.40 thf(fact_200_power__increasing,axiom,
% 0.17/0.40 ! [N: nat,N3: nat,A: rat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ N @ N3 )
% 0.17/0.40 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.17/0.40 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_increasing
% 0.17/0.40 thf(fact_201_power__increasing,axiom,
% 0.17/0.40 ! [N: nat,N3: nat,A: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ N @ N3 )
% 0.17/0.40 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.17/0.40 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_increasing
% 0.17/0.40 thf(fact_202_power__increasing,axiom,
% 0.17/0.40 ! [N: nat,N3: nat,A: int] :
% 0.17/0.40 ( ( ord_less_eq_nat @ N @ N3 )
% 0.17/0.40 => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.17/0.40 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_increasing
% 0.17/0.40 thf(fact_203_one__le__numeral,axiom,
% 0.17/0.40 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_le_numeral
% 0.17/0.40 thf(fact_204_one__le__numeral,axiom,
% 0.17/0.40 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_le_numeral
% 0.17/0.40 thf(fact_205_one__le__numeral,axiom,
% 0.17/0.40 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_le_numeral
% 0.17/0.40 thf(fact_206_one__le__numeral,axiom,
% 0.17/0.40 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 0.17/0.40
% 0.17/0.40 % one_le_numeral
% 0.17/0.40 thf(fact_207_not__numeral__less__one,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 0.17/0.40
% 0.17/0.40 % not_numeral_less_one
% 0.17/0.40 thf(fact_208_not__numeral__less__one,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 0.17/0.40
% 0.17/0.40 % not_numeral_less_one
% 0.17/0.40 thf(fact_209_not__numeral__less__one,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 0.17/0.40
% 0.17/0.40 % not_numeral_less_one
% 0.17/0.40 thf(fact_210_not__numeral__less__one,axiom,
% 0.17/0.40 ! [N: num] :
% 0.17/0.40 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 0.17/0.40
% 0.17/0.40 % not_numeral_less_one
% 0.17/0.40 thf(fact_211_numeral__One,axiom,
% 0.17/0.40 ( ( numera6690914467698888265omplex @ one )
% 0.17/0.40 = one_one_complex ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_One
% 0.17/0.40 thf(fact_212_numeral__One,axiom,
% 0.17/0.40 ( ( numeral_numeral_real @ one )
% 0.17/0.40 = one_one_real ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_One
% 0.17/0.40 thf(fact_213_numeral__One,axiom,
% 0.17/0.40 ( ( numeral_numeral_rat @ one )
% 0.17/0.40 = one_one_rat ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_One
% 0.17/0.40 thf(fact_214_numeral__One,axiom,
% 0.17/0.40 ( ( numeral_numeral_nat @ one )
% 0.17/0.40 = one_one_nat ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_One
% 0.17/0.40 thf(fact_215_numeral__One,axiom,
% 0.17/0.40 ( ( numeral_numeral_int @ one )
% 0.17/0.40 = one_one_int ) ).
% 0.17/0.40
% 0.17/0.40 % numeral_One
% 0.17/0.40 thf(fact_216_numerals_I1_J,axiom,
% 0.17/0.40 ( ( numeral_numeral_nat @ one )
% 0.17/0.40 = one_one_nat ) ).
% 0.17/0.40
% 0.17/0.40 % numerals(1)
% 0.17/0.40 thf(fact_217_ex__power__ivl1,axiom,
% 0.17/0.40 ! [B2: nat,K: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.17/0.40 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 0.17/0.40 => ? [N4: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N4 ) @ K )
% 0.17/0.40 & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % ex_power_ivl1
% 0.17/0.40 thf(fact_218_ex__power__ivl2,axiom,
% 0.17/0.40 ! [B2: nat,K: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.17/0.40 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.17/0.40 => ? [N4: nat] :
% 0.17/0.40 ( ( ord_less_nat @ ( power_power_nat @ B2 @ N4 ) @ K )
% 0.17/0.40 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % ex_power_ivl2
% 0.17/0.40 thf(fact_219_power2__commute,axiom,
% 0.17/0.40 ! [X4: complex,Y3: complex] :
% 0.17/0.40 ( ( power_power_complex @ ( minus_minus_complex @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = ( power_power_complex @ ( minus_minus_complex @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power2_commute
% 0.17/0.40 thf(fact_220_power2__commute,axiom,
% 0.17/0.40 ! [X4: real,Y3: real] :
% 0.17/0.40 ( ( power_power_real @ ( minus_minus_real @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = ( power_power_real @ ( minus_minus_real @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power2_commute
% 0.17/0.40 thf(fact_221_power2__commute,axiom,
% 0.17/0.40 ! [X4: rat,Y3: rat] :
% 0.17/0.40 ( ( power_power_rat @ ( minus_minus_rat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = ( power_power_rat @ ( minus_minus_rat @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power2_commute
% 0.17/0.40 thf(fact_222_power2__commute,axiom,
% 0.17/0.40 ! [X4: int,Y3: int] :
% 0.17/0.40 ( ( power_power_int @ ( minus_minus_int @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = ( power_power_int @ ( minus_minus_int @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power2_commute
% 0.17/0.40 thf(fact_223_power__le__imp__le__exp,axiom,
% 0.17/0.40 ! [A: real,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_real @ one_one_real @ A )
% 0.17/0.40 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.17/0.40 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_le_imp_le_exp
% 0.17/0.40 thf(fact_224_power__le__imp__le__exp,axiom,
% 0.17/0.40 ! [A: rat,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_rat @ one_one_rat @ A )
% 0.17/0.40 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.17/0.40 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_le_imp_le_exp
% 0.17/0.40 thf(fact_225_power__le__imp__le__exp,axiom,
% 0.17/0.40 ! [A: nat,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_nat @ one_one_nat @ A )
% 0.17/0.40 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.17/0.40 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_le_imp_le_exp
% 0.17/0.40 thf(fact_226_power__le__imp__le__exp,axiom,
% 0.17/0.40 ! [A: int,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_int @ one_one_int @ A )
% 0.17/0.40 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.17/0.40 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_le_imp_le_exp
% 0.17/0.40 thf(fact_227_one__power2,axiom,
% 0.17/0.40 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = one_one_rat ) ).
% 0.17/0.40
% 0.17/0.40 % one_power2
% 0.17/0.40 thf(fact_228_one__power2,axiom,
% 0.17/0.40 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = one_one_nat ) ).
% 0.17/0.40
% 0.17/0.40 % one_power2
% 0.17/0.40 thf(fact_229_one__power2,axiom,
% 0.17/0.40 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = one_one_real ) ).
% 0.17/0.40
% 0.17/0.40 % one_power2
% 0.17/0.40 thf(fact_230_one__power2,axiom,
% 0.17/0.40 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = one_one_int ) ).
% 0.17/0.40
% 0.17/0.40 % one_power2
% 0.17/0.40 thf(fact_231_one__power2,axiom,
% 0.17/0.40 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.40 = one_one_complex ) ).
% 0.17/0.40
% 0.17/0.40 % one_power2
% 0.17/0.40 thf(fact_232_diff__le__diff__pow,axiom,
% 0.17/0.40 ! [K: nat,M: nat,N: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.17/0.40 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % diff_le_diff_pow
% 0.17/0.40 thf(fact_233_power__divide,axiom,
% 0.17/0.40 ! [A: complex,B2: complex,N: nat] :
% 0.17/0.40 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B2 ) @ N )
% 0.17/0.40 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_divide
% 0.17/0.40 thf(fact_234_power__divide,axiom,
% 0.17/0.40 ! [A: real,B2: real,N: nat] :
% 0.17/0.40 ( ( power_power_real @ ( divide_divide_real @ A @ B2 ) @ N )
% 0.17/0.40 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_divide
% 0.17/0.40 thf(fact_235_power__divide,axiom,
% 0.17/0.40 ! [A: rat,B2: rat,N: nat] :
% 0.17/0.40 ( ( power_power_rat @ ( divide_divide_rat @ A @ B2 ) @ N )
% 0.17/0.40 = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % power_divide
% 0.17/0.40 thf(fact_236_Nat_Oadd__diff__assoc,axiom,
% 0.17/0.40 ! [K: nat,J: nat,I: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.40 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.17/0.40 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % Nat.add_diff_assoc
% 0.17/0.40 thf(fact_237_Nat_Oadd__diff__assoc2,axiom,
% 0.17/0.40 ! [K: nat,J: nat,I: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.40 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.17/0.40 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.17/0.40
% 0.17/0.40 % Nat.add_diff_assoc2
% 0.17/0.40 thf(fact_238_Nat_Odiff__diff__right,axiom,
% 0.17/0.40 ! [K: nat,J: nat,I: nat] :
% 0.17/0.40 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.40 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.17/0.41 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Nat.diff_diff_right
% 0.17/0.41 thf(fact_239__092_060open_062high_Ax_An_A_060_A2_A_094_Am_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_092_060close_062,axiom,
% 0.17/0.41 ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.17/0.41 & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % \<open>high x n < 2 ^ m \<and> low x n < 2 ^ n\<close>
% 0.17/0.41 thf(fact_240_le__add__diff__inverse,axiom,
% 0.17/0.41 ! [B2: real,A: real] :
% 0.17/0.41 ( ( ord_less_eq_real @ B2 @ A )
% 0.17/0.41 => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_add_diff_inverse
% 0.17/0.41 thf(fact_241_le__add__diff__inverse,axiom,
% 0.17/0.41 ! [B2: rat,A: rat] :
% 0.17/0.41 ( ( ord_less_eq_rat @ B2 @ A )
% 0.17/0.41 => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_add_diff_inverse
% 0.17/0.41 thf(fact_242_le__add__diff__inverse,axiom,
% 0.17/0.41 ! [B2: nat,A: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ B2 @ A )
% 0.17/0.41 => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_add_diff_inverse
% 0.17/0.41 thf(fact_243_le__add__diff__inverse,axiom,
% 0.17/0.41 ! [B2: int,A: int] :
% 0.17/0.41 ( ( ord_less_eq_int @ B2 @ A )
% 0.17/0.41 => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_add_diff_inverse
% 0.17/0.41 thf(fact_244_le__add__diff__inverse2,axiom,
% 0.17/0.41 ! [B2: real,A: real] :
% 0.17/0.41 ( ( ord_less_eq_real @ B2 @ A )
% 0.17/0.41 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_add_diff_inverse2
% 0.17/0.41 thf(fact_245_le__add__diff__inverse2,axiom,
% 0.17/0.41 ! [B2: rat,A: rat] :
% 0.17/0.41 ( ( ord_less_eq_rat @ B2 @ A )
% 0.17/0.41 => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_add_diff_inverse2
% 0.17/0.41 thf(fact_246_le__add__diff__inverse2,axiom,
% 0.17/0.41 ! [B2: nat,A: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ B2 @ A )
% 0.17/0.41 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_add_diff_inverse2
% 0.17/0.41 thf(fact_247_le__add__diff__inverse2,axiom,
% 0.17/0.41 ! [B2: int,A: int] :
% 0.17/0.41 ( ( ord_less_eq_int @ B2 @ A )
% 0.17/0.41 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_add_diff_inverse2
% 0.17/0.41 thf(fact_248_div__exp__eq,axiom,
% 0.17/0.41 ! [A: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( 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 ) )
% 0.17/0.41 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % div_exp_eq
% 0.17/0.41 thf(fact_249_div__exp__eq,axiom,
% 0.17/0.41 ! [A: int,M: nat,N: nat] :
% 0.17/0.41 ( ( 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 ) )
% 0.17/0.41 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % div_exp_eq
% 0.17/0.41 thf(fact_250_field__less__half__sum,axiom,
% 0.17/0.41 ! [X4: real,Y3: real] :
% 0.17/0.41 ( ( ord_less_real @ X4 @ Y3 )
% 0.17/0.41 => ( ord_less_real @ X4 @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % field_less_half_sum
% 0.17/0.41 thf(fact_251_field__less__half__sum,axiom,
% 0.17/0.41 ! [X4: rat,Y3: rat] :
% 0.17/0.41 ( ( ord_less_rat @ X4 @ Y3 )
% 0.17/0.41 => ( ord_less_rat @ X4 @ ( divide_divide_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % field_less_half_sum
% 0.17/0.41 thf(fact_252_high__inv,axiom,
% 0.17/0.41 ! [X4: nat,N: nat,Y3: nat] :
% 0.17/0.41 ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.41 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X4 ) @ N )
% 0.17/0.41 = Y3 ) ) ).
% 0.17/0.41
% 0.17/0.41 % high_inv
% 0.17/0.41 thf(fact_253_diff__diff__left,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.17/0.41 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_diff_left
% 0.17/0.41 thf(fact_254_diff__diff__cancel,axiom,
% 0.17/0.41 ! [I: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ N )
% 0.17/0.41 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.17/0.41 = I ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_diff_cancel
% 0.17/0.41 thf(fact_255_nat__add__left__cancel__le,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.17/0.41 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_add_left_cancel_le
% 0.17/0.41 thf(fact_256_even__odd__cases,axiom,
% 0.17/0.41 ! [X4: nat] :
% 0.17/0.41 ( ! [N4: nat] :
% 0.17/0.41 ( X4
% 0.17/0.41 != ( plus_plus_nat @ N4 @ N4 ) )
% 0.17/0.41 => ~ ! [N4: nat] :
% 0.17/0.41 ( X4
% 0.17/0.41 != ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % even_odd_cases
% 0.17/0.41 thf(fact_257_bit__split__inv,axiom,
% 0.17/0.41 ! [X4: nat,D: nat] :
% 0.17/0.41 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X4 @ D ) @ ( vEBT_VEBT_low @ X4 @ D ) @ D )
% 0.17/0.41 = X4 ) ).
% 0.17/0.41
% 0.17/0.41 % bit_split_inv
% 0.17/0.41 thf(fact_258_nat_Oinject,axiom,
% 0.17/0.41 ! [X2: nat,Y2: nat] :
% 0.17/0.41 ( ( ( suc @ X2 )
% 0.17/0.41 = ( suc @ Y2 ) )
% 0.17/0.41 = ( X2 = Y2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat.inject
% 0.17/0.41 thf(fact_259_old_Onat_Oinject,axiom,
% 0.17/0.41 ! [Nat: nat,Nat2: nat] :
% 0.17/0.41 ( ( ( suc @ Nat )
% 0.17/0.41 = ( suc @ Nat2 ) )
% 0.17/0.41 = ( Nat = Nat2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % old.nat.inject
% 0.17/0.41 thf(fact_260_numeral__times__numeral,axiom,
% 0.17/0.41 ! [M: num,N: num] :
% 0.17/0.41 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.17/0.41 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % numeral_times_numeral
% 0.17/0.41 thf(fact_261_numeral__times__numeral,axiom,
% 0.17/0.41 ! [M: num,N: num] :
% 0.17/0.41 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.17/0.41 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % numeral_times_numeral
% 0.17/0.41 thf(fact_262_numeral__times__numeral,axiom,
% 0.17/0.41 ! [M: num,N: num] :
% 0.17/0.41 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.17/0.41 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % numeral_times_numeral
% 0.17/0.41 thf(fact_263_numeral__times__numeral,axiom,
% 0.17/0.41 ! [M: num,N: num] :
% 0.17/0.41 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.17/0.41 = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % numeral_times_numeral
% 0.17/0.41 thf(fact_264_numeral__times__numeral,axiom,
% 0.17/0.41 ! [M: num,N: num] :
% 0.17/0.41 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.17/0.41 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % numeral_times_numeral
% 0.17/0.41 thf(fact_265_mult__numeral__left__semiring__numeral,axiom,
% 0.17/0.41 ! [V: num,W: num,Z: complex] :
% 0.17/0.41 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.17/0.41 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_left_semiring_numeral
% 0.17/0.41 thf(fact_266_mult__numeral__left__semiring__numeral,axiom,
% 0.17/0.41 ! [V: num,W: num,Z: real] :
% 0.17/0.41 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.17/0.41 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_left_semiring_numeral
% 0.17/0.41 thf(fact_267_mult__numeral__left__semiring__numeral,axiom,
% 0.17/0.41 ! [V: num,W: num,Z: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.17/0.41 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_left_semiring_numeral
% 0.17/0.41 thf(fact_268_mult__numeral__left__semiring__numeral,axiom,
% 0.17/0.41 ! [V: num,W: num,Z: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.17/0.41 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_left_semiring_numeral
% 0.17/0.41 thf(fact_269_mult__numeral__left__semiring__numeral,axiom,
% 0.17/0.41 ! [V: num,W: num,Z: int] :
% 0.17/0.41 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.17/0.41 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_left_semiring_numeral
% 0.17/0.41 thf(fact_270_low__inv,axiom,
% 0.17/0.41 ! [X4: nat,N: nat,Y3: nat] :
% 0.17/0.41 ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.41 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X4 ) @ N )
% 0.17/0.41 = X4 ) ) ).
% 0.17/0.41
% 0.17/0.41 % low_inv
% 0.17/0.41 thf(fact_271_bits__div__by__1,axiom,
% 0.17/0.41 ! [A: nat] :
% 0.17/0.41 ( ( divide_divide_nat @ A @ one_one_nat )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % bits_div_by_1
% 0.17/0.41 thf(fact_272_bits__div__by__1,axiom,
% 0.17/0.41 ! [A: int] :
% 0.17/0.41 ( ( divide_divide_int @ A @ one_one_int )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % bits_div_by_1
% 0.17/0.41 thf(fact_273_div__by__1,axiom,
% 0.17/0.41 ! [A: complex] :
% 0.17/0.41 ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % div_by_1
% 0.17/0.41 thf(fact_274_div__by__1,axiom,
% 0.17/0.41 ! [A: real] :
% 0.17/0.41 ( ( divide_divide_real @ A @ one_one_real )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % div_by_1
% 0.17/0.41 thf(fact_275_div__by__1,axiom,
% 0.17/0.41 ! [A: rat] :
% 0.17/0.41 ( ( divide_divide_rat @ A @ one_one_rat )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % div_by_1
% 0.17/0.41 thf(fact_276_div__by__1,axiom,
% 0.17/0.41 ! [A: nat] :
% 0.17/0.41 ( ( divide_divide_nat @ A @ one_one_nat )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % div_by_1
% 0.17/0.41 thf(fact_277_div__by__1,axiom,
% 0.17/0.41 ! [A: int] :
% 0.17/0.41 ( ( divide_divide_int @ A @ one_one_int )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % div_by_1
% 0.17/0.41 thf(fact_278_lessI,axiom,
% 0.17/0.41 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % lessI
% 0.17/0.41 thf(fact_279_Suc__mono,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ N )
% 0.17/0.41 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_mono
% 0.17/0.41 thf(fact_280_Suc__less__eq,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.17/0.41 = ( ord_less_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_less_eq
% 0.17/0.41 thf(fact_281_Suc__le__mono,axiom,
% 0.17/0.41 ! [N: nat,M: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.17/0.41 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_le_mono
% 0.17/0.41 thf(fact_282_add__Suc__right,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.17/0.41 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_Suc_right
% 0.17/0.41 thf(fact_283_nat__add__left__cancel__less,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.17/0.41 = ( ord_less_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_add_left_cancel_less
% 0.17/0.41 thf(fact_284_diff__Suc__Suc,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.17/0.41 = ( minus_minus_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_Suc_Suc
% 0.17/0.41 thf(fact_285_Suc__diff__diff,axiom,
% 0.17/0.41 ! [M: nat,N: nat,K: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.17/0.41 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_diff_diff
% 0.17/0.41 thf(fact_286_nat__1__eq__mult__iff,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( one_one_nat
% 0.17/0.41 = ( times_times_nat @ M @ N ) )
% 0.17/0.41 = ( ( M = one_one_nat )
% 0.17/0.41 & ( N = one_one_nat ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_1_eq_mult_iff
% 0.17/0.41 thf(fact_287_nat__mult__eq__1__iff,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ( times_times_nat @ M @ N )
% 0.17/0.41 = one_one_nat )
% 0.17/0.41 = ( ( M = one_one_nat )
% 0.17/0.41 & ( N = one_one_nat ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_mult_eq_1_iff
% 0.17/0.41 thf(fact_288_semiring__norm_I6_J,axiom,
% 0.17/0.41 ! [M: num,N: num] :
% 0.17/0.41 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.17/0.41 = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % semiring_norm(6)
% 0.17/0.41 thf(fact_289_bit__concat__def,axiom,
% 0.17/0.41 ( vEBT_VEBT_bit_concat
% 0.17/0.41 = ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % bit_concat_def
% 0.17/0.41 thf(fact_290_distrib__left__numeral,axiom,
% 0.17/0.41 ! [V: num,B2: complex,C: complex] :
% 0.17/0.41 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_left_numeral
% 0.17/0.41 thf(fact_291_distrib__left__numeral,axiom,
% 0.17/0.41 ! [V: num,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_left_numeral
% 0.17/0.41 thf(fact_292_distrib__left__numeral,axiom,
% 0.17/0.41 ! [V: num,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_left_numeral
% 0.17/0.41 thf(fact_293_distrib__left__numeral,axiom,
% 0.17/0.41 ! [V: num,B2: nat,C: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_left_numeral
% 0.17/0.41 thf(fact_294_distrib__left__numeral,axiom,
% 0.17/0.41 ! [V: num,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_left_numeral
% 0.17/0.41 thf(fact_295_distrib__right__numeral,axiom,
% 0.17/0.41 ! [A: complex,B2: complex,V: num] :
% 0.17/0.41 ( ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
% 0.17/0.41 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_right_numeral
% 0.17/0.41 thf(fact_296_distrib__right__numeral,axiom,
% 0.17/0.41 ! [A: real,B2: real,V: num] :
% 0.17/0.41 ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_right_numeral
% 0.17/0.41 thf(fact_297_distrib__right__numeral,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,V: num] :
% 0.17/0.41 ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_right_numeral
% 0.17/0.41 thf(fact_298_distrib__right__numeral,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,V: num] :
% 0.17/0.41 ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ ( numeral_numeral_nat @ V ) )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_right_numeral
% 0.17/0.41 thf(fact_299_distrib__right__numeral,axiom,
% 0.17/0.41 ! [A: int,B2: int,V: num] :
% 0.17/0.41 ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_right_numeral
% 0.17/0.41 thf(fact_300_left__diff__distrib__numeral,axiom,
% 0.17/0.41 ! [A: complex,B2: complex,V: num] :
% 0.17/0.41 ( ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
% 0.17/0.41 = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_diff_distrib_numeral
% 0.17/0.41 thf(fact_301_left__diff__distrib__numeral,axiom,
% 0.17/0.41 ! [A: real,B2: real,V: num] :
% 0.17/0.41 ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
% 0.17/0.41 = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_diff_distrib_numeral
% 0.17/0.41 thf(fact_302_left__diff__distrib__numeral,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,V: num] :
% 0.17/0.41 ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
% 0.17/0.41 = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_diff_distrib_numeral
% 0.17/0.41 thf(fact_303_left__diff__distrib__numeral,axiom,
% 0.17/0.41 ! [A: int,B2: int,V: num] :
% 0.17/0.41 ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
% 0.17/0.41 = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_diff_distrib_numeral
% 0.17/0.41 thf(fact_304_right__diff__distrib__numeral,axiom,
% 0.17/0.41 ! [V: num,B2: complex,C: complex] :
% 0.17/0.41 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % right_diff_distrib_numeral
% 0.17/0.41 thf(fact_305_right__diff__distrib__numeral,axiom,
% 0.17/0.41 ! [V: num,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % right_diff_distrib_numeral
% 0.17/0.41 thf(fact_306_right__diff__distrib__numeral,axiom,
% 0.17/0.41 ! [V: num,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % right_diff_distrib_numeral
% 0.17/0.41 thf(fact_307_right__diff__distrib__numeral,axiom,
% 0.17/0.41 ! [V: num,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % right_diff_distrib_numeral
% 0.17/0.41 thf(fact_308_mult__Suc__right,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( times_times_nat @ M @ ( suc @ N ) )
% 0.17/0.41 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_Suc_right
% 0.17/0.41 thf(fact_309_diff__Suc__1,axiom,
% 0.17/0.41 ! [N: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 0.17/0.41 = N ) ).
% 0.17/0.41
% 0.17/0.41 % diff_Suc_1
% 0.17/0.41 thf(fact_310_semiring__norm_I2_J,axiom,
% 0.17/0.41 ( ( plus_plus_num @ one @ one )
% 0.17/0.41 = ( bit0 @ one ) ) ).
% 0.17/0.41
% 0.17/0.41 % semiring_norm(2)
% 0.17/0.41 thf(fact_311_le__divide__eq__numeral1_I1_J,axiom,
% 0.17/0.41 ! [A: real,B2: real,W: num] :
% 0.17/0.41 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 0.17/0.41 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_divide_eq_numeral1(1)
% 0.17/0.41 thf(fact_312_le__divide__eq__numeral1_I1_J,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,W: num] :
% 0.17/0.41 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 0.17/0.41 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_divide_eq_numeral1(1)
% 0.17/0.41 thf(fact_313_divide__le__eq__numeral1_I1_J,axiom,
% 0.17/0.41 ! [B2: real,W: num,A: real] :
% 0.17/0.41 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
% 0.17/0.41 = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % divide_le_eq_numeral1(1)
% 0.17/0.41 thf(fact_314_divide__le__eq__numeral1_I1_J,axiom,
% 0.17/0.41 ! [B2: rat,W: num,A: rat] :
% 0.17/0.41 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.17/0.41 = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % divide_le_eq_numeral1(1)
% 0.17/0.41 thf(fact_315_less__divide__eq__numeral1_I1_J,axiom,
% 0.17/0.41 ! [A: real,B2: real,W: num] :
% 0.17/0.41 ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 0.17/0.41 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_divide_eq_numeral1(1)
% 0.17/0.41 thf(fact_316_less__divide__eq__numeral1_I1_J,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,W: num] :
% 0.17/0.41 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 0.17/0.41 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_divide_eq_numeral1(1)
% 0.17/0.41 thf(fact_317_divide__less__eq__numeral1_I1_J,axiom,
% 0.17/0.41 ! [B2: real,W: num,A: real] :
% 0.17/0.41 ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
% 0.17/0.41 = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % divide_less_eq_numeral1(1)
% 0.17/0.41 thf(fact_318_divide__less__eq__numeral1_I1_J,axiom,
% 0.17/0.41 ! [B2: rat,W: num,A: rat] :
% 0.17/0.41 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.17/0.41 = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % divide_less_eq_numeral1(1)
% 0.17/0.41 thf(fact_319_diff__Suc__diff__eq2,axiom,
% 0.17/0.41 ! [K: nat,J: nat,I: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.41 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 0.17/0.41 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_Suc_diff_eq2
% 0.17/0.41 thf(fact_320_diff__Suc__diff__eq1,axiom,
% 0.17/0.41 ! [K: nat,J: nat,I: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.41 => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 0.17/0.41 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_Suc_diff_eq1
% 0.17/0.41 thf(fact_321_power__add__numeral,axiom,
% 0.17/0.41 ! [A: complex,M: num,N: num] :
% 0.17/0.41 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.17/0.41 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add_numeral
% 0.17/0.41 thf(fact_322_power__add__numeral,axiom,
% 0.17/0.41 ! [A: real,M: num,N: num] :
% 0.17/0.41 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.17/0.41 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add_numeral
% 0.17/0.41 thf(fact_323_power__add__numeral,axiom,
% 0.17/0.41 ! [A: rat,M: num,N: num] :
% 0.17/0.41 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.17/0.41 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add_numeral
% 0.17/0.41 thf(fact_324_power__add__numeral,axiom,
% 0.17/0.41 ! [A: nat,M: num,N: num] :
% 0.17/0.41 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.17/0.41 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add_numeral
% 0.17/0.41 thf(fact_325_power__add__numeral,axiom,
% 0.17/0.41 ! [A: int,M: num,N: num] :
% 0.17/0.41 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.17/0.41 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add_numeral
% 0.17/0.41 thf(fact_326_power__add__numeral2,axiom,
% 0.17/0.41 ! [A: complex,M: num,N: num,B2: complex] :
% 0.17/0.41 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 0.17/0.41 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add_numeral2
% 0.17/0.41 thf(fact_327_power__add__numeral2,axiom,
% 0.17/0.41 ! [A: real,M: num,N: num,B2: real] :
% 0.17/0.41 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 0.17/0.41 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add_numeral2
% 0.17/0.41 thf(fact_328_power__add__numeral2,axiom,
% 0.17/0.41 ! [A: rat,M: num,N: num,B2: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 0.17/0.41 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add_numeral2
% 0.17/0.41 thf(fact_329_power__add__numeral2,axiom,
% 0.17/0.41 ! [A: nat,M: num,N: num,B2: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 0.17/0.41 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add_numeral2
% 0.17/0.41 thf(fact_330_power__add__numeral2,axiom,
% 0.17/0.41 ! [A: int,M: num,N: num,B2: int] :
% 0.17/0.41 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 0.17/0.41 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add_numeral2
% 0.17/0.41 thf(fact_331_Suc__numeral,axiom,
% 0.17/0.41 ! [N: num] :
% 0.17/0.41 ( ( suc @ ( numeral_numeral_nat @ N ) )
% 0.17/0.41 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_numeral
% 0.17/0.41 thf(fact_332_add__2__eq__Suc_H,axiom,
% 0.17/0.41 ! [N: nat] :
% 0.17/0.41 ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( suc @ ( suc @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_2_eq_Suc'
% 0.17/0.41 thf(fact_333_add__2__eq__Suc,axiom,
% 0.17/0.41 ! [N: nat] :
% 0.17/0.41 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.41 = ( suc @ ( suc @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_2_eq_Suc
% 0.17/0.41 thf(fact_334_div2__Suc__Suc,axiom,
% 0.17/0.41 ! [M: nat] :
% 0.17/0.41 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % div2_Suc_Suc
% 0.17/0.41 thf(fact_335_Suc__1,axiom,
% 0.17/0.41 ( ( suc @ one_one_nat )
% 0.17/0.41 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_1
% 0.17/0.41 thf(fact_336_Suc__mult__le__cancel1,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.17/0.41 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_mult_le_cancel1
% 0.17/0.41 thf(fact_337_Suc__mult__less__cancel1,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.17/0.41 = ( ord_less_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_mult_less_cancel1
% 0.17/0.41 thf(fact_338_mult__Suc,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( suc @ M ) @ N )
% 0.17/0.41 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_Suc
% 0.17/0.41 thf(fact_339_Suc__inject,axiom,
% 0.17/0.41 ! [X4: nat,Y3: nat] :
% 0.17/0.41 ( ( ( suc @ X4 )
% 0.17/0.41 = ( suc @ Y3 ) )
% 0.17/0.41 => ( X4 = Y3 ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_inject
% 0.17/0.41 thf(fact_340_n__not__Suc__n,axiom,
% 0.17/0.41 ! [N: nat] :
% 0.17/0.41 ( N
% 0.17/0.41 != ( suc @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % n_not_Suc_n
% 0.17/0.41 thf(fact_341_Suc__mult__cancel1,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.17/0.41 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.17/0.41 = ( M = N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_mult_cancel1
% 0.17/0.41 thf(fact_342_power__Suc2,axiom,
% 0.17/0.41 ! [A: complex,N: nat] :
% 0.17/0.41 ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.17/0.41 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_Suc2
% 0.17/0.41 thf(fact_343_power__Suc2,axiom,
% 0.17/0.41 ! [A: real,N: nat] :
% 0.17/0.41 ( ( power_power_real @ A @ ( suc @ N ) )
% 0.17/0.41 = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_Suc2
% 0.17/0.41 thf(fact_344_power__Suc2,axiom,
% 0.17/0.41 ! [A: rat,N: nat] :
% 0.17/0.41 ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.17/0.41 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_Suc2
% 0.17/0.41 thf(fact_345_power__Suc2,axiom,
% 0.17/0.41 ! [A: nat,N: nat] :
% 0.17/0.41 ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.17/0.41 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_Suc2
% 0.17/0.41 thf(fact_346_power__Suc2,axiom,
% 0.17/0.41 ! [A: int,N: nat] :
% 0.17/0.41 ( ( power_power_int @ A @ ( suc @ N ) )
% 0.17/0.41 = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_Suc2
% 0.17/0.41 thf(fact_347_power__Suc,axiom,
% 0.17/0.41 ! [A: complex,N: nat] :
% 0.17/0.41 ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.17/0.41 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_Suc
% 0.17/0.41 thf(fact_348_power__Suc,axiom,
% 0.17/0.41 ! [A: real,N: nat] :
% 0.17/0.41 ( ( power_power_real @ A @ ( suc @ N ) )
% 0.17/0.41 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_Suc
% 0.17/0.41 thf(fact_349_power__Suc,axiom,
% 0.17/0.41 ! [A: rat,N: nat] :
% 0.17/0.41 ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.17/0.41 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_Suc
% 0.17/0.41 thf(fact_350_power__Suc,axiom,
% 0.17/0.41 ! [A: nat,N: nat] :
% 0.17/0.41 ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.17/0.41 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_Suc
% 0.17/0.41 thf(fact_351_power__Suc,axiom,
% 0.17/0.41 ! [A: int,N: nat] :
% 0.17/0.41 ( ( power_power_int @ A @ ( suc @ N ) )
% 0.17/0.41 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_Suc
% 0.17/0.41 thf(fact_352_ring__class_Oring__distribs_I2_J,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ring_class.ring_distribs(2)
% 0.17/0.41 thf(fact_353_ring__class_Oring__distribs_I2_J,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ring_class.ring_distribs(2)
% 0.17/0.41 thf(fact_354_ring__class_Oring__distribs_I2_J,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ring_class.ring_distribs(2)
% 0.17/0.41 thf(fact_355_ring__class_Oring__distribs_I1_J,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ring_class.ring_distribs(1)
% 0.17/0.41 thf(fact_356_ring__class_Oring__distribs_I1_J,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ring_class.ring_distribs(1)
% 0.17/0.41 thf(fact_357_ring__class_Oring__distribs_I1_J,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ring_class.ring_distribs(1)
% 0.17/0.41 thf(fact_358_comm__semiring__class_Odistrib,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % comm_semiring_class.distrib
% 0.17/0.41 thf(fact_359_comm__semiring__class_Odistrib,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % comm_semiring_class.distrib
% 0.17/0.41 thf(fact_360_comm__semiring__class_Odistrib,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % comm_semiring_class.distrib
% 0.17/0.41 thf(fact_361_comm__semiring__class_Odistrib,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % comm_semiring_class.distrib
% 0.17/0.41 thf(fact_362_distrib__left,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_left
% 0.17/0.41 thf(fact_363_distrib__left,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_left
% 0.17/0.41 thf(fact_364_distrib__left,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.41 ( ( times_times_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_left
% 0.17/0.41 thf(fact_365_distrib__left,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_left
% 0.17/0.41 thf(fact_366_distrib__right,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_right
% 0.17/0.41 thf(fact_367_distrib__right,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_right
% 0.17/0.41 thf(fact_368_distrib__right,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_right
% 0.17/0.41 thf(fact_369_distrib__right,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % distrib_right
% 0.17/0.41 thf(fact_370_combine__common__factor,axiom,
% 0.17/0.41 ! [A: real,E: real,B2: real,C: real] :
% 0.17/0.41 ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ C ) )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ E ) @ C ) ) ).
% 0.17/0.41
% 0.17/0.41 % combine_common_factor
% 0.17/0.41 thf(fact_371_combine__common__factor,axiom,
% 0.17/0.41 ! [A: rat,E: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ C ) )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ E ) @ C ) ) ).
% 0.17/0.41
% 0.17/0.41 % combine_common_factor
% 0.17/0.41 thf(fact_372_combine__common__factor,axiom,
% 0.17/0.41 ! [A: nat,E: nat,B2: nat,C: nat] :
% 0.17/0.41 ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ E ) @ C ) ) ).
% 0.17/0.41
% 0.17/0.41 % combine_common_factor
% 0.17/0.41 thf(fact_373_combine__common__factor,axiom,
% 0.17/0.41 ! [A: int,E: int,B2: int,C: int] :
% 0.17/0.41 ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ E ) @ C ) ) ).
% 0.17/0.41
% 0.17/0.41 % combine_common_factor
% 0.17/0.41 thf(fact_374_left__diff__distrib,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.17/0.41 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_diff_distrib
% 0.17/0.41 thf(fact_375_left__diff__distrib,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.17/0.41 = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_diff_distrib
% 0.17/0.41 thf(fact_376_left__diff__distrib,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.17/0.41 = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_diff_distrib
% 0.17/0.41 thf(fact_377_right__diff__distrib,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % right_diff_distrib
% 0.17/0.41 thf(fact_378_right__diff__distrib,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % right_diff_distrib
% 0.17/0.41 thf(fact_379_right__diff__distrib,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % right_diff_distrib
% 0.17/0.41 thf(fact_380_left__diff__distrib_H,axiom,
% 0.17/0.41 ! [B2: real,C: real,A: real] :
% 0.17/0.41 ( ( times_times_real @ ( minus_minus_real @ B2 @ C ) @ A )
% 0.17/0.41 = ( minus_minus_real @ ( times_times_real @ B2 @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_diff_distrib'
% 0.17/0.41 thf(fact_381_left__diff__distrib_H,axiom,
% 0.17/0.41 ! [B2: rat,C: rat,A: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( minus_minus_rat @ B2 @ C ) @ A )
% 0.17/0.41 = ( minus_minus_rat @ ( times_times_rat @ B2 @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_diff_distrib'
% 0.17/0.41 thf(fact_382_left__diff__distrib_H,axiom,
% 0.17/0.41 ! [B2: nat,C: nat,A: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A )
% 0.17/0.41 = ( minus_minus_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_diff_distrib'
% 0.17/0.41 thf(fact_383_left__diff__distrib_H,axiom,
% 0.17/0.41 ! [B2: int,C: int,A: int] :
% 0.17/0.41 ( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A )
% 0.17/0.41 = ( minus_minus_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_diff_distrib'
% 0.17/0.41 thf(fact_384_right__diff__distrib_H,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % right_diff_distrib'
% 0.17/0.41 thf(fact_385_right__diff__distrib_H,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % right_diff_distrib'
% 0.17/0.41 thf(fact_386_right__diff__distrib_H,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.41 ( ( times_times_nat @ A @ ( minus_minus_nat @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % right_diff_distrib'
% 0.17/0.41 thf(fact_387_right__diff__distrib_H,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % right_diff_distrib'
% 0.17/0.41 thf(fact_388_Nat_OlessE,axiom,
% 0.17/0.41 ! [I: nat,K: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I @ K )
% 0.17/0.41 => ( ( K
% 0.17/0.41 != ( suc @ I ) )
% 0.17/0.41 => ~ ! [J2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I @ J2 )
% 0.17/0.41 => ( K
% 0.17/0.41 != ( suc @ J2 ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Nat.lessE
% 0.17/0.41 thf(fact_389_Suc__lessD,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.17/0.41 => ( ord_less_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_lessD
% 0.17/0.41 thf(fact_390_Suc__lessE,axiom,
% 0.17/0.41 ! [I: nat,K: nat] :
% 0.17/0.41 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 0.17/0.41 => ~ ! [J2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I @ J2 )
% 0.17/0.41 => ( K
% 0.17/0.41 != ( suc @ J2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_lessE
% 0.17/0.41 thf(fact_391_Suc__lessI,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ N )
% 0.17/0.41 => ( ( ( suc @ M )
% 0.17/0.41 != N )
% 0.17/0.41 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_lessI
% 0.17/0.41 thf(fact_392_less__SucE,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.17/0.41 => ( ~ ( ord_less_nat @ M @ N )
% 0.17/0.41 => ( M = N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_SucE
% 0.17/0.41 thf(fact_393_less__SucI,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ N )
% 0.17/0.41 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_SucI
% 0.17/0.41 thf(fact_394_Ex__less__Suc,axiom,
% 0.17/0.41 ! [N: nat,P: nat > $o] :
% 0.17/0.41 ( ( ? [I2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.17/0.41 & ( P @ I2 ) ) )
% 0.17/0.41 = ( ( P @ N )
% 0.17/0.41 | ? [I2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I2 @ N )
% 0.17/0.41 & ( P @ I2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Ex_less_Suc
% 0.17/0.41 thf(fact_395_less__Suc__eq,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.17/0.41 = ( ( ord_less_nat @ M @ N )
% 0.17/0.41 | ( M = N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_Suc_eq
% 0.17/0.41 thf(fact_396_not__less__eq,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.17/0.41 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % not_less_eq
% 0.17/0.41 thf(fact_397_All__less__Suc,axiom,
% 0.17/0.41 ! [N: nat,P: nat > $o] :
% 0.17/0.41 ( ( ! [I2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.17/0.41 => ( P @ I2 ) ) )
% 0.17/0.41 = ( ( P @ N )
% 0.17/0.41 & ! [I2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I2 @ N )
% 0.17/0.41 => ( P @ I2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % All_less_Suc
% 0.17/0.41 thf(fact_398_Suc__less__eq2,axiom,
% 0.17/0.41 ! [N: nat,M: nat] :
% 0.17/0.41 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.17/0.41 = ( ? [M2: nat] :
% 0.17/0.41 ( ( M
% 0.17/0.41 = ( suc @ M2 ) )
% 0.17/0.41 & ( ord_less_nat @ N @ M2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_less_eq2
% 0.17/0.41 thf(fact_399_less__antisym,axiom,
% 0.17/0.41 ! [N: nat,M: nat] :
% 0.17/0.41 ( ~ ( ord_less_nat @ N @ M )
% 0.17/0.41 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.17/0.41 => ( M = N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_antisym
% 0.17/0.41 thf(fact_400_Suc__less__SucD,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.17/0.41 => ( ord_less_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_less_SucD
% 0.17/0.41 thf(fact_401_less__trans__Suc,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I @ J )
% 0.17/0.41 => ( ( ord_less_nat @ J @ K )
% 0.17/0.41 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_trans_Suc
% 0.17/0.41 thf(fact_402_less__Suc__induct,axiom,
% 0.17/0.41 ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.17/0.41 ( ( ord_less_nat @ I @ J )
% 0.17/0.41 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 0.17/0.41 => ( ! [I3: nat,J2: nat,K2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I3 @ J2 )
% 0.17/0.41 => ( ( ord_less_nat @ J2 @ K2 )
% 0.17/0.41 => ( ( P @ I3 @ J2 )
% 0.17/0.41 => ( ( P @ J2 @ K2 )
% 0.17/0.41 => ( P @ I3 @ K2 ) ) ) ) )
% 0.17/0.41 => ( P @ I @ J ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_Suc_induct
% 0.17/0.41 thf(fact_403_strict__inc__induct,axiom,
% 0.17/0.41 ! [I: nat,J: nat,P: nat > $o] :
% 0.17/0.41 ( ( ord_less_nat @ I @ J )
% 0.17/0.41 => ( ! [I3: nat] :
% 0.17/0.41 ( ( J
% 0.17/0.41 = ( suc @ I3 ) )
% 0.17/0.41 => ( P @ I3 ) )
% 0.17/0.41 => ( ! [I3: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I3 @ J )
% 0.17/0.41 => ( ( P @ ( suc @ I3 ) )
% 0.17/0.41 => ( P @ I3 ) ) )
% 0.17/0.41 => ( P @ I ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % strict_inc_induct
% 0.17/0.41 thf(fact_404_not__less__less__Suc__eq,axiom,
% 0.17/0.41 ! [N: nat,M: nat] :
% 0.17/0.41 ( ~ ( ord_less_nat @ N @ M )
% 0.17/0.41 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.17/0.41 = ( N = M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % not_less_less_Suc_eq
% 0.17/0.41 thf(fact_405_Suc__leD,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.17/0.41 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_leD
% 0.17/0.41 thf(fact_406_le__SucE,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.17/0.41 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 => ( M
% 0.17/0.41 = ( suc @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_SucE
% 0.17/0.41 thf(fact_407_le__SucI,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_SucI
% 0.17/0.41 thf(fact_408_Suc__le__D,axiom,
% 0.17/0.41 ! [N: nat,M3: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
% 0.17/0.41 => ? [M4: nat] :
% 0.17/0.41 ( M3
% 0.17/0.41 = ( suc @ M4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_le_D
% 0.17/0.41 thf(fact_409_le__Suc__eq,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.17/0.41 = ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 | ( M
% 0.17/0.41 = ( suc @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_Suc_eq
% 0.17/0.41 thf(fact_410_Suc__n__not__le__n,axiom,
% 0.17/0.41 ! [N: nat] :
% 0.17/0.41 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_n_not_le_n
% 0.17/0.41 thf(fact_411_not__less__eq__eq,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.17/0.41 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.17/0.41
% 0.17/0.41 % not_less_eq_eq
% 0.17/0.41 thf(fact_412_full__nat__induct,axiom,
% 0.17/0.41 ! [P: nat > $o,N: nat] :
% 0.17/0.41 ( ! [N4: nat] :
% 0.17/0.41 ( ! [M5: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
% 0.17/0.41 => ( P @ M5 ) )
% 0.17/0.41 => ( P @ N4 ) )
% 0.17/0.41 => ( P @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % full_nat_induct
% 0.17/0.41 thf(fact_413_nat__induct__at__least,axiom,
% 0.17/0.41 ! [M: nat,N: nat,P: nat > $o] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 => ( ( P @ M )
% 0.17/0.41 => ( ! [N4: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ N4 )
% 0.17/0.41 => ( ( P @ N4 )
% 0.17/0.41 => ( P @ ( suc @ N4 ) ) ) )
% 0.17/0.41 => ( P @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_induct_at_least
% 0.17/0.41 thf(fact_414_transitive__stepwise__le,axiom,
% 0.17/0.41 ! [M: nat,N: nat,R: nat > nat > $o] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 => ( ! [X3: nat] : ( R @ X3 @ X3 )
% 0.17/0.41 => ( ! [X3: nat,Y4: nat,Z2: nat] :
% 0.17/0.41 ( ( R @ X3 @ Y4 )
% 0.17/0.41 => ( ( R @ Y4 @ Z2 )
% 0.17/0.41 => ( R @ X3 @ Z2 ) ) )
% 0.17/0.41 => ( ! [N4: nat] : ( R @ N4 @ ( suc @ N4 ) )
% 0.17/0.41 => ( R @ M @ N ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % transitive_stepwise_le
% 0.17/0.41 thf(fact_415_le__cube,axiom,
% 0.17/0.41 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_cube
% 0.17/0.41 thf(fact_416_le__square,axiom,
% 0.17/0.41 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_square
% 0.17/0.41 thf(fact_417_mult__le__mono,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ( ord_less_eq_nat @ K @ L2 )
% 0.17/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_le_mono
% 0.17/0.41 thf(fact_418_mult__le__mono1,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_le_mono1
% 0.17/0.41 thf(fact_419_mult__le__mono2,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_le_mono2
% 0.17/0.41 thf(fact_420_nat__arith_Osuc1,axiom,
% 0.17/0.41 ! [A3: nat,K: nat,A: nat] :
% 0.17/0.41 ( ( A3
% 0.17/0.41 = ( plus_plus_nat @ K @ A ) )
% 0.17/0.41 => ( ( suc @ A3 )
% 0.17/0.41 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_arith.suc1
% 0.17/0.41 thf(fact_421_add__Suc,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.17/0.41 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_Suc
% 0.17/0.41 thf(fact_422_add__Suc__shift,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.17/0.41 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_Suc_shift
% 0.17/0.41 thf(fact_423_add__mult__distrib,axiom,
% 0.17/0.41 ! [M: nat,N: nat,K: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_mult_distrib
% 0.17/0.41 thf(fact_424_add__mult__distrib2,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_mult_distrib2
% 0.17/0.41 thf(fact_425_zero__induct__lemma,axiom,
% 0.17/0.41 ! [P: nat > $o,K: nat,I: nat] :
% 0.17/0.41 ( ( P @ K )
% 0.17/0.41 => ( ! [N4: nat] :
% 0.17/0.41 ( ( P @ ( suc @ N4 ) )
% 0.17/0.41 => ( P @ N4 ) )
% 0.17/0.41 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % zero_induct_lemma
% 0.17/0.41 thf(fact_426_diff__mult__distrib,axiom,
% 0.17/0.41 ! [M: nat,N: nat,K: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 0.17/0.41 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_mult_distrib
% 0.17/0.41 thf(fact_427_diff__mult__distrib2,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.17/0.41 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_mult_distrib2
% 0.17/0.41 thf(fact_428_nat__mult__1,axiom,
% 0.17/0.41 ! [N: nat] :
% 0.17/0.41 ( ( times_times_nat @ one_one_nat @ N )
% 0.17/0.41 = N ) ).
% 0.17/0.41
% 0.17/0.41 % nat_mult_1
% 0.17/0.41 thf(fact_429_nat__mult__1__right,axiom,
% 0.17/0.41 ! [N: nat] :
% 0.17/0.41 ( ( times_times_nat @ N @ one_one_nat )
% 0.17/0.41 = N ) ).
% 0.17/0.41
% 0.17/0.41 % nat_mult_1_right
% 0.17/0.41 thf(fact_430_two__realpow__ge__one,axiom,
% 0.17/0.41 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % two_realpow_ge_one
% 0.17/0.41 thf(fact_431_add__One__commute,axiom,
% 0.17/0.41 ! [N: num] :
% 0.17/0.41 ( ( plus_plus_num @ one @ N )
% 0.17/0.41 = ( plus_plus_num @ N @ one ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_One_commute
% 0.17/0.41 thf(fact_432_power__commutes,axiom,
% 0.17/0.41 ! [A: complex,N: nat] :
% 0.17/0.41 ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 0.17/0.41 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_commutes
% 0.17/0.41 thf(fact_433_power__commutes,axiom,
% 0.17/0.41 ! [A: real,N: nat] :
% 0.17/0.41 ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 0.17/0.41 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_commutes
% 0.17/0.41 thf(fact_434_power__commutes,axiom,
% 0.17/0.41 ! [A: rat,N: nat] :
% 0.17/0.41 ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 0.17/0.41 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_commutes
% 0.17/0.41 thf(fact_435_power__commutes,axiom,
% 0.17/0.41 ! [A: nat,N: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 0.17/0.41 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_commutes
% 0.17/0.41 thf(fact_436_power__commutes,axiom,
% 0.17/0.41 ! [A: int,N: nat] :
% 0.17/0.41 ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 0.17/0.41 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_commutes
% 0.17/0.41 thf(fact_437_power__mult__distrib,axiom,
% 0.17/0.41 ! [A: complex,B2: complex,N: nat] :
% 0.17/0.41 ( ( power_power_complex @ ( times_times_complex @ A @ B2 ) @ N )
% 0.17/0.41 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult_distrib
% 0.17/0.41 thf(fact_438_power__mult__distrib,axiom,
% 0.17/0.41 ! [A: real,B2: real,N: nat] :
% 0.17/0.41 ( ( power_power_real @ ( times_times_real @ A @ B2 ) @ N )
% 0.17/0.41 = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult_distrib
% 0.17/0.41 thf(fact_439_power__mult__distrib,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,N: nat] :
% 0.17/0.41 ( ( power_power_rat @ ( times_times_rat @ A @ B2 ) @ N )
% 0.17/0.41 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult_distrib
% 0.17/0.41 thf(fact_440_power__mult__distrib,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,N: nat] :
% 0.17/0.41 ( ( power_power_nat @ ( times_times_nat @ A @ B2 ) @ N )
% 0.17/0.41 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult_distrib
% 0.17/0.41 thf(fact_441_power__mult__distrib,axiom,
% 0.17/0.41 ! [A: int,B2: int,N: nat] :
% 0.17/0.41 ( ( power_power_int @ ( times_times_int @ A @ B2 ) @ N )
% 0.17/0.41 = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult_distrib
% 0.17/0.41 thf(fact_442_power__commuting__commutes,axiom,
% 0.17/0.41 ! [X4: complex,Y3: complex,N: nat] :
% 0.17/0.41 ( ( ( times_times_complex @ X4 @ Y3 )
% 0.17/0.41 = ( times_times_complex @ Y3 @ X4 ) )
% 0.17/0.41 => ( ( times_times_complex @ ( power_power_complex @ X4 @ N ) @ Y3 )
% 0.17/0.41 = ( times_times_complex @ Y3 @ ( power_power_complex @ X4 @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_commuting_commutes
% 0.17/0.41 thf(fact_443_power__commuting__commutes,axiom,
% 0.17/0.41 ! [X4: real,Y3: real,N: nat] :
% 0.17/0.41 ( ( ( times_times_real @ X4 @ Y3 )
% 0.17/0.41 = ( times_times_real @ Y3 @ X4 ) )
% 0.17/0.41 => ( ( times_times_real @ ( power_power_real @ X4 @ N ) @ Y3 )
% 0.17/0.41 = ( times_times_real @ Y3 @ ( power_power_real @ X4 @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_commuting_commutes
% 0.17/0.41 thf(fact_444_power__commuting__commutes,axiom,
% 0.17/0.41 ! [X4: rat,Y3: rat,N: nat] :
% 0.17/0.41 ( ( ( times_times_rat @ X4 @ Y3 )
% 0.17/0.41 = ( times_times_rat @ Y3 @ X4 ) )
% 0.17/0.41 => ( ( times_times_rat @ ( power_power_rat @ X4 @ N ) @ Y3 )
% 0.17/0.41 = ( times_times_rat @ Y3 @ ( power_power_rat @ X4 @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_commuting_commutes
% 0.17/0.41 thf(fact_445_power__commuting__commutes,axiom,
% 0.17/0.41 ! [X4: nat,Y3: nat,N: nat] :
% 0.17/0.41 ( ( ( times_times_nat @ X4 @ Y3 )
% 0.17/0.41 = ( times_times_nat @ Y3 @ X4 ) )
% 0.17/0.41 => ( ( times_times_nat @ ( power_power_nat @ X4 @ N ) @ Y3 )
% 0.17/0.41 = ( times_times_nat @ Y3 @ ( power_power_nat @ X4 @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_commuting_commutes
% 0.17/0.41 thf(fact_446_power__commuting__commutes,axiom,
% 0.17/0.41 ! [X4: int,Y3: int,N: nat] :
% 0.17/0.41 ( ( ( times_times_int @ X4 @ Y3 )
% 0.17/0.41 = ( times_times_int @ Y3 @ X4 ) )
% 0.17/0.41 => ( ( times_times_int @ ( power_power_int @ X4 @ N ) @ Y3 )
% 0.17/0.41 = ( times_times_int @ Y3 @ ( power_power_int @ X4 @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_commuting_commutes
% 0.17/0.41 thf(fact_447_power__mult,axiom,
% 0.17/0.41 ! [A: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 0.17/0.41 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult
% 0.17/0.41 thf(fact_448_power__mult,axiom,
% 0.17/0.41 ! [A: real,M: nat,N: nat] :
% 0.17/0.41 ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 0.17/0.41 = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult
% 0.17/0.41 thf(fact_449_power__mult,axiom,
% 0.17/0.41 ! [A: int,M: nat,N: nat] :
% 0.17/0.41 ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 0.17/0.41 = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult
% 0.17/0.41 thf(fact_450_power__mult,axiom,
% 0.17/0.41 ! [A: complex,M: nat,N: nat] :
% 0.17/0.41 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 0.17/0.41 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult
% 0.17/0.41 thf(fact_451_left__add__mult__distrib,axiom,
% 0.17/0.41 ! [I: nat,U: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_add_mult_distrib
% 0.17/0.41 thf(fact_452_div__mult2__eq,axiom,
% 0.17/0.41 ! [M: nat,N: nat,Q2: nat] :
% 0.17/0.41 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 0.17/0.41 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % div_mult2_eq
% 0.17/0.41 thf(fact_453_power__odd__eq,axiom,
% 0.17/0.41 ! [A: complex,N: nat] :
% 0.17/0.41 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.17/0.41 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_odd_eq
% 0.17/0.41 thf(fact_454_power__odd__eq,axiom,
% 0.17/0.41 ! [A: real,N: nat] :
% 0.17/0.41 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.17/0.41 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_odd_eq
% 0.17/0.41 thf(fact_455_power__odd__eq,axiom,
% 0.17/0.41 ! [A: rat,N: nat] :
% 0.17/0.41 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.17/0.41 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_odd_eq
% 0.17/0.41 thf(fact_456_power__odd__eq,axiom,
% 0.17/0.41 ! [A: nat,N: nat] :
% 0.17/0.41 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.17/0.41 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_odd_eq
% 0.17/0.41 thf(fact_457_power__odd__eq,axiom,
% 0.17/0.41 ! [A: int,N: nat] :
% 0.17/0.41 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.17/0.41 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_odd_eq
% 0.17/0.41 thf(fact_458_less__1__mult,axiom,
% 0.17/0.41 ! [M: real,N: real] :
% 0.17/0.41 ( ( ord_less_real @ one_one_real @ M )
% 0.17/0.41 => ( ( ord_less_real @ one_one_real @ N )
% 0.17/0.41 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_1_mult
% 0.17/0.41 thf(fact_459_less__1__mult,axiom,
% 0.17/0.41 ! [M: rat,N: rat] :
% 0.17/0.41 ( ( ord_less_rat @ one_one_rat @ M )
% 0.17/0.41 => ( ( ord_less_rat @ one_one_rat @ N )
% 0.17/0.41 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_1_mult
% 0.17/0.41 thf(fact_460_less__1__mult,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ one_one_nat @ M )
% 0.17/0.41 => ( ( ord_less_nat @ one_one_nat @ N )
% 0.17/0.41 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_1_mult
% 0.17/0.41 thf(fact_461_less__1__mult,axiom,
% 0.17/0.41 ! [M: int,N: int] :
% 0.17/0.41 ( ( ord_less_int @ one_one_int @ M )
% 0.17/0.41 => ( ( ord_less_int @ one_one_int @ N )
% 0.17/0.41 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_1_mult
% 0.17/0.41 thf(fact_462_eq__add__iff1,axiom,
% 0.17/0.41 ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.17/0.41 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C )
% 0.17/0.41 = D ) ) ).
% 0.17/0.41
% 0.17/0.41 % eq_add_iff1
% 0.17/0.41 thf(fact_463_eq__add__iff1,axiom,
% 0.17/0.41 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.17/0.41 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C )
% 0.17/0.41 = D ) ) ).
% 0.17/0.41
% 0.17/0.41 % eq_add_iff1
% 0.17/0.41 thf(fact_464_eq__add__iff1,axiom,
% 0.17/0.41 ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.17/0.41 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C )
% 0.17/0.41 = D ) ) ).
% 0.17/0.41
% 0.17/0.41 % eq_add_iff1
% 0.17/0.41 thf(fact_465_eq__add__iff2,axiom,
% 0.17/0.41 ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.17/0.41 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( C
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % eq_add_iff2
% 0.17/0.41 thf(fact_466_eq__add__iff2,axiom,
% 0.17/0.41 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.17/0.41 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( C
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % eq_add_iff2
% 0.17/0.41 thf(fact_467_eq__add__iff2,axiom,
% 0.17/0.41 ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.17/0.41 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( C
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % eq_add_iff2
% 0.17/0.41 thf(fact_468_square__diff__square__factored,axiom,
% 0.17/0.41 ! [X4: real,Y3: real] :
% 0.17/0.41 ( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 0.17/0.41 = ( times_times_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( minus_minus_real @ X4 @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % square_diff_square_factored
% 0.17/0.41 thf(fact_469_square__diff__square__factored,axiom,
% 0.17/0.41 ! [X4: rat,Y3: rat] :
% 0.17/0.41 ( ( minus_minus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 0.17/0.41 = ( times_times_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( minus_minus_rat @ X4 @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % square_diff_square_factored
% 0.17/0.41 thf(fact_470_square__diff__square__factored,axiom,
% 0.17/0.41 ! [X4: int,Y3: int] :
% 0.17/0.41 ( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 0.17/0.41 = ( times_times_int @ ( plus_plus_int @ X4 @ Y3 ) @ ( minus_minus_int @ X4 @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % square_diff_square_factored
% 0.17/0.41 thf(fact_471_mult__diff__mult,axiom,
% 0.17/0.41 ! [X4: real,Y3: real,A: real,B2: real] :
% 0.17/0.41 ( ( minus_minus_real @ ( times_times_real @ X4 @ Y3 ) @ ( times_times_real @ A @ B2 ) )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ X4 @ ( minus_minus_real @ Y3 @ B2 ) ) @ ( times_times_real @ ( minus_minus_real @ X4 @ A ) @ B2 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_diff_mult
% 0.17/0.41 thf(fact_472_mult__diff__mult,axiom,
% 0.17/0.41 ! [X4: rat,Y3: rat,A: rat,B2: rat] :
% 0.17/0.41 ( ( minus_minus_rat @ ( times_times_rat @ X4 @ Y3 ) @ ( times_times_rat @ A @ B2 ) )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ X4 @ ( minus_minus_rat @ Y3 @ B2 ) ) @ ( times_times_rat @ ( minus_minus_rat @ X4 @ A ) @ B2 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_diff_mult
% 0.17/0.41 thf(fact_473_mult__diff__mult,axiom,
% 0.17/0.41 ! [X4: int,Y3: int,A: int,B2: int] :
% 0.17/0.41 ( ( minus_minus_int @ ( times_times_int @ X4 @ Y3 ) @ ( times_times_int @ A @ B2 ) )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ X4 @ ( minus_minus_int @ Y3 @ B2 ) ) @ ( times_times_int @ ( minus_minus_int @ X4 @ A ) @ B2 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_diff_mult
% 0.17/0.41 thf(fact_474_lift__Suc__mono__less,axiom,
% 0.17/0.41 ! [F: nat > real,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_less
% 0.17/0.41 thf(fact_475_lift__Suc__mono__less,axiom,
% 0.17/0.41 ! [F: nat > rat,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_less
% 0.17/0.41 thf(fact_476_lift__Suc__mono__less,axiom,
% 0.17/0.41 ! [F: nat > num,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_less
% 0.17/0.41 thf(fact_477_lift__Suc__mono__less,axiom,
% 0.17/0.41 ! [F: nat > nat,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_less
% 0.17/0.41 thf(fact_478_lift__Suc__mono__less,axiom,
% 0.17/0.41 ! [F: nat > int,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_less
% 0.17/0.41 thf(fact_479_lift__Suc__mono__less__iff,axiom,
% 0.17/0.41 ! [F: nat > real,N: nat,M: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 0.17/0.41 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_less_iff
% 0.17/0.41 thf(fact_480_lift__Suc__mono__less__iff,axiom,
% 0.17/0.41 ! [F: nat > rat,N: nat,M: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 0.17/0.41 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_less_iff
% 0.17/0.41 thf(fact_481_lift__Suc__mono__less__iff,axiom,
% 0.17/0.41 ! [F: nat > num,N: nat,M: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 0.17/0.41 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_less_iff
% 0.17/0.41 thf(fact_482_lift__Suc__mono__less__iff,axiom,
% 0.17/0.41 ! [F: nat > nat,N: nat,M: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.17/0.41 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_less_iff
% 0.17/0.41 thf(fact_483_lift__Suc__mono__less__iff,axiom,
% 0.17/0.41 ! [F: nat > int,N: nat,M: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 0.17/0.41 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_less_iff
% 0.17/0.41 thf(fact_484_lift__Suc__mono__le,axiom,
% 0.17/0.41 ! [F: nat > set_nat,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_le
% 0.17/0.41 thf(fact_485_lift__Suc__mono__le,axiom,
% 0.17/0.41 ! [F: nat > rat,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_le
% 0.17/0.41 thf(fact_486_lift__Suc__mono__le,axiom,
% 0.17/0.41 ! [F: nat > num,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_le
% 0.17/0.41 thf(fact_487_lift__Suc__mono__le,axiom,
% 0.17/0.41 ! [F: nat > nat,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_le
% 0.17/0.41 thf(fact_488_lift__Suc__mono__le,axiom,
% 0.17/0.41 ! [F: nat > int,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_mono_le
% 0.17/0.41 thf(fact_489_lift__Suc__antimono__le,axiom,
% 0.17/0.41 ! [F: nat > set_nat,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_antimono_le
% 0.17/0.41 thf(fact_490_lift__Suc__antimono__le,axiom,
% 0.17/0.41 ! [F: nat > rat,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_antimono_le
% 0.17/0.41 thf(fact_491_lift__Suc__antimono__le,axiom,
% 0.17/0.41 ! [F: nat > num,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_antimono_le
% 0.17/0.41 thf(fact_492_lift__Suc__antimono__le,axiom,
% 0.17/0.41 ! [F: nat > nat,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_antimono_le
% 0.17/0.41 thf(fact_493_lift__Suc__antimono__le,axiom,
% 0.17/0.41 ! [F: nat > int,N: nat,N5: nat] :
% 0.17/0.41 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.17/0.41 => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % lift_Suc_antimono_le
% 0.17/0.41 thf(fact_494_Suc__leI,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ N )
% 0.17/0.41 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_leI
% 0.17/0.41 thf(fact_495_Suc__le__eq,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.17/0.41 = ( ord_less_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_le_eq
% 0.17/0.41 thf(fact_496_dec__induct,axiom,
% 0.17/0.41 ! [I: nat,J: nat,P: nat > $o] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ( P @ I )
% 0.17/0.41 => ( ! [N4: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ N4 )
% 0.17/0.41 => ( ( ord_less_nat @ N4 @ J )
% 0.17/0.41 => ( ( P @ N4 )
% 0.17/0.41 => ( P @ ( suc @ N4 ) ) ) ) )
% 0.17/0.41 => ( P @ J ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % dec_induct
% 0.17/0.41 thf(fact_497_inc__induct,axiom,
% 0.17/0.41 ! [I: nat,J: nat,P: nat > $o] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ( P @ J )
% 0.17/0.41 => ( ! [N4: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ N4 )
% 0.17/0.41 => ( ( ord_less_nat @ N4 @ J )
% 0.17/0.41 => ( ( P @ ( suc @ N4 ) )
% 0.17/0.41 => ( P @ N4 ) ) ) )
% 0.17/0.41 => ( P @ I ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % inc_induct
% 0.17/0.41 thf(fact_498_Suc__le__lessD,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.17/0.41 => ( ord_less_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_le_lessD
% 0.17/0.41 thf(fact_499_le__less__Suc__eq,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.17/0.41 = ( N = M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_less_Suc_eq
% 0.17/0.41 thf(fact_500_less__Suc__eq__le,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.17/0.41 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_Suc_eq_le
% 0.17/0.41 thf(fact_501_less__eq__Suc__le,axiom,
% 0.17/0.41 ( ord_less_nat
% 0.17/0.41 = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_eq_Suc_le
% 0.17/0.41 thf(fact_502_le__imp__less__Suc,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_imp_less_Suc
% 0.17/0.41 thf(fact_503_less__imp__Suc__add,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ N )
% 0.17/0.41 => ? [K2: nat] :
% 0.17/0.41 ( N
% 0.17/0.41 = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_imp_Suc_add
% 0.17/0.41 thf(fact_504_less__iff__Suc__add,axiom,
% 0.17/0.41 ( ord_less_nat
% 0.17/0.41 = ( ^ [M6: nat,N2: nat] :
% 0.17/0.41 ? [K3: nat] :
% 0.17/0.41 ( N2
% 0.17/0.41 = ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_iff_Suc_add
% 0.17/0.41 thf(fact_505_less__add__Suc2,axiom,
% 0.17/0.41 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_Suc2
% 0.17/0.41 thf(fact_506_less__add__Suc1,axiom,
% 0.17/0.41 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_Suc1
% 0.17/0.41 thf(fact_507_less__natE,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ N )
% 0.17/0.41 => ~ ! [Q3: nat] :
% 0.17/0.41 ( N
% 0.17/0.41 != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_natE
% 0.17/0.41 thf(fact_508_diff__less__Suc,axiom,
% 0.17/0.41 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_less_Suc
% 0.17/0.41 thf(fact_509_Suc__diff__Suc,axiom,
% 0.17/0.41 ! [N: nat,M: nat] :
% 0.17/0.41 ( ( ord_less_nat @ N @ M )
% 0.17/0.41 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 0.17/0.41 = ( minus_minus_nat @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_diff_Suc
% 0.17/0.41 thf(fact_510_Suc__diff__le,axiom,
% 0.17/0.41 ! [N: nat,M: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.41 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.17/0.41 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_diff_le
% 0.17/0.41 thf(fact_511_Suc__eq__plus1,axiom,
% 0.17/0.41 ( suc
% 0.17/0.41 = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_eq_plus1
% 0.17/0.41 thf(fact_512_plus__1__eq__Suc,axiom,
% 0.17/0.41 ( ( plus_plus_nat @ one_one_nat )
% 0.17/0.41 = suc ) ).
% 0.17/0.41
% 0.17/0.41 % plus_1_eq_Suc
% 0.17/0.41 thf(fact_513_Suc__eq__plus1__left,axiom,
% 0.17/0.41 ( suc
% 0.17/0.41 = ( plus_plus_nat @ one_one_nat ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_eq_plus1_left
% 0.17/0.41 thf(fact_514_diff__Suc__eq__diff__pred,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.17/0.41 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_Suc_eq_diff_pred
% 0.17/0.41 thf(fact_515_Suc__nat__number__of__add,axiom,
% 0.17/0.41 ! [V: num,N: nat] :
% 0.17/0.41 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 0.17/0.41 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_nat_number_of_add
% 0.17/0.41 thf(fact_516_div__nat__eqI,axiom,
% 0.17/0.41 ! [N: nat,Q2: nat,M: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 0.17/0.41 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 0.17/0.41 => ( ( divide_divide_nat @ M @ N )
% 0.17/0.41 = Q2 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % div_nat_eqI
% 0.17/0.41 thf(fact_517_ordered__ring__class_Ole__add__iff2,axiom,
% 0.17/0.41 ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.17/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ordered_ring_class.le_add_iff2
% 0.17/0.41 thf(fact_518_ordered__ring__class_Ole__add__iff2,axiom,
% 0.17/0.41 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.17/0.41 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ordered_ring_class.le_add_iff2
% 0.17/0.41 thf(fact_519_ordered__ring__class_Ole__add__iff2,axiom,
% 0.17/0.41 ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.17/0.41 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ordered_ring_class.le_add_iff2
% 0.17/0.41 thf(fact_520_ordered__ring__class_Ole__add__iff1,axiom,
% 0.17/0.41 ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.17/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.17/0.41
% 0.17/0.41 % ordered_ring_class.le_add_iff1
% 0.17/0.41 thf(fact_521_ordered__ring__class_Ole__add__iff1,axiom,
% 0.17/0.41 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.17/0.41 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.17/0.41
% 0.17/0.41 % ordered_ring_class.le_add_iff1
% 0.17/0.41 thf(fact_522_ordered__ring__class_Ole__add__iff1,axiom,
% 0.17/0.41 ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.17/0.41 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.17/0.41
% 0.17/0.41 % ordered_ring_class.le_add_iff1
% 0.17/0.41 thf(fact_523_less__add__iff2,axiom,
% 0.17/0.41 ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.17/0.41 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_iff2
% 0.17/0.41 thf(fact_524_less__add__iff2,axiom,
% 0.17/0.41 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.17/0.41 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_iff2
% 0.17/0.41 thf(fact_525_less__add__iff2,axiom,
% 0.17/0.41 ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.17/0.41 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_iff2
% 0.17/0.41 thf(fact_526_less__add__iff1,axiom,
% 0.17/0.41 ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.17/0.41 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_iff1
% 0.17/0.41 thf(fact_527_less__add__iff1,axiom,
% 0.17/0.41 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.17/0.41 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_iff1
% 0.17/0.41 thf(fact_528_less__add__iff1,axiom,
% 0.17/0.41 ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.17/0.41 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.17/0.41 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_iff1
% 0.17/0.41 thf(fact_529_square__diff__one__factored,axiom,
% 0.17/0.41 ! [X4: complex] :
% 0.17/0.41 ( ( minus_minus_complex @ ( times_times_complex @ X4 @ X4 ) @ one_one_complex )
% 0.17/0.41 = ( times_times_complex @ ( plus_plus_complex @ X4 @ one_one_complex ) @ ( minus_minus_complex @ X4 @ one_one_complex ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % square_diff_one_factored
% 0.17/0.41 thf(fact_530_square__diff__one__factored,axiom,
% 0.17/0.41 ! [X4: real] :
% 0.17/0.41 ( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ one_one_real )
% 0.17/0.41 = ( times_times_real @ ( plus_plus_real @ X4 @ one_one_real ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % square_diff_one_factored
% 0.17/0.41 thf(fact_531_square__diff__one__factored,axiom,
% 0.17/0.41 ! [X4: rat] :
% 0.17/0.41 ( ( minus_minus_rat @ ( times_times_rat @ X4 @ X4 ) @ one_one_rat )
% 0.17/0.41 = ( times_times_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) @ ( minus_minus_rat @ X4 @ one_one_rat ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % square_diff_one_factored
% 0.17/0.41 thf(fact_532_square__diff__one__factored,axiom,
% 0.17/0.41 ! [X4: int] :
% 0.17/0.41 ( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ one_one_int )
% 0.17/0.41 = ( times_times_int @ ( plus_plus_int @ X4 @ one_one_int ) @ ( minus_minus_int @ X4 @ one_one_int ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % square_diff_one_factored
% 0.17/0.41 thf(fact_533_mult__numeral__1__right,axiom,
% 0.17/0.41 ! [A: complex] :
% 0.17/0.41 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_1_right
% 0.17/0.41 thf(fact_534_mult__numeral__1__right,axiom,
% 0.17/0.41 ! [A: real] :
% 0.17/0.41 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_1_right
% 0.17/0.41 thf(fact_535_mult__numeral__1__right,axiom,
% 0.17/0.41 ! [A: rat] :
% 0.17/0.41 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_1_right
% 0.17/0.41 thf(fact_536_mult__numeral__1__right,axiom,
% 0.17/0.41 ! [A: nat] :
% 0.17/0.41 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_1_right
% 0.17/0.41 thf(fact_537_mult__numeral__1__right,axiom,
% 0.17/0.41 ! [A: int] :
% 0.17/0.41 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_1_right
% 0.17/0.41 thf(fact_538_mult__numeral__1,axiom,
% 0.17/0.41 ! [A: complex] :
% 0.17/0.41 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_1
% 0.17/0.41 thf(fact_539_mult__numeral__1,axiom,
% 0.17/0.41 ! [A: real] :
% 0.17/0.41 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_1
% 0.17/0.41 thf(fact_540_mult__numeral__1,axiom,
% 0.17/0.41 ! [A: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_1
% 0.17/0.41 thf(fact_541_mult__numeral__1,axiom,
% 0.17/0.41 ! [A: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_1
% 0.17/0.41 thf(fact_542_mult__numeral__1,axiom,
% 0.17/0.41 ! [A: int] :
% 0.17/0.41 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_numeral_1
% 0.17/0.41 thf(fact_543_left__right__inverse__power,axiom,
% 0.17/0.41 ! [X4: complex,Y3: complex,N: nat] :
% 0.17/0.41 ( ( ( times_times_complex @ X4 @ Y3 )
% 0.17/0.41 = one_one_complex )
% 0.17/0.41 => ( ( times_times_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 0.17/0.41 = one_one_complex ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_right_inverse_power
% 0.17/0.41 thf(fact_544_left__right__inverse__power,axiom,
% 0.17/0.41 ! [X4: real,Y3: real,N: nat] :
% 0.17/0.41 ( ( ( times_times_real @ X4 @ Y3 )
% 0.17/0.41 = one_one_real )
% 0.17/0.41 => ( ( times_times_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 0.17/0.41 = one_one_real ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_right_inverse_power
% 0.17/0.41 thf(fact_545_left__right__inverse__power,axiom,
% 0.17/0.41 ! [X4: rat,Y3: rat,N: nat] :
% 0.17/0.41 ( ( ( times_times_rat @ X4 @ Y3 )
% 0.17/0.41 = one_one_rat )
% 0.17/0.41 => ( ( times_times_rat @ ( power_power_rat @ X4 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 0.17/0.41 = one_one_rat ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_right_inverse_power
% 0.17/0.41 thf(fact_546_left__right__inverse__power,axiom,
% 0.17/0.41 ! [X4: nat,Y3: nat,N: nat] :
% 0.17/0.41 ( ( ( times_times_nat @ X4 @ Y3 )
% 0.17/0.41 = one_one_nat )
% 0.17/0.41 => ( ( times_times_nat @ ( power_power_nat @ X4 @ N ) @ ( power_power_nat @ Y3 @ N ) )
% 0.17/0.41 = one_one_nat ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_right_inverse_power
% 0.17/0.41 thf(fact_547_left__right__inverse__power,axiom,
% 0.17/0.41 ! [X4: int,Y3: int,N: nat] :
% 0.17/0.41 ( ( ( times_times_int @ X4 @ Y3 )
% 0.17/0.41 = one_one_int )
% 0.17/0.41 => ( ( times_times_int @ ( power_power_int @ X4 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 0.17/0.41 = one_one_int ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_right_inverse_power
% 0.17/0.41 thf(fact_548_power__add,axiom,
% 0.17/0.41 ! [A: complex,M: nat,N: nat] :
% 0.17/0.41 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 0.17/0.41 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add
% 0.17/0.41 thf(fact_549_power__add,axiom,
% 0.17/0.41 ! [A: real,M: nat,N: nat] :
% 0.17/0.41 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 0.17/0.41 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add
% 0.17/0.41 thf(fact_550_power__add,axiom,
% 0.17/0.41 ! [A: rat,M: nat,N: nat] :
% 0.17/0.41 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.17/0.41 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add
% 0.17/0.41 thf(fact_551_power__add,axiom,
% 0.17/0.41 ! [A: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.17/0.41 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add
% 0.17/0.41 thf(fact_552_power__add,axiom,
% 0.17/0.41 ! [A: int,M: nat,N: nat] :
% 0.17/0.41 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 0.17/0.41 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_add
% 0.17/0.41 thf(fact_553_Suc__div__le__mono,axiom,
% 0.17/0.41 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_div_le_mono
% 0.17/0.41 thf(fact_554_less__mult__imp__div__less,axiom,
% 0.17/0.41 ! [M: nat,I: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 0.17/0.41 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_mult_imp_div_less
% 0.17/0.41 thf(fact_555_times__div__less__eq__dividend,axiom,
% 0.17/0.41 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 0.17/0.41
% 0.17/0.41 % times_div_less_eq_dividend
% 0.17/0.41 thf(fact_556_div__times__less__eq__dividend,axiom,
% 0.17/0.41 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 0.17/0.41
% 0.17/0.41 % div_times_less_eq_dividend
% 0.17/0.41 thf(fact_557_power__gt1__lemma,axiom,
% 0.17/0.41 ! [A: real,N: nat] :
% 0.17/0.41 ( ( ord_less_real @ one_one_real @ A )
% 0.17/0.41 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_gt1_lemma
% 0.17/0.41 thf(fact_558_power__gt1__lemma,axiom,
% 0.17/0.41 ! [A: rat,N: nat] :
% 0.17/0.41 ( ( ord_less_rat @ one_one_rat @ A )
% 0.17/0.41 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_gt1_lemma
% 0.17/0.41 thf(fact_559_power__gt1__lemma,axiom,
% 0.17/0.41 ! [A: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ one_one_nat @ A )
% 0.17/0.41 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_gt1_lemma
% 0.17/0.41 thf(fact_560_power__gt1__lemma,axiom,
% 0.17/0.41 ! [A: int,N: nat] :
% 0.17/0.41 ( ( ord_less_int @ one_one_int @ A )
% 0.17/0.41 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_gt1_lemma
% 0.17/0.41 thf(fact_561_power__less__power__Suc,axiom,
% 0.17/0.41 ! [A: real,N: nat] :
% 0.17/0.41 ( ( ord_less_real @ one_one_real @ A )
% 0.17/0.41 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_less_power_Suc
% 0.17/0.41 thf(fact_562_power__less__power__Suc,axiom,
% 0.17/0.41 ! [A: rat,N: nat] :
% 0.17/0.41 ( ( ord_less_rat @ one_one_rat @ A )
% 0.17/0.41 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_less_power_Suc
% 0.17/0.41 thf(fact_563_power__less__power__Suc,axiom,
% 0.17/0.41 ! [A: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ one_one_nat @ A )
% 0.17/0.41 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_less_power_Suc
% 0.17/0.41 thf(fact_564_power__less__power__Suc,axiom,
% 0.17/0.41 ! [A: int,N: nat] :
% 0.17/0.41 ( ( ord_less_int @ one_one_int @ A )
% 0.17/0.41 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_less_power_Suc
% 0.17/0.41 thf(fact_565_power__gt1,axiom,
% 0.17/0.41 ! [A: real,N: nat] :
% 0.17/0.41 ( ( ord_less_real @ one_one_real @ A )
% 0.17/0.41 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_gt1
% 0.17/0.41 thf(fact_566_power__gt1,axiom,
% 0.17/0.41 ! [A: rat,N: nat] :
% 0.17/0.41 ( ( ord_less_rat @ one_one_rat @ A )
% 0.17/0.41 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_gt1
% 0.17/0.41 thf(fact_567_power__gt1,axiom,
% 0.17/0.41 ! [A: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ one_one_nat @ A )
% 0.17/0.41 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_gt1
% 0.17/0.41 thf(fact_568_power__gt1,axiom,
% 0.17/0.41 ! [A: int,N: nat] :
% 0.17/0.41 ( ( ord_less_int @ one_one_int @ A )
% 0.17/0.41 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_gt1
% 0.17/0.41 thf(fact_569_linorder__neqE__linordered__idom,axiom,
% 0.17/0.41 ! [X4: real,Y3: real] :
% 0.17/0.41 ( ( X4 != Y3 )
% 0.17/0.41 => ( ~ ( ord_less_real @ X4 @ Y3 )
% 0.17/0.41 => ( ord_less_real @ Y3 @ X4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % linorder_neqE_linordered_idom
% 0.17/0.41 thf(fact_570_linorder__neqE__linordered__idom,axiom,
% 0.17/0.41 ! [X4: rat,Y3: rat] :
% 0.17/0.41 ( ( X4 != Y3 )
% 0.17/0.41 => ( ~ ( ord_less_rat @ X4 @ Y3 )
% 0.17/0.41 => ( ord_less_rat @ Y3 @ X4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % linorder_neqE_linordered_idom
% 0.17/0.41 thf(fact_571_linorder__neqE__linordered__idom,axiom,
% 0.17/0.41 ! [X4: int,Y3: int] :
% 0.17/0.41 ( ( X4 != Y3 )
% 0.17/0.41 => ( ~ ( ord_less_int @ X4 @ Y3 )
% 0.17/0.41 => ( ord_less_int @ Y3 @ X4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % linorder_neqE_linordered_idom
% 0.17/0.41 thf(fact_572_nat__diff__add__eq2,axiom,
% 0.17/0.41 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.17/0.41 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_diff_add_eq2
% 0.17/0.41 thf(fact_573_nat__diff__add__eq1,axiom,
% 0.17/0.41 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ J @ I )
% 0.17/0.41 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.17/0.41 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_diff_add_eq1
% 0.17/0.41 thf(fact_574_nat__le__add__iff2,axiom,
% 0.17/0.41 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.17/0.41 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_le_add_iff2
% 0.17/0.41 thf(fact_575_nat__le__add__iff1,axiom,
% 0.17/0.41 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ J @ I )
% 0.17/0.41 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.17/0.41 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_le_add_iff1
% 0.17/0.41 thf(fact_576_nat__eq__add__iff2,axiom,
% 0.17/0.41 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.17/0.41 = ( M
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_eq_add_iff2
% 0.17/0.41 thf(fact_577_nat__eq__add__iff1,axiom,
% 0.17/0.41 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ J @ I )
% 0.17/0.41 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.17/0.41 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 0.17/0.41 = N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_eq_add_iff1
% 0.17/0.41 thf(fact_578_nat__neq__iff,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( M != N )
% 0.17/0.41 = ( ( ord_less_nat @ M @ N )
% 0.17/0.41 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_neq_iff
% 0.17/0.41 thf(fact_579_less__not__refl,axiom,
% 0.17/0.41 ! [N: nat] :
% 0.17/0.41 ~ ( ord_less_nat @ N @ N ) ).
% 0.17/0.41
% 0.17/0.41 % less_not_refl
% 0.17/0.41 thf(fact_580_less__not__refl2,axiom,
% 0.17/0.41 ! [N: nat,M: nat] :
% 0.17/0.41 ( ( ord_less_nat @ N @ M )
% 0.17/0.41 => ( M != N ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_not_refl2
% 0.17/0.41 thf(fact_581_less__not__refl3,axiom,
% 0.17/0.41 ! [S: nat,T: nat] :
% 0.17/0.41 ( ( ord_less_nat @ S @ T )
% 0.17/0.41 => ( S != T ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_not_refl3
% 0.17/0.41 thf(fact_582_less__irrefl__nat,axiom,
% 0.17/0.41 ! [N: nat] :
% 0.17/0.41 ~ ( ord_less_nat @ N @ N ) ).
% 0.17/0.41
% 0.17/0.41 % less_irrefl_nat
% 0.17/0.41 thf(fact_583_nat__less__induct,axiom,
% 0.17/0.41 ! [P: nat > $o,N: nat] :
% 0.17/0.41 ( ! [N4: nat] :
% 0.17/0.41 ( ! [M5: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M5 @ N4 )
% 0.17/0.41 => ( P @ M5 ) )
% 0.17/0.41 => ( P @ N4 ) )
% 0.17/0.41 => ( P @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_less_induct
% 0.17/0.41 thf(fact_584_infinite__descent,axiom,
% 0.17/0.41 ! [P: nat > $o,N: nat] :
% 0.17/0.41 ( ! [N4: nat] :
% 0.17/0.41 ( ~ ( P @ N4 )
% 0.17/0.41 => ? [M5: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M5 @ N4 )
% 0.17/0.41 & ~ ( P @ M5 ) ) )
% 0.17/0.41 => ( P @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % infinite_descent
% 0.17/0.41 thf(fact_585_linorder__neqE__nat,axiom,
% 0.17/0.41 ! [X4: nat,Y3: nat] :
% 0.17/0.41 ( ( X4 != Y3 )
% 0.17/0.41 => ( ~ ( ord_less_nat @ X4 @ Y3 )
% 0.17/0.41 => ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % linorder_neqE_nat
% 0.17/0.41 thf(fact_586_le__refl,axiom,
% 0.17/0.41 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.17/0.41
% 0.17/0.41 % le_refl
% 0.17/0.41 thf(fact_587_le__trans,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ( ord_less_eq_nat @ J @ K )
% 0.17/0.41 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_trans
% 0.17/0.41 thf(fact_588_eq__imp__le,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( M = N )
% 0.17/0.41 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % eq_imp_le
% 0.17/0.41 thf(fact_589_le__antisym,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.41 => ( M = N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_antisym
% 0.17/0.41 thf(fact_590_nat__le__linear,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_le_linear
% 0.17/0.41 thf(fact_591_Nat_Oex__has__greatest__nat,axiom,
% 0.17/0.41 ! [P: nat > $o,K: nat,B2: nat] :
% 0.17/0.41 ( ( P @ K )
% 0.17/0.41 => ( ! [Y4: nat] :
% 0.17/0.41 ( ( P @ Y4 )
% 0.17/0.41 => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 0.17/0.41 => ? [X3: nat] :
% 0.17/0.41 ( ( P @ X3 )
% 0.17/0.41 & ! [Y5: nat] :
% 0.17/0.41 ( ( P @ Y5 )
% 0.17/0.41 => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Nat.ex_has_greatest_nat
% 0.17/0.41 thf(fact_592_diff__commute,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.17/0.41 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_commute
% 0.17/0.41 thf(fact_593_left__add__twice,axiom,
% 0.17/0.41 ! [A: complex,B2: complex] :
% 0.17/0.41 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B2 ) )
% 0.17/0.41 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_add_twice
% 0.17/0.41 thf(fact_594_left__add__twice,axiom,
% 0.17/0.41 ! [A: real,B2: real] :
% 0.17/0.41 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 0.17/0.41 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_add_twice
% 0.17/0.41 thf(fact_595_left__add__twice,axiom,
% 0.17/0.41 ! [A: rat,B2: rat] :
% 0.17/0.41 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 0.17/0.41 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_add_twice
% 0.17/0.41 thf(fact_596_left__add__twice,axiom,
% 0.17/0.41 ! [A: nat,B2: nat] :
% 0.17/0.41 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_add_twice
% 0.17/0.41 thf(fact_597_left__add__twice,axiom,
% 0.17/0.41 ! [A: int,B2: int] :
% 0.17/0.41 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 0.17/0.41 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % left_add_twice
% 0.17/0.41 thf(fact_598_mult__2__right,axiom,
% 0.17/0.41 ! [Z: complex] :
% 0.17/0.41 ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_2_right
% 0.17/0.41 thf(fact_599_mult__2__right,axiom,
% 0.17/0.41 ! [Z: real] :
% 0.17/0.41 ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( plus_plus_real @ Z @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_2_right
% 0.17/0.41 thf(fact_600_mult__2__right,axiom,
% 0.17/0.41 ! [Z: rat] :
% 0.17/0.41 ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_2_right
% 0.17/0.41 thf(fact_601_mult__2__right,axiom,
% 0.17/0.41 ! [Z: nat] :
% 0.17/0.41 ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_2_right
% 0.17/0.41 thf(fact_602_mult__2__right,axiom,
% 0.17/0.41 ! [Z: int] :
% 0.17/0.41 ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( plus_plus_int @ Z @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_2_right
% 0.17/0.41 thf(fact_603_mult__2,axiom,
% 0.17/0.41 ! [Z: complex] :
% 0.17/0.41 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 0.17/0.41 = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_2
% 0.17/0.41 thf(fact_604_mult__2,axiom,
% 0.17/0.41 ! [Z: real] :
% 0.17/0.41 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 0.17/0.41 = ( plus_plus_real @ Z @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_2
% 0.17/0.41 thf(fact_605_mult__2,axiom,
% 0.17/0.41 ! [Z: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 0.17/0.41 = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_2
% 0.17/0.41 thf(fact_606_mult__2,axiom,
% 0.17/0.41 ! [Z: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 0.17/0.41 = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_2
% 0.17/0.41 thf(fact_607_mult__2,axiom,
% 0.17/0.41 ! [Z: int] :
% 0.17/0.41 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 0.17/0.41 = ( plus_plus_int @ Z @ Z ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult_2
% 0.17/0.41 thf(fact_608_power2__eq__square,axiom,
% 0.17/0.41 ! [A: complex] :
% 0.17/0.41 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( times_times_complex @ A @ A ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_eq_square
% 0.17/0.41 thf(fact_609_power2__eq__square,axiom,
% 0.17/0.41 ! [A: real] :
% 0.17/0.41 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( times_times_real @ A @ A ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_eq_square
% 0.17/0.41 thf(fact_610_power2__eq__square,axiom,
% 0.17/0.41 ! [A: rat] :
% 0.17/0.41 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( times_times_rat @ A @ A ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_eq_square
% 0.17/0.41 thf(fact_611_power2__eq__square,axiom,
% 0.17/0.41 ! [A: nat] :
% 0.17/0.41 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( times_times_nat @ A @ A ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_eq_square
% 0.17/0.41 thf(fact_612_power2__eq__square,axiom,
% 0.17/0.41 ! [A: int] :
% 0.17/0.41 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( times_times_int @ A @ A ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_eq_square
% 0.17/0.41 thf(fact_613_power4__eq__xxxx,axiom,
% 0.17/0.41 ! [X4: complex] :
% 0.17/0.41 ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.17/0.41 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).
% 0.17/0.41
% 0.17/0.41 % power4_eq_xxxx
% 0.17/0.41 thf(fact_614_power4__eq__xxxx,axiom,
% 0.17/0.41 ! [X4: real] :
% 0.17/0.41 ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.17/0.41 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).
% 0.17/0.41
% 0.17/0.41 % power4_eq_xxxx
% 0.17/0.41 thf(fact_615_power4__eq__xxxx,axiom,
% 0.17/0.41 ! [X4: rat] :
% 0.17/0.41 ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.17/0.41 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).
% 0.17/0.41
% 0.17/0.41 % power4_eq_xxxx
% 0.17/0.41 thf(fact_616_power4__eq__xxxx,axiom,
% 0.17/0.41 ! [X4: nat] :
% 0.17/0.41 ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.17/0.41 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).
% 0.17/0.41
% 0.17/0.41 % power4_eq_xxxx
% 0.17/0.41 thf(fact_617_power4__eq__xxxx,axiom,
% 0.17/0.41 ! [X4: int] :
% 0.17/0.41 ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.17/0.41 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).
% 0.17/0.41
% 0.17/0.41 % power4_eq_xxxx
% 0.17/0.41 thf(fact_618_power__even__eq,axiom,
% 0.17/0.41 ! [A: nat,N: nat] :
% 0.17/0.41 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.41 = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_even_eq
% 0.17/0.41 thf(fact_619_power__even__eq,axiom,
% 0.17/0.41 ! [A: real,N: nat] :
% 0.17/0.41 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.41 = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_even_eq
% 0.17/0.41 thf(fact_620_power__even__eq,axiom,
% 0.17/0.41 ! [A: int,N: nat] :
% 0.17/0.41 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.41 = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_even_eq
% 0.17/0.41 thf(fact_621_power__even__eq,axiom,
% 0.17/0.41 ! [A: complex,N: nat] :
% 0.17/0.41 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.41 = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_even_eq
% 0.17/0.41 thf(fact_622_nat__less__add__iff1,axiom,
% 0.17/0.41 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ J @ I )
% 0.17/0.41 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.17/0.41 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_less_add_iff1
% 0.17/0.41 thf(fact_623_nat__less__add__iff2,axiom,
% 0.17/0.41 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.17/0.41 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_less_add_iff2
% 0.17/0.41 thf(fact_624_power2__sum,axiom,
% 0.17/0.41 ! [X4: complex,Y3: complex] :
% 0.17/0.41 ( ( power_power_complex @ ( plus_plus_complex @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_sum
% 0.17/0.41 thf(fact_625_power2__sum,axiom,
% 0.17/0.41 ! [X4: real,Y3: real] :
% 0.17/0.41 ( ( power_power_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_sum
% 0.17/0.41 thf(fact_626_power2__sum,axiom,
% 0.17/0.41 ! [X4: rat,Y3: rat] :
% 0.17/0.41 ( ( power_power_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_sum
% 0.17/0.41 thf(fact_627_power2__sum,axiom,
% 0.17/0.41 ! [X4: nat,Y3: nat] :
% 0.17/0.41 ( ( power_power_nat @ ( plus_plus_nat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_sum
% 0.17/0.41 thf(fact_628_power2__sum,axiom,
% 0.17/0.41 ! [X4: int,Y3: int] :
% 0.17/0.41 ( ( power_power_int @ ( plus_plus_int @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_sum
% 0.17/0.41 thf(fact_629_add__diff__add,axiom,
% 0.17/0.41 ! [A: real,C: real,B2: real,D: real] :
% 0.17/0.41 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) )
% 0.17/0.41 = ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_add
% 0.17/0.41 thf(fact_630_add__diff__add,axiom,
% 0.17/0.41 ! [A: rat,C: rat,B2: rat,D: rat] :
% 0.17/0.41 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) )
% 0.17/0.41 = ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_add
% 0.17/0.41 thf(fact_631_add__diff__add,axiom,
% 0.17/0.41 ! [A: int,C: int,B2: int,D: int] :
% 0.17/0.41 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) )
% 0.17/0.41 = ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_add
% 0.17/0.41 thf(fact_632_nat__less__le,axiom,
% 0.17/0.41 ( ord_less_nat
% 0.17/0.41 = ( ^ [M6: nat,N2: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M6 @ N2 )
% 0.17/0.41 & ( M6 != N2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_less_le
% 0.17/0.41 thf(fact_633_less__imp__le__nat,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ N )
% 0.17/0.41 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_imp_le_nat
% 0.17/0.41 thf(fact_634_le__eq__less__or__eq,axiom,
% 0.17/0.41 ( ord_less_eq_nat
% 0.17/0.41 = ( ^ [M6: nat,N2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M6 @ N2 )
% 0.17/0.41 | ( M6 = N2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_eq_less_or_eq
% 0.17/0.41 thf(fact_635_less__or__eq__imp__le,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ( ord_less_nat @ M @ N )
% 0.17/0.41 | ( M = N ) )
% 0.17/0.41 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_or_eq_imp_le
% 0.17/0.41 thf(fact_636_le__neq__implies__less,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 => ( ( M != N )
% 0.17/0.41 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_neq_implies_less
% 0.17/0.41 thf(fact_637_less__mono__imp__le__mono,axiom,
% 0.17/0.41 ! [F: nat > nat,I: nat,J: nat] :
% 0.17/0.41 ( ! [I3: nat,J2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I3 @ J2 )
% 0.17/0.41 => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_mono_imp_le_mono
% 0.17/0.41 thf(fact_638_add__lessD1,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.17/0.41 => ( ord_less_nat @ I @ K ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_lessD1
% 0.17/0.41 thf(fact_639_add__less__mono,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I @ J )
% 0.17/0.41 => ( ( ord_less_nat @ K @ L2 )
% 0.17/0.41 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_less_mono
% 0.17/0.41 thf(fact_640_not__add__less1,axiom,
% 0.17/0.41 ! [I: nat,J: nat] :
% 0.17/0.41 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.17/0.41
% 0.17/0.41 % not_add_less1
% 0.17/0.41 thf(fact_641_not__add__less2,axiom,
% 0.17/0.41 ! [J: nat,I: nat] :
% 0.17/0.41 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.17/0.41
% 0.17/0.41 % not_add_less2
% 0.17/0.41 thf(fact_642_add__less__mono1,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I @ J )
% 0.17/0.41 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_less_mono1
% 0.17/0.41 thf(fact_643_trans__less__add1,axiom,
% 0.17/0.41 ! [I: nat,J: nat,M: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I @ J )
% 0.17/0.41 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % trans_less_add1
% 0.17/0.41 thf(fact_644_trans__less__add2,axiom,
% 0.17/0.41 ! [I: nat,J: nat,M: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I @ J )
% 0.17/0.41 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % trans_less_add2
% 0.17/0.41 thf(fact_645_less__add__eq__less,axiom,
% 0.17/0.41 ! [K: nat,L2: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ K @ L2 )
% 0.17/0.41 => ( ( ( plus_plus_nat @ M @ L2 )
% 0.17/0.41 = ( plus_plus_nat @ K @ N ) )
% 0.17/0.41 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_eq_less
% 0.17/0.41 thf(fact_646_add__leE,axiom,
% 0.17/0.41 ! [M: nat,K: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.17/0.41 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_leE
% 0.17/0.41 thf(fact_647_le__add1,axiom,
% 0.17/0.41 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_add1
% 0.17/0.41 thf(fact_648_le__add2,axiom,
% 0.17/0.41 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_add2
% 0.17/0.41 thf(fact_649_add__leD1,axiom,
% 0.17/0.41 ! [M: nat,K: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.17/0.41 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_leD1
% 0.17/0.41 thf(fact_650_add__leD2,axiom,
% 0.17/0.41 ! [M: nat,K: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.17/0.41 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_leD2
% 0.17/0.41 thf(fact_651_le__Suc__ex,axiom,
% 0.17/0.41 ! [K: nat,L2: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ K @ L2 )
% 0.17/0.41 => ? [N4: nat] :
% 0.17/0.41 ( L2
% 0.17/0.41 = ( plus_plus_nat @ K @ N4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_Suc_ex
% 0.17/0.41 thf(fact_652_add__le__mono,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ( ord_less_eq_nat @ K @ L2 )
% 0.17/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_mono
% 0.17/0.41 thf(fact_653_add__le__mono1,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_mono1
% 0.17/0.41 thf(fact_654_trans__le__add1,axiom,
% 0.17/0.41 ! [I: nat,J: nat,M: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % trans_le_add1
% 0.17/0.41 thf(fact_655_trans__le__add2,axiom,
% 0.17/0.41 ! [I: nat,J: nat,M: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % trans_le_add2
% 0.17/0.41 thf(fact_656_nat__le__iff__add,axiom,
% 0.17/0.41 ( ord_less_eq_nat
% 0.17/0.41 = ( ^ [M6: nat,N2: nat] :
% 0.17/0.41 ? [K3: nat] :
% 0.17/0.41 ( N2
% 0.17/0.41 = ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % nat_le_iff_add
% 0.17/0.41 thf(fact_657_diff__less__mono2,axiom,
% 0.17/0.41 ! [M: nat,N: nat,L2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M @ N )
% 0.17/0.41 => ( ( ord_less_nat @ M @ L2 )
% 0.17/0.41 => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_less_mono2
% 0.17/0.41 thf(fact_658_less__imp__diff__less,axiom,
% 0.17/0.41 ! [J: nat,K: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_nat @ J @ K )
% 0.17/0.41 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_imp_diff_less
% 0.17/0.41 thf(fact_659_eq__diff__iff,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ K @ M )
% 0.17/0.41 => ( ( ord_less_eq_nat @ K @ N )
% 0.17/0.41 => ( ( ( minus_minus_nat @ M @ K )
% 0.17/0.41 = ( minus_minus_nat @ N @ K ) )
% 0.17/0.41 = ( M = N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % eq_diff_iff
% 0.17/0.41 thf(fact_660_le__diff__iff,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ K @ M )
% 0.17/0.41 => ( ( ord_less_eq_nat @ K @ N )
% 0.17/0.41 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.17/0.41 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_diff_iff
% 0.17/0.41 thf(fact_661_Nat_Odiff__diff__eq,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ K @ M )
% 0.17/0.41 => ( ( ord_less_eq_nat @ K @ N )
% 0.17/0.41 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.17/0.41 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Nat.diff_diff_eq
% 0.17/0.41 thf(fact_662_diff__le__mono,axiom,
% 0.17/0.41 ! [M: nat,N: nat,L2: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_le_mono
% 0.17/0.41 thf(fact_663_diff__le__self,axiom,
% 0.17/0.41 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.17/0.41
% 0.17/0.41 % diff_le_self
% 0.17/0.41 thf(fact_664_le__diff__iff_H,axiom,
% 0.17/0.41 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ A @ C )
% 0.17/0.41 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.17/0.41 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
% 0.17/0.41 = ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_diff_iff'
% 0.17/0.41 thf(fact_665_diff__le__mono2,axiom,
% 0.17/0.41 ! [M: nat,N: nat,L2: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.41 => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_le_mono2
% 0.17/0.41 thf(fact_666_power2__diff,axiom,
% 0.17/0.41 ! [X4: complex,Y3: complex] :
% 0.17/0.41 ( ( power_power_complex @ ( minus_minus_complex @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_diff
% 0.17/0.41 thf(fact_667_power2__diff,axiom,
% 0.17/0.41 ! [X4: real,Y3: real] :
% 0.17/0.41 ( ( power_power_real @ ( minus_minus_real @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_diff
% 0.17/0.41 thf(fact_668_power2__diff,axiom,
% 0.17/0.41 ! [X4: rat,Y3: rat] :
% 0.17/0.41 ( ( power_power_rat @ ( minus_minus_rat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_diff
% 0.17/0.41 thf(fact_669_power2__diff,axiom,
% 0.17/0.41 ! [X4: int,Y3: int] :
% 0.17/0.41 ( ( power_power_int @ ( minus_minus_int @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.41 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power2_diff
% 0.17/0.41 thf(fact_670_Nat_Odiff__cancel,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.17/0.41 = ( minus_minus_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Nat.diff_cancel
% 0.17/0.41 thf(fact_671_diff__cancel2,axiom,
% 0.17/0.41 ! [M: nat,K: nat,N: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.17/0.41 = ( minus_minus_nat @ M @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_cancel2
% 0.17/0.41 thf(fact_672_diff__add__inverse,axiom,
% 0.17/0.41 ! [N: nat,M: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.17/0.41 = M ) ).
% 0.17/0.41
% 0.17/0.41 % diff_add_inverse
% 0.17/0.41 thf(fact_673_diff__add__inverse2,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.17/0.41 = M ) ).
% 0.17/0.41
% 0.17/0.41 % diff_add_inverse2
% 0.17/0.41 thf(fact_674_add__le__add__imp__diff__le,axiom,
% 0.17/0.41 ! [I: real,K: real,N: real,J: real] :
% 0.17/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.17/0.41 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.17/0.41 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.17/0.41 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.17/0.41 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_add_imp_diff_le
% 0.17/0.41 thf(fact_675_add__le__add__imp__diff__le,axiom,
% 0.17/0.41 ! [I: rat,K: rat,N: rat,J: rat] :
% 0.17/0.41 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.17/0.41 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.17/0.41 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.17/0.41 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.17/0.41 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_add_imp_diff_le
% 0.17/0.41 thf(fact_676_add__le__add__imp__diff__le,axiom,
% 0.17/0.41 ! [I: nat,K: nat,N: nat,J: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.17/0.41 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.17/0.41 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.17/0.41 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_add_imp_diff_le
% 0.17/0.41 thf(fact_677_add__le__add__imp__diff__le,axiom,
% 0.17/0.41 ! [I: int,K: int,N: int,J: int] :
% 0.17/0.41 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.17/0.41 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.17/0.41 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.17/0.41 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.17/0.41 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_add_imp_diff_le
% 0.17/0.41 thf(fact_678_add__le__imp__le__diff,axiom,
% 0.17/0.41 ! [I: real,K: real,N: real] :
% 0.17/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.17/0.41 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_imp_le_diff
% 0.17/0.41 thf(fact_679_add__le__imp__le__diff,axiom,
% 0.17/0.41 ! [I: rat,K: rat,N: rat] :
% 0.17/0.41 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.17/0.41 => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_imp_le_diff
% 0.17/0.41 thf(fact_680_add__le__imp__le__diff,axiom,
% 0.17/0.41 ! [I: nat,K: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.17/0.41 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_imp_le_diff
% 0.17/0.41 thf(fact_681_add__le__imp__le__diff,axiom,
% 0.17/0.41 ! [I: int,K: int,N: int] :
% 0.17/0.41 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.17/0.41 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_imp_le_diff
% 0.17/0.41 thf(fact_682_add__mono1,axiom,
% 0.17/0.41 ! [A: real,B2: real] :
% 0.17/0.41 ( ( ord_less_real @ A @ B2 )
% 0.17/0.41 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B2 @ one_one_real ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_mono1
% 0.17/0.41 thf(fact_683_add__mono1,axiom,
% 0.17/0.41 ! [A: rat,B2: rat] :
% 0.17/0.41 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.41 => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B2 @ one_one_rat ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_mono1
% 0.17/0.41 thf(fact_684_add__mono1,axiom,
% 0.17/0.41 ! [A: nat,B2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.41 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_mono1
% 0.17/0.41 thf(fact_685_add__mono1,axiom,
% 0.17/0.41 ! [A: int,B2: int] :
% 0.17/0.41 ( ( ord_less_int @ A @ B2 )
% 0.17/0.41 => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_mono1
% 0.17/0.41 thf(fact_686_less__add__one,axiom,
% 0.17/0.41 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_one
% 0.17/0.41 thf(fact_687_less__add__one,axiom,
% 0.17/0.41 ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_one
% 0.17/0.41 thf(fact_688_less__add__one,axiom,
% 0.17/0.41 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_one
% 0.17/0.41 thf(fact_689_less__add__one,axiom,
% 0.17/0.41 ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_add_one
% 0.17/0.41 thf(fact_690_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.17/0.41 ! [A: real,B2: real] :
% 0.17/0.41 ( ~ ( ord_less_real @ A @ B2 )
% 0.17/0.41 => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % linordered_semidom_class.add_diff_inverse
% 0.17/0.41 thf(fact_691_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.17/0.41 ! [A: rat,B2: rat] :
% 0.17/0.41 ( ~ ( ord_less_rat @ A @ B2 )
% 0.17/0.41 => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % linordered_semidom_class.add_diff_inverse
% 0.17/0.41 thf(fact_692_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.17/0.41 ! [A: nat,B2: nat] :
% 0.17/0.41 ( ~ ( ord_less_nat @ A @ B2 )
% 0.17/0.41 => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % linordered_semidom_class.add_diff_inverse
% 0.17/0.41 thf(fact_693_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.17/0.41 ! [A: int,B2: int] :
% 0.17/0.41 ( ~ ( ord_less_int @ A @ B2 )
% 0.17/0.41 => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
% 0.17/0.41 = A ) ) ).
% 0.17/0.41
% 0.17/0.41 % linordered_semidom_class.add_diff_inverse
% 0.17/0.41 thf(fact_694_mono__nat__linear__lb,axiom,
% 0.17/0.41 ! [F: nat > nat,M: nat,K: nat] :
% 0.17/0.41 ( ! [M4: nat,N4: nat] :
% 0.17/0.41 ( ( ord_less_nat @ M4 @ N4 )
% 0.17/0.41 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
% 0.17/0.41 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mono_nat_linear_lb
% 0.17/0.41 thf(fact_695_less__diff__iff,axiom,
% 0.17/0.41 ! [K: nat,M: nat,N: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ K @ M )
% 0.17/0.41 => ( ( ord_less_eq_nat @ K @ N )
% 0.17/0.41 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.17/0.41 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_diff_iff
% 0.17/0.41 thf(fact_696_diff__less__mono,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.41 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.41 => ( ( ord_less_eq_nat @ C @ A )
% 0.17/0.41 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % diff_less_mono
% 0.17/0.41 thf(fact_697_add__diff__inverse__nat,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ~ ( ord_less_nat @ M @ N )
% 0.17/0.41 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.17/0.41 = M ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_inverse_nat
% 0.17/0.41 thf(fact_698_less__diff__conv,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.17/0.41 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_diff_conv
% 0.17/0.41 thf(fact_699_Nat_Ole__imp__diff__is__add,axiom,
% 0.17/0.41 ! [I: nat,J: nat,K: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.41 => ( ( ( minus_minus_nat @ J @ I )
% 0.17/0.41 = K )
% 0.17/0.41 = ( J
% 0.17/0.41 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Nat.le_imp_diff_is_add
% 0.17/0.41 thf(fact_700_Nat_Odiff__add__assoc2,axiom,
% 0.17/0.41 ! [K: nat,J: nat,I: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.41 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.17/0.41 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Nat.diff_add_assoc2
% 0.17/0.41 thf(fact_701_Nat_Odiff__add__assoc,axiom,
% 0.17/0.41 ! [K: nat,J: nat,I: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.41 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.17/0.41 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Nat.diff_add_assoc
% 0.17/0.41 thf(fact_702_Nat_Ole__diff__conv2,axiom,
% 0.17/0.41 ! [K: nat,J: nat,I: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.41 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.17/0.41 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % Nat.le_diff_conv2
% 0.17/0.41 thf(fact_703_le__diff__conv,axiom,
% 0.17/0.41 ! [J: nat,K: nat,I: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.17/0.41 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % le_diff_conv
% 0.17/0.41 thf(fact_704_less__diff__conv2,axiom,
% 0.17/0.41 ! [K: nat,J: nat,I: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ K @ J )
% 0.17/0.41 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.17/0.41 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_diff_conv2
% 0.17/0.41 thf(fact_705_field__sum__of__halves,axiom,
% 0.17/0.41 ! [X4: real] :
% 0.17/0.41 ( ( plus_plus_real @ ( divide_divide_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.17/0.41 = X4 ) ).
% 0.17/0.41
% 0.17/0.41 % field_sum_of_halves
% 0.17/0.41 thf(fact_706_field__sum__of__halves,axiom,
% 0.17/0.41 ! [X4: rat] :
% 0.17/0.41 ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.17/0.41 = X4 ) ).
% 0.17/0.41
% 0.17/0.41 % field_sum_of_halves
% 0.17/0.41 thf(fact_707__092_060open_062res_A_061_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Asc_A_L_Aminy_092_060close_062,axiom,
% 0.17/0.41 ( res
% 0.17/0.41 = ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ) @ miny ) ) ).
% 0.17/0.41
% 0.17/0.41 % \<open>res = 2 ^ (deg div 2) * sc + miny\<close>
% 0.17/0.41 thf(fact_708_sum__squares__bound,axiom,
% 0.17/0.41 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % sum_squares_bound
% 0.17/0.41 thf(fact_709_sum__squares__bound,axiom,
% 0.17/0.41 ! [X4: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % sum_squares_bound
% 0.17/0.41 thf(fact_710_enat__ord__number_I1_J,axiom,
% 0.17/0.41 ! [M: num,N: num] :
% 0.17/0.41 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.17/0.41 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % enat_ord_number(1)
% 0.17/0.41 thf(fact_711_enat__ord__number_I2_J,axiom,
% 0.17/0.41 ! [M: num,N: num] :
% 0.17/0.41 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.17/0.41 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % enat_ord_number(2)
% 0.17/0.41 thf(fact_712_double__not__eq__Suc__double,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.17/0.41 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % double_not_eq_Suc_double
% 0.17/0.41 thf(fact_713_Suc__double__not__eq__double,axiom,
% 0.17/0.41 ! [M: nat,N: nat] :
% 0.17/0.41 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.17/0.41 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % Suc_double_not_eq_double
% 0.17/0.41 thf(fact_714_add__diff__cancel,axiom,
% 0.17/0.41 ! [A: real,B2: real] :
% 0.17/0.41 ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel
% 0.17/0.41 thf(fact_715_add__diff__cancel,axiom,
% 0.17/0.41 ! [A: rat,B2: rat] :
% 0.17/0.41 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel
% 0.17/0.41 thf(fact_716_add__diff__cancel,axiom,
% 0.17/0.41 ! [A: int,B2: int] :
% 0.17/0.41 ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel
% 0.17/0.41 thf(fact_717_diff__add__cancel,axiom,
% 0.17/0.41 ! [A: real,B2: real] :
% 0.17/0.41 ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % diff_add_cancel
% 0.17/0.41 thf(fact_718_diff__add__cancel,axiom,
% 0.17/0.41 ! [A: rat,B2: rat] :
% 0.17/0.41 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % diff_add_cancel
% 0.17/0.41 thf(fact_719_diff__add__cancel,axiom,
% 0.17/0.41 ! [A: int,B2: int] :
% 0.17/0.41 ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % diff_add_cancel
% 0.17/0.41 thf(fact_720_add__diff__cancel__left,axiom,
% 0.17/0.41 ! [C: real,A: real,B2: real] :
% 0.17/0.41 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 0.17/0.41 = ( minus_minus_real @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_left
% 0.17/0.41 thf(fact_721_add__diff__cancel__left,axiom,
% 0.17/0.41 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.41 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 0.17/0.41 = ( minus_minus_rat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_left
% 0.17/0.41 thf(fact_722_add__diff__cancel__left,axiom,
% 0.17/0.41 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 0.17/0.41 = ( minus_minus_nat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_left
% 0.17/0.41 thf(fact_723_add__diff__cancel__left,axiom,
% 0.17/0.41 ! [C: int,A: int,B2: int] :
% 0.17/0.41 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 0.17/0.41 = ( minus_minus_int @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_left
% 0.17/0.41 thf(fact_724_add__diff__cancel__left_H,axiom,
% 0.17/0.41 ! [A: real,B2: real] :
% 0.17/0.41 ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ A )
% 0.17/0.41 = B2 ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_left'
% 0.17/0.41 thf(fact_725_add__diff__cancel__left_H,axiom,
% 0.17/0.41 ! [A: rat,B2: rat] :
% 0.17/0.41 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ A )
% 0.17/0.41 = B2 ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_left'
% 0.17/0.41 thf(fact_726_add__diff__cancel__left_H,axiom,
% 0.17/0.41 ! [A: nat,B2: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
% 0.17/0.41 = B2 ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_left'
% 0.17/0.41 thf(fact_727_add__diff__cancel__left_H,axiom,
% 0.17/0.41 ! [A: int,B2: int] :
% 0.17/0.41 ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ A )
% 0.17/0.41 = B2 ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_left'
% 0.17/0.41 thf(fact_728_add__diff__cancel__right,axiom,
% 0.17/0.41 ! [A: real,C: real,B2: real] :
% 0.17/0.41 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_real @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_right
% 0.17/0.41 thf(fact_729_add__diff__cancel__right,axiom,
% 0.17/0.41 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.41 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_rat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_right
% 0.17/0.41 thf(fact_730_add__diff__cancel__right,axiom,
% 0.17/0.41 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_nat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_right
% 0.17/0.41 thf(fact_731_add__diff__cancel__right,axiom,
% 0.17/0.41 ! [A: int,C: int,B2: int] :
% 0.17/0.41 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 0.17/0.41 = ( minus_minus_int @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_right
% 0.17/0.41 thf(fact_732_add__diff__cancel__right_H,axiom,
% 0.17/0.41 ! [A: real,B2: real] :
% 0.17/0.41 ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_right'
% 0.17/0.41 thf(fact_733_add__diff__cancel__right_H,axiom,
% 0.17/0.41 ! [A: rat,B2: rat] :
% 0.17/0.41 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_right'
% 0.17/0.41 thf(fact_734_add__diff__cancel__right_H,axiom,
% 0.17/0.41 ! [A: nat,B2: nat] :
% 0.17/0.41 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_right'
% 0.17/0.41 thf(fact_735_add__diff__cancel__right_H,axiom,
% 0.17/0.41 ! [A: int,B2: int] :
% 0.17/0.41 ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_cancel_right'
% 0.17/0.41 thf(fact_736_add__right__cancel,axiom,
% 0.17/0.41 ! [B2: real,A: real,C: real] :
% 0.17/0.41 ( ( ( plus_plus_real @ B2 @ A )
% 0.17/0.41 = ( plus_plus_real @ C @ A ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_right_cancel
% 0.17/0.41 thf(fact_737_add__right__cancel,axiom,
% 0.17/0.41 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.41 ( ( ( plus_plus_rat @ B2 @ A )
% 0.17/0.41 = ( plus_plus_rat @ C @ A ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_right_cancel
% 0.17/0.41 thf(fact_738_add__right__cancel,axiom,
% 0.17/0.41 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.41 ( ( ( plus_plus_nat @ B2 @ A )
% 0.17/0.41 = ( plus_plus_nat @ C @ A ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_right_cancel
% 0.17/0.41 thf(fact_739_add__right__cancel,axiom,
% 0.17/0.41 ! [B2: int,A: int,C: int] :
% 0.17/0.41 ( ( ( plus_plus_int @ B2 @ A )
% 0.17/0.41 = ( plus_plus_int @ C @ A ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_right_cancel
% 0.17/0.41 thf(fact_740_add__left__cancel,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( ( plus_plus_real @ A @ B2 )
% 0.17/0.41 = ( plus_plus_real @ A @ C ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_left_cancel
% 0.17/0.41 thf(fact_741_add__left__cancel,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( ( plus_plus_rat @ A @ B2 )
% 0.17/0.41 = ( plus_plus_rat @ A @ C ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_left_cancel
% 0.17/0.41 thf(fact_742_add__left__cancel,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.41 ( ( ( plus_plus_nat @ A @ B2 )
% 0.17/0.41 = ( plus_plus_nat @ A @ C ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_left_cancel
% 0.17/0.41 thf(fact_743_add__left__cancel,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( ( plus_plus_int @ A @ B2 )
% 0.17/0.41 = ( plus_plus_int @ A @ C ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_left_cancel
% 0.17/0.41 thf(fact_744_add__le__cancel__right,axiom,
% 0.17/0.41 ! [A: real,C: real,B2: real] :
% 0.17/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 0.17/0.41 = ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_cancel_right
% 0.17/0.41 thf(fact_745_add__le__cancel__right,axiom,
% 0.17/0.41 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.41 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.17/0.41 = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_cancel_right
% 0.17/0.41 thf(fact_746_add__le__cancel__right,axiom,
% 0.17/0.41 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.17/0.41 = ( ord_less_eq_nat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_cancel_right
% 0.17/0.41 thf(fact_747_add__le__cancel__right,axiom,
% 0.17/0.41 ! [A: int,C: int,B2: int] :
% 0.17/0.41 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 0.17/0.41 = ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_cancel_right
% 0.17/0.41 thf(fact_748_add__le__cancel__left,axiom,
% 0.17/0.41 ! [C: real,A: real,B2: real] :
% 0.17/0.41 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 0.17/0.41 = ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_cancel_left
% 0.17/0.41 thf(fact_749_add__le__cancel__left,axiom,
% 0.17/0.41 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.41 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 0.17/0.41 = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_cancel_left
% 0.17/0.41 thf(fact_750_add__le__cancel__left,axiom,
% 0.17/0.41 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.41 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 0.17/0.41 = ( ord_less_eq_nat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_cancel_left
% 0.17/0.41 thf(fact_751_add__le__cancel__left,axiom,
% 0.17/0.41 ! [C: int,A: int,B2: int] :
% 0.17/0.41 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 0.17/0.41 = ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_le_cancel_left
% 0.17/0.41 thf(fact_752_add__less__cancel__right,axiom,
% 0.17/0.41 ! [A: real,C: real,B2: real] :
% 0.17/0.41 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 0.17/0.41 = ( ord_less_real @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_less_cancel_right
% 0.17/0.41 thf(fact_753_add__less__cancel__right,axiom,
% 0.17/0.41 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.41 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.17/0.41 = ( ord_less_rat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_less_cancel_right
% 0.17/0.41 thf(fact_754_add__less__cancel__right,axiom,
% 0.17/0.41 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.17/0.41 = ( ord_less_nat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_less_cancel_right
% 0.17/0.41 thf(fact_755_add__less__cancel__right,axiom,
% 0.17/0.41 ! [A: int,C: int,B2: int] :
% 0.17/0.41 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 0.17/0.41 = ( ord_less_int @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_less_cancel_right
% 0.17/0.41 thf(fact_756_add__less__cancel__left,axiom,
% 0.17/0.41 ! [C: real,A: real,B2: real] :
% 0.17/0.41 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 0.17/0.41 = ( ord_less_real @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_less_cancel_left
% 0.17/0.41 thf(fact_757_add__less__cancel__left,axiom,
% 0.17/0.41 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.41 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 0.17/0.41 = ( ord_less_rat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_less_cancel_left
% 0.17/0.41 thf(fact_758_add__less__cancel__left,axiom,
% 0.17/0.41 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.41 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 0.17/0.41 = ( ord_less_nat @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_less_cancel_left
% 0.17/0.41 thf(fact_759_add__less__cancel__left,axiom,
% 0.17/0.41 ! [C: int,A: int,B2: int] :
% 0.17/0.41 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 0.17/0.41 = ( ord_less_int @ A @ B2 ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_less_cancel_left
% 0.17/0.41 thf(fact_760_mult__1,axiom,
% 0.17/0.41 ! [A: complex] :
% 0.17/0.41 ( ( times_times_complex @ one_one_complex @ A )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_1
% 0.17/0.41 thf(fact_761_mult__1,axiom,
% 0.17/0.41 ! [A: real] :
% 0.17/0.41 ( ( times_times_real @ one_one_real @ A )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_1
% 0.17/0.41 thf(fact_762_mult__1,axiom,
% 0.17/0.41 ! [A: rat] :
% 0.17/0.41 ( ( times_times_rat @ one_one_rat @ A )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_1
% 0.17/0.41 thf(fact_763_mult__1,axiom,
% 0.17/0.41 ! [A: nat] :
% 0.17/0.41 ( ( times_times_nat @ one_one_nat @ A )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_1
% 0.17/0.41 thf(fact_764_mult__1,axiom,
% 0.17/0.41 ! [A: int] :
% 0.17/0.41 ( ( times_times_int @ one_one_int @ A )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult_1
% 0.17/0.41 thf(fact_765_mult_Oright__neutral,axiom,
% 0.17/0.41 ! [A: complex] :
% 0.17/0.41 ( ( times_times_complex @ A @ one_one_complex )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult.right_neutral
% 0.17/0.41 thf(fact_766_mult_Oright__neutral,axiom,
% 0.17/0.41 ! [A: real] :
% 0.17/0.41 ( ( times_times_real @ A @ one_one_real )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult.right_neutral
% 0.17/0.41 thf(fact_767_mult_Oright__neutral,axiom,
% 0.17/0.41 ! [A: rat] :
% 0.17/0.41 ( ( times_times_rat @ A @ one_one_rat )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult.right_neutral
% 0.17/0.41 thf(fact_768_mult_Oright__neutral,axiom,
% 0.17/0.41 ! [A: nat] :
% 0.17/0.41 ( ( times_times_nat @ A @ one_one_nat )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult.right_neutral
% 0.17/0.41 thf(fact_769_mult_Oright__neutral,axiom,
% 0.17/0.41 ! [A: int] :
% 0.17/0.41 ( ( times_times_int @ A @ one_one_int )
% 0.17/0.41 = A ) ).
% 0.17/0.41
% 0.17/0.41 % mult.right_neutral
% 0.17/0.41 thf(fact_770_semiring__norm_I13_J,axiom,
% 0.17/0.41 ! [M: num,N: num] :
% 0.17/0.41 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.17/0.41 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % semiring_norm(13)
% 0.17/0.41 thf(fact_771_semiring__norm_I12_J,axiom,
% 0.17/0.41 ! [N: num] :
% 0.17/0.41 ( ( times_times_num @ one @ N )
% 0.17/0.41 = N ) ).
% 0.17/0.41
% 0.17/0.41 % semiring_norm(12)
% 0.17/0.41 thf(fact_772_semiring__norm_I11_J,axiom,
% 0.17/0.41 ! [M: num] :
% 0.17/0.41 ( ( times_times_num @ M @ one )
% 0.17/0.41 = M ) ).
% 0.17/0.41
% 0.17/0.41 % semiring_norm(11)
% 0.17/0.41 thf(fact_773_num__double,axiom,
% 0.17/0.41 ! [N: num] :
% 0.17/0.41 ( ( times_times_num @ ( bit0 @ one ) @ N )
% 0.17/0.41 = ( bit0 @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % num_double
% 0.17/0.41 thf(fact_774_power__mult__numeral,axiom,
% 0.17/0.41 ! [A: nat,M: num,N: num] :
% 0.17/0.41 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.17/0.41 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult_numeral
% 0.17/0.41 thf(fact_775_power__mult__numeral,axiom,
% 0.17/0.41 ! [A: real,M: num,N: num] :
% 0.17/0.41 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.17/0.41 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult_numeral
% 0.17/0.41 thf(fact_776_power__mult__numeral,axiom,
% 0.17/0.41 ! [A: int,M: num,N: num] :
% 0.17/0.41 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.17/0.41 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult_numeral
% 0.17/0.41 thf(fact_777_power__mult__numeral,axiom,
% 0.17/0.41 ! [A: complex,M: num,N: num] :
% 0.17/0.41 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.17/0.41 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % power_mult_numeral
% 0.17/0.41 thf(fact_778_complete__real,axiom,
% 0.17/0.41 ! [S2: set_real] :
% 0.17/0.41 ( ? [X5: real] : ( member_real @ X5 @ S2 )
% 0.17/0.41 => ( ? [Z3: real] :
% 0.17/0.41 ! [X3: real] :
% 0.17/0.41 ( ( member_real @ X3 @ S2 )
% 0.17/0.41 => ( ord_less_eq_real @ X3 @ Z3 ) )
% 0.17/0.41 => ? [Y4: real] :
% 0.17/0.41 ( ! [X5: real] :
% 0.17/0.41 ( ( member_real @ X5 @ S2 )
% 0.17/0.41 => ( ord_less_eq_real @ X5 @ Y4 ) )
% 0.17/0.41 & ! [Z3: real] :
% 0.17/0.41 ( ! [X3: real] :
% 0.17/0.41 ( ( member_real @ X3 @ S2 )
% 0.17/0.41 => ( ord_less_eq_real @ X3 @ Z3 ) )
% 0.17/0.41 => ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % complete_real
% 0.17/0.41 thf(fact_779_real__arch__pow,axiom,
% 0.17/0.41 ! [X4: real,Y3: real] :
% 0.17/0.41 ( ( ord_less_real @ one_one_real @ X4 )
% 0.17/0.41 => ? [N4: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X4 @ N4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % real_arch_pow
% 0.17/0.41 thf(fact_780_less__eq__real__def,axiom,
% 0.17/0.41 ( ord_less_eq_real
% 0.17/0.41 = ( ^ [X: real,Y: real] :
% 0.17/0.41 ( ( ord_less_real @ X @ Y )
% 0.17/0.41 | ( X = Y ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % less_eq_real_def
% 0.17/0.41 thf(fact_781_add__diff__assoc__enat,axiom,
% 0.17/0.41 ! [Z: extended_enat,Y3: extended_enat,X4: extended_enat] :
% 0.17/0.41 ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
% 0.17/0.41 => ( ( plus_p3455044024723400733d_enat @ X4 @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
% 0.17/0.41 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y3 ) @ Z ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_diff_assoc_enat
% 0.17/0.41 thf(fact_782_enat__less__induct,axiom,
% 0.17/0.41 ! [P: extended_enat > $o,N: extended_enat] :
% 0.17/0.41 ( ! [N4: extended_enat] :
% 0.17/0.41 ( ! [M5: extended_enat] :
% 0.17/0.41 ( ( ord_le72135733267957522d_enat @ M5 @ N4 )
% 0.17/0.41 => ( P @ M5 ) )
% 0.17/0.41 => ( P @ N4 ) )
% 0.17/0.41 => ( P @ N ) ) ).
% 0.17/0.41
% 0.17/0.41 % enat_less_induct
% 0.17/0.41 thf(fact_783_four__x__squared,axiom,
% 0.17/0.41 ! [X4: real] :
% 0.17/0.41 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.41 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % four_x_squared
% 0.17/0.41 thf(fact_784_L2__set__mult__ineq__lemma,axiom,
% 0.17/0.41 ! [A: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % L2_set_mult_ineq_lemma
% 0.17/0.41 thf(fact_785_div__mult2__numeral__eq,axiom,
% 0.17/0.41 ! [A: nat,K: num,L2: num] :
% 0.17/0.41 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
% 0.17/0.41 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % div_mult2_numeral_eq
% 0.17/0.41 thf(fact_786_div__mult2__numeral__eq,axiom,
% 0.17/0.41 ! [A: int,K: num,L2: num] :
% 0.17/0.41 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
% 0.17/0.41 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % div_mult2_numeral_eq
% 0.17/0.41 thf(fact_787_mult_Oleft__commute,axiom,
% 0.17/0.41 ! [B2: real,A: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ B2 @ ( times_times_real @ A @ C ) )
% 0.17/0.41 = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.left_commute
% 0.17/0.41 thf(fact_788_mult_Oleft__commute,axiom,
% 0.17/0.41 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ B2 @ ( times_times_rat @ A @ C ) )
% 0.17/0.41 = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.left_commute
% 0.17/0.41 thf(fact_789_mult_Oleft__commute,axiom,
% 0.17/0.41 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.41 ( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
% 0.17/0.41 = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.left_commute
% 0.17/0.41 thf(fact_790_mult_Oleft__commute,axiom,
% 0.17/0.41 ! [B2: int,A: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ B2 @ ( times_times_int @ A @ C ) )
% 0.17/0.41 = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.left_commute
% 0.17/0.41 thf(fact_791_mult_Ocommute,axiom,
% 0.17/0.41 ( times_times_real
% 0.17/0.41 = ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.commute
% 0.17/0.41 thf(fact_792_mult_Ocommute,axiom,
% 0.17/0.41 ( times_times_rat
% 0.17/0.41 = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ B3 @ A4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.commute
% 0.17/0.41 thf(fact_793_mult_Ocommute,axiom,
% 0.17/0.41 ( times_times_nat
% 0.17/0.41 = ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.commute
% 0.17/0.41 thf(fact_794_mult_Ocommute,axiom,
% 0.17/0.41 ( times_times_int
% 0.17/0.41 = ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.commute
% 0.17/0.41 thf(fact_795_mult_Oassoc,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
% 0.17/0.41 = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.assoc
% 0.17/0.41 thf(fact_796_mult_Oassoc,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 0.17/0.41 = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.assoc
% 0.17/0.41 thf(fact_797_mult_Oassoc,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.17/0.41 = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.assoc
% 0.17/0.41 thf(fact_798_mult_Oassoc,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.17/0.41 = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % mult.assoc
% 0.17/0.41 thf(fact_799_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
% 0.17/0.41 = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ab_semigroup_mult_class.mult_ac(1)
% 0.17/0.41 thf(fact_800_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 0.17/0.41 = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ab_semigroup_mult_class.mult_ac(1)
% 0.17/0.41 thf(fact_801_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.41 ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.17/0.41 = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ab_semigroup_mult_class.mult_ac(1)
% 0.17/0.41 thf(fact_802_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.17/0.41 = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % ab_semigroup_mult_class.mult_ac(1)
% 0.17/0.41 thf(fact_803_one__reorient,axiom,
% 0.17/0.41 ! [X4: complex] :
% 0.17/0.41 ( ( one_one_complex = X4 )
% 0.17/0.41 = ( X4 = one_one_complex ) ) ).
% 0.17/0.41
% 0.17/0.41 % one_reorient
% 0.17/0.41 thf(fact_804_one__reorient,axiom,
% 0.17/0.41 ! [X4: real] :
% 0.17/0.41 ( ( one_one_real = X4 )
% 0.17/0.41 = ( X4 = one_one_real ) ) ).
% 0.17/0.41
% 0.17/0.41 % one_reorient
% 0.17/0.41 thf(fact_805_one__reorient,axiom,
% 0.17/0.41 ! [X4: rat] :
% 0.17/0.41 ( ( one_one_rat = X4 )
% 0.17/0.41 = ( X4 = one_one_rat ) ) ).
% 0.17/0.41
% 0.17/0.41 % one_reorient
% 0.17/0.41 thf(fact_806_one__reorient,axiom,
% 0.17/0.41 ! [X4: nat] :
% 0.17/0.41 ( ( one_one_nat = X4 )
% 0.17/0.41 = ( X4 = one_one_nat ) ) ).
% 0.17/0.41
% 0.17/0.41 % one_reorient
% 0.17/0.41 thf(fact_807_one__reorient,axiom,
% 0.17/0.41 ! [X4: int] :
% 0.17/0.41 ( ( one_one_int = X4 )
% 0.17/0.41 = ( X4 = one_one_int ) ) ).
% 0.17/0.41
% 0.17/0.41 % one_reorient
% 0.17/0.41 thf(fact_808_add__right__imp__eq,axiom,
% 0.17/0.41 ! [B2: real,A: real,C: real] :
% 0.17/0.41 ( ( ( plus_plus_real @ B2 @ A )
% 0.17/0.41 = ( plus_plus_real @ C @ A ) )
% 0.17/0.41 => ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_right_imp_eq
% 0.17/0.41 thf(fact_809_add__right__imp__eq,axiom,
% 0.17/0.41 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.41 ( ( ( plus_plus_rat @ B2 @ A )
% 0.17/0.41 = ( plus_plus_rat @ C @ A ) )
% 0.17/0.41 => ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_right_imp_eq
% 0.17/0.41 thf(fact_810_add__right__imp__eq,axiom,
% 0.17/0.41 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.41 ( ( ( plus_plus_nat @ B2 @ A )
% 0.17/0.41 = ( plus_plus_nat @ C @ A ) )
% 0.17/0.41 => ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_right_imp_eq
% 0.17/0.41 thf(fact_811_add__right__imp__eq,axiom,
% 0.17/0.41 ! [B2: int,A: int,C: int] :
% 0.17/0.41 ( ( ( plus_plus_int @ B2 @ A )
% 0.17/0.41 = ( plus_plus_int @ C @ A ) )
% 0.17/0.41 => ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_right_imp_eq
% 0.17/0.41 thf(fact_812_add__left__imp__eq,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( ( plus_plus_real @ A @ B2 )
% 0.17/0.41 = ( plus_plus_real @ A @ C ) )
% 0.17/0.41 => ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_left_imp_eq
% 0.17/0.41 thf(fact_813_add__left__imp__eq,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( ( plus_plus_rat @ A @ B2 )
% 0.17/0.41 = ( plus_plus_rat @ A @ C ) )
% 0.17/0.41 => ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_left_imp_eq
% 0.17/0.41 thf(fact_814_add__left__imp__eq,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.41 ( ( ( plus_plus_nat @ A @ B2 )
% 0.17/0.41 = ( plus_plus_nat @ A @ C ) )
% 0.17/0.41 => ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_left_imp_eq
% 0.17/0.41 thf(fact_815_add__left__imp__eq,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( ( plus_plus_int @ A @ B2 )
% 0.17/0.41 = ( plus_plus_int @ A @ C ) )
% 0.17/0.41 => ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add_left_imp_eq
% 0.17/0.41 thf(fact_816_add_Oleft__commute,axiom,
% 0.17/0.41 ! [B2: real,A: real,C: real] :
% 0.17/0.41 ( ( plus_plus_real @ B2 @ ( plus_plus_real @ A @ C ) )
% 0.17/0.41 = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.left_commute
% 0.17/0.41 thf(fact_817_add_Oleft__commute,axiom,
% 0.17/0.41 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.41 ( ( plus_plus_rat @ B2 @ ( plus_plus_rat @ A @ C ) )
% 0.17/0.41 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.left_commute
% 0.17/0.41 thf(fact_818_add_Oleft__commute,axiom,
% 0.17/0.41 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.41 ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
% 0.17/0.41 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.left_commute
% 0.17/0.41 thf(fact_819_add_Oleft__commute,axiom,
% 0.17/0.41 ! [B2: int,A: int,C: int] :
% 0.17/0.41 ( ( plus_plus_int @ B2 @ ( plus_plus_int @ A @ C ) )
% 0.17/0.41 = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.left_commute
% 0.17/0.41 thf(fact_820_add_Ocommute,axiom,
% 0.17/0.41 ( plus_plus_real
% 0.17/0.41 = ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.commute
% 0.17/0.41 thf(fact_821_add_Ocommute,axiom,
% 0.17/0.41 ( plus_plus_rat
% 0.17/0.41 = ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ B3 @ A4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.commute
% 0.17/0.41 thf(fact_822_add_Ocommute,axiom,
% 0.17/0.41 ( plus_plus_nat
% 0.17/0.41 = ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.commute
% 0.17/0.41 thf(fact_823_add_Ocommute,axiom,
% 0.17/0.41 ( plus_plus_int
% 0.17/0.41 = ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.commute
% 0.17/0.41 thf(fact_824_add_Oright__cancel,axiom,
% 0.17/0.41 ! [B2: real,A: real,C: real] :
% 0.17/0.41 ( ( ( plus_plus_real @ B2 @ A )
% 0.17/0.41 = ( plus_plus_real @ C @ A ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.right_cancel
% 0.17/0.41 thf(fact_825_add_Oright__cancel,axiom,
% 0.17/0.41 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.41 ( ( ( plus_plus_rat @ B2 @ A )
% 0.17/0.41 = ( plus_plus_rat @ C @ A ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.right_cancel
% 0.17/0.41 thf(fact_826_add_Oright__cancel,axiom,
% 0.17/0.41 ! [B2: int,A: int,C: int] :
% 0.17/0.41 ( ( ( plus_plus_int @ B2 @ A )
% 0.17/0.41 = ( plus_plus_int @ C @ A ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.right_cancel
% 0.17/0.41 thf(fact_827_add_Oleft__cancel,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( ( plus_plus_real @ A @ B2 )
% 0.17/0.41 = ( plus_plus_real @ A @ C ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.left_cancel
% 0.17/0.41 thf(fact_828_add_Oleft__cancel,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( ( plus_plus_rat @ A @ B2 )
% 0.17/0.41 = ( plus_plus_rat @ A @ C ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.left_cancel
% 0.17/0.41 thf(fact_829_add_Oleft__cancel,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( ( plus_plus_int @ A @ B2 )
% 0.17/0.41 = ( plus_plus_int @ A @ C ) )
% 0.17/0.41 = ( B2 = C ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.left_cancel
% 0.17/0.41 thf(fact_830_add_Oassoc,axiom,
% 0.17/0.41 ! [A: real,B2: real,C: real] :
% 0.17/0.41 ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.assoc
% 0.17/0.41 thf(fact_831_add_Oassoc,axiom,
% 0.17/0.41 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.41 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.assoc
% 0.17/0.41 thf(fact_832_add_Oassoc,axiom,
% 0.17/0.41 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.41 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.assoc
% 0.17/0.41 thf(fact_833_add_Oassoc,axiom,
% 0.17/0.41 ! [A: int,B2: int,C: int] :
% 0.17/0.41 ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.17/0.41 = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % add.assoc
% 0.17/0.41 thf(fact_834_group__cancel_Oadd2,axiom,
% 0.17/0.41 ! [B4: real,K: real,B2: real,A: real] :
% 0.17/0.41 ( ( B4
% 0.17/0.41 = ( plus_plus_real @ K @ B2 ) )
% 0.17/0.41 => ( ( plus_plus_real @ A @ B4 )
% 0.17/0.41 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % group_cancel.add2
% 0.17/0.41 thf(fact_835_group__cancel_Oadd2,axiom,
% 0.17/0.41 ! [B4: rat,K: rat,B2: rat,A: rat] :
% 0.17/0.41 ( ( B4
% 0.17/0.41 = ( plus_plus_rat @ K @ B2 ) )
% 0.17/0.41 => ( ( plus_plus_rat @ A @ B4 )
% 0.17/0.41 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % group_cancel.add2
% 0.17/0.41 thf(fact_836_group__cancel_Oadd2,axiom,
% 0.17/0.41 ! [B4: nat,K: nat,B2: nat,A: nat] :
% 0.17/0.41 ( ( B4
% 0.17/0.41 = ( plus_plus_nat @ K @ B2 ) )
% 0.17/0.41 => ( ( plus_plus_nat @ A @ B4 )
% 0.17/0.41 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % group_cancel.add2
% 0.17/0.41 thf(fact_837_group__cancel_Oadd2,axiom,
% 0.17/0.41 ! [B4: int,K: int,B2: int,A: int] :
% 0.17/0.41 ( ( B4
% 0.17/0.41 = ( plus_plus_int @ K @ B2 ) )
% 0.17/0.41 => ( ( plus_plus_int @ A @ B4 )
% 0.17/0.41 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % group_cancel.add2
% 0.17/0.41 thf(fact_838_group__cancel_Oadd1,axiom,
% 0.17/0.41 ! [A3: real,K: real,A: real,B2: real] :
% 0.17/0.41 ( ( A3
% 0.17/0.41 = ( plus_plus_real @ K @ A ) )
% 0.17/0.41 => ( ( plus_plus_real @ A3 @ B2 )
% 0.17/0.41 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % group_cancel.add1
% 0.17/0.41 thf(fact_839_group__cancel_Oadd1,axiom,
% 0.17/0.41 ! [A3: rat,K: rat,A: rat,B2: rat] :
% 0.17/0.41 ( ( A3
% 0.17/0.41 = ( plus_plus_rat @ K @ A ) )
% 0.17/0.41 => ( ( plus_plus_rat @ A3 @ B2 )
% 0.17/0.41 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.41
% 0.17/0.41 % group_cancel.add1
% 0.17/0.41 thf(fact_840_group__cancel_Oadd1,axiom,
% 0.17/0.41 ! [A3: nat,K: nat,A: nat,B2: nat] :
% 0.17/0.41 ( ( A3
% 0.17/0.41 = ( plus_plus_nat @ K @ A ) )
% 0.17/0.41 => ( ( plus_plus_nat @ A3 @ B2 )
% 0.17/0.41 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % group_cancel.add1
% 0.17/0.42 thf(fact_841_group__cancel_Oadd1,axiom,
% 0.17/0.42 ! [A3: int,K: int,A: int,B2: int] :
% 0.17/0.42 ( ( A3
% 0.17/0.42 = ( plus_plus_int @ K @ A ) )
% 0.17/0.42 => ( ( plus_plus_int @ A3 @ B2 )
% 0.17/0.42 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % group_cancel.add1
% 0.17/0.42 thf(fact_842_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.17/0.42 ! [I: real,J: real,K: real,L2: real] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ( plus_plus_real @ I @ K )
% 0.17/0.42 = ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(4)
% 0.17/0.42 thf(fact_843_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.17/0.42 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ( plus_plus_rat @ I @ K )
% 0.17/0.42 = ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(4)
% 0.17/0.42 thf(fact_844_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.17/0.42 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ( plus_plus_nat @ I @ K )
% 0.17/0.42 = ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(4)
% 0.17/0.42 thf(fact_845_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.17/0.42 ! [I: int,J: int,K: int,L2: int] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ( plus_plus_int @ I @ K )
% 0.17/0.42 = ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(4)
% 0.17/0.42 thf(fact_846_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % ab_semigroup_add_class.add_ac(1)
% 0.17/0.42 thf(fact_847_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % ab_semigroup_add_class.add_ac(1)
% 0.17/0.42 thf(fact_848_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % ab_semigroup_add_class.add_ac(1)
% 0.17/0.42 thf(fact_849_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % ab_semigroup_add_class.add_ac(1)
% 0.17/0.42 thf(fact_850_diff__right__commute,axiom,
% 0.17/0.42 ! [A: real,C: real,B2: real] :
% 0.17/0.42 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 )
% 0.17/0.42 = ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_right_commute
% 0.17/0.42 thf(fact_851_diff__right__commute,axiom,
% 0.17/0.42 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.42 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 )
% 0.17/0.42 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_right_commute
% 0.17/0.42 thf(fact_852_diff__right__commute,axiom,
% 0.17/0.42 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.42 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
% 0.17/0.42 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_right_commute
% 0.17/0.42 thf(fact_853_diff__right__commute,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 )
% 0.17/0.42 = ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_right_commute
% 0.17/0.42 thf(fact_854_diff__eq__diff__eq,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.42 ( ( ( minus_minus_real @ A @ B2 )
% 0.17/0.42 = ( minus_minus_real @ C @ D ) )
% 0.17/0.42 => ( ( A = B2 )
% 0.17/0.42 = ( C = D ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_diff_eq
% 0.17/0.42 thf(fact_855_diff__eq__diff__eq,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.42 ( ( ( minus_minus_rat @ A @ B2 )
% 0.17/0.42 = ( minus_minus_rat @ C @ D ) )
% 0.17/0.42 => ( ( A = B2 )
% 0.17/0.42 = ( C = D ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_diff_eq
% 0.17/0.42 thf(fact_856_diff__eq__diff__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.42 ( ( ( minus_minus_int @ A @ B2 )
% 0.17/0.42 = ( minus_minus_int @ C @ D ) )
% 0.17/0.42 => ( ( A = B2 )
% 0.17/0.42 = ( C = D ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_diff_eq
% 0.17/0.42 thf(fact_857_add__le__imp__le__right,axiom,
% 0.17/0.42 ! [A: real,C: real,B2: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 0.17/0.42 => ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_imp_le_right
% 0.17/0.42 thf(fact_858_add__le__imp__le__right,axiom,
% 0.17/0.42 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.17/0.42 => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_imp_le_right
% 0.17/0.42 thf(fact_859_add__le__imp__le__right,axiom,
% 0.17/0.42 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.17/0.42 => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_imp_le_right
% 0.17/0.42 thf(fact_860_add__le__imp__le__right,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 0.17/0.42 => ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_imp_le_right
% 0.17/0.42 thf(fact_861_add__le__imp__le__left,axiom,
% 0.17/0.42 ! [C: real,A: real,B2: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 0.17/0.42 => ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_imp_le_left
% 0.17/0.42 thf(fact_862_add__le__imp__le__left,axiom,
% 0.17/0.42 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 0.17/0.42 => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_imp_le_left
% 0.17/0.42 thf(fact_863_add__le__imp__le__left,axiom,
% 0.17/0.42 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 0.17/0.42 => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_imp_le_left
% 0.17/0.42 thf(fact_864_add__le__imp__le__left,axiom,
% 0.17/0.42 ! [C: int,A: int,B2: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 0.17/0.42 => ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_imp_le_left
% 0.17/0.42 thf(fact_865_le__iff__add,axiom,
% 0.17/0.42 ( ord_less_eq_nat
% 0.17/0.42 = ( ^ [A4: nat,B3: nat] :
% 0.17/0.42 ? [C2: nat] :
% 0.17/0.42 ( B3
% 0.17/0.42 = ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_iff_add
% 0.17/0.42 thf(fact_866_add__right__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_right_mono
% 0.17/0.42 thf(fact_867_add__right__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_right_mono
% 0.17/0.42 thf(fact_868_add__right__mono,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_right_mono
% 0.17/0.42 thf(fact_869_add__right__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_right_mono
% 0.17/0.42 thf(fact_870_less__eqE,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ~ ! [C3: nat] :
% 0.17/0.42 ( B2
% 0.17/0.42 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % less_eqE
% 0.17/0.42 thf(fact_871_add__left__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_left_mono
% 0.17/0.42 thf(fact_872_add__left__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_left_mono
% 0.17/0.42 thf(fact_873_add__left__mono,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_left_mono
% 0.17/0.42 thf(fact_874_add__left__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_left_mono
% 0.17/0.42 thf(fact_875_add__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_real @ C @ D )
% 0.17/0.42 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono
% 0.17/0.42 thf(fact_876_add__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_rat @ C @ D )
% 0.17/0.42 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono
% 0.17/0.42 thf(fact_877_add__mono,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_nat @ C @ D )
% 0.17/0.42 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono
% 0.17/0.42 thf(fact_878_add__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_int @ C @ D )
% 0.17/0.42 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono
% 0.17/0.42 thf(fact_879_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.17/0.42 ! [I: real,J: real,K: real,L2: real] :
% 0.17/0.42 ( ( ( ord_less_eq_real @ I @ J )
% 0.17/0.42 & ( ord_less_eq_real @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(1)
% 0.17/0.42 thf(fact_880_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.17/0.42 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.17/0.42 ( ( ( ord_less_eq_rat @ I @ J )
% 0.17/0.42 & ( ord_less_eq_rat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(1)
% 0.17/0.42 thf(fact_881_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.17/0.42 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.42 ( ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.42 & ( ord_less_eq_nat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(1)
% 0.17/0.42 thf(fact_882_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.17/0.42 ! [I: int,J: int,K: int,L2: int] :
% 0.17/0.42 ( ( ( ord_less_eq_int @ I @ J )
% 0.17/0.42 & ( ord_less_eq_int @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(1)
% 0.17/0.42 thf(fact_883_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.17/0.42 ! [I: real,J: real,K: real,L2: real] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( ord_less_eq_real @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(2)
% 0.17/0.42 thf(fact_884_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.17/0.42 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( ord_less_eq_rat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(2)
% 0.17/0.42 thf(fact_885_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.17/0.42 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( ord_less_eq_nat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(2)
% 0.17/0.42 thf(fact_886_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.17/0.42 ! [I: int,J: int,K: int,L2: int] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( ord_less_eq_int @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(2)
% 0.17/0.42 thf(fact_887_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.17/0.42 ! [I: real,J: real,K: real,L2: real] :
% 0.17/0.42 ( ( ( ord_less_eq_real @ I @ J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(3)
% 0.17/0.42 thf(fact_888_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.17/0.42 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.17/0.42 ( ( ( ord_less_eq_rat @ I @ J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(3)
% 0.17/0.42 thf(fact_889_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.17/0.42 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.42 ( ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(3)
% 0.17/0.42 thf(fact_890_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.17/0.42 ! [I: int,J: int,K: int,L2: int] :
% 0.17/0.42 ( ( ( ord_less_eq_int @ I @ J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_semiring(3)
% 0.17/0.42 thf(fact_891_diff__eq__diff__less__eq,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.42 ( ( ( minus_minus_real @ A @ B2 )
% 0.17/0.42 = ( minus_minus_real @ C @ D ) )
% 0.17/0.42 => ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.42 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_diff_less_eq
% 0.17/0.42 thf(fact_892_diff__eq__diff__less__eq,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.42 ( ( ( minus_minus_rat @ A @ B2 )
% 0.17/0.42 = ( minus_minus_rat @ C @ D ) )
% 0.17/0.42 => ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.42 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_diff_less_eq
% 0.17/0.42 thf(fact_893_diff__eq__diff__less__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.42 ( ( ( minus_minus_int @ A @ B2 )
% 0.17/0.42 = ( minus_minus_int @ C @ D ) )
% 0.17/0.42 => ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.42 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_diff_less_eq
% 0.17/0.42 thf(fact_894_diff__right__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_right_mono
% 0.17/0.42 thf(fact_895_diff__right__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_right_mono
% 0.17/0.42 thf(fact_896_diff__right__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_right_mono
% 0.17/0.42 thf(fact_897_diff__left__mono,axiom,
% 0.17/0.42 ! [B2: real,A: real,C: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ B2 @ A )
% 0.17/0.42 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_left_mono
% 0.17/0.42 thf(fact_898_diff__left__mono,axiom,
% 0.17/0.42 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ B2 @ A )
% 0.17/0.42 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_left_mono
% 0.17/0.42 thf(fact_899_diff__left__mono,axiom,
% 0.17/0.42 ! [B2: int,A: int,C: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ B2 @ A )
% 0.17/0.42 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_left_mono
% 0.17/0.42 thf(fact_900_diff__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,D: real,C: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_real @ D @ C )
% 0.17/0.42 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_mono
% 0.17/0.42 thf(fact_901_diff__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,D: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_rat @ D @ C )
% 0.17/0.42 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_mono
% 0.17/0.42 thf(fact_902_diff__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,D: int,C: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_int @ D @ C )
% 0.17/0.42 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_mono
% 0.17/0.42 thf(fact_903_add__less__imp__less__right,axiom,
% 0.17/0.42 ! [A: real,C: real,B2: real] :
% 0.17/0.42 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 0.17/0.42 => ( ord_less_real @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_imp_less_right
% 0.17/0.42 thf(fact_904_add__less__imp__less__right,axiom,
% 0.17/0.42 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.42 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.17/0.42 => ( ord_less_rat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_imp_less_right
% 0.17/0.42 thf(fact_905_add__less__imp__less__right,axiom,
% 0.17/0.42 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.42 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.17/0.42 => ( ord_less_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_imp_less_right
% 0.17/0.42 thf(fact_906_add__less__imp__less__right,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 0.17/0.42 => ( ord_less_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_imp_less_right
% 0.17/0.42 thf(fact_907_add__less__imp__less__left,axiom,
% 0.17/0.42 ! [C: real,A: real,B2: real] :
% 0.17/0.42 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 0.17/0.42 => ( ord_less_real @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_imp_less_left
% 0.17/0.42 thf(fact_908_add__less__imp__less__left,axiom,
% 0.17/0.42 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.42 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 0.17/0.42 => ( ord_less_rat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_imp_less_left
% 0.17/0.42 thf(fact_909_add__less__imp__less__left,axiom,
% 0.17/0.42 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.42 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 0.17/0.42 => ( ord_less_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_imp_less_left
% 0.17/0.42 thf(fact_910_add__less__imp__less__left,axiom,
% 0.17/0.42 ! [C: int,A: int,B2: int] :
% 0.17/0.42 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 0.17/0.42 => ( ord_less_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_imp_less_left
% 0.17/0.42 thf(fact_911_add__strict__right__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( ord_less_real @ A @ B2 )
% 0.17/0.42 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_right_mono
% 0.17/0.42 thf(fact_912_add__strict__right__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.42 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_right_mono
% 0.17/0.42 thf(fact_913_add__strict__right__mono,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.42 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_right_mono
% 0.17/0.42 thf(fact_914_add__strict__right__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( ord_less_int @ A @ B2 )
% 0.17/0.42 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_right_mono
% 0.17/0.42 thf(fact_915_add__strict__left__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( ord_less_real @ A @ B2 )
% 0.17/0.42 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_left_mono
% 0.17/0.42 thf(fact_916_add__strict__left__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.42 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_left_mono
% 0.17/0.42 thf(fact_917_add__strict__left__mono,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.42 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_left_mono
% 0.17/0.42 thf(fact_918_add__strict__left__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( ord_less_int @ A @ B2 )
% 0.17/0.42 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_left_mono
% 0.17/0.42 thf(fact_919_add__strict__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.42 ( ( ord_less_real @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_real @ C @ D )
% 0.17/0.42 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_mono
% 0.17/0.42 thf(fact_920_add__strict__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.42 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_rat @ C @ D )
% 0.17/0.42 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_mono
% 0.17/0.42 thf(fact_921_add__strict__mono,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.17/0.42 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_nat @ C @ D )
% 0.17/0.42 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_mono
% 0.17/0.42 thf(fact_922_add__strict__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.42 ( ( ord_less_int @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_int @ C @ D )
% 0.17/0.42 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_strict_mono
% 0.17/0.42 thf(fact_923_add__mono__thms__linordered__field_I1_J,axiom,
% 0.17/0.42 ! [I: real,J: real,K: real,L2: real] :
% 0.17/0.42 ( ( ( ord_less_real @ I @ J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(1)
% 0.17/0.42 thf(fact_924_add__mono__thms__linordered__field_I1_J,axiom,
% 0.17/0.42 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.17/0.42 ( ( ( ord_less_rat @ I @ J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(1)
% 0.17/0.42 thf(fact_925_add__mono__thms__linordered__field_I1_J,axiom,
% 0.17/0.42 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.42 ( ( ( ord_less_nat @ I @ J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(1)
% 0.17/0.42 thf(fact_926_add__mono__thms__linordered__field_I1_J,axiom,
% 0.17/0.42 ! [I: int,J: int,K: int,L2: int] :
% 0.17/0.42 ( ( ( ord_less_int @ I @ J )
% 0.17/0.42 & ( K = L2 ) )
% 0.17/0.42 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(1)
% 0.17/0.42 thf(fact_927_add__mono__thms__linordered__field_I2_J,axiom,
% 0.17/0.42 ! [I: real,J: real,K: real,L2: real] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( ord_less_real @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(2)
% 0.17/0.42 thf(fact_928_add__mono__thms__linordered__field_I2_J,axiom,
% 0.17/0.42 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( ord_less_rat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(2)
% 0.17/0.42 thf(fact_929_add__mono__thms__linordered__field_I2_J,axiom,
% 0.17/0.42 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( ord_less_nat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(2)
% 0.17/0.42 thf(fact_930_add__mono__thms__linordered__field_I2_J,axiom,
% 0.17/0.42 ! [I: int,J: int,K: int,L2: int] :
% 0.17/0.42 ( ( ( I = J )
% 0.17/0.42 & ( ord_less_int @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(2)
% 0.17/0.42 thf(fact_931_add__mono__thms__linordered__field_I5_J,axiom,
% 0.17/0.42 ! [I: real,J: real,K: real,L2: real] :
% 0.17/0.42 ( ( ( ord_less_real @ I @ J )
% 0.17/0.42 & ( ord_less_real @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(5)
% 0.17/0.42 thf(fact_932_add__mono__thms__linordered__field_I5_J,axiom,
% 0.17/0.42 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.17/0.42 ( ( ( ord_less_rat @ I @ J )
% 0.17/0.42 & ( ord_less_rat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(5)
% 0.17/0.42 thf(fact_933_add__mono__thms__linordered__field_I5_J,axiom,
% 0.17/0.42 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.42 ( ( ( ord_less_nat @ I @ J )
% 0.17/0.42 & ( ord_less_nat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(5)
% 0.17/0.42 thf(fact_934_add__mono__thms__linordered__field_I5_J,axiom,
% 0.17/0.42 ! [I: int,J: int,K: int,L2: int] :
% 0.17/0.42 ( ( ( ord_less_int @ I @ J )
% 0.17/0.42 & ( ord_less_int @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(5)
% 0.17/0.42 thf(fact_935_diff__strict__right__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( ord_less_real @ A @ B2 )
% 0.17/0.42 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_strict_right_mono
% 0.17/0.42 thf(fact_936_diff__strict__right__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.42 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_strict_right_mono
% 0.17/0.42 thf(fact_937_diff__strict__right__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( ord_less_int @ A @ B2 )
% 0.17/0.42 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_strict_right_mono
% 0.17/0.42 thf(fact_938_diff__strict__left__mono,axiom,
% 0.17/0.42 ! [B2: real,A: real,C: real] :
% 0.17/0.42 ( ( ord_less_real @ B2 @ A )
% 0.17/0.42 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_strict_left_mono
% 0.17/0.42 thf(fact_939_diff__strict__left__mono,axiom,
% 0.17/0.42 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_rat @ B2 @ A )
% 0.17/0.42 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_strict_left_mono
% 0.17/0.42 thf(fact_940_diff__strict__left__mono,axiom,
% 0.17/0.42 ! [B2: int,A: int,C: int] :
% 0.17/0.42 ( ( ord_less_int @ B2 @ A )
% 0.17/0.42 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_strict_left_mono
% 0.17/0.42 thf(fact_941_diff__eq__diff__less,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.42 ( ( ( minus_minus_real @ A @ B2 )
% 0.17/0.42 = ( minus_minus_real @ C @ D ) )
% 0.17/0.42 => ( ( ord_less_real @ A @ B2 )
% 0.17/0.42 = ( ord_less_real @ C @ D ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_diff_less
% 0.17/0.42 thf(fact_942_diff__eq__diff__less,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.42 ( ( ( minus_minus_rat @ A @ B2 )
% 0.17/0.42 = ( minus_minus_rat @ C @ D ) )
% 0.17/0.42 => ( ( ord_less_rat @ A @ B2 )
% 0.17/0.42 = ( ord_less_rat @ C @ D ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_diff_less
% 0.17/0.42 thf(fact_943_diff__eq__diff__less,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.42 ( ( ( minus_minus_int @ A @ B2 )
% 0.17/0.42 = ( minus_minus_int @ C @ D ) )
% 0.17/0.42 => ( ( ord_less_int @ A @ B2 )
% 0.17/0.42 = ( ord_less_int @ C @ D ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_diff_less
% 0.17/0.42 thf(fact_944_diff__strict__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,D: real,C: real] :
% 0.17/0.42 ( ( ord_less_real @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_real @ D @ C )
% 0.17/0.42 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_strict_mono
% 0.17/0.42 thf(fact_945_diff__strict__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,D: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_rat @ D @ C )
% 0.17/0.42 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_strict_mono
% 0.17/0.42 thf(fact_946_diff__strict__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,D: int,C: int] :
% 0.17/0.42 ( ( ord_less_int @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_int @ D @ C )
% 0.17/0.42 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_strict_mono
% 0.17/0.42 thf(fact_947_mult_Ocomm__neutral,axiom,
% 0.17/0.42 ! [A: complex] :
% 0.17/0.42 ( ( times_times_complex @ A @ one_one_complex )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mult.comm_neutral
% 0.17/0.42 thf(fact_948_mult_Ocomm__neutral,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( times_times_real @ A @ one_one_real )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mult.comm_neutral
% 0.17/0.42 thf(fact_949_mult_Ocomm__neutral,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( times_times_rat @ A @ one_one_rat )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mult.comm_neutral
% 0.17/0.42 thf(fact_950_mult_Ocomm__neutral,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( times_times_nat @ A @ one_one_nat )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mult.comm_neutral
% 0.17/0.42 thf(fact_951_mult_Ocomm__neutral,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( times_times_int @ A @ one_one_int )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mult.comm_neutral
% 0.17/0.42 thf(fact_952_comm__monoid__mult__class_Omult__1,axiom,
% 0.17/0.42 ! [A: complex] :
% 0.17/0.42 ( ( times_times_complex @ one_one_complex @ A )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % comm_monoid_mult_class.mult_1
% 0.17/0.42 thf(fact_953_comm__monoid__mult__class_Omult__1,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( times_times_real @ one_one_real @ A )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % comm_monoid_mult_class.mult_1
% 0.17/0.42 thf(fact_954_comm__monoid__mult__class_Omult__1,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( times_times_rat @ one_one_rat @ A )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % comm_monoid_mult_class.mult_1
% 0.17/0.42 thf(fact_955_comm__monoid__mult__class_Omult__1,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( times_times_nat @ one_one_nat @ A )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % comm_monoid_mult_class.mult_1
% 0.17/0.42 thf(fact_956_comm__monoid__mult__class_Omult__1,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( times_times_int @ one_one_int @ A )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % comm_monoid_mult_class.mult_1
% 0.17/0.42 thf(fact_957_diff__diff__eq,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.17/0.42 = ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_diff_eq
% 0.17/0.42 thf(fact_958_diff__diff__eq,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.17/0.42 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_diff_eq
% 0.17/0.42 thf(fact_959_diff__diff__eq,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
% 0.17/0.42 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_diff_eq
% 0.17/0.42 thf(fact_960_diff__diff__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.17/0.42 = ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_diff_eq
% 0.17/0.42 thf(fact_961_add__implies__diff,axiom,
% 0.17/0.42 ! [C: real,B2: real,A: real] :
% 0.17/0.42 ( ( ( plus_plus_real @ C @ B2 )
% 0.17/0.42 = A )
% 0.17/0.42 => ( C
% 0.17/0.42 = ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_implies_diff
% 0.17/0.42 thf(fact_962_add__implies__diff,axiom,
% 0.17/0.42 ! [C: rat,B2: rat,A: rat] :
% 0.17/0.42 ( ( ( plus_plus_rat @ C @ B2 )
% 0.17/0.42 = A )
% 0.17/0.42 => ( C
% 0.17/0.42 = ( minus_minus_rat @ A @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_implies_diff
% 0.17/0.42 thf(fact_963_add__implies__diff,axiom,
% 0.17/0.42 ! [C: nat,B2: nat,A: nat] :
% 0.17/0.42 ( ( ( plus_plus_nat @ C @ B2 )
% 0.17/0.42 = A )
% 0.17/0.42 => ( C
% 0.17/0.42 = ( minus_minus_nat @ A @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_implies_diff
% 0.17/0.42 thf(fact_964_add__implies__diff,axiom,
% 0.17/0.42 ! [C: int,B2: int,A: int] :
% 0.17/0.42 ( ( ( plus_plus_int @ C @ B2 )
% 0.17/0.42 = A )
% 0.17/0.42 => ( C
% 0.17/0.42 = ( minus_minus_int @ A @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_implies_diff
% 0.17/0.42 thf(fact_965_diff__add__eq__diff__diff__swap,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 0.17/0.42 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_add_eq_diff_diff_swap
% 0.17/0.42 thf(fact_966_diff__add__eq__diff__diff__swap,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 0.17/0.42 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_add_eq_diff_diff_swap
% 0.17/0.42 thf(fact_967_diff__add__eq__diff__diff__swap,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 0.17/0.42 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_add_eq_diff_diff_swap
% 0.17/0.42 thf(fact_968_diff__add__eq,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.17/0.42 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_add_eq
% 0.17/0.42 thf(fact_969_diff__add__eq,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.17/0.42 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_add_eq
% 0.17/0.42 thf(fact_970_diff__add__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.17/0.42 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_add_eq
% 0.17/0.42 thf(fact_971_diff__diff__eq2,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( minus_minus_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 0.17/0.42 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_diff_eq2
% 0.17/0.42 thf(fact_972_diff__diff__eq2,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 0.17/0.42 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_diff_eq2
% 0.17/0.42 thf(fact_973_diff__diff__eq2,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( minus_minus_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 0.17/0.42 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_diff_eq2
% 0.17/0.42 thf(fact_974_add__diff__eq,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( plus_plus_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 0.17/0.42 = ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_diff_eq
% 0.17/0.42 thf(fact_975_add__diff__eq,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 0.17/0.42 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_diff_eq
% 0.17/0.42 thf(fact_976_add__diff__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( plus_plus_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 0.17/0.42 = ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_diff_eq
% 0.17/0.42 thf(fact_977_eq__diff__eq,axiom,
% 0.17/0.42 ! [A: real,C: real,B2: real] :
% 0.17/0.42 ( ( A
% 0.17/0.42 = ( minus_minus_real @ C @ B2 ) )
% 0.17/0.42 = ( ( plus_plus_real @ A @ B2 )
% 0.17/0.42 = C ) ) ).
% 0.17/0.42
% 0.17/0.42 % eq_diff_eq
% 0.17/0.42 thf(fact_978_eq__diff__eq,axiom,
% 0.17/0.42 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.42 ( ( A
% 0.17/0.42 = ( minus_minus_rat @ C @ B2 ) )
% 0.17/0.42 = ( ( plus_plus_rat @ A @ B2 )
% 0.17/0.42 = C ) ) ).
% 0.17/0.42
% 0.17/0.42 % eq_diff_eq
% 0.17/0.42 thf(fact_979_eq__diff__eq,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( A
% 0.17/0.42 = ( minus_minus_int @ C @ B2 ) )
% 0.17/0.42 = ( ( plus_plus_int @ A @ B2 )
% 0.17/0.42 = C ) ) ).
% 0.17/0.42
% 0.17/0.42 % eq_diff_eq
% 0.17/0.42 thf(fact_980_diff__eq__eq,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( ( minus_minus_real @ A @ B2 )
% 0.17/0.42 = C )
% 0.17/0.42 = ( A
% 0.17/0.42 = ( plus_plus_real @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_eq
% 0.17/0.42 thf(fact_981_diff__eq__eq,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( ( minus_minus_rat @ A @ B2 )
% 0.17/0.42 = C )
% 0.17/0.42 = ( A
% 0.17/0.42 = ( plus_plus_rat @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_eq
% 0.17/0.42 thf(fact_982_diff__eq__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( ( minus_minus_int @ A @ B2 )
% 0.17/0.42 = C )
% 0.17/0.42 = ( A
% 0.17/0.42 = ( plus_plus_int @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_eq_eq
% 0.17/0.42 thf(fact_983_group__cancel_Osub1,axiom,
% 0.17/0.42 ! [A3: real,K: real,A: real,B2: real] :
% 0.17/0.42 ( ( A3
% 0.17/0.42 = ( plus_plus_real @ K @ A ) )
% 0.17/0.42 => ( ( minus_minus_real @ A3 @ B2 )
% 0.17/0.42 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % group_cancel.sub1
% 0.17/0.42 thf(fact_984_group__cancel_Osub1,axiom,
% 0.17/0.42 ! [A3: rat,K: rat,A: rat,B2: rat] :
% 0.17/0.42 ( ( A3
% 0.17/0.42 = ( plus_plus_rat @ K @ A ) )
% 0.17/0.42 => ( ( minus_minus_rat @ A3 @ B2 )
% 0.17/0.42 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % group_cancel.sub1
% 0.17/0.42 thf(fact_985_group__cancel_Osub1,axiom,
% 0.17/0.42 ! [A3: int,K: int,A: int,B2: int] :
% 0.17/0.42 ( ( A3
% 0.17/0.42 = ( plus_plus_int @ K @ A ) )
% 0.17/0.42 => ( ( minus_minus_int @ A3 @ B2 )
% 0.17/0.42 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % group_cancel.sub1
% 0.17/0.42 thf(fact_986_add__less__le__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.42 ( ( ord_less_real @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_real @ C @ D )
% 0.17/0.42 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_le_mono
% 0.17/0.42 thf(fact_987_add__less__le__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.42 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_rat @ C @ D )
% 0.17/0.42 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_le_mono
% 0.17/0.42 thf(fact_988_add__less__le__mono,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.17/0.42 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_nat @ C @ D )
% 0.17/0.42 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_le_mono
% 0.17/0.42 thf(fact_989_add__less__le__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.42 ( ( ord_less_int @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_int @ C @ D )
% 0.17/0.42 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_less_le_mono
% 0.17/0.42 thf(fact_990_add__le__less__mono,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_real @ C @ D )
% 0.17/0.42 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_less_mono
% 0.17/0.42 thf(fact_991_add__le__less__mono,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_rat @ C @ D )
% 0.17/0.42 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_less_mono
% 0.17/0.42 thf(fact_992_add__le__less__mono,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_nat @ C @ D )
% 0.17/0.42 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_less_mono
% 0.17/0.42 thf(fact_993_add__le__less__mono,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_int @ C @ D )
% 0.17/0.42 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_less_mono
% 0.17/0.42 thf(fact_994_add__mono__thms__linordered__field_I3_J,axiom,
% 0.17/0.42 ! [I: real,J: real,K: real,L2: real] :
% 0.17/0.42 ( ( ( ord_less_real @ I @ J )
% 0.17/0.42 & ( ord_less_eq_real @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(3)
% 0.17/0.42 thf(fact_995_add__mono__thms__linordered__field_I3_J,axiom,
% 0.17/0.42 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.17/0.42 ( ( ( ord_less_rat @ I @ J )
% 0.17/0.42 & ( ord_less_eq_rat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(3)
% 0.17/0.42 thf(fact_996_add__mono__thms__linordered__field_I3_J,axiom,
% 0.17/0.42 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.42 ( ( ( ord_less_nat @ I @ J )
% 0.17/0.42 & ( ord_less_eq_nat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(3)
% 0.17/0.42 thf(fact_997_add__mono__thms__linordered__field_I3_J,axiom,
% 0.17/0.42 ! [I: int,J: int,K: int,L2: int] :
% 0.17/0.42 ( ( ( ord_less_int @ I @ J )
% 0.17/0.42 & ( ord_less_eq_int @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(3)
% 0.17/0.42 thf(fact_998_add__mono__thms__linordered__field_I4_J,axiom,
% 0.17/0.42 ! [I: real,J: real,K: real,L2: real] :
% 0.17/0.42 ( ( ( ord_less_eq_real @ I @ J )
% 0.17/0.42 & ( ord_less_real @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(4)
% 0.17/0.42 thf(fact_999_add__mono__thms__linordered__field_I4_J,axiom,
% 0.17/0.42 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.17/0.42 ( ( ( ord_less_eq_rat @ I @ J )
% 0.17/0.42 & ( ord_less_rat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(4)
% 0.17/0.42 thf(fact_1000_add__mono__thms__linordered__field_I4_J,axiom,
% 0.17/0.42 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.17/0.42 ( ( ( ord_less_eq_nat @ I @ J )
% 0.17/0.42 & ( ord_less_nat @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(4)
% 0.17/0.42 thf(fact_1001_add__mono__thms__linordered__field_I4_J,axiom,
% 0.17/0.42 ! [I: int,J: int,K: int,L2: int] :
% 0.17/0.42 ( ( ( ord_less_eq_int @ I @ J )
% 0.17/0.42 & ( ord_less_int @ K @ L2 ) )
% 0.17/0.42 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_mono_thms_linordered_field(4)
% 0.17/0.42 thf(fact_1002_diff__le__eq,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.17/0.42 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_le_eq
% 0.17/0.42 thf(fact_1003_diff__le__eq,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.17/0.42 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_le_eq
% 0.17/0.42 thf(fact_1004_diff__le__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.17/0.42 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_le_eq
% 0.17/0.42 thf(fact_1005_le__diff__eq,axiom,
% 0.17/0.42 ! [A: real,C: real,B2: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B2 ) )
% 0.17/0.42 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_diff_eq
% 0.17/0.42 thf(fact_1006_le__diff__eq,axiom,
% 0.17/0.42 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
% 0.17/0.42 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_diff_eq
% 0.17/0.42 thf(fact_1007_le__diff__eq,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 0.17/0.42 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_diff_eq
% 0.17/0.42 thf(fact_1008_diff__add,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
% 0.17/0.42 = B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_add
% 0.17/0.42 thf(fact_1009_le__add__diff,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_add_diff
% 0.17/0.42 thf(fact_1010_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 0.17/0.42 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.17/0.42 thf(fact_1011_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 0.17/0.42 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.17/0.42 thf(fact_1012_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
% 0.17/0.42 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.17/0.42 thf(fact_1013_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
% 0.17/0.42 = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.17/0.42 thf(fact_1014_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
% 0.17/0.42 = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.17/0.42 thf(fact_1015_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 0.17/0.42 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.17/0.42 thf(fact_1016_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
% 0.17/0.42 = B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.17/0.42 thf(fact_1017_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.42 => ( ( ( minus_minus_nat @ B2 @ A )
% 0.17/0.42 = C )
% 0.17/0.42 = ( B2
% 0.17/0.42 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.17/0.42 thf(fact_1018_diff__less__eq,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( ord_less_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.17/0.42 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_less_eq
% 0.17/0.42 thf(fact_1019_diff__less__eq,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.17/0.42 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_less_eq
% 0.17/0.42 thf(fact_1020_diff__less__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( ord_less_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.17/0.42 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_less_eq
% 0.17/0.42 thf(fact_1021_less__diff__eq,axiom,
% 0.17/0.42 ! [A: real,C: real,B2: real] :
% 0.17/0.42 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B2 ) )
% 0.17/0.42 = ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % less_diff_eq
% 0.17/0.42 thf(fact_1022_less__diff__eq,axiom,
% 0.17/0.42 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.42 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
% 0.17/0.42 = ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % less_diff_eq
% 0.17/0.42 thf(fact_1023_less__diff__eq,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 0.17/0.42 = ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % less_diff_eq
% 0.17/0.42 thf(fact_1024_real__average__minus__second,axiom,
% 0.17/0.42 ! [B2: real,A: real] :
% 0.17/0.42 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.17/0.42 = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % real_average_minus_second
% 0.17/0.42 thf(fact_1025_real__average__minus__first,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.17/0.42 = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % real_average_minus_first
% 0.17/0.42 thf(fact_1026_times__divide__eq__left,axiom,
% 0.17/0.42 ! [B2: complex,C: complex,A: complex] :
% 0.17/0.42 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B2 @ C ) @ A )
% 0.17/0.42 = ( divide1717551699836669952omplex @ ( times_times_complex @ B2 @ A ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % times_divide_eq_left
% 0.17/0.42 thf(fact_1027_times__divide__eq__left,axiom,
% 0.17/0.42 ! [B2: real,C: real,A: real] :
% 0.17/0.42 ( ( times_times_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.17/0.42 = ( divide_divide_real @ ( times_times_real @ B2 @ A ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % times_divide_eq_left
% 0.17/0.42 thf(fact_1028_times__divide__eq__left,axiom,
% 0.17/0.42 ! [B2: rat,C: rat,A: rat] :
% 0.17/0.42 ( ( times_times_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.17/0.42 = ( divide_divide_rat @ ( times_times_rat @ B2 @ A ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % times_divide_eq_left
% 0.17/0.42 thf(fact_1029_divide__divide__eq__left,axiom,
% 0.17/0.42 ! [A: complex,B2: complex,C: complex] :
% 0.17/0.42 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
% 0.17/0.42 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_eq_left
% 0.17/0.42 thf(fact_1030_divide__divide__eq__left,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
% 0.17/0.42 = ( divide_divide_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_eq_left
% 0.17/0.42 thf(fact_1031_divide__divide__eq__left,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
% 0.17/0.42 = ( divide_divide_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_eq_left
% 0.17/0.42 thf(fact_1032_divide__divide__eq__right,axiom,
% 0.17/0.42 ! [A: complex,B2: complex,C: complex] :
% 0.17/0.42 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.17/0.42 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_eq_right
% 0.17/0.42 thf(fact_1033_divide__divide__eq__right,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( divide_divide_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.17/0.42 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_eq_right
% 0.17/0.42 thf(fact_1034_divide__divide__eq__right,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.42 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_eq_right
% 0.17/0.42 thf(fact_1035_times__divide__eq__right,axiom,
% 0.17/0.42 ! [A: complex,B2: complex,C: complex] :
% 0.17/0.42 ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.17/0.42 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % times_divide_eq_right
% 0.17/0.42 thf(fact_1036_times__divide__eq__right,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( times_times_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.17/0.42 = ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % times_divide_eq_right
% 0.17/0.42 thf(fact_1037_times__divide__eq__right,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( times_times_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.42 = ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % times_divide_eq_right
% 0.17/0.42 thf(fact_1038_less__half__sum,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( ord_less_real @ A @ B2 )
% 0.17/0.42 => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % less_half_sum
% 0.17/0.42 thf(fact_1039_less__half__sum,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.42 => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % less_half_sum
% 0.17/0.42 thf(fact_1040_gt__half__sum,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( ord_less_real @ A @ B2 )
% 0.17/0.42 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % gt_half_sum
% 0.17/0.42 thf(fact_1041_gt__half__sum,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.42 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % gt_half_sum
% 0.17/0.42 thf(fact_1042_discrete,axiom,
% 0.17/0.42 ( ord_less_nat
% 0.17/0.42 = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % discrete
% 0.17/0.42 thf(fact_1043_discrete,axiom,
% 0.17/0.42 ( ord_less_int
% 0.17/0.42 = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % discrete
% 0.17/0.42 thf(fact_1044_low__def,axiom,
% 0.17/0.42 ( vEBT_VEBT_low
% 0.17/0.42 = ( ^ [X: nat,N2: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % low_def
% 0.17/0.42 thf(fact_1045_zdiv__numeral__Bit0,axiom,
% 0.17/0.42 ! [V: num,W: num] :
% 0.17/0.42 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.17/0.42 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % zdiv_numeral_Bit0
% 0.17/0.42 thf(fact_1046_mod__mod__trivial,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mod_trivial
% 0.17/0.42 thf(fact_1047_mod__mod__trivial,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mod_trivial
% 0.17/0.42 thf(fact_1048_mod__mod__trivial,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mod_trivial
% 0.17/0.42 thf(fact_1049_mod__add__self2,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_self2
% 0.17/0.42 thf(fact_1050_mod__add__self2,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_self2
% 0.17/0.42 thf(fact_1051_mod__add__self2,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_self2
% 0.17/0.42 thf(fact_1052_mod__add__self1,axiom,
% 0.17/0.42 ! [B2: nat,A: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_self1
% 0.17/0.42 thf(fact_1053_mod__add__self1,axiom,
% 0.17/0.42 ! [B2: int,A: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_self1
% 0.17/0.42 thf(fact_1054_mod__add__self1,axiom,
% 0.17/0.42 ! [B2: code_integer,A: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B2 @ A ) @ B2 )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_self1
% 0.17/0.42 thf(fact_1055_minus__mod__self2,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_mod_self2
% 0.17/0.42 thf(fact_1056_minus__mod__self2,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_mod_self2
% 0.17/0.42 thf(fact_1057_real__divide__square__eq,axiom,
% 0.17/0.42 ! [R2: real,A: real] :
% 0.17/0.42 ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
% 0.17/0.42 = ( divide_divide_real @ A @ R2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % real_divide_square_eq
% 0.17/0.42 thf(fact_1058_mod__less,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( ord_less_nat @ M @ N )
% 0.17/0.42 => ( ( modulo_modulo_nat @ M @ N )
% 0.17/0.42 = M ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_less
% 0.17/0.42 thf(fact_1059_mod__mult__self4,axiom,
% 0.17/0.42 ! [B2: nat,C: nat,A: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self4
% 0.17/0.42 thf(fact_1060_mod__mult__self4,axiom,
% 0.17/0.42 ! [B2: int,C: int,A: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self4
% 0.17/0.42 thf(fact_1061_mod__mult__self4,axiom,
% 0.17/0.42 ! [B2: code_integer,C: code_integer,A: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ C ) @ A ) @ B2 )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self4
% 0.17/0.42 thf(fact_1062_mod__mult__self3,axiom,
% 0.17/0.42 ! [C: nat,B2: nat,A: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self3
% 0.17/0.42 thf(fact_1063_mod__mult__self3,axiom,
% 0.17/0.42 ! [C: int,B2: int,A: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self3
% 0.17/0.42 thf(fact_1064_mod__mult__self3,axiom,
% 0.17/0.42 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B2 ) @ A ) @ B2 )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self3
% 0.17/0.42 thf(fact_1065_mod__mult__self2,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self2
% 0.17/0.42 thf(fact_1066_mod__mult__self2,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self2
% 0.17/0.42 thf(fact_1067_mod__mult__self2,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) @ B2 )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self2
% 0.17/0.42 thf(fact_1068_mod__mult__self1,axiom,
% 0.17/0.42 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self1
% 0.17/0.42 thf(fact_1069_mod__mult__self1,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self1
% 0.17/0.42 thf(fact_1070_mod__mult__self1,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) @ B2 )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_self1
% 0.17/0.42 thf(fact_1071_Suc__mod__mult__self1,axiom,
% 0.17/0.42 ! [M: nat,K: nat,N: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 0.17/0.42 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.17/0.42
% 0.17/0.42 % Suc_mod_mult_self1
% 0.17/0.42 thf(fact_1072_Suc__mod__mult__self2,axiom,
% 0.17/0.42 ! [M: nat,N: nat,K: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 0.17/0.42 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.17/0.42
% 0.17/0.42 % Suc_mod_mult_self2
% 0.17/0.42 thf(fact_1073_Suc__mod__mult__self3,axiom,
% 0.17/0.42 ! [K: nat,N: nat,M: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 0.17/0.42 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.17/0.42
% 0.17/0.42 % Suc_mod_mult_self3
% 0.17/0.42 thf(fact_1074_Suc__mod__mult__self4,axiom,
% 0.17/0.42 ! [N: nat,K: nat,M: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 0.17/0.42 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.17/0.42
% 0.17/0.42 % Suc_mod_mult_self4
% 0.17/0.42 thf(fact_1075_bits__one__mod__two__eq__one,axiom,
% 0.17/0.42 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.42 = one_one_nat ) ).
% 0.17/0.42
% 0.17/0.42 % bits_one_mod_two_eq_one
% 0.17/0.42 thf(fact_1076_bits__one__mod__two__eq__one,axiom,
% 0.17/0.42 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.42 = one_one_int ) ).
% 0.17/0.42
% 0.17/0.42 % bits_one_mod_two_eq_one
% 0.17/0.42 thf(fact_1077_bits__one__mod__two__eq__one,axiom,
% 0.17/0.42 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.42 = one_one_Code_integer ) ).
% 0.17/0.42
% 0.17/0.42 % bits_one_mod_two_eq_one
% 0.17/0.42 thf(fact_1078_one__mod__two__eq__one,axiom,
% 0.17/0.42 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.42 = one_one_nat ) ).
% 0.17/0.42
% 0.17/0.42 % one_mod_two_eq_one
% 0.17/0.42 thf(fact_1079_one__mod__two__eq__one,axiom,
% 0.17/0.42 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.42 = one_one_int ) ).
% 0.17/0.42
% 0.17/0.42 % one_mod_two_eq_one
% 0.17/0.42 thf(fact_1080_one__mod__two__eq__one,axiom,
% 0.17/0.42 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.42 = one_one_Code_integer ) ).
% 0.17/0.42
% 0.17/0.42 % one_mod_two_eq_one
% 0.17/0.42 thf(fact_1081_mod2__Suc__Suc,axiom,
% 0.17/0.42 ! [M: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.42 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod2_Suc_Suc
% 0.17/0.42 thf(fact_1082_Suc__times__numeral__mod__eq,axiom,
% 0.17/0.42 ! [K: num,N: nat] :
% 0.17/0.42 ( ( ( numeral_numeral_nat @ K )
% 0.17/0.42 != one_one_nat )
% 0.17/0.42 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 0.17/0.42 = one_one_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % Suc_times_numeral_mod_eq
% 0.17/0.42 thf(fact_1083_mod__mult__right__eq,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_right_eq
% 0.17/0.42 thf(fact_1084_mod__mult__right__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_right_eq
% 0.17/0.42 thf(fact_1085_mod__mult__right__eq,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_right_eq
% 0.17/0.42 thf(fact_1086_mod__mult__left__eq,axiom,
% 0.17/0.42 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_left_eq
% 0.17/0.42 thf(fact_1087_mod__mult__left__eq,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_left_eq
% 0.17/0.42 thf(fact_1088_mod__mult__left__eq,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_left_eq
% 0.17/0.42 thf(fact_1089_mult__mod__right,axiom,
% 0.17/0.42 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.42 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.17/0.42 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_mod_right
% 0.17/0.42 thf(fact_1090_mult__mod__right,axiom,
% 0.17/0.42 ! [C: int,A: int,B2: int] :
% 0.17/0.42 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 0.17/0.42 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_mod_right
% 0.17/0.42 thf(fact_1091_mult__mod__right,axiom,
% 0.17/0.42 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_mod_right
% 0.17/0.42 thf(fact_1092_mod__mult__mult2,axiom,
% 0.17/0.42 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 0.17/0.42 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_mult2
% 0.17/0.42 thf(fact_1093_mod__mult__mult2,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.17/0.42 = ( times_times_int @ ( modulo_modulo_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_mult2
% 0.17/0.42 thf(fact_1094_mod__mult__mult2,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.17/0.42 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_mult2
% 0.17/0.42 thf(fact_1095_mod__mult__cong,axiom,
% 0.17/0.42 ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
% 0.17/0.42 ( ( ( modulo_modulo_nat @ A @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ A2 @ C ) )
% 0.17/0.42 => ( ( ( modulo_modulo_nat @ B2 @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ B @ C ) )
% 0.17/0.42 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B ) @ C ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_cong
% 0.17/0.42 thf(fact_1096_mod__mult__cong,axiom,
% 0.17/0.42 ! [A: int,C: int,A2: int,B2: int,B: int] :
% 0.17/0.42 ( ( ( modulo_modulo_int @ A @ C )
% 0.17/0.42 = ( modulo_modulo_int @ A2 @ C ) )
% 0.17/0.42 => ( ( ( modulo_modulo_int @ B2 @ C )
% 0.17/0.42 = ( modulo_modulo_int @ B @ C ) )
% 0.17/0.42 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( times_times_int @ A2 @ B ) @ C ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_cong
% 0.17/0.42 thf(fact_1097_mod__mult__cong,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 0.17/0.42 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A2 @ C ) )
% 0.17/0.42 => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.17/0.42 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ C ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_cong
% 0.17/0.42 thf(fact_1098_mod__mult__eq,axiom,
% 0.17/0.42 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_eq
% 0.17/0.42 thf(fact_1099_mod__mult__eq,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_eq
% 0.17/0.42 thf(fact_1100_mod__mult__eq,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_eq
% 0.17/0.42 thf(fact_1101_mod__add__right__eq,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_right_eq
% 0.17/0.42 thf(fact_1102_mod__add__right__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_right_eq
% 0.17/0.42 thf(fact_1103_mod__add__right__eq,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_right_eq
% 0.17/0.42 thf(fact_1104_mod__add__left__eq,axiom,
% 0.17/0.42 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_left_eq
% 0.17/0.42 thf(fact_1105_mod__add__left__eq,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_left_eq
% 0.17/0.42 thf(fact_1106_mod__add__left__eq,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_left_eq
% 0.17/0.42 thf(fact_1107_mod__add__cong,axiom,
% 0.17/0.42 ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
% 0.17/0.42 ( ( ( modulo_modulo_nat @ A @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ A2 @ C ) )
% 0.17/0.42 => ( ( ( modulo_modulo_nat @ B2 @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ B @ C ) )
% 0.17/0.42 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B ) @ C ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_cong
% 0.17/0.42 thf(fact_1108_mod__add__cong,axiom,
% 0.17/0.42 ! [A: int,C: int,A2: int,B2: int,B: int] :
% 0.17/0.42 ( ( ( modulo_modulo_int @ A @ C )
% 0.17/0.42 = ( modulo_modulo_int @ A2 @ C ) )
% 0.17/0.42 => ( ( ( modulo_modulo_int @ B2 @ C )
% 0.17/0.42 = ( modulo_modulo_int @ B @ C ) )
% 0.17/0.42 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_cong
% 0.17/0.42 thf(fact_1109_mod__add__cong,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 0.17/0.42 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A2 @ C ) )
% 0.17/0.42 => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.17/0.42 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_cong
% 0.17/0.42 thf(fact_1110_mod__add__eq,axiom,
% 0.17/0.42 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_eq
% 0.17/0.42 thf(fact_1111_mod__add__eq,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_eq
% 0.17/0.42 thf(fact_1112_mod__add__eq,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_add_eq
% 0.17/0.42 thf(fact_1113_mod__diff__right__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_diff_right_eq
% 0.17/0.42 thf(fact_1114_mod__diff__right__eq,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_diff_right_eq
% 0.17/0.42 thf(fact_1115_mod__diff__left__eq,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_diff_left_eq
% 0.17/0.42 thf(fact_1116_mod__diff__left__eq,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_diff_left_eq
% 0.17/0.42 thf(fact_1117_mod__diff__cong,axiom,
% 0.17/0.42 ! [A: int,C: int,A2: int,B2: int,B: int] :
% 0.17/0.42 ( ( ( modulo_modulo_int @ A @ C )
% 0.17/0.42 = ( modulo_modulo_int @ A2 @ C ) )
% 0.17/0.42 => ( ( ( modulo_modulo_int @ B2 @ C )
% 0.17/0.42 = ( modulo_modulo_int @ B @ C ) )
% 0.17/0.42 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_diff_cong
% 0.17/0.42 thf(fact_1118_mod__diff__cong,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 0.17/0.42 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A2 @ C ) )
% 0.17/0.42 => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.17/0.42 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_diff_cong
% 0.17/0.42 thf(fact_1119_mod__diff__eq,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_diff_eq
% 0.17/0.42 thf(fact_1120_mod__diff__eq,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_diff_eq
% 0.17/0.42 thf(fact_1121_power__mod,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,N: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B2 ) @ N ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % power_mod
% 0.17/0.42 thf(fact_1122_power__mod,axiom,
% 0.17/0.42 ! [A: int,B2: int,N: nat] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B2 ) @ N ) @ B2 )
% 0.17/0.42 = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % power_mod
% 0.17/0.42 thf(fact_1123_power__mod,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer,N: nat] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ N ) @ B2 )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % power_mod
% 0.17/0.42 thf(fact_1124_mod__Suc__Suc__eq,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 0.17/0.42 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_Suc_Suc_eq
% 0.17/0.42 thf(fact_1125_mod__Suc__eq,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 0.17/0.42 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_Suc_eq
% 0.17/0.42 thf(fact_1126_mod__less__eq__dividend,axiom,
% 0.17/0.42 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 0.17/0.42
% 0.17/0.42 % mod_less_eq_dividend
% 0.17/0.42 thf(fact_1127_cong__exp__iff__simps_I9_J,axiom,
% 0.17/0.42 ! [M: num,Q2: num,N: num] :
% 0.17/0.42 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.17/0.42 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.17/0.42 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.17/0.42 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(9)
% 0.17/0.42 thf(fact_1128_cong__exp__iff__simps_I9_J,axiom,
% 0.17/0.42 ! [M: num,Q2: num,N: num] :
% 0.17/0.42 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.17/0.42 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.17/0.42 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.17/0.42 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(9)
% 0.17/0.42 thf(fact_1129_cong__exp__iff__simps_I9_J,axiom,
% 0.17/0.42 ! [M: num,Q2: num,N: num] :
% 0.17/0.42 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.17/0.42 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(9)
% 0.17/0.42 thf(fact_1130_cong__exp__iff__simps_I4_J,axiom,
% 0.17/0.42 ! [M: num,N: num] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 0.17/0.42 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(4)
% 0.17/0.42 thf(fact_1131_cong__exp__iff__simps_I4_J,axiom,
% 0.17/0.42 ! [M: num,N: num] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 0.17/0.42 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(4)
% 0.17/0.42 thf(fact_1132_cong__exp__iff__simps_I4_J,axiom,
% 0.17/0.42 ! [M: num,N: num] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 0.17/0.42 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(4)
% 0.17/0.42 thf(fact_1133_mod__geq,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ~ ( ord_less_nat @ M @ N )
% 0.17/0.42 => ( ( modulo_modulo_nat @ M @ N )
% 0.17/0.42 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_geq
% 0.17/0.42 thf(fact_1134_nat__mod__eq__iff,axiom,
% 0.17/0.42 ! [X4: nat,N: nat,Y3: nat] :
% 0.17/0.42 ( ( ( modulo_modulo_nat @ X4 @ N )
% 0.17/0.42 = ( modulo_modulo_nat @ Y3 @ N ) )
% 0.17/0.42 = ( ? [Q1: nat,Q22: nat] :
% 0.17/0.42 ( ( plus_plus_nat @ X4 @ ( times_times_nat @ N @ Q1 ) )
% 0.17/0.42 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % nat_mod_eq_iff
% 0.17/0.42 thf(fact_1135_cong__exp__iff__simps_I6_J,axiom,
% 0.17/0.42 ! [Q2: num,N: num] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.17/0.42 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(6)
% 0.17/0.42 thf(fact_1136_cong__exp__iff__simps_I6_J,axiom,
% 0.17/0.42 ! [Q2: num,N: num] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.17/0.42 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(6)
% 0.17/0.42 thf(fact_1137_cong__exp__iff__simps_I6_J,axiom,
% 0.17/0.42 ! [Q2: num,N: num] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.17/0.42 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(6)
% 0.17/0.42 thf(fact_1138_cong__exp__iff__simps_I8_J,axiom,
% 0.17/0.42 ! [M: num,Q2: num] :
% 0.17/0.42 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.17/0.42 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(8)
% 0.17/0.42 thf(fact_1139_cong__exp__iff__simps_I8_J,axiom,
% 0.17/0.42 ! [M: num,Q2: num] :
% 0.17/0.42 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.17/0.42 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(8)
% 0.17/0.42 thf(fact_1140_cong__exp__iff__simps_I8_J,axiom,
% 0.17/0.42 ! [M: num,Q2: num] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.17/0.42 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % cong_exp_iff_simps(8)
% 0.17/0.42 thf(fact_1141_mod__eqE,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( ( modulo_modulo_int @ A @ C )
% 0.17/0.42 = ( modulo_modulo_int @ B2 @ C ) )
% 0.17/0.42 => ~ ! [D3: int] :
% 0.17/0.42 ( B2
% 0.17/0.42 != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_eqE
% 0.17/0.42 thf(fact_1142_mod__eqE,axiom,
% 0.17/0.42 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.17/0.42 = ( modulo364778990260209775nteger @ B2 @ C ) )
% 0.17/0.42 => ~ ! [D3: code_integer] :
% 0.17/0.42 ( B2
% 0.17/0.42 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_eqE
% 0.17/0.42 thf(fact_1143_div__add1__eq,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_add1_eq
% 0.17/0.42 thf(fact_1144_div__add1__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_add1_eq
% 0.17/0.42 thf(fact_1145_div__add1__eq,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.42 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_add1_eq
% 0.17/0.42 thf(fact_1146_mod__induct,axiom,
% 0.17/0.42 ! [P: nat > $o,N: nat,P2: nat,M: nat] :
% 0.17/0.42 ( ( P @ N )
% 0.17/0.42 => ( ( ord_less_nat @ N @ P2 )
% 0.17/0.42 => ( ( ord_less_nat @ M @ P2 )
% 0.17/0.42 => ( ! [N4: nat] :
% 0.17/0.42 ( ( ord_less_nat @ N4 @ P2 )
% 0.17/0.42 => ( ( P @ N4 )
% 0.17/0.42 => ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P2 ) ) ) )
% 0.17/0.42 => ( P @ M ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_induct
% 0.17/0.42 thf(fact_1147_mod__Suc__le__divisor,axiom,
% 0.17/0.42 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 0.17/0.42
% 0.17/0.42 % mod_Suc_le_divisor
% 0.17/0.42 thf(fact_1148_nat__mod__eq__lemma,axiom,
% 0.17/0.42 ! [X4: nat,N: nat,Y3: nat] :
% 0.17/0.42 ( ( ( modulo_modulo_nat @ X4 @ N )
% 0.17/0.42 = ( modulo_modulo_nat @ Y3 @ N ) )
% 0.17/0.42 => ( ( ord_less_eq_nat @ Y3 @ X4 )
% 0.17/0.42 => ? [Q3: nat] :
% 0.17/0.42 ( X4
% 0.17/0.42 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % nat_mod_eq_lemma
% 0.17/0.42 thf(fact_1149_mod__eq__nat2E,axiom,
% 0.17/0.42 ! [M: nat,Q2: nat,N: nat] :
% 0.17/0.42 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.17/0.42 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.17/0.42 => ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.42 => ~ ! [S3: nat] :
% 0.17/0.42 ( N
% 0.17/0.42 != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_eq_nat2E
% 0.17/0.42 thf(fact_1150_mod__eq__nat1E,axiom,
% 0.17/0.42 ! [M: nat,Q2: nat,N: nat] :
% 0.17/0.42 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.17/0.42 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.17/0.42 => ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.42 => ~ ! [S3: nat] :
% 0.17/0.42 ( M
% 0.17/0.42 != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_eq_nat1E
% 0.17/0.42 thf(fact_1151_mod__if,axiom,
% 0.17/0.42 ( modulo_modulo_nat
% 0.17/0.42 = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N2 ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_if
% 0.17/0.42 thf(fact_1152_le__mod__geq,axiom,
% 0.17/0.42 ! [N: nat,M: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.42 => ( ( modulo_modulo_nat @ M @ N )
% 0.17/0.42 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_mod_geq
% 0.17/0.42 thf(fact_1153_mult__div__mod__eq,axiom,
% 0.17/0.42 ! [B2: nat,A: nat] :
% 0.17/0.42 ( ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mult_div_mod_eq
% 0.17/0.42 thf(fact_1154_mult__div__mod__eq,axiom,
% 0.17/0.42 ! [B2: int,A: int] :
% 0.17/0.42 ( ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mult_div_mod_eq
% 0.17/0.42 thf(fact_1155_mult__div__mod__eq,axiom,
% 0.17/0.42 ! [B2: code_integer,A: code_integer] :
% 0.17/0.42 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mult_div_mod_eq
% 0.17/0.42 thf(fact_1156_mod__mult__div__eq,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_div_eq
% 0.17/0.42 thf(fact_1157_mod__mult__div__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_div_eq
% 0.17/0.42 thf(fact_1158_mod__mult__div__eq,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult_div_eq
% 0.17/0.42 thf(fact_1159_mod__div__mult__eq,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mod_div_mult_eq
% 0.17/0.42 thf(fact_1160_mod__div__mult__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mod_div_mult_eq
% 0.17/0.42 thf(fact_1161_mod__div__mult__eq,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mod_div_mult_eq
% 0.17/0.42 thf(fact_1162_div__mult__mod__eq,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % div_mult_mod_eq
% 0.17/0.42 thf(fact_1163_div__mult__mod__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % div_mult_mod_eq
% 0.17/0.42 thf(fact_1164_div__mult__mod__eq,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % div_mult_mod_eq
% 0.17/0.42 thf(fact_1165_mod__div__decomp,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( A
% 0.17/0.42 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_div_decomp
% 0.17/0.42 thf(fact_1166_mod__div__decomp,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( A
% 0.17/0.42 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_div_decomp
% 0.17/0.42 thf(fact_1167_mod__div__decomp,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( A
% 0.17/0.42 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_div_decomp
% 0.17/0.42 thf(fact_1168_cancel__div__mod__rules_I1_J,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
% 0.17/0.42 = ( plus_plus_nat @ A @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % cancel_div_mod_rules(1)
% 0.17/0.42 thf(fact_1169_cancel__div__mod__rules_I1_J,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
% 0.17/0.42 = ( plus_plus_int @ A @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % cancel_div_mod_rules(1)
% 0.17/0.42 thf(fact_1170_cancel__div__mod__rules_I1_J,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.42 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
% 0.17/0.42 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % cancel_div_mod_rules(1)
% 0.17/0.42 thf(fact_1171_cancel__div__mod__rules_I2_J,axiom,
% 0.17/0.42 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.42 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
% 0.17/0.42 = ( plus_plus_nat @ A @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % cancel_div_mod_rules(2)
% 0.17/0.42 thf(fact_1172_cancel__div__mod__rules_I2_J,axiom,
% 0.17/0.42 ! [B2: int,A: int,C: int] :
% 0.17/0.42 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
% 0.17/0.42 = ( plus_plus_int @ A @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % cancel_div_mod_rules(2)
% 0.17/0.42 thf(fact_1173_cancel__div__mod__rules_I2_J,axiom,
% 0.17/0.42 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.17/0.42 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
% 0.17/0.42 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.17/0.42
% 0.17/0.42 % cancel_div_mod_rules(2)
% 0.17/0.42 thf(fact_1174_div__mult1__eq,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_mult1_eq
% 0.17/0.42 thf(fact_1175_div__mult1__eq,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_mult1_eq
% 0.17/0.42 thf(fact_1176_div__mult1__eq,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.42 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_mult1_eq
% 0.17/0.42 thf(fact_1177_minus__mult__div__eq__mod,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( minus_minus_nat @ A @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
% 0.17/0.42 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_mult_div_eq_mod
% 0.17/0.42 thf(fact_1178_minus__mult__div__eq__mod,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( minus_minus_int @ A @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
% 0.17/0.42 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_mult_div_eq_mod
% 0.17/0.42 thf(fact_1179_minus__mult__div__eq__mod,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_mult_div_eq_mod
% 0.17/0.42 thf(fact_1180_minus__mod__eq__mult__div,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.17/0.42 = ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_mod_eq_mult_div
% 0.17/0.42 thf(fact_1181_minus__mod__eq__mult__div,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
% 0.17/0.42 = ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_mod_eq_mult_div
% 0.17/0.42 thf(fact_1182_minus__mod__eq__mult__div,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.17/0.42 = ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_mod_eq_mult_div
% 0.17/0.42 thf(fact_1183_minus__mod__eq__div__mult,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.17/0.42 = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_mod_eq_div_mult
% 0.17/0.42 thf(fact_1184_minus__mod__eq__div__mult,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
% 0.17/0.42 = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_mod_eq_div_mult
% 0.17/0.42 thf(fact_1185_minus__mod__eq__div__mult,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.17/0.42 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_mod_eq_div_mult
% 0.17/0.42 thf(fact_1186_minus__div__mult__eq__mod,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
% 0.17/0.42 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_div_mult_eq_mod
% 0.17/0.42 thf(fact_1187_minus__div__mult__eq__mod,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
% 0.17/0.42 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_div_mult_eq_mod
% 0.17/0.42 thf(fact_1188_minus__div__mult__eq__mod,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
% 0.17/0.42 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % minus_div_mult_eq_mod
% 0.17/0.42 thf(fact_1189_mod__mult2__eq,axiom,
% 0.17/0.42 ! [M: nat,N: nat,Q2: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 0.17/0.42 = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mod_mult2_eq
% 0.17/0.42 thf(fact_1190_div__mod__decomp,axiom,
% 0.17/0.42 ! [A3: nat,N: nat] :
% 0.17/0.42 ( A3
% 0.17/0.42 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ N ) @ N ) @ ( modulo_modulo_nat @ A3 @ N ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_mod_decomp
% 0.17/0.42 thf(fact_1191_modulo__nat__def,axiom,
% 0.17/0.42 ( modulo_modulo_nat
% 0.17/0.42 = ( ^ [M6: nat,N2: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N2 ) @ N2 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % modulo_nat_def
% 0.17/0.42 thf(fact_1192_linordered__field__no__lb,axiom,
% 0.17/0.42 ! [X5: real] :
% 0.17/0.42 ? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% 0.17/0.42
% 0.17/0.42 % linordered_field_no_lb
% 0.17/0.42 thf(fact_1193_linordered__field__no__lb,axiom,
% 0.17/0.42 ! [X5: rat] :
% 0.17/0.42 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X5 ) ).
% 0.17/0.42
% 0.17/0.42 % linordered_field_no_lb
% 0.17/0.42 thf(fact_1194_linordered__field__no__ub,axiom,
% 0.17/0.42 ! [X5: real] :
% 0.17/0.42 ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% 0.17/0.42
% 0.17/0.42 % linordered_field_no_ub
% 0.17/0.42 thf(fact_1195_linordered__field__no__ub,axiom,
% 0.17/0.42 ! [X5: rat] :
% 0.17/0.42 ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% 0.17/0.42
% 0.17/0.42 % linordered_field_no_ub
% 0.17/0.42 thf(fact_1196_bounded__Max__nat,axiom,
% 0.17/0.42 ! [P: nat > $o,X4: nat,M7: nat] :
% 0.17/0.42 ( ( P @ X4 )
% 0.17/0.42 => ( ! [X3: nat] :
% 0.17/0.42 ( ( P @ X3 )
% 0.17/0.42 => ( ord_less_eq_nat @ X3 @ M7 ) )
% 0.17/0.42 => ~ ! [M4: nat] :
% 0.17/0.42 ( ( P @ M4 )
% 0.17/0.42 => ~ ! [X5: nat] :
% 0.17/0.42 ( ( P @ X5 )
% 0.17/0.42 => ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % bounded_Max_nat
% 0.17/0.42 thf(fact_1197_div__exp__mod__exp__eq,axiom,
% 0.17/0.42 ! [A: nat,N: nat,M: nat] :
% 0.17/0.42 ( ( 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 ) )
% 0.17/0.42 = ( 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 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_exp_mod_exp_eq
% 0.17/0.42 thf(fact_1198_div__exp__mod__exp__eq,axiom,
% 0.17/0.42 ! [A: int,N: nat,M: nat] :
% 0.17/0.42 ( ( 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 ) )
% 0.17/0.42 = ( 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 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_exp_mod_exp_eq
% 0.17/0.42 thf(fact_1199_div__exp__mod__exp__eq,axiom,
% 0.17/0.42 ! [A: code_integer,N: nat,M: nat] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.17/0.42 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_exp_mod_exp_eq
% 0.17/0.42 thf(fact_1200_mult__exp__mod__exp__eq,axiom,
% 0.17/0.42 ! [M: nat,N: nat,A: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.42 => ( ( 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 ) )
% 0.17/0.42 = ( 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 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_exp_mod_exp_eq
% 0.17/0.42 thf(fact_1201_mult__exp__mod__exp__eq,axiom,
% 0.17/0.42 ! [M: nat,N: nat,A: int] :
% 0.17/0.42 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.42 => ( ( 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 ) )
% 0.17/0.42 = ( 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 ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_exp_mod_exp_eq
% 0.17/0.42 thf(fact_1202_mult__exp__mod__exp__eq,axiom,
% 0.17/0.42 ! [M: nat,N: nat,A: code_integer] :
% 0.17/0.42 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.42 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.42 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_exp_mod_exp_eq
% 0.17/0.42 thf(fact_1203_divide__divide__eq__left_H,axiom,
% 0.17/0.42 ! [A: complex,B2: complex,C: complex] :
% 0.17/0.42 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
% 0.17/0.42 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_eq_left'
% 0.17/0.42 thf(fact_1204_divide__divide__eq__left_H,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
% 0.17/0.42 = ( divide_divide_real @ A @ ( times_times_real @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_eq_left'
% 0.17/0.42 thf(fact_1205_divide__divide__eq__left_H,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
% 0.17/0.42 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_eq_left'
% 0.17/0.42 thf(fact_1206_divide__divide__times__eq,axiom,
% 0.17/0.42 ! [X4: complex,Y3: complex,Z: complex,W: complex] :
% 0.17/0.42 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.17/0.42 = ( divide1717551699836669952omplex @ ( times_times_complex @ X4 @ W ) @ ( times_times_complex @ Y3 @ Z ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_times_eq
% 0.17/0.42 thf(fact_1207_divide__divide__times__eq,axiom,
% 0.17/0.42 ! [X4: real,Y3: real,Z: real,W: real] :
% 0.17/0.42 ( ( divide_divide_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 0.17/0.42 = ( divide_divide_real @ ( times_times_real @ X4 @ W ) @ ( times_times_real @ Y3 @ Z ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_times_eq
% 0.17/0.42 thf(fact_1208_divide__divide__times__eq,axiom,
% 0.17/0.42 ! [X4: rat,Y3: rat,Z: rat,W: rat] :
% 0.17/0.42 ( ( divide_divide_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 0.17/0.42 = ( divide_divide_rat @ ( times_times_rat @ X4 @ W ) @ ( times_times_rat @ Y3 @ Z ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_divide_times_eq
% 0.17/0.42 thf(fact_1209_times__divide__times__eq,axiom,
% 0.17/0.42 ! [X4: complex,Y3: complex,Z: complex,W: complex] :
% 0.17/0.42 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.17/0.42 = ( divide1717551699836669952omplex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ Y3 @ W ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % times_divide_times_eq
% 0.17/0.42 thf(fact_1210_times__divide__times__eq,axiom,
% 0.17/0.42 ! [X4: real,Y3: real,Z: real,W: real] :
% 0.17/0.42 ( ( times_times_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 0.17/0.42 = ( divide_divide_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y3 @ W ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % times_divide_times_eq
% 0.17/0.42 thf(fact_1211_times__divide__times__eq,axiom,
% 0.17/0.42 ! [X4: rat,Y3: rat,Z: rat,W: rat] :
% 0.17/0.42 ( ( times_times_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 0.17/0.42 = ( divide_divide_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y3 @ W ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % times_divide_times_eq
% 0.17/0.42 thf(fact_1212_add__divide__distrib,axiom,
% 0.17/0.42 ! [A: complex,B2: complex,C: complex] :
% 0.17/0.42 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_divide_distrib
% 0.17/0.42 thf(fact_1213_add__divide__distrib,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_divide_distrib
% 0.17/0.42 thf(fact_1214_add__divide__distrib,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.17/0.42 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_divide_distrib
% 0.17/0.42 thf(fact_1215_diff__divide__distrib,axiom,
% 0.17/0.42 ! [A: complex,B2: complex,C: complex] :
% 0.17/0.42 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B2 ) @ C )
% 0.17/0.42 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_divide_distrib
% 0.17/0.42 thf(fact_1216_diff__divide__distrib,axiom,
% 0.17/0.42 ! [A: real,B2: real,C: real] :
% 0.17/0.42 ( ( divide_divide_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.17/0.42 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_divide_distrib
% 0.17/0.42 thf(fact_1217_diff__divide__distrib,axiom,
% 0.17/0.42 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.42 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.17/0.42 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_divide_distrib
% 0.17/0.42 thf(fact_1218_unset__bit__Suc,axiom,
% 0.17/0.42 ! [N: nat,A: code_integer] :
% 0.17/0.42 ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 0.17/0.42 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % unset_bit_Suc
% 0.17/0.42 thf(fact_1219_unset__bit__Suc,axiom,
% 0.17/0.42 ! [N: nat,A: int] :
% 0.17/0.42 ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 0.17/0.42 = ( 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 ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % unset_bit_Suc
% 0.17/0.42 thf(fact_1220_unset__bit__Suc,axiom,
% 0.17/0.42 ! [N: nat,A: nat] :
% 0.17/0.42 ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 0.17/0.42 = ( 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 ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % unset_bit_Suc
% 0.17/0.42 thf(fact_1221_flip__bit__Suc,axiom,
% 0.17/0.42 ! [N: nat,A: code_integer] :
% 0.17/0.42 ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 0.17/0.42 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % flip_bit_Suc
% 0.17/0.42 thf(fact_1222_flip__bit__Suc,axiom,
% 0.17/0.42 ! [N: nat,A: int] :
% 0.17/0.42 ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 0.17/0.42 = ( 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 ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % flip_bit_Suc
% 0.17/0.42 thf(fact_1223_flip__bit__Suc,axiom,
% 0.17/0.42 ! [N: nat,A: nat] :
% 0.17/0.42 ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 0.17/0.42 = ( 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 ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % flip_bit_Suc
% 0.17/0.42 thf(fact_1224_set__bit__Suc,axiom,
% 0.17/0.42 ! [N: nat,A: code_integer] :
% 0.17/0.42 ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 0.17/0.42 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % set_bit_Suc
% 0.17/0.42 thf(fact_1225_set__bit__Suc,axiom,
% 0.17/0.42 ! [N: nat,A: int] :
% 0.17/0.42 ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 0.17/0.42 = ( 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 ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % set_bit_Suc
% 0.17/0.42 thf(fact_1226_set__bit__Suc,axiom,
% 0.17/0.42 ! [N: nat,A: nat] :
% 0.17/0.42 ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 0.17/0.42 = ( 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 ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % set_bit_Suc
% 0.17/0.42 thf(fact_1227_dbl__simps_I3_J,axiom,
% 0.17/0.42 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 0.17/0.42 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % dbl_simps(3)
% 0.17/0.42 thf(fact_1228_dbl__simps_I3_J,axiom,
% 0.17/0.42 ( ( neg_numeral_dbl_real @ one_one_real )
% 0.17/0.42 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % dbl_simps(3)
% 0.17/0.42 thf(fact_1229_dbl__simps_I3_J,axiom,
% 0.17/0.42 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 0.17/0.42 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % dbl_simps(3)
% 0.17/0.42 thf(fact_1230_dbl__simps_I3_J,axiom,
% 0.17/0.42 ( ( neg_numeral_dbl_int @ one_one_int )
% 0.17/0.42 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % dbl_simps(3)
% 0.17/0.42 thf(fact_1231_divmod__digit__1_I1_J,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.17/0.42 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.17/0.42 => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.17/0.42 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_Code_integer )
% 0.17/0.42 = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divmod_digit_1(1)
% 0.17/0.42 thf(fact_1232_divmod__digit__1_I1_J,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.42 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.17/0.42 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_nat )
% 0.17/0.42 = ( divide_divide_nat @ A @ B2 ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divmod_digit_1(1)
% 0.17/0.42 thf(fact_1233_divmod__digit__1_I1_J,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.42 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.42 => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.17/0.42 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_int )
% 0.17/0.42 = ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divmod_digit_1(1)
% 0.17/0.42 thf(fact_1234_signed__take__bit__Suc,axiom,
% 0.17/0.42 ! [N: nat,A: code_integer] :
% 0.17/0.42 ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 0.17/0.42 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % signed_take_bit_Suc
% 0.17/0.42 thf(fact_1235_signed__take__bit__Suc,axiom,
% 0.17/0.42 ! [N: nat,A: int] :
% 0.17/0.42 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 0.17/0.42 = ( 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 ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % signed_take_bit_Suc
% 0.17/0.42 thf(fact_1236_power__numeral,axiom,
% 0.17/0.42 ! [K: num,L2: num] :
% 0.17/0.42 ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L2 ) )
% 0.17/0.42 = ( numera6690914467698888265omplex @ ( pow @ K @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % power_numeral
% 0.17/0.42 thf(fact_1237_power__numeral,axiom,
% 0.17/0.42 ! [K: num,L2: num] :
% 0.17/0.42 ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L2 ) )
% 0.17/0.42 = ( numeral_numeral_real @ ( pow @ K @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % power_numeral
% 0.17/0.42 thf(fact_1238_power__numeral,axiom,
% 0.17/0.42 ! [K: num,L2: num] :
% 0.17/0.42 ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L2 ) )
% 0.17/0.42 = ( numeral_numeral_rat @ ( pow @ K @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % power_numeral
% 0.17/0.42 thf(fact_1239_power__numeral,axiom,
% 0.17/0.42 ! [K: num,L2: num] :
% 0.17/0.42 ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L2 ) )
% 0.17/0.42 = ( numeral_numeral_nat @ ( pow @ K @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % power_numeral
% 0.17/0.42 thf(fact_1240_power__numeral,axiom,
% 0.17/0.42 ! [K: num,L2: num] :
% 0.17/0.42 ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L2 ) )
% 0.17/0.42 = ( numeral_numeral_int @ ( pow @ K @ L2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % power_numeral
% 0.17/0.42 thf(fact_1241_arith__geo__mean,axiom,
% 0.17/0.42 ! [U: real,X4: real,Y3: real] :
% 0.17/0.42 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.42 = ( times_times_real @ X4 @ Y3 ) )
% 0.17/0.42 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.17/0.42 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.17/0.42 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % arith_geo_mean
% 0.17/0.42 thf(fact_1242_arith__geo__mean,axiom,
% 0.17/0.42 ! [U: rat,X4: rat,Y3: rat] :
% 0.17/0.42 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.42 = ( times_times_rat @ X4 @ Y3 ) )
% 0.17/0.42 => ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.17/0.42 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.17/0.42 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % arith_geo_mean
% 0.17/0.42 thf(fact_1243_odd__two__times__div__two__nat,axiom,
% 0.17/0.42 ! [N: nat] :
% 0.17/0.42 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.42 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.42 = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % odd_two_times_div_two_nat
% 0.17/0.42 thf(fact_1244_less__nat__zero__code,axiom,
% 0.17/0.42 ! [N: nat] :
% 0.17/0.42 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % less_nat_zero_code
% 0.17/0.42 thf(fact_1245_neq0__conv,axiom,
% 0.17/0.42 ! [N: nat] :
% 0.17/0.42 ( ( N != zero_zero_nat )
% 0.17/0.42 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.17/0.42
% 0.17/0.42 % neq0_conv
% 0.17/0.42 thf(fact_1246_bot__nat__0_Onot__eq__extremum,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( A != zero_zero_nat )
% 0.17/0.42 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.17/0.42
% 0.17/0.42 % bot_nat_0.not_eq_extremum
% 0.17/0.42 thf(fact_1247_le0,axiom,
% 0.17/0.42 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.17/0.42
% 0.17/0.42 % le0
% 0.17/0.42 thf(fact_1248_bot__nat__0_Oextremum,axiom,
% 0.17/0.42 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.17/0.42
% 0.17/0.42 % bot_nat_0.extremum
% 0.17/0.42 thf(fact_1249_Nat_Oadd__0__right,axiom,
% 0.17/0.42 ! [M: nat] :
% 0.17/0.42 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.17/0.42 = M ) ).
% 0.17/0.42
% 0.17/0.42 % Nat.add_0_right
% 0.17/0.42 thf(fact_1250_add__is__0,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( ( plus_plus_nat @ M @ N )
% 0.17/0.42 = zero_zero_nat )
% 0.17/0.42 = ( ( M = zero_zero_nat )
% 0.17/0.42 & ( N = zero_zero_nat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_is_0
% 0.17/0.42 thf(fact_1251_diff__self__eq__0,axiom,
% 0.17/0.42 ! [M: nat] :
% 0.17/0.42 ( ( minus_minus_nat @ M @ M )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % diff_self_eq_0
% 0.17/0.42 thf(fact_1252_diff__0__eq__0,axiom,
% 0.17/0.42 ! [N: nat] :
% 0.17/0.42 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % diff_0_eq_0
% 0.17/0.42 thf(fact_1253_mult__cancel2,axiom,
% 0.17/0.42 ! [M: nat,K: nat,N: nat] :
% 0.17/0.42 ( ( ( times_times_nat @ M @ K )
% 0.17/0.42 = ( times_times_nat @ N @ K ) )
% 0.17/0.42 = ( ( M = N )
% 0.17/0.42 | ( K = zero_zero_nat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_cancel2
% 0.17/0.42 thf(fact_1254_mult__cancel1,axiom,
% 0.17/0.42 ! [K: nat,M: nat,N: nat] :
% 0.17/0.42 ( ( ( times_times_nat @ K @ M )
% 0.17/0.42 = ( times_times_nat @ K @ N ) )
% 0.17/0.42 = ( ( M = N )
% 0.17/0.42 | ( K = zero_zero_nat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_cancel1
% 0.17/0.42 thf(fact_1255_mult__0__right,axiom,
% 0.17/0.42 ! [M: nat] :
% 0.17/0.42 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % mult_0_right
% 0.17/0.42 thf(fact_1256_mult__is__0,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( ( times_times_nat @ M @ N )
% 0.17/0.42 = zero_zero_nat )
% 0.17/0.42 = ( ( M = zero_zero_nat )
% 0.17/0.42 | ( N = zero_zero_nat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_is_0
% 0.17/0.42 thf(fact_1257_div__pos__pos__trivial,axiom,
% 0.17/0.42 ! [K: int,L2: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.17/0.42 => ( ( ord_less_int @ K @ L2 )
% 0.17/0.42 => ( ( divide_divide_int @ K @ L2 )
% 0.17/0.42 = zero_zero_int ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_pos_pos_trivial
% 0.17/0.42 thf(fact_1258_div__neg__neg__trivial,axiom,
% 0.17/0.42 ! [K: int,L2: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.17/0.42 => ( ( ord_less_int @ L2 @ K )
% 0.17/0.42 => ( ( divide_divide_int @ K @ L2 )
% 0.17/0.42 = zero_zero_int ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_neg_neg_trivial
% 0.17/0.42 thf(fact_1259_i0__less,axiom,
% 0.17/0.42 ! [N: extended_enat] :
% 0.17/0.42 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.17/0.42 = ( N != zero_z5237406670263579293d_enat ) ) ).
% 0.17/0.42
% 0.17/0.42 % i0_less
% 0.17/0.42 thf(fact_1260_idiff__0,axiom,
% 0.17/0.42 ! [N: extended_enat] :
% 0.17/0.42 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.17/0.42 = zero_z5237406670263579293d_enat ) ).
% 0.17/0.42
% 0.17/0.42 % idiff_0
% 0.17/0.42 thf(fact_1261_idiff__0__right,axiom,
% 0.17/0.42 ! [N: extended_enat] :
% 0.17/0.42 ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.17/0.42 = N ) ).
% 0.17/0.42
% 0.17/0.42 % idiff_0_right
% 0.17/0.42 thf(fact_1262_le__zero__eq,axiom,
% 0.17/0.42 ! [N: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.17/0.42 = ( N = zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_zero_eq
% 0.17/0.42 thf(fact_1263_not__gr__zero,axiom,
% 0.17/0.42 ! [N: nat] :
% 0.17/0.42 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.17/0.42 = ( N = zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % not_gr_zero
% 0.17/0.42 thf(fact_1264_mult__cancel__right,axiom,
% 0.17/0.42 ! [A: real,C: real,B2: real] :
% 0.17/0.42 ( ( ( times_times_real @ A @ C )
% 0.17/0.42 = ( times_times_real @ B2 @ C ) )
% 0.17/0.42 = ( ( C = zero_zero_real )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_cancel_right
% 0.17/0.42 thf(fact_1265_mult__cancel__right,axiom,
% 0.17/0.42 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.42 ( ( ( times_times_rat @ A @ C )
% 0.17/0.42 = ( times_times_rat @ B2 @ C ) )
% 0.17/0.42 = ( ( C = zero_zero_rat )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_cancel_right
% 0.17/0.42 thf(fact_1266_mult__cancel__right,axiom,
% 0.17/0.42 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.42 ( ( ( times_times_nat @ A @ C )
% 0.17/0.42 = ( times_times_nat @ B2 @ C ) )
% 0.17/0.42 = ( ( C = zero_zero_nat )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_cancel_right
% 0.17/0.42 thf(fact_1267_mult__cancel__right,axiom,
% 0.17/0.42 ! [A: int,C: int,B2: int] :
% 0.17/0.42 ( ( ( times_times_int @ A @ C )
% 0.17/0.42 = ( times_times_int @ B2 @ C ) )
% 0.17/0.42 = ( ( C = zero_zero_int )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_cancel_right
% 0.17/0.42 thf(fact_1268_mult__cancel__left,axiom,
% 0.17/0.42 ! [C: real,A: real,B2: real] :
% 0.17/0.42 ( ( ( times_times_real @ C @ A )
% 0.17/0.42 = ( times_times_real @ C @ B2 ) )
% 0.17/0.42 = ( ( C = zero_zero_real )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_cancel_left
% 0.17/0.42 thf(fact_1269_mult__cancel__left,axiom,
% 0.17/0.42 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.42 ( ( ( times_times_rat @ C @ A )
% 0.17/0.42 = ( times_times_rat @ C @ B2 ) )
% 0.17/0.42 = ( ( C = zero_zero_rat )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_cancel_left
% 0.17/0.42 thf(fact_1270_mult__cancel__left,axiom,
% 0.17/0.42 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.42 ( ( ( times_times_nat @ C @ A )
% 0.17/0.42 = ( times_times_nat @ C @ B2 ) )
% 0.17/0.42 = ( ( C = zero_zero_nat )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_cancel_left
% 0.17/0.42 thf(fact_1271_mult__cancel__left,axiom,
% 0.17/0.42 ! [C: int,A: int,B2: int] :
% 0.17/0.42 ( ( ( times_times_int @ C @ A )
% 0.17/0.42 = ( times_times_int @ C @ B2 ) )
% 0.17/0.42 = ( ( C = zero_zero_int )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_cancel_left
% 0.17/0.42 thf(fact_1272_mult__eq__0__iff,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( ( times_times_real @ A @ B2 )
% 0.17/0.42 = zero_zero_real )
% 0.17/0.42 = ( ( A = zero_zero_real )
% 0.17/0.42 | ( B2 = zero_zero_real ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_eq_0_iff
% 0.17/0.42 thf(fact_1273_mult__eq__0__iff,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( ( times_times_rat @ A @ B2 )
% 0.17/0.42 = zero_zero_rat )
% 0.17/0.42 = ( ( A = zero_zero_rat )
% 0.17/0.42 | ( B2 = zero_zero_rat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_eq_0_iff
% 0.17/0.42 thf(fact_1274_mult__eq__0__iff,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( ( times_times_nat @ A @ B2 )
% 0.17/0.42 = zero_zero_nat )
% 0.17/0.42 = ( ( A = zero_zero_nat )
% 0.17/0.42 | ( B2 = zero_zero_nat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_eq_0_iff
% 0.17/0.42 thf(fact_1275_mult__eq__0__iff,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( ( times_times_int @ A @ B2 )
% 0.17/0.42 = zero_zero_int )
% 0.17/0.42 = ( ( A = zero_zero_int )
% 0.17/0.42 | ( B2 = zero_zero_int ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_eq_0_iff
% 0.17/0.42 thf(fact_1276_mult__zero__right,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( times_times_real @ A @ zero_zero_real )
% 0.17/0.42 = zero_zero_real ) ).
% 0.17/0.42
% 0.17/0.42 % mult_zero_right
% 0.17/0.42 thf(fact_1277_mult__zero__right,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( times_times_rat @ A @ zero_zero_rat )
% 0.17/0.42 = zero_zero_rat ) ).
% 0.17/0.42
% 0.17/0.42 % mult_zero_right
% 0.17/0.42 thf(fact_1278_mult__zero__right,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( times_times_nat @ A @ zero_zero_nat )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % mult_zero_right
% 0.17/0.42 thf(fact_1279_mult__zero__right,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( times_times_int @ A @ zero_zero_int )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % mult_zero_right
% 0.17/0.42 thf(fact_1280_mult__zero__left,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( times_times_real @ zero_zero_real @ A )
% 0.17/0.42 = zero_zero_real ) ).
% 0.17/0.42
% 0.17/0.42 % mult_zero_left
% 0.17/0.42 thf(fact_1281_mult__zero__left,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( times_times_rat @ zero_zero_rat @ A )
% 0.17/0.42 = zero_zero_rat ) ).
% 0.17/0.42
% 0.17/0.42 % mult_zero_left
% 0.17/0.42 thf(fact_1282_mult__zero__left,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( times_times_nat @ zero_zero_nat @ A )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % mult_zero_left
% 0.17/0.42 thf(fact_1283_mult__zero__left,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( times_times_int @ zero_zero_int @ A )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % mult_zero_left
% 0.17/0.42 thf(fact_1284_add__0,axiom,
% 0.17/0.42 ! [A: literal] :
% 0.17/0.42 ( ( plus_plus_literal @ zero_zero_literal @ A )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % add_0
% 0.17/0.42 thf(fact_1285_add__0,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % add_0
% 0.17/0.42 thf(fact_1286_add__0,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % add_0
% 0.17/0.42 thf(fact_1287_add__0,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % add_0
% 0.17/0.42 thf(fact_1288_add__0,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % add_0
% 0.17/0.42 thf(fact_1289_zero__eq__add__iff__both__eq__0,axiom,
% 0.17/0.42 ! [X4: nat,Y3: nat] :
% 0.17/0.42 ( ( zero_zero_nat
% 0.17/0.42 = ( plus_plus_nat @ X4 @ Y3 ) )
% 0.17/0.42 = ( ( X4 = zero_zero_nat )
% 0.17/0.42 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % zero_eq_add_iff_both_eq_0
% 0.17/0.42 thf(fact_1290_add__eq__0__iff__both__eq__0,axiom,
% 0.17/0.42 ! [X4: nat,Y3: nat] :
% 0.17/0.42 ( ( ( plus_plus_nat @ X4 @ Y3 )
% 0.17/0.42 = zero_zero_nat )
% 0.17/0.42 = ( ( X4 = zero_zero_nat )
% 0.17/0.42 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_eq_0_iff_both_eq_0
% 0.17/0.42 thf(fact_1291_add__cancel__right__right,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( A
% 0.17/0.42 = ( plus_plus_real @ A @ B2 ) )
% 0.17/0.42 = ( B2 = zero_zero_real ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_right_right
% 0.17/0.42 thf(fact_1292_add__cancel__right__right,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( A
% 0.17/0.42 = ( plus_plus_rat @ A @ B2 ) )
% 0.17/0.42 = ( B2 = zero_zero_rat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_right_right
% 0.17/0.42 thf(fact_1293_add__cancel__right__right,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( A
% 0.17/0.42 = ( plus_plus_nat @ A @ B2 ) )
% 0.17/0.42 = ( B2 = zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_right_right
% 0.17/0.42 thf(fact_1294_add__cancel__right__right,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( A
% 0.17/0.42 = ( plus_plus_int @ A @ B2 ) )
% 0.17/0.42 = ( B2 = zero_zero_int ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_right_right
% 0.17/0.42 thf(fact_1295_add__cancel__right__left,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( A
% 0.17/0.42 = ( plus_plus_real @ B2 @ A ) )
% 0.17/0.42 = ( B2 = zero_zero_real ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_right_left
% 0.17/0.42 thf(fact_1296_add__cancel__right__left,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( A
% 0.17/0.42 = ( plus_plus_rat @ B2 @ A ) )
% 0.17/0.42 = ( B2 = zero_zero_rat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_right_left
% 0.17/0.42 thf(fact_1297_add__cancel__right__left,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( A
% 0.17/0.42 = ( plus_plus_nat @ B2 @ A ) )
% 0.17/0.42 = ( B2 = zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_right_left
% 0.17/0.42 thf(fact_1298_add__cancel__right__left,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( A
% 0.17/0.42 = ( plus_plus_int @ B2 @ A ) )
% 0.17/0.42 = ( B2 = zero_zero_int ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_right_left
% 0.17/0.42 thf(fact_1299_add__cancel__left__right,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( ( plus_plus_real @ A @ B2 )
% 0.17/0.42 = A )
% 0.17/0.42 = ( B2 = zero_zero_real ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_left_right
% 0.17/0.42 thf(fact_1300_add__cancel__left__right,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( ( plus_plus_rat @ A @ B2 )
% 0.17/0.42 = A )
% 0.17/0.42 = ( B2 = zero_zero_rat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_left_right
% 0.17/0.42 thf(fact_1301_add__cancel__left__right,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( ( plus_plus_nat @ A @ B2 )
% 0.17/0.42 = A )
% 0.17/0.42 = ( B2 = zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_left_right
% 0.17/0.42 thf(fact_1302_add__cancel__left__right,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( ( plus_plus_int @ A @ B2 )
% 0.17/0.42 = A )
% 0.17/0.42 = ( B2 = zero_zero_int ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_left_right
% 0.17/0.42 thf(fact_1303_add__cancel__left__left,axiom,
% 0.17/0.42 ! [B2: real,A: real] :
% 0.17/0.42 ( ( ( plus_plus_real @ B2 @ A )
% 0.17/0.42 = A )
% 0.17/0.42 = ( B2 = zero_zero_real ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_left_left
% 0.17/0.42 thf(fact_1304_add__cancel__left__left,axiom,
% 0.17/0.42 ! [B2: rat,A: rat] :
% 0.17/0.42 ( ( ( plus_plus_rat @ B2 @ A )
% 0.17/0.42 = A )
% 0.17/0.42 = ( B2 = zero_zero_rat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_left_left
% 0.17/0.42 thf(fact_1305_add__cancel__left__left,axiom,
% 0.17/0.42 ! [B2: nat,A: nat] :
% 0.17/0.42 ( ( ( plus_plus_nat @ B2 @ A )
% 0.17/0.42 = A )
% 0.17/0.42 = ( B2 = zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_left_left
% 0.17/0.42 thf(fact_1306_add__cancel__left__left,axiom,
% 0.17/0.42 ! [B2: int,A: int] :
% 0.17/0.42 ( ( ( plus_plus_int @ B2 @ A )
% 0.17/0.42 = A )
% 0.17/0.42 = ( B2 = zero_zero_int ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_cancel_left_left
% 0.17/0.42 thf(fact_1307_double__zero__sym,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( zero_zero_real
% 0.17/0.42 = ( plus_plus_real @ A @ A ) )
% 0.17/0.42 = ( A = zero_zero_real ) ) ).
% 0.17/0.42
% 0.17/0.42 % double_zero_sym
% 0.17/0.42 thf(fact_1308_double__zero__sym,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( zero_zero_rat
% 0.17/0.42 = ( plus_plus_rat @ A @ A ) )
% 0.17/0.42 = ( A = zero_zero_rat ) ) ).
% 0.17/0.42
% 0.17/0.42 % double_zero_sym
% 0.17/0.42 thf(fact_1309_double__zero__sym,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( zero_zero_int
% 0.17/0.42 = ( plus_plus_int @ A @ A ) )
% 0.17/0.42 = ( A = zero_zero_int ) ) ).
% 0.17/0.42
% 0.17/0.42 % double_zero_sym
% 0.17/0.42 thf(fact_1310_add_Oright__neutral,axiom,
% 0.17/0.42 ! [A: literal] :
% 0.17/0.42 ( ( plus_plus_literal @ A @ zero_zero_literal )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % add.right_neutral
% 0.17/0.42 thf(fact_1311_add_Oright__neutral,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % add.right_neutral
% 0.17/0.42 thf(fact_1312_add_Oright__neutral,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % add.right_neutral
% 0.17/0.42 thf(fact_1313_add_Oright__neutral,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % add.right_neutral
% 0.17/0.42 thf(fact_1314_add_Oright__neutral,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % add.right_neutral
% 0.17/0.42 thf(fact_1315_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( minus_minus_real @ A @ A )
% 0.17/0.42 = zero_zero_real ) ).
% 0.17/0.42
% 0.17/0.42 % cancel_comm_monoid_add_class.diff_cancel
% 0.17/0.42 thf(fact_1316_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( minus_minus_rat @ A @ A )
% 0.17/0.42 = zero_zero_rat ) ).
% 0.17/0.42
% 0.17/0.42 % cancel_comm_monoid_add_class.diff_cancel
% 0.17/0.42 thf(fact_1317_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( minus_minus_nat @ A @ A )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % cancel_comm_monoid_add_class.diff_cancel
% 0.17/0.42 thf(fact_1318_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( minus_minus_int @ A @ A )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % cancel_comm_monoid_add_class.diff_cancel
% 0.17/0.42 thf(fact_1319_diff__zero,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % diff_zero
% 0.17/0.42 thf(fact_1320_diff__zero,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % diff_zero
% 0.17/0.42 thf(fact_1321_diff__zero,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % diff_zero
% 0.17/0.42 thf(fact_1322_diff__zero,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % diff_zero
% 0.17/0.42 thf(fact_1323_zero__diff,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % zero_diff
% 0.17/0.42 thf(fact_1324_diff__0__right,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % diff_0_right
% 0.17/0.42 thf(fact_1325_diff__0__right,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % diff_0_right
% 0.17/0.42 thf(fact_1326_diff__0__right,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % diff_0_right
% 0.17/0.42 thf(fact_1327_diff__self,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( minus_minus_real @ A @ A )
% 0.17/0.42 = zero_zero_real ) ).
% 0.17/0.42
% 0.17/0.42 % diff_self
% 0.17/0.42 thf(fact_1328_diff__self,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( minus_minus_rat @ A @ A )
% 0.17/0.42 = zero_zero_rat ) ).
% 0.17/0.42
% 0.17/0.42 % diff_self
% 0.17/0.42 thf(fact_1329_diff__self,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( minus_minus_int @ A @ A )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % diff_self
% 0.17/0.42 thf(fact_1330_div__0,axiom,
% 0.17/0.42 ! [A: complex] :
% 0.17/0.42 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 0.17/0.42 = zero_zero_complex ) ).
% 0.17/0.42
% 0.17/0.42 % div_0
% 0.17/0.42 thf(fact_1331_div__0,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( divide_divide_real @ zero_zero_real @ A )
% 0.17/0.42 = zero_zero_real ) ).
% 0.17/0.42
% 0.17/0.42 % div_0
% 0.17/0.42 thf(fact_1332_div__0,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 0.17/0.42 = zero_zero_rat ) ).
% 0.17/0.42
% 0.17/0.42 % div_0
% 0.17/0.42 thf(fact_1333_div__0,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % div_0
% 0.17/0.42 thf(fact_1334_div__0,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % div_0
% 0.17/0.42 thf(fact_1335_div__by__0,axiom,
% 0.17/0.42 ! [A: complex] :
% 0.17/0.42 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.17/0.42 = zero_zero_complex ) ).
% 0.17/0.42
% 0.17/0.42 % div_by_0
% 0.17/0.42 thf(fact_1336_div__by__0,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.17/0.42 = zero_zero_real ) ).
% 0.17/0.42
% 0.17/0.42 % div_by_0
% 0.17/0.42 thf(fact_1337_div__by__0,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.17/0.42 = zero_zero_rat ) ).
% 0.17/0.42
% 0.17/0.42 % div_by_0
% 0.17/0.42 thf(fact_1338_div__by__0,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % div_by_0
% 0.17/0.42 thf(fact_1339_div__by__0,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % div_by_0
% 0.17/0.42 thf(fact_1340_bits__div__0,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % bits_div_0
% 0.17/0.42 thf(fact_1341_bits__div__0,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % bits_div_0
% 0.17/0.42 thf(fact_1342_bits__div__by__0,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % bits_div_by_0
% 0.17/0.42 thf(fact_1343_bits__div__by__0,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % bits_div_by_0
% 0.17/0.42 thf(fact_1344_divide__eq__0__iff,axiom,
% 0.17/0.42 ! [A: complex,B2: complex] :
% 0.17/0.42 ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 0.17/0.42 = zero_zero_complex )
% 0.17/0.42 = ( ( A = zero_zero_complex )
% 0.17/0.42 | ( B2 = zero_zero_complex ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_eq_0_iff
% 0.17/0.42 thf(fact_1345_divide__eq__0__iff,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( ( divide_divide_real @ A @ B2 )
% 0.17/0.42 = zero_zero_real )
% 0.17/0.42 = ( ( A = zero_zero_real )
% 0.17/0.42 | ( B2 = zero_zero_real ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_eq_0_iff
% 0.17/0.42 thf(fact_1346_divide__eq__0__iff,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( ( divide_divide_rat @ A @ B2 )
% 0.17/0.42 = zero_zero_rat )
% 0.17/0.42 = ( ( A = zero_zero_rat )
% 0.17/0.42 | ( B2 = zero_zero_rat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_eq_0_iff
% 0.17/0.42 thf(fact_1347_divide__cancel__left,axiom,
% 0.17/0.42 ! [C: complex,A: complex,B2: complex] :
% 0.17/0.42 ( ( ( divide1717551699836669952omplex @ C @ A )
% 0.17/0.42 = ( divide1717551699836669952omplex @ C @ B2 ) )
% 0.17/0.42 = ( ( C = zero_zero_complex )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_cancel_left
% 0.17/0.42 thf(fact_1348_divide__cancel__left,axiom,
% 0.17/0.42 ! [C: real,A: real,B2: real] :
% 0.17/0.42 ( ( ( divide_divide_real @ C @ A )
% 0.17/0.42 = ( divide_divide_real @ C @ B2 ) )
% 0.17/0.42 = ( ( C = zero_zero_real )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_cancel_left
% 0.17/0.42 thf(fact_1349_divide__cancel__left,axiom,
% 0.17/0.42 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.42 ( ( ( divide_divide_rat @ C @ A )
% 0.17/0.42 = ( divide_divide_rat @ C @ B2 ) )
% 0.17/0.42 = ( ( C = zero_zero_rat )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_cancel_left
% 0.17/0.42 thf(fact_1350_divide__cancel__right,axiom,
% 0.17/0.42 ! [A: complex,C: complex,B2: complex] :
% 0.17/0.42 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.17/0.42 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.17/0.42 = ( ( C = zero_zero_complex )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_cancel_right
% 0.17/0.42 thf(fact_1351_divide__cancel__right,axiom,
% 0.17/0.42 ! [A: real,C: real,B2: real] :
% 0.17/0.42 ( ( ( divide_divide_real @ A @ C )
% 0.17/0.42 = ( divide_divide_real @ B2 @ C ) )
% 0.17/0.42 = ( ( C = zero_zero_real )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_cancel_right
% 0.17/0.42 thf(fact_1352_divide__cancel__right,axiom,
% 0.17/0.42 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.42 ( ( ( divide_divide_rat @ A @ C )
% 0.17/0.42 = ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.42 = ( ( C = zero_zero_rat )
% 0.17/0.42 | ( A = B2 ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % divide_cancel_right
% 0.17/0.42 thf(fact_1353_division__ring__divide__zero,axiom,
% 0.17/0.42 ! [A: complex] :
% 0.17/0.42 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.17/0.42 = zero_zero_complex ) ).
% 0.17/0.42
% 0.17/0.42 % division_ring_divide_zero
% 0.17/0.42 thf(fact_1354_division__ring__divide__zero,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.17/0.42 = zero_zero_real ) ).
% 0.17/0.42
% 0.17/0.42 % division_ring_divide_zero
% 0.17/0.42 thf(fact_1355_division__ring__divide__zero,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.17/0.42 = zero_zero_rat ) ).
% 0.17/0.42
% 0.17/0.42 % division_ring_divide_zero
% 0.17/0.42 thf(fact_1356_dvd__0__right,axiom,
% 0.17/0.42 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_0_right
% 0.17/0.42 thf(fact_1357_dvd__0__right,axiom,
% 0.17/0.42 ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_0_right
% 0.17/0.42 thf(fact_1358_dvd__0__right,axiom,
% 0.17/0.42 ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_0_right
% 0.17/0.42 thf(fact_1359_dvd__0__right,axiom,
% 0.17/0.42 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_0_right
% 0.17/0.42 thf(fact_1360_dvd__0__right,axiom,
% 0.17/0.42 ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_0_right
% 0.17/0.42 thf(fact_1361_dvd__0__left__iff,axiom,
% 0.17/0.42 ! [A: code_integer] :
% 0.17/0.42 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.17/0.42 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_0_left_iff
% 0.17/0.42 thf(fact_1362_dvd__0__left__iff,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.17/0.42 = ( A = zero_zero_real ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_0_left_iff
% 0.17/0.42 thf(fact_1363_dvd__0__left__iff,axiom,
% 0.17/0.42 ! [A: rat] :
% 0.17/0.42 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.17/0.42 = ( A = zero_zero_rat ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_0_left_iff
% 0.17/0.42 thf(fact_1364_dvd__0__left__iff,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.17/0.42 = ( A = zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_0_left_iff
% 0.17/0.42 thf(fact_1365_dvd__0__left__iff,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.17/0.42 = ( A = zero_zero_int ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_0_left_iff
% 0.17/0.42 thf(fact_1366_dvd__add__triv__right__iff,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ A ) )
% 0.17/0.42 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_add_triv_right_iff
% 0.17/0.42 thf(fact_1367_dvd__add__triv__right__iff,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 0.17/0.42 = ( dvd_dvd_real @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_add_triv_right_iff
% 0.17/0.42 thf(fact_1368_dvd__add__triv__right__iff,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 0.17/0.42 = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_add_triv_right_iff
% 0.17/0.42 thf(fact_1369_dvd__add__triv__right__iff,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 0.17/0.42 = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_add_triv_right_iff
% 0.17/0.42 thf(fact_1370_dvd__add__triv__right__iff,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 0.17/0.42 = ( dvd_dvd_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_add_triv_right_iff
% 0.17/0.42 thf(fact_1371_dvd__add__triv__left__iff,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer] :
% 0.17/0.42 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.17/0.42 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_add_triv_left_iff
% 0.17/0.42 thf(fact_1372_dvd__add__triv__left__iff,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 0.17/0.42 = ( dvd_dvd_real @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_add_triv_left_iff
% 0.17/0.42 thf(fact_1373_dvd__add__triv__left__iff,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 0.17/0.42 = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_add_triv_left_iff
% 0.17/0.42 thf(fact_1374_dvd__add__triv__left__iff,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 0.17/0.42 = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_add_triv_left_iff
% 0.17/0.42 thf(fact_1375_dvd__add__triv__left__iff,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 0.17/0.42 = ( dvd_dvd_int @ A @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_add_triv_left_iff
% 0.17/0.42 thf(fact_1376_power__Suc0__right,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % power_Suc0_right
% 0.17/0.42 thf(fact_1377_power__Suc0__right,axiom,
% 0.17/0.42 ! [A: real] :
% 0.17/0.42 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % power_Suc0_right
% 0.17/0.42 thf(fact_1378_power__Suc0__right,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % power_Suc0_right
% 0.17/0.42 thf(fact_1379_power__Suc0__right,axiom,
% 0.17/0.42 ! [A: complex] :
% 0.17/0.42 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % power_Suc0_right
% 0.17/0.42 thf(fact_1380_mod__0,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % mod_0
% 0.17/0.42 thf(fact_1381_mod__0,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % mod_0
% 0.17/0.42 thf(fact_1382_mod__0,axiom,
% 0.17/0.42 ! [A: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.17/0.42 = zero_z3403309356797280102nteger ) ).
% 0.17/0.42
% 0.17/0.42 % mod_0
% 0.17/0.42 thf(fact_1383_mod__by__0,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mod_by_0
% 0.17/0.42 thf(fact_1384_mod__by__0,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ A @ zero_zero_int )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mod_by_0
% 0.17/0.42 thf(fact_1385_mod__by__0,axiom,
% 0.17/0.42 ! [A: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 0.17/0.42 = A ) ).
% 0.17/0.42
% 0.17/0.42 % mod_by_0
% 0.17/0.42 thf(fact_1386_mod__self,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ A @ A )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % mod_self
% 0.17/0.42 thf(fact_1387_mod__self,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ A @ A )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % mod_self
% 0.17/0.42 thf(fact_1388_mod__self,axiom,
% 0.17/0.42 ! [A: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ A @ A )
% 0.17/0.42 = zero_z3403309356797280102nteger ) ).
% 0.17/0.42
% 0.17/0.42 % mod_self
% 0.17/0.42 thf(fact_1389_bits__mod__0,axiom,
% 0.17/0.42 ! [A: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % bits_mod_0
% 0.17/0.42 thf(fact_1390_bits__mod__0,axiom,
% 0.17/0.42 ! [A: int] :
% 0.17/0.42 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % bits_mod_0
% 0.17/0.42 thf(fact_1391_bits__mod__0,axiom,
% 0.17/0.42 ! [A: code_integer] :
% 0.17/0.42 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.17/0.42 = zero_z3403309356797280102nteger ) ).
% 0.17/0.42
% 0.17/0.42 % bits_mod_0
% 0.17/0.42 thf(fact_1392_div__dvd__div,axiom,
% 0.17/0.42 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.42 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.42 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.17/0.42 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B2 @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 0.17/0.42 = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_dvd_div
% 0.17/0.42 thf(fact_1393_div__dvd__div,axiom,
% 0.17/0.42 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.42 ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.42 => ( ( dvd_dvd_nat @ A @ C )
% 0.17/0.42 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A ) @ ( divide_divide_nat @ C @ A ) )
% 0.17/0.42 = ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_dvd_div
% 0.17/0.42 thf(fact_1394_div__dvd__div,axiom,
% 0.17/0.42 ! [A: int,B2: int,C: int] :
% 0.17/0.42 ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.42 => ( ( dvd_dvd_int @ A @ C )
% 0.17/0.42 => ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A ) @ ( divide_divide_int @ C @ A ) )
% 0.17/0.42 = ( dvd_dvd_int @ B2 @ C ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_dvd_div
% 0.17/0.42 thf(fact_1395_zero__less__Suc,axiom,
% 0.17/0.42 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.17/0.42
% 0.17/0.42 % zero_less_Suc
% 0.17/0.42 thf(fact_1396_less__Suc0,axiom,
% 0.17/0.42 ! [N: nat] :
% 0.17/0.42 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.17/0.42 = ( N = zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % less_Suc0
% 0.17/0.42 thf(fact_1397_add__gr__0,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.17/0.42 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.17/0.42 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_gr_0
% 0.17/0.42 thf(fact_1398_one__eq__mult__iff,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( ( suc @ zero_zero_nat )
% 0.17/0.42 = ( times_times_nat @ M @ N ) )
% 0.17/0.42 = ( ( M
% 0.17/0.42 = ( suc @ zero_zero_nat ) )
% 0.17/0.42 & ( N
% 0.17/0.42 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % one_eq_mult_iff
% 0.17/0.42 thf(fact_1399_mult__eq__1__iff,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( ( times_times_nat @ M @ N )
% 0.17/0.42 = ( suc @ zero_zero_nat ) )
% 0.17/0.42 = ( ( M
% 0.17/0.42 = ( suc @ zero_zero_nat ) )
% 0.17/0.42 & ( N
% 0.17/0.42 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_eq_1_iff
% 0.17/0.42 thf(fact_1400_div__by__Suc__0,axiom,
% 0.17/0.42 ! [M: nat] :
% 0.17/0.42 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.17/0.42 = M ) ).
% 0.17/0.42
% 0.17/0.42 % div_by_Suc_0
% 0.17/0.42 thf(fact_1401_zero__less__diff,axiom,
% 0.17/0.42 ! [N: nat,M: nat] :
% 0.17/0.42 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.17/0.42 = ( ord_less_nat @ M @ N ) ) ).
% 0.17/0.42
% 0.17/0.42 % zero_less_diff
% 0.17/0.42 thf(fact_1402_nat__mult__less__cancel__disj,axiom,
% 0.17/0.42 ! [K: nat,M: nat,N: nat] :
% 0.17/0.42 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.17/0.42 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.42 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % nat_mult_less_cancel_disj
% 0.17/0.42 thf(fact_1403_nat__0__less__mult__iff,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.17/0.42 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.17/0.42 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % nat_0_less_mult_iff
% 0.17/0.42 thf(fact_1404_mult__less__cancel2,axiom,
% 0.17/0.42 ! [M: nat,K: nat,N: nat] :
% 0.17/0.42 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.17/0.42 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.42 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % mult_less_cancel2
% 0.17/0.42 thf(fact_1405_div__less,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( ord_less_nat @ M @ N )
% 0.17/0.42 => ( ( divide_divide_nat @ M @ N )
% 0.17/0.42 = zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % div_less
% 0.17/0.42 thf(fact_1406_not__real__square__gt__zero,axiom,
% 0.17/0.42 ! [X4: real] :
% 0.17/0.42 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X4 @ X4 ) ) )
% 0.17/0.42 = ( X4 = zero_zero_real ) ) ).
% 0.17/0.42
% 0.17/0.42 % not_real_square_gt_zero
% 0.17/0.42 thf(fact_1407_diff__is__0__eq,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( ( minus_minus_nat @ M @ N )
% 0.17/0.42 = zero_zero_nat )
% 0.17/0.42 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_is_0_eq
% 0.17/0.42 thf(fact_1408_diff__is__0__eq_H,axiom,
% 0.17/0.42 ! [M: nat,N: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.42 => ( ( minus_minus_nat @ M @ N )
% 0.17/0.42 = zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % diff_is_0_eq'
% 0.17/0.42 thf(fact_1409_dvd__1__iff__1,axiom,
% 0.17/0.42 ! [M: nat] :
% 0.17/0.42 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.17/0.42 = ( M
% 0.17/0.42 = ( suc @ zero_zero_nat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_1_iff_1
% 0.17/0.42 thf(fact_1410_dvd__1__left,axiom,
% 0.17/0.42 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.17/0.42
% 0.17/0.42 % dvd_1_left
% 0.17/0.42 thf(fact_1411_less__one,axiom,
% 0.17/0.42 ! [N: nat] :
% 0.17/0.42 ( ( ord_less_nat @ N @ one_one_nat )
% 0.17/0.42 = ( N = zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % less_one
% 0.17/0.42 thf(fact_1412_power__Suc__0,axiom,
% 0.17/0.42 ! [N: nat] :
% 0.17/0.42 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.17/0.42 = ( suc @ zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % power_Suc_0
% 0.17/0.42 thf(fact_1413_nat__power__eq__Suc__0__iff,axiom,
% 0.17/0.42 ! [X4: nat,M: nat] :
% 0.17/0.42 ( ( ( power_power_nat @ X4 @ M )
% 0.17/0.42 = ( suc @ zero_zero_nat ) )
% 0.17/0.42 = ( ( M = zero_zero_nat )
% 0.17/0.42 | ( X4
% 0.17/0.42 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % nat_power_eq_Suc_0_iff
% 0.17/0.42 thf(fact_1414_nat__zero__less__power__iff,axiom,
% 0.17/0.42 ! [X4: nat,N: nat] :
% 0.17/0.42 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N ) )
% 0.17/0.42 = ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 0.17/0.42 | ( N = zero_zero_nat ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % nat_zero_less_power_iff
% 0.17/0.42 thf(fact_1415_nat__mult__div__cancel__disj,axiom,
% 0.17/0.42 ! [K: nat,M: nat,N: nat] :
% 0.17/0.42 ( ( ( K = zero_zero_nat )
% 0.17/0.42 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.17/0.42 = zero_zero_nat ) )
% 0.17/0.42 & ( ( K != zero_zero_nat )
% 0.17/0.42 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.17/0.42 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % nat_mult_div_cancel_disj
% 0.17/0.42 thf(fact_1416_mod__by__Suc__0,axiom,
% 0.17/0.42 ! [M: nat] :
% 0.17/0.42 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.17/0.42 = zero_zero_nat ) ).
% 0.17/0.42
% 0.17/0.42 % mod_by_Suc_0
% 0.17/0.42 thf(fact_1417_nat__mult__dvd__cancel__disj,axiom,
% 0.17/0.42 ! [K: nat,M: nat,N: nat] :
% 0.17/0.42 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.17/0.42 = ( ( K = zero_zero_nat )
% 0.17/0.42 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.17/0.42
% 0.17/0.42 % nat_mult_dvd_cancel_disj
% 0.17/0.42 thf(fact_1418_nat__dvd__1__iff__1,axiom,
% 0.17/0.42 ! [M: nat] :
% 0.17/0.42 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.17/0.42 = ( M = one_one_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % nat_dvd_1_iff_1
% 0.17/0.42 thf(fact_1419_signed__take__bit__of__0,axiom,
% 0.17/0.42 ! [N: nat] :
% 0.17/0.42 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % signed_take_bit_of_0
% 0.17/0.42 thf(fact_1420_dbl__simps_I2_J,axiom,
% 0.17/0.42 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 0.17/0.42 = zero_zero_real ) ).
% 0.17/0.42
% 0.17/0.42 % dbl_simps(2)
% 0.17/0.42 thf(fact_1421_dbl__simps_I2_J,axiom,
% 0.17/0.42 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 0.17/0.42 = zero_zero_rat ) ).
% 0.17/0.42
% 0.17/0.42 % dbl_simps(2)
% 0.17/0.42 thf(fact_1422_dbl__simps_I2_J,axiom,
% 0.17/0.42 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 0.17/0.42 = zero_zero_int ) ).
% 0.17/0.42
% 0.17/0.42 % dbl_simps(2)
% 0.17/0.42 thf(fact_1423_add__le__same__cancel1,axiom,
% 0.17/0.42 ! [B2: real,A: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
% 0.17/0.42 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_same_cancel1
% 0.17/0.42 thf(fact_1424_add__le__same__cancel1,axiom,
% 0.17/0.42 ! [B2: rat,A: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
% 0.17/0.42 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_same_cancel1
% 0.17/0.42 thf(fact_1425_add__le__same__cancel1,axiom,
% 0.17/0.42 ! [B2: nat,A: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 0.17/0.42 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_same_cancel1
% 0.17/0.42 thf(fact_1426_add__le__same__cancel1,axiom,
% 0.17/0.42 ! [B2: int,A: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 0.17/0.42 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_same_cancel1
% 0.17/0.42 thf(fact_1427_add__le__same__cancel2,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_same_cancel2
% 0.17/0.42 thf(fact_1428_add__le__same__cancel2,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_same_cancel2
% 0.17/0.42 thf(fact_1429_add__le__same__cancel2,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_same_cancel2
% 0.17/0.42 thf(fact_1430_add__le__same__cancel2,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.17/0.42 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.17/0.42
% 0.17/0.42 % add_le_same_cancel2
% 0.17/0.42 thf(fact_1431_le__add__same__cancel1,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 0.17/0.42 = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_add_same_cancel1
% 0.17/0.42 thf(fact_1432_le__add__same__cancel1,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 0.17/0.42 = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_add_same_cancel1
% 0.17/0.42 thf(fact_1433_le__add__same__cancel1,axiom,
% 0.17/0.42 ! [A: nat,B2: nat] :
% 0.17/0.42 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 0.17/0.42 = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_add_same_cancel1
% 0.17/0.42 thf(fact_1434_le__add__same__cancel1,axiom,
% 0.17/0.42 ! [A: int,B2: int] :
% 0.17/0.42 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 0.17/0.42 = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_add_same_cancel1
% 0.17/0.42 thf(fact_1435_le__add__same__cancel2,axiom,
% 0.17/0.42 ! [A: real,B2: real] :
% 0.17/0.42 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 0.17/0.42 = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% 0.17/0.42
% 0.17/0.42 % le_add_same_cancel2
% 0.17/0.42 thf(fact_1436_le__add__same__cancel2,axiom,
% 0.17/0.42 ! [A: rat,B2: rat] :
% 0.17/0.42 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 0.17/0.42 = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % le_add_same_cancel2
% 0.17/0.43 thf(fact_1437_le__add__same__cancel2,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % le_add_same_cancel2
% 0.17/0.43 thf(fact_1438_le__add__same__cancel2,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % le_add_same_cancel2
% 0.17/0.43 thf(fact_1439_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.17/0.43 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % double_add_le_zero_iff_single_add_le_zero
% 0.17/0.43 thf(fact_1440_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.17/0.43 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % double_add_le_zero_iff_single_add_le_zero
% 0.17/0.43 thf(fact_1441_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.17/0.43 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % double_add_le_zero_iff_single_add_le_zero
% 0.17/0.43 thf(fact_1442_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.17/0.43 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_le_double_add_iff_zero_le_single_add
% 0.17/0.43 thf(fact_1443_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.17/0.43 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_le_double_add_iff_zero_le_single_add
% 0.17/0.43 thf(fact_1444_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.17/0.43 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_le_double_add_iff_zero_le_single_add
% 0.17/0.43 thf(fact_1445_diff__ge__0__iff__ge,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_eq_real @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % diff_ge_0_iff_ge
% 0.17/0.43 thf(fact_1446_diff__ge__0__iff__ge,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_eq_rat @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % diff_ge_0_iff_ge
% 0.17/0.43 thf(fact_1447_diff__ge__0__iff__ge,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_eq_int @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % diff_ge_0_iff_ge
% 0.17/0.43 thf(fact_1448_add__less__same__cancel1,axiom,
% 0.17/0.43 ! [B2: real,A: real] :
% 0.17/0.43 ( ( ord_less_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
% 0.17/0.43 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % add_less_same_cancel1
% 0.17/0.43 thf(fact_1449_add__less__same__cancel1,axiom,
% 0.17/0.43 ! [B2: rat,A: rat] :
% 0.17/0.43 ( ( ord_less_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
% 0.17/0.43 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % add_less_same_cancel1
% 0.17/0.43 thf(fact_1450_add__less__same__cancel1,axiom,
% 0.17/0.43 ! [B2: nat,A: nat] :
% 0.17/0.43 ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 0.17/0.43 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % add_less_same_cancel1
% 0.17/0.43 thf(fact_1451_add__less__same__cancel1,axiom,
% 0.17/0.43 ! [B2: int,A: int] :
% 0.17/0.43 ( ( ord_less_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 0.17/0.43 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % add_less_same_cancel1
% 0.17/0.43 thf(fact_1452_add__less__same__cancel2,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 0.17/0.43 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % add_less_same_cancel2
% 0.17/0.43 thf(fact_1453_add__less__same__cancel2,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 0.17/0.43 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % add_less_same_cancel2
% 0.17/0.43 thf(fact_1454_add__less__same__cancel2,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 0.17/0.43 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % add_less_same_cancel2
% 0.17/0.43 thf(fact_1455_add__less__same__cancel2,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.17/0.43 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % add_less_same_cancel2
% 0.17/0.43 thf(fact_1456_less__add__same__cancel1,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_add_same_cancel1
% 0.17/0.43 thf(fact_1457_less__add__same__cancel1,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_add_same_cancel1
% 0.17/0.43 thf(fact_1458_less__add__same__cancel1,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_add_same_cancel1
% 0.17/0.43 thf(fact_1459_less__add__same__cancel1,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_add_same_cancel1
% 0.17/0.43 thf(fact_1460_less__add__same__cancel2,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_add_same_cancel2
% 0.17/0.43 thf(fact_1461_less__add__same__cancel2,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_add_same_cancel2
% 0.17/0.43 thf(fact_1462_less__add__same__cancel2,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_add_same_cancel2
% 0.17/0.43 thf(fact_1463_less__add__same__cancel2,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_add_same_cancel2
% 0.17/0.43 thf(fact_1464_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.17/0.43 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % double_add_less_zero_iff_single_add_less_zero
% 0.17/0.43 thf(fact_1465_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.17/0.43 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % double_add_less_zero_iff_single_add_less_zero
% 0.17/0.43 thf(fact_1466_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.17/0.43 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % double_add_less_zero_iff_single_add_less_zero
% 0.17/0.43 thf(fact_1467_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.17/0.43 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_less_double_add_iff_zero_less_single_add
% 0.17/0.43 thf(fact_1468_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.17/0.43 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_less_double_add_iff_zero_less_single_add
% 0.17/0.43 thf(fact_1469_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.17/0.43 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_less_double_add_iff_zero_less_single_add
% 0.17/0.43 thf(fact_1470_diff__gt__0__iff__gt,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_real @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % diff_gt_0_iff_gt
% 0.17/0.43 thf(fact_1471_diff__gt__0__iff__gt,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_rat @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % diff_gt_0_iff_gt
% 0.17/0.43 thf(fact_1472_diff__gt__0__iff__gt,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_int @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % diff_gt_0_iff_gt
% 0.17/0.43 thf(fact_1473_mult__cancel__left1,axiom,
% 0.17/0.43 ! [C: complex,B2: complex] :
% 0.17/0.43 ( ( C
% 0.17/0.43 = ( times_times_complex @ C @ B2 ) )
% 0.17/0.43 = ( ( C = zero_zero_complex )
% 0.17/0.43 | ( B2 = one_one_complex ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_left1
% 0.17/0.43 thf(fact_1474_mult__cancel__left1,axiom,
% 0.17/0.43 ! [C: real,B2: real] :
% 0.17/0.43 ( ( C
% 0.17/0.43 = ( times_times_real @ C @ B2 ) )
% 0.17/0.43 = ( ( C = zero_zero_real )
% 0.17/0.43 | ( B2 = one_one_real ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_left1
% 0.17/0.43 thf(fact_1475_mult__cancel__left1,axiom,
% 0.17/0.43 ! [C: rat,B2: rat] :
% 0.17/0.43 ( ( C
% 0.17/0.43 = ( times_times_rat @ C @ B2 ) )
% 0.17/0.43 = ( ( C = zero_zero_rat )
% 0.17/0.43 | ( B2 = one_one_rat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_left1
% 0.17/0.43 thf(fact_1476_mult__cancel__left1,axiom,
% 0.17/0.43 ! [C: int,B2: int] :
% 0.17/0.43 ( ( C
% 0.17/0.43 = ( times_times_int @ C @ B2 ) )
% 0.17/0.43 = ( ( C = zero_zero_int )
% 0.17/0.43 | ( B2 = one_one_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_left1
% 0.17/0.43 thf(fact_1477_mult__cancel__left2,axiom,
% 0.17/0.43 ! [C: complex,A: complex] :
% 0.17/0.43 ( ( ( times_times_complex @ C @ A )
% 0.17/0.43 = C )
% 0.17/0.43 = ( ( C = zero_zero_complex )
% 0.17/0.43 | ( A = one_one_complex ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_left2
% 0.17/0.43 thf(fact_1478_mult__cancel__left2,axiom,
% 0.17/0.43 ! [C: real,A: real] :
% 0.17/0.43 ( ( ( times_times_real @ C @ A )
% 0.17/0.43 = C )
% 0.17/0.43 = ( ( C = zero_zero_real )
% 0.17/0.43 | ( A = one_one_real ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_left2
% 0.17/0.43 thf(fact_1479_mult__cancel__left2,axiom,
% 0.17/0.43 ! [C: rat,A: rat] :
% 0.17/0.43 ( ( ( times_times_rat @ C @ A )
% 0.17/0.43 = C )
% 0.17/0.43 = ( ( C = zero_zero_rat )
% 0.17/0.43 | ( A = one_one_rat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_left2
% 0.17/0.43 thf(fact_1480_mult__cancel__left2,axiom,
% 0.17/0.43 ! [C: int,A: int] :
% 0.17/0.43 ( ( ( times_times_int @ C @ A )
% 0.17/0.43 = C )
% 0.17/0.43 = ( ( C = zero_zero_int )
% 0.17/0.43 | ( A = one_one_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_left2
% 0.17/0.43 thf(fact_1481_mult__cancel__right1,axiom,
% 0.17/0.43 ! [C: complex,B2: complex] :
% 0.17/0.43 ( ( C
% 0.17/0.43 = ( times_times_complex @ B2 @ C ) )
% 0.17/0.43 = ( ( C = zero_zero_complex )
% 0.17/0.43 | ( B2 = one_one_complex ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_right1
% 0.17/0.43 thf(fact_1482_mult__cancel__right1,axiom,
% 0.17/0.43 ! [C: real,B2: real] :
% 0.17/0.43 ( ( C
% 0.17/0.43 = ( times_times_real @ B2 @ C ) )
% 0.17/0.43 = ( ( C = zero_zero_real )
% 0.17/0.43 | ( B2 = one_one_real ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_right1
% 0.17/0.43 thf(fact_1483_mult__cancel__right1,axiom,
% 0.17/0.43 ! [C: rat,B2: rat] :
% 0.17/0.43 ( ( C
% 0.17/0.43 = ( times_times_rat @ B2 @ C ) )
% 0.17/0.43 = ( ( C = zero_zero_rat )
% 0.17/0.43 | ( B2 = one_one_rat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_right1
% 0.17/0.43 thf(fact_1484_mult__cancel__right1,axiom,
% 0.17/0.43 ! [C: int,B2: int] :
% 0.17/0.43 ( ( C
% 0.17/0.43 = ( times_times_int @ B2 @ C ) )
% 0.17/0.43 = ( ( C = zero_zero_int )
% 0.17/0.43 | ( B2 = one_one_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_right1
% 0.17/0.43 thf(fact_1485_mult__cancel__right2,axiom,
% 0.17/0.43 ! [A: complex,C: complex] :
% 0.17/0.43 ( ( ( times_times_complex @ A @ C )
% 0.17/0.43 = C )
% 0.17/0.43 = ( ( C = zero_zero_complex )
% 0.17/0.43 | ( A = one_one_complex ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_right2
% 0.17/0.43 thf(fact_1486_mult__cancel__right2,axiom,
% 0.17/0.43 ! [A: real,C: real] :
% 0.17/0.43 ( ( ( times_times_real @ A @ C )
% 0.17/0.43 = C )
% 0.17/0.43 = ( ( C = zero_zero_real )
% 0.17/0.43 | ( A = one_one_real ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_right2
% 0.17/0.43 thf(fact_1487_mult__cancel__right2,axiom,
% 0.17/0.43 ! [A: rat,C: rat] :
% 0.17/0.43 ( ( ( times_times_rat @ A @ C )
% 0.17/0.43 = C )
% 0.17/0.43 = ( ( C = zero_zero_rat )
% 0.17/0.43 | ( A = one_one_rat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_right2
% 0.17/0.43 thf(fact_1488_mult__cancel__right2,axiom,
% 0.17/0.43 ! [A: int,C: int] :
% 0.17/0.43 ( ( ( times_times_int @ A @ C )
% 0.17/0.43 = C )
% 0.17/0.43 = ( ( C = zero_zero_int )
% 0.17/0.43 | ( A = one_one_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_cancel_right2
% 0.17/0.43 thf(fact_1489_sum__squares__eq__zero__iff,axiom,
% 0.17/0.43 ! [X4: real,Y3: real] :
% 0.17/0.43 ( ( ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 0.17/0.43 = zero_zero_real )
% 0.17/0.43 = ( ( X4 = zero_zero_real )
% 0.17/0.43 & ( Y3 = zero_zero_real ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % sum_squares_eq_zero_iff
% 0.17/0.43 thf(fact_1490_sum__squares__eq__zero__iff,axiom,
% 0.17/0.43 ! [X4: rat,Y3: rat] :
% 0.17/0.43 ( ( ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 0.17/0.43 = zero_zero_rat )
% 0.17/0.43 = ( ( X4 = zero_zero_rat )
% 0.17/0.43 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % sum_squares_eq_zero_iff
% 0.17/0.43 thf(fact_1491_sum__squares__eq__zero__iff,axiom,
% 0.17/0.43 ! [X4: int,Y3: int] :
% 0.17/0.43 ( ( ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 = ( ( X4 = zero_zero_int )
% 0.17/0.43 & ( Y3 = zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % sum_squares_eq_zero_iff
% 0.17/0.43 thf(fact_1492_div__mult__mult1__if,axiom,
% 0.17/0.43 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.43 ( ( ( C = zero_zero_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 0.17/0.43 = zero_zero_nat ) )
% 0.17/0.43 & ( ( C != zero_zero_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 0.17/0.43 = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_mult1_if
% 0.17/0.43 thf(fact_1493_div__mult__mult1__if,axiom,
% 0.17/0.43 ! [C: int,A: int,B2: int] :
% 0.17/0.43 ( ( ( C = zero_zero_int )
% 0.17/0.43 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.43 = zero_zero_int ) )
% 0.17/0.43 & ( ( C != zero_zero_int )
% 0.17/0.43 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.43 = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_mult1_if
% 0.17/0.43 thf(fact_1494_div__mult__mult2,axiom,
% 0.17/0.43 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.43 ( ( C != zero_zero_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 0.17/0.43 = ( divide_divide_nat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_mult2
% 0.17/0.43 thf(fact_1495_div__mult__mult2,axiom,
% 0.17/0.43 ! [C: int,A: int,B2: int] :
% 0.17/0.43 ( ( C != zero_zero_int )
% 0.17/0.43 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.17/0.43 = ( divide_divide_int @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_mult2
% 0.17/0.43 thf(fact_1496_div__mult__mult1,axiom,
% 0.17/0.43 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.43 ( ( C != zero_zero_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 0.17/0.43 = ( divide_divide_nat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_mult1
% 0.17/0.43 thf(fact_1497_div__mult__mult1,axiom,
% 0.17/0.43 ! [C: int,A: int,B2: int] :
% 0.17/0.43 ( ( C != zero_zero_int )
% 0.17/0.43 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.43 = ( divide_divide_int @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_mult1
% 0.17/0.43 thf(fact_1498_nonzero__mult__div__cancel__left,axiom,
% 0.17/0.43 ! [A: complex,B2: complex] :
% 0.17/0.43 ( ( A != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ A )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_div_cancel_left
% 0.17/0.43 thf(fact_1499_nonzero__mult__div__cancel__left,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( A != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ A )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_div_cancel_left
% 0.17/0.43 thf(fact_1500_nonzero__mult__div__cancel__left,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( A != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ A )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_div_cancel_left
% 0.17/0.43 thf(fact_1501_nonzero__mult__div__cancel__left,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( A != zero_zero_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ A )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_div_cancel_left
% 0.17/0.43 thf(fact_1502_nonzero__mult__div__cancel__left,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( A != zero_zero_int )
% 0.17/0.43 => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ A )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_div_cancel_left
% 0.17/0.43 thf(fact_1503_nonzero__mult__div__cancel__right,axiom,
% 0.17/0.43 ! [B2: complex,A: complex] :
% 0.17/0.43 ( ( B2 != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ B2 )
% 0.17/0.43 = A ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_div_cancel_right
% 0.17/0.43 thf(fact_1504_nonzero__mult__div__cancel__right,axiom,
% 0.17/0.43 ! [B2: real,A: real] :
% 0.17/0.43 ( ( B2 != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ B2 )
% 0.17/0.43 = A ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_div_cancel_right
% 0.17/0.43 thf(fact_1505_nonzero__mult__div__cancel__right,axiom,
% 0.17/0.43 ! [B2: rat,A: rat] :
% 0.17/0.43 ( ( B2 != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ B2 )
% 0.17/0.43 = A ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_div_cancel_right
% 0.17/0.43 thf(fact_1506_nonzero__mult__div__cancel__right,axiom,
% 0.17/0.43 ! [B2: nat,A: nat] :
% 0.17/0.43 ( ( B2 != zero_zero_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
% 0.17/0.43 = A ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_div_cancel_right
% 0.17/0.43 thf(fact_1507_nonzero__mult__div__cancel__right,axiom,
% 0.17/0.43 ! [B2: int,A: int] :
% 0.17/0.43 ( ( B2 != zero_zero_int )
% 0.17/0.43 => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ B2 )
% 0.17/0.43 = A ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_div_cancel_right
% 0.17/0.43 thf(fact_1508_mult__divide__mult__cancel__left__if,axiom,
% 0.17/0.43 ! [C: complex,A: complex,B2: complex] :
% 0.17/0.43 ( ( ( C = zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 0.17/0.43 = zero_zero_complex ) )
% 0.17/0.43 & ( ( C != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 0.17/0.43 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_divide_mult_cancel_left_if
% 0.17/0.43 thf(fact_1509_mult__divide__mult__cancel__left__if,axiom,
% 0.17/0.43 ! [C: real,A: real,B2: real] :
% 0.17/0.43 ( ( ( C = zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.43 = zero_zero_real ) )
% 0.17/0.43 & ( ( C != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.43 = ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_divide_mult_cancel_left_if
% 0.17/0.43 thf(fact_1510_mult__divide__mult__cancel__left__if,axiom,
% 0.17/0.43 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.43 ( ( ( C = zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.43 = zero_zero_rat ) )
% 0.17/0.43 & ( ( C != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.43 = ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_divide_mult_cancel_left_if
% 0.17/0.43 thf(fact_1511_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.17/0.43 ! [C: complex,A: complex,B2: complex] :
% 0.17/0.43 ( ( C != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 0.17/0.43 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_left
% 0.17/0.43 thf(fact_1512_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.17/0.43 ! [C: real,A: real,B2: real] :
% 0.17/0.43 ( ( C != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.43 = ( divide_divide_real @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_left
% 0.17/0.43 thf(fact_1513_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.17/0.43 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.43 ( ( C != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.43 = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_left
% 0.17/0.43 thf(fact_1514_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.17/0.43 ! [C: complex,A: complex,B2: complex] :
% 0.17/0.43 ( ( C != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B2 @ C ) )
% 0.17/0.43 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_left2
% 0.17/0.43 thf(fact_1515_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.17/0.43 ! [C: real,A: real,B2: real] :
% 0.17/0.43 ( ( C != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B2 @ C ) )
% 0.17/0.43 = ( divide_divide_real @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_left2
% 0.17/0.43 thf(fact_1516_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.17/0.43 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.43 ( ( C != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B2 @ C ) )
% 0.17/0.43 = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_left2
% 0.17/0.43 thf(fact_1517_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.17/0.43 ! [C: complex,A: complex,B2: complex] :
% 0.17/0.43 ( ( C != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) )
% 0.17/0.43 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_right
% 0.17/0.43 thf(fact_1518_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.17/0.43 ! [C: real,A: real,B2: real] :
% 0.17/0.43 ( ( C != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.17/0.43 = ( divide_divide_real @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_right
% 0.17/0.43 thf(fact_1519_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.17/0.43 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.43 ( ( C != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.17/0.43 = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_right
% 0.17/0.43 thf(fact_1520_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.17/0.43 ! [C: complex,A: complex,B2: complex] :
% 0.17/0.43 ( ( C != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B2 ) )
% 0.17/0.43 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_right2
% 0.17/0.43 thf(fact_1521_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.17/0.43 ! [C: real,A: real,B2: real] :
% 0.17/0.43 ( ( C != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.43 = ( divide_divide_real @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_right2
% 0.17/0.43 thf(fact_1522_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.17/0.43 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.43 ( ( C != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.43 = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_mult_divide_mult_cancel_right2
% 0.17/0.43 thf(fact_1523_diff__numeral__special_I9_J,axiom,
% 0.17/0.43 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 0.17/0.43 = zero_zero_complex ) ).
% 0.17/0.43
% 0.17/0.43 % diff_numeral_special(9)
% 0.17/0.43 thf(fact_1524_diff__numeral__special_I9_J,axiom,
% 0.17/0.43 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.17/0.43 = zero_zero_real ) ).
% 0.17/0.43
% 0.17/0.43 % diff_numeral_special(9)
% 0.17/0.43 thf(fact_1525_diff__numeral__special_I9_J,axiom,
% 0.17/0.43 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.17/0.43 = zero_zero_rat ) ).
% 0.17/0.43
% 0.17/0.43 % diff_numeral_special(9)
% 0.17/0.43 thf(fact_1526_diff__numeral__special_I9_J,axiom,
% 0.17/0.43 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.17/0.43 = zero_zero_int ) ).
% 0.17/0.43
% 0.17/0.43 % diff_numeral_special(9)
% 0.17/0.43 thf(fact_1527_diff__add__zero,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % diff_add_zero
% 0.17/0.43 thf(fact_1528_div__self,axiom,
% 0.17/0.43 ! [A: complex] :
% 0.17/0.43 ( ( A != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.17/0.43 = one_one_complex ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_self
% 0.17/0.43 thf(fact_1529_div__self,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( A != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ A @ A )
% 0.17/0.43 = one_one_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_self
% 0.17/0.43 thf(fact_1530_div__self,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( A != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ A @ A )
% 0.17/0.43 = one_one_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_self
% 0.17/0.43 thf(fact_1531_div__self,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( A != zero_zero_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ A @ A )
% 0.17/0.43 = one_one_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_self
% 0.17/0.43 thf(fact_1532_div__self,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( A != zero_zero_int )
% 0.17/0.43 => ( ( divide_divide_int @ A @ A )
% 0.17/0.43 = one_one_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_self
% 0.17/0.43 thf(fact_1533_divide__eq__1__iff,axiom,
% 0.17/0.43 ! [A: complex,B2: complex] :
% 0.17/0.43 ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 0.17/0.43 = one_one_complex )
% 0.17/0.43 = ( ( B2 != zero_zero_complex )
% 0.17/0.43 & ( A = B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_eq_1_iff
% 0.17/0.43 thf(fact_1534_divide__eq__1__iff,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ( divide_divide_real @ A @ B2 )
% 0.17/0.43 = one_one_real )
% 0.17/0.43 = ( ( B2 != zero_zero_real )
% 0.17/0.43 & ( A = B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_eq_1_iff
% 0.17/0.43 thf(fact_1535_divide__eq__1__iff,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ( divide_divide_rat @ A @ B2 )
% 0.17/0.43 = one_one_rat )
% 0.17/0.43 = ( ( B2 != zero_zero_rat )
% 0.17/0.43 & ( A = B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_eq_1_iff
% 0.17/0.43 thf(fact_1536_one__eq__divide__iff,axiom,
% 0.17/0.43 ! [A: complex,B2: complex] :
% 0.17/0.43 ( ( one_one_complex
% 0.17/0.43 = ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.17/0.43 = ( ( B2 != zero_zero_complex )
% 0.17/0.43 & ( A = B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % one_eq_divide_iff
% 0.17/0.43 thf(fact_1537_one__eq__divide__iff,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( one_one_real
% 0.17/0.43 = ( divide_divide_real @ A @ B2 ) )
% 0.17/0.43 = ( ( B2 != zero_zero_real )
% 0.17/0.43 & ( A = B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % one_eq_divide_iff
% 0.17/0.43 thf(fact_1538_one__eq__divide__iff,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( one_one_rat
% 0.17/0.43 = ( divide_divide_rat @ A @ B2 ) )
% 0.17/0.43 = ( ( B2 != zero_zero_rat )
% 0.17/0.43 & ( A = B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % one_eq_divide_iff
% 0.17/0.43 thf(fact_1539_divide__self,axiom,
% 0.17/0.43 ! [A: complex] :
% 0.17/0.43 ( ( A != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.17/0.43 = one_one_complex ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_self
% 0.17/0.43 thf(fact_1540_divide__self,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( A != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ A @ A )
% 0.17/0.43 = one_one_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_self
% 0.17/0.43 thf(fact_1541_divide__self,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( A != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ A @ A )
% 0.17/0.43 = one_one_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_self
% 0.17/0.43 thf(fact_1542_divide__self__if,axiom,
% 0.17/0.43 ! [A: complex] :
% 0.17/0.43 ( ( ( A = zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.17/0.43 = zero_zero_complex ) )
% 0.17/0.43 & ( ( A != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.17/0.43 = one_one_complex ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_self_if
% 0.17/0.43 thf(fact_1543_divide__self__if,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ( A = zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ A @ A )
% 0.17/0.43 = zero_zero_real ) )
% 0.17/0.43 & ( ( A != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ A @ A )
% 0.17/0.43 = one_one_real ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_self_if
% 0.17/0.43 thf(fact_1544_divide__self__if,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ( A = zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ A @ A )
% 0.17/0.43 = zero_zero_rat ) )
% 0.17/0.43 & ( ( A != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ A @ A )
% 0.17/0.43 = one_one_rat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_self_if
% 0.17/0.43 thf(fact_1545_divide__eq__eq__1,axiom,
% 0.17/0.43 ! [B2: real,A: real] :
% 0.17/0.43 ( ( ( divide_divide_real @ B2 @ A )
% 0.17/0.43 = one_one_real )
% 0.17/0.43 = ( ( A != zero_zero_real )
% 0.17/0.43 & ( A = B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_eq_eq_1
% 0.17/0.43 thf(fact_1546_divide__eq__eq__1,axiom,
% 0.17/0.43 ! [B2: rat,A: rat] :
% 0.17/0.43 ( ( ( divide_divide_rat @ B2 @ A )
% 0.17/0.43 = one_one_rat )
% 0.17/0.43 = ( ( A != zero_zero_rat )
% 0.17/0.43 & ( A = B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_eq_eq_1
% 0.17/0.43 thf(fact_1547_eq__divide__eq__1,axiom,
% 0.17/0.43 ! [B2: real,A: real] :
% 0.17/0.43 ( ( one_one_real
% 0.17/0.43 = ( divide_divide_real @ B2 @ A ) )
% 0.17/0.43 = ( ( A != zero_zero_real )
% 0.17/0.43 & ( A = B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % eq_divide_eq_1
% 0.17/0.43 thf(fact_1548_eq__divide__eq__1,axiom,
% 0.17/0.43 ! [B2: rat,A: rat] :
% 0.17/0.43 ( ( one_one_rat
% 0.17/0.43 = ( divide_divide_rat @ B2 @ A ) )
% 0.17/0.43 = ( ( A != zero_zero_rat )
% 0.17/0.43 & ( A = B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % eq_divide_eq_1
% 0.17/0.43 thf(fact_1549_one__divide__eq__0__iff,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ( divide_divide_real @ one_one_real @ A )
% 0.17/0.43 = zero_zero_real )
% 0.17/0.43 = ( A = zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % one_divide_eq_0_iff
% 0.17/0.43 thf(fact_1550_one__divide__eq__0__iff,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 0.17/0.43 = zero_zero_rat )
% 0.17/0.43 = ( A = zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % one_divide_eq_0_iff
% 0.17/0.43 thf(fact_1551_zero__eq__1__divide__iff,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( zero_zero_real
% 0.17/0.43 = ( divide_divide_real @ one_one_real @ A ) )
% 0.17/0.43 = ( A = zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_eq_1_divide_iff
% 0.17/0.43 thf(fact_1552_zero__eq__1__divide__iff,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( zero_zero_rat
% 0.17/0.43 = ( divide_divide_rat @ one_one_rat @ A ) )
% 0.17/0.43 = ( A = zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_eq_1_divide_iff
% 0.17/0.43 thf(fact_1553_dvd__times__right__cancel__iff,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.43 ( ( A != zero_z3403309356797280102nteger )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 0.17/0.43 = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_times_right_cancel_iff
% 0.17/0.43 thf(fact_1554_dvd__times__right__cancel__iff,axiom,
% 0.17/0.43 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.43 ( ( A != zero_zero_nat )
% 0.17/0.43 => ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) )
% 0.17/0.43 = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_times_right_cancel_iff
% 0.17/0.43 thf(fact_1555_dvd__times__right__cancel__iff,axiom,
% 0.17/0.43 ! [A: int,B2: int,C: int] :
% 0.17/0.43 ( ( A != zero_zero_int )
% 0.17/0.43 => ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) )
% 0.17/0.43 = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_times_right_cancel_iff
% 0.17/0.43 thf(fact_1556_dvd__times__left__cancel__iff,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.43 ( ( A != zero_z3403309356797280102nteger )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 0.17/0.43 = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_times_left_cancel_iff
% 0.17/0.43 thf(fact_1557_dvd__times__left__cancel__iff,axiom,
% 0.17/0.43 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.43 ( ( A != zero_zero_nat )
% 0.17/0.43 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) )
% 0.17/0.43 = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_times_left_cancel_iff
% 0.17/0.43 thf(fact_1558_dvd__times__left__cancel__iff,axiom,
% 0.17/0.43 ! [A: int,B2: int,C: int] :
% 0.17/0.43 ( ( A != zero_zero_int )
% 0.17/0.43 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) )
% 0.17/0.43 = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_times_left_cancel_iff
% 0.17/0.43 thf(fact_1559_dvd__mult__cancel__right,axiom,
% 0.17/0.43 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.17/0.43 = ( ( C = zero_z3403309356797280102nteger )
% 0.17/0.43 | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_cancel_right
% 0.17/0.43 thf(fact_1560_dvd__mult__cancel__right,axiom,
% 0.17/0.43 ! [A: real,C: real,B2: real] :
% 0.17/0.43 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.17/0.43 = ( ( C = zero_zero_real )
% 0.17/0.43 | ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_cancel_right
% 0.17/0.43 thf(fact_1561_dvd__mult__cancel__right,axiom,
% 0.17/0.43 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.43 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.17/0.43 = ( ( C = zero_zero_rat )
% 0.17/0.43 | ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_cancel_right
% 0.17/0.43 thf(fact_1562_dvd__mult__cancel__right,axiom,
% 0.17/0.43 ! [A: int,C: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.17/0.43 = ( ( C = zero_zero_int )
% 0.17/0.43 | ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_cancel_right
% 0.17/0.43 thf(fact_1563_dvd__mult__cancel__left,axiom,
% 0.17/0.43 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
% 0.17/0.43 = ( ( C = zero_z3403309356797280102nteger )
% 0.17/0.43 | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_cancel_left
% 0.17/0.43 thf(fact_1564_dvd__mult__cancel__left,axiom,
% 0.17/0.43 ! [C: real,A: real,B2: real] :
% 0.17/0.43 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.43 = ( ( C = zero_zero_real )
% 0.17/0.43 | ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_cancel_left
% 0.17/0.43 thf(fact_1565_dvd__mult__cancel__left,axiom,
% 0.17/0.43 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.43 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.43 = ( ( C = zero_zero_rat )
% 0.17/0.43 | ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_cancel_left
% 0.17/0.43 thf(fact_1566_dvd__mult__cancel__left,axiom,
% 0.17/0.43 ! [C: int,A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.43 = ( ( C = zero_zero_int )
% 0.17/0.43 | ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_cancel_left
% 0.17/0.43 thf(fact_1567_power__0__Suc,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 0.17/0.43 = zero_zero_rat ) ).
% 0.17/0.43
% 0.17/0.43 % power_0_Suc
% 0.17/0.43 thf(fact_1568_power__0__Suc,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % power_0_Suc
% 0.17/0.43 thf(fact_1569_power__0__Suc,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.17/0.43 = zero_zero_real ) ).
% 0.17/0.43
% 0.17/0.43 % power_0_Suc
% 0.17/0.43 thf(fact_1570_power__0__Suc,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.17/0.43 = zero_zero_int ) ).
% 0.17/0.43
% 0.17/0.43 % power_0_Suc
% 0.17/0.43 thf(fact_1571_power__0__Suc,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 0.17/0.43 = zero_zero_complex ) ).
% 0.17/0.43
% 0.17/0.43 % power_0_Suc
% 0.17/0.43 thf(fact_1572_power__zero__numeral,axiom,
% 0.17/0.43 ! [K: num] :
% 0.17/0.43 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 0.17/0.43 = zero_zero_rat ) ).
% 0.17/0.43
% 0.17/0.43 % power_zero_numeral
% 0.17/0.43 thf(fact_1573_power__zero__numeral,axiom,
% 0.17/0.43 ! [K: num] :
% 0.17/0.43 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % power_zero_numeral
% 0.17/0.43 thf(fact_1574_power__zero__numeral,axiom,
% 0.17/0.43 ! [K: num] :
% 0.17/0.43 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.17/0.43 = zero_zero_real ) ).
% 0.17/0.43
% 0.17/0.43 % power_zero_numeral
% 0.17/0.43 thf(fact_1575_power__zero__numeral,axiom,
% 0.17/0.43 ! [K: num] :
% 0.17/0.43 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.17/0.43 = zero_zero_int ) ).
% 0.17/0.43
% 0.17/0.43 % power_zero_numeral
% 0.17/0.43 thf(fact_1576_power__zero__numeral,axiom,
% 0.17/0.43 ! [K: num] :
% 0.17/0.43 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 0.17/0.43 = zero_zero_complex ) ).
% 0.17/0.43
% 0.17/0.43 % power_zero_numeral
% 0.17/0.43 thf(fact_1577_power__eq__0__iff,axiom,
% 0.17/0.43 ! [A: rat,N: nat] :
% 0.17/0.43 ( ( ( power_power_rat @ A @ N )
% 0.17/0.43 = zero_zero_rat )
% 0.17/0.43 = ( ( A = zero_zero_rat )
% 0.17/0.43 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_eq_0_iff
% 0.17/0.43 thf(fact_1578_power__eq__0__iff,axiom,
% 0.17/0.43 ! [A: nat,N: nat] :
% 0.17/0.43 ( ( ( power_power_nat @ A @ N )
% 0.17/0.43 = zero_zero_nat )
% 0.17/0.43 = ( ( A = zero_zero_nat )
% 0.17/0.43 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_eq_0_iff
% 0.17/0.43 thf(fact_1579_power__eq__0__iff,axiom,
% 0.17/0.43 ! [A: real,N: nat] :
% 0.17/0.43 ( ( ( power_power_real @ A @ N )
% 0.17/0.43 = zero_zero_real )
% 0.17/0.43 = ( ( A = zero_zero_real )
% 0.17/0.43 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_eq_0_iff
% 0.17/0.43 thf(fact_1580_power__eq__0__iff,axiom,
% 0.17/0.43 ! [A: int,N: nat] :
% 0.17/0.43 ( ( ( power_power_int @ A @ N )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 = ( ( A = zero_zero_int )
% 0.17/0.43 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_eq_0_iff
% 0.17/0.43 thf(fact_1581_power__eq__0__iff,axiom,
% 0.17/0.43 ! [A: complex,N: nat] :
% 0.17/0.43 ( ( ( power_power_complex @ A @ N )
% 0.17/0.43 = zero_zero_complex )
% 0.17/0.43 = ( ( A = zero_zero_complex )
% 0.17/0.43 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_eq_0_iff
% 0.17/0.43 thf(fact_1582_unit__prod,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.43 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_prod
% 0.17/0.43 thf(fact_1583_unit__prod,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.43 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.43 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_prod
% 0.17/0.43 thf(fact_1584_unit__prod,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.43 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.43 => ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_prod
% 0.17/0.43 thf(fact_1585_dvd__add__times__triv__left__iff,axiom,
% 0.17/0.43 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B2 ) )
% 0.17/0.43 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_add_times_triv_left_iff
% 0.17/0.43 thf(fact_1586_dvd__add__times__triv__left__iff,axiom,
% 0.17/0.43 ! [A: real,C: real,B2: real] :
% 0.17/0.43 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B2 ) )
% 0.17/0.43 = ( dvd_dvd_real @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_add_times_triv_left_iff
% 0.17/0.43 thf(fact_1587_dvd__add__times__triv__left__iff,axiom,
% 0.17/0.43 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.43 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B2 ) )
% 0.17/0.43 = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_add_times_triv_left_iff
% 0.17/0.43 thf(fact_1588_dvd__add__times__triv__left__iff,axiom,
% 0.17/0.43 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B2 ) )
% 0.17/0.43 = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_add_times_triv_left_iff
% 0.17/0.43 thf(fact_1589_dvd__add__times__triv__left__iff,axiom,
% 0.17/0.43 ! [A: int,C: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B2 ) )
% 0.17/0.43 = ( dvd_dvd_int @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_add_times_triv_left_iff
% 0.17/0.43 thf(fact_1590_dvd__add__times__triv__right__iff,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 0.17/0.43 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_add_times_triv_right_iff
% 0.17/0.43 thf(fact_1591_dvd__add__times__triv__right__iff,axiom,
% 0.17/0.43 ! [A: real,B2: real,C: real] :
% 0.17/0.43 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ ( times_times_real @ C @ A ) ) )
% 0.17/0.43 = ( dvd_dvd_real @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_add_times_triv_right_iff
% 0.17/0.43 thf(fact_1592_dvd__add__times__triv__right__iff,axiom,
% 0.17/0.43 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.43 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ ( times_times_rat @ C @ A ) ) )
% 0.17/0.43 = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_add_times_triv_right_iff
% 0.17/0.43 thf(fact_1593_dvd__add__times__triv__right__iff,axiom,
% 0.17/0.43 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A ) ) )
% 0.17/0.43 = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_add_times_triv_right_iff
% 0.17/0.43 thf(fact_1594_dvd__add__times__triv__right__iff,axiom,
% 0.17/0.43 ! [A: int,B2: int,C: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A ) ) )
% 0.17/0.43 = ( dvd_dvd_int @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_add_times_triv_right_iff
% 0.17/0.43 thf(fact_1595_mod__mult__self1__is__0,axiom,
% 0.17/0.43 ! [B2: nat,A: nat] :
% 0.17/0.43 ( ( modulo_modulo_nat @ ( times_times_nat @ B2 @ A ) @ B2 )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % mod_mult_self1_is_0
% 0.17/0.43 thf(fact_1596_mod__mult__self1__is__0,axiom,
% 0.17/0.43 ! [B2: int,A: int] :
% 0.17/0.43 ( ( modulo_modulo_int @ ( times_times_int @ B2 @ A ) @ B2 )
% 0.17/0.43 = zero_zero_int ) ).
% 0.17/0.43
% 0.17/0.43 % mod_mult_self1_is_0
% 0.17/0.43 thf(fact_1597_mod__mult__self1__is__0,axiom,
% 0.17/0.43 ! [B2: code_integer,A: code_integer] :
% 0.17/0.43 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ B2 )
% 0.17/0.43 = zero_z3403309356797280102nteger ) ).
% 0.17/0.43
% 0.17/0.43 % mod_mult_self1_is_0
% 0.17/0.43 thf(fact_1598_mod__mult__self2__is__0,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % mod_mult_self2_is_0
% 0.17/0.43 thf(fact_1599_mod__mult__self2__is__0,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ B2 )
% 0.17/0.43 = zero_zero_int ) ).
% 0.17/0.43
% 0.17/0.43 % mod_mult_self2_is_0
% 0.17/0.43 thf(fact_1600_mod__mult__self2__is__0,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ B2 )
% 0.17/0.43 = zero_z3403309356797280102nteger ) ).
% 0.17/0.43
% 0.17/0.43 % mod_mult_self2_is_0
% 0.17/0.43 thf(fact_1601_dvd__div__mult__self,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.43 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_mult_self
% 0.17/0.43 thf(fact_1602_dvd__div__mult__self,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.43 => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_mult_self
% 0.17/0.43 thf(fact_1603_dvd__div__mult__self,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.43 => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_mult_self
% 0.17/0.43 thf(fact_1604_dvd__mult__div__cancel,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.43 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ A ) )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_div_cancel
% 0.17/0.43 thf(fact_1605_dvd__mult__div__cancel,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.43 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ A ) )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_div_cancel
% 0.17/0.43 thf(fact_1606_dvd__mult__div__cancel,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.43 => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ A ) )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_div_cancel
% 0.17/0.43 thf(fact_1607_mod__by__1,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % mod_by_1
% 0.17/0.43 thf(fact_1608_mod__by__1,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.17/0.43 = zero_zero_int ) ).
% 0.17/0.43
% 0.17/0.43 % mod_by_1
% 0.17/0.43 thf(fact_1609_mod__by__1,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.17/0.43 = zero_z3403309356797280102nteger ) ).
% 0.17/0.43
% 0.17/0.43 % mod_by_1
% 0.17/0.43 thf(fact_1610_bits__mod__by__1,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % bits_mod_by_1
% 0.17/0.43 thf(fact_1611_bits__mod__by__1,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.17/0.43 = zero_zero_int ) ).
% 0.17/0.43
% 0.17/0.43 % bits_mod_by_1
% 0.17/0.43 thf(fact_1612_bits__mod__by__1,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.17/0.43 = zero_z3403309356797280102nteger ) ).
% 0.17/0.43
% 0.17/0.43 % bits_mod_by_1
% 0.17/0.43 thf(fact_1613_unit__div,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.43 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div
% 0.17/0.43 thf(fact_1614_unit__div,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.43 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.43 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div
% 0.17/0.43 thf(fact_1615_unit__div,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.43 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.43 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div
% 0.17/0.43 thf(fact_1616_unit__div__1__unit,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.43 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_1_unit
% 0.17/0.43 thf(fact_1617_unit__div__1__unit,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.43 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_1_unit
% 0.17/0.43 thf(fact_1618_unit__div__1__unit,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.43 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_1_unit
% 0.17/0.43 thf(fact_1619_unit__div__1__div__1,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.43 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.17/0.43 = A ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_1_div_1
% 0.17/0.43 thf(fact_1620_unit__div__1__div__1,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.17/0.43 = A ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_1_div_1
% 0.17/0.43 thf(fact_1621_unit__div__1__div__1,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.43 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 0.17/0.43 = A ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_1_div_1
% 0.17/0.43 thf(fact_1622_div__add,axiom,
% 0.17/0.43 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.43 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 0.17/0.43 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_add
% 0.17/0.43 thf(fact_1623_div__add,axiom,
% 0.17/0.43 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ C @ A )
% 0.17/0.43 => ( ( dvd_dvd_nat @ C @ B2 )
% 0.17/0.43 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.17/0.43 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_add
% 0.17/0.43 thf(fact_1624_div__add,axiom,
% 0.17/0.43 ! [C: int,A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ C @ A )
% 0.17/0.43 => ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.43 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.17/0.43 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_add
% 0.17/0.43 thf(fact_1625_div__diff,axiom,
% 0.17/0.43 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.43 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
% 0.17/0.43 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_diff
% 0.17/0.43 thf(fact_1626_div__diff,axiom,
% 0.17/0.43 ! [C: int,A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ C @ A )
% 0.17/0.43 => ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.43 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.17/0.43 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_diff
% 0.17/0.43 thf(fact_1627_mod__div__trivial,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % mod_div_trivial
% 0.17/0.43 thf(fact_1628_mod__div__trivial,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 0.17/0.43 = zero_zero_int ) ).
% 0.17/0.43
% 0.17/0.43 % mod_div_trivial
% 0.17/0.43 thf(fact_1629_mod__div__trivial,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 0.17/0.43 = zero_z3403309356797280102nteger ) ).
% 0.17/0.43
% 0.17/0.43 % mod_div_trivial
% 0.17/0.43 thf(fact_1630_bits__mod__div__trivial,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % bits_mod_div_trivial
% 0.17/0.43 thf(fact_1631_bits__mod__div__trivial,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 0.17/0.43 = zero_zero_int ) ).
% 0.17/0.43
% 0.17/0.43 % bits_mod_div_trivial
% 0.17/0.43 thf(fact_1632_bits__mod__div__trivial,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 0.17/0.43 = zero_z3403309356797280102nteger ) ).
% 0.17/0.43
% 0.17/0.43 % bits_mod_div_trivial
% 0.17/0.43 thf(fact_1633_zmod__numeral__Bit0,axiom,
% 0.17/0.43 ! [V: num,W: num] :
% 0.17/0.43 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.17/0.43 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zmod_numeral_Bit0
% 0.17/0.43 thf(fact_1634_dvd__imp__mod__0,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.43 => ( ( modulo_modulo_nat @ B2 @ A )
% 0.17/0.43 = zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_imp_mod_0
% 0.17/0.43 thf(fact_1635_dvd__imp__mod__0,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.43 => ( ( modulo_modulo_int @ B2 @ A )
% 0.17/0.43 = zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_imp_mod_0
% 0.17/0.43 thf(fact_1636_dvd__imp__mod__0,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.43 => ( ( modulo364778990260209775nteger @ B2 @ A )
% 0.17/0.43 = zero_z3403309356797280102nteger ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_imp_mod_0
% 0.17/0.43 thf(fact_1637_Suc__pred,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.17/0.43 = N ) ) ).
% 0.17/0.43
% 0.17/0.43 % Suc_pred
% 0.17/0.43 thf(fact_1638_one__le__mult__iff,axiom,
% 0.17/0.43 ! [M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 0.17/0.43 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.17/0.43 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % one_le_mult_iff
% 0.17/0.43 thf(fact_1639_nat__mult__le__cancel__disj,axiom,
% 0.17/0.43 ! [K: nat,M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.17/0.43 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.43 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nat_mult_le_cancel_disj
% 0.17/0.43 thf(fact_1640_mult__le__cancel2,axiom,
% 0.17/0.43 ! [M: nat,K: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.17/0.43 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.43 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_le_cancel2
% 0.17/0.43 thf(fact_1641_div__mult__self1__is__m,axiom,
% 0.17/0.43 ! [N: nat,M: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 0.17/0.43 = M ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_self1_is_m
% 0.17/0.43 thf(fact_1642_div__mult__self__is__m,axiom,
% 0.17/0.43 ! [N: nat,M: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 0.17/0.43 = M ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_self_is_m
% 0.17/0.43 thf(fact_1643_signed__take__bit__Suc__1,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 0.17/0.43 = one_one_int ) ).
% 0.17/0.43
% 0.17/0.43 % signed_take_bit_Suc_1
% 0.17/0.43 thf(fact_1644_signed__take__bit__numeral__of__1,axiom,
% 0.17/0.43 ! [K: num] :
% 0.17/0.43 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 0.17/0.43 = one_one_int ) ).
% 0.17/0.43
% 0.17/0.43 % signed_take_bit_numeral_of_1
% 0.17/0.43 thf(fact_1645_dbl__simps_I5_J,axiom,
% 0.17/0.43 ! [K: num] :
% 0.17/0.43 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.17/0.43 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dbl_simps(5)
% 0.17/0.43 thf(fact_1646_dbl__simps_I5_J,axiom,
% 0.17/0.43 ! [K: num] :
% 0.17/0.43 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 0.17/0.43 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dbl_simps(5)
% 0.17/0.43 thf(fact_1647_dbl__simps_I5_J,axiom,
% 0.17/0.43 ! [K: num] :
% 0.17/0.43 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 0.17/0.43 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dbl_simps(5)
% 0.17/0.43 thf(fact_1648_dbl__simps_I5_J,axiom,
% 0.17/0.43 ! [K: num] :
% 0.17/0.43 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 0.17/0.43 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dbl_simps(5)
% 0.17/0.43 thf(fact_1649_zero__le__divide__1__iff,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.17/0.43 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_le_divide_1_iff
% 0.17/0.43 thf(fact_1650_zero__le__divide__1__iff,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.17/0.43 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_le_divide_1_iff
% 0.17/0.43 thf(fact_1651_divide__le__0__1__iff,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.17/0.43 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_le_0_1_iff
% 0.17/0.43 thf(fact_1652_divide__le__0__1__iff,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.17/0.43 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_le_0_1_iff
% 0.17/0.43 thf(fact_1653_zero__less__divide__1__iff,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.17/0.43 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_less_divide_1_iff
% 0.17/0.43 thf(fact_1654_zero__less__divide__1__iff,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.17/0.43 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_less_divide_1_iff
% 0.17/0.43 thf(fact_1655_less__divide__eq__1__pos,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.43 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_real @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_divide_eq_1_pos
% 0.17/0.43 thf(fact_1656_less__divide__eq__1__pos,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.43 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_divide_eq_1_pos
% 0.17/0.43 thf(fact_1657_less__divide__eq__1__neg,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.43 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_real @ B2 @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_divide_eq_1_neg
% 0.17/0.43 thf(fact_1658_less__divide__eq__1__neg,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.43 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_rat @ B2 @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % less_divide_eq_1_neg
% 0.17/0.43 thf(fact_1659_divide__less__eq__1__pos,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.43 => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.17/0.43 = ( ord_less_real @ B2 @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_less_eq_1_pos
% 0.17/0.43 thf(fact_1660_divide__less__eq__1__pos,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.43 => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.17/0.43 = ( ord_less_rat @ B2 @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_less_eq_1_pos
% 0.17/0.43 thf(fact_1661_divide__less__eq__1__neg,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.43 => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.17/0.43 = ( ord_less_real @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_less_eq_1_neg
% 0.17/0.43 thf(fact_1662_divide__less__eq__1__neg,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.43 => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.17/0.43 = ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_less_eq_1_neg
% 0.17/0.43 thf(fact_1663_divide__less__0__1__iff,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.17/0.43 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_less_0_1_iff
% 0.17/0.43 thf(fact_1664_divide__less__0__1__iff,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.17/0.43 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_less_0_1_iff
% 0.17/0.43 thf(fact_1665_divide__eq__eq__numeral1_I1_J,axiom,
% 0.17/0.43 ! [B2: complex,W: num,A: complex] :
% 0.17/0.43 ( ( ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) )
% 0.17/0.43 = A )
% 0.17/0.43 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.17/0.43 != zero_zero_complex )
% 0.17/0.43 => ( B2
% 0.17/0.43 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.17/0.43 & ( ( ( numera6690914467698888265omplex @ W )
% 0.17/0.43 = zero_zero_complex )
% 0.17/0.43 => ( A = zero_zero_complex ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_eq_eq_numeral1(1)
% 0.17/0.43 thf(fact_1666_divide__eq__eq__numeral1_I1_J,axiom,
% 0.17/0.43 ! [B2: real,W: num,A: real] :
% 0.17/0.43 ( ( ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) )
% 0.17/0.43 = A )
% 0.17/0.43 = ( ( ( ( numeral_numeral_real @ W )
% 0.17/0.43 != zero_zero_real )
% 0.17/0.43 => ( B2
% 0.17/0.43 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 0.17/0.43 & ( ( ( numeral_numeral_real @ W )
% 0.17/0.43 = zero_zero_real )
% 0.17/0.43 => ( A = zero_zero_real ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_eq_eq_numeral1(1)
% 0.17/0.43 thf(fact_1667_divide__eq__eq__numeral1_I1_J,axiom,
% 0.17/0.43 ! [B2: rat,W: num,A: rat] :
% 0.17/0.43 ( ( ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) )
% 0.17/0.43 = A )
% 0.17/0.43 = ( ( ( ( numeral_numeral_rat @ W )
% 0.17/0.43 != zero_zero_rat )
% 0.17/0.43 => ( B2
% 0.17/0.43 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 0.17/0.43 & ( ( ( numeral_numeral_rat @ W )
% 0.17/0.43 = zero_zero_rat )
% 0.17/0.43 => ( A = zero_zero_rat ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_eq_eq_numeral1(1)
% 0.17/0.43 thf(fact_1668_eq__divide__eq__numeral1_I1_J,axiom,
% 0.17/0.43 ! [A: complex,B2: complex,W: num] :
% 0.17/0.43 ( ( A
% 0.17/0.43 = ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) ) )
% 0.17/0.43 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.17/0.43 != zero_zero_complex )
% 0.17/0.43 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 0.17/0.43 = B2 ) )
% 0.17/0.43 & ( ( ( numera6690914467698888265omplex @ W )
% 0.17/0.43 = zero_zero_complex )
% 0.17/0.43 => ( A = zero_zero_complex ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % eq_divide_eq_numeral1(1)
% 0.17/0.43 thf(fact_1669_eq__divide__eq__numeral1_I1_J,axiom,
% 0.17/0.43 ! [A: real,B2: real,W: num] :
% 0.17/0.43 ( ( A
% 0.17/0.43 = ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 0.17/0.43 = ( ( ( ( numeral_numeral_real @ W )
% 0.17/0.43 != zero_zero_real )
% 0.17/0.43 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 0.17/0.43 = B2 ) )
% 0.17/0.43 & ( ( ( numeral_numeral_real @ W )
% 0.17/0.43 = zero_zero_real )
% 0.17/0.43 => ( A = zero_zero_real ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % eq_divide_eq_numeral1(1)
% 0.17/0.43 thf(fact_1670_eq__divide__eq__numeral1_I1_J,axiom,
% 0.17/0.43 ! [A: rat,B2: rat,W: num] :
% 0.17/0.43 ( ( A
% 0.17/0.43 = ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 0.17/0.43 = ( ( ( ( numeral_numeral_rat @ W )
% 0.17/0.43 != zero_zero_rat )
% 0.17/0.43 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 0.17/0.43 = B2 ) )
% 0.17/0.43 & ( ( ( numeral_numeral_rat @ W )
% 0.17/0.43 = zero_zero_rat )
% 0.17/0.43 => ( A = zero_zero_rat ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % eq_divide_eq_numeral1(1)
% 0.17/0.43 thf(fact_1671_nonzero__divide__mult__cancel__left,axiom,
% 0.17/0.43 ! [A: complex,B2: complex] :
% 0.17/0.43 ( ( A != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B2 ) )
% 0.17/0.43 = ( divide1717551699836669952omplex @ one_one_complex @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_divide_mult_cancel_left
% 0.17/0.43 thf(fact_1672_nonzero__divide__mult__cancel__left,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( A != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B2 ) )
% 0.17/0.43 = ( divide_divide_real @ one_one_real @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_divide_mult_cancel_left
% 0.17/0.43 thf(fact_1673_nonzero__divide__mult__cancel__left,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( A != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B2 ) )
% 0.17/0.43 = ( divide_divide_rat @ one_one_rat @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_divide_mult_cancel_left
% 0.17/0.43 thf(fact_1674_nonzero__divide__mult__cancel__right,axiom,
% 0.17/0.43 ! [B2: complex,A: complex] :
% 0.17/0.43 ( ( B2 != zero_zero_complex )
% 0.17/0.43 => ( ( divide1717551699836669952omplex @ B2 @ ( times_times_complex @ A @ B2 ) )
% 0.17/0.43 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_divide_mult_cancel_right
% 0.17/0.43 thf(fact_1675_nonzero__divide__mult__cancel__right,axiom,
% 0.17/0.43 ! [B2: real,A: real] :
% 0.17/0.43 ( ( B2 != zero_zero_real )
% 0.17/0.43 => ( ( divide_divide_real @ B2 @ ( times_times_real @ A @ B2 ) )
% 0.17/0.43 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_divide_mult_cancel_right
% 0.17/0.43 thf(fact_1676_nonzero__divide__mult__cancel__right,axiom,
% 0.17/0.43 ! [B2: rat,A: rat] :
% 0.17/0.43 ( ( B2 != zero_zero_rat )
% 0.17/0.43 => ( ( divide_divide_rat @ B2 @ ( times_times_rat @ A @ B2 ) )
% 0.17/0.43 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonzero_divide_mult_cancel_right
% 0.17/0.43 thf(fact_1677_div__mult__self1,axiom,
% 0.17/0.43 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.43 ( ( B2 != zero_zero_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
% 0.17/0.43 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_self1
% 0.17/0.43 thf(fact_1678_div__mult__self1,axiom,
% 0.17/0.43 ! [B2: int,A: int,C: int] :
% 0.17/0.43 ( ( B2 != zero_zero_int )
% 0.17/0.43 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
% 0.17/0.43 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_self1
% 0.17/0.43 thf(fact_1679_div__mult__self2,axiom,
% 0.17/0.43 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.43 ( ( B2 != zero_zero_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
% 0.17/0.43 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_self2
% 0.17/0.43 thf(fact_1680_div__mult__self2,axiom,
% 0.17/0.43 ! [B2: int,A: int,C: int] :
% 0.17/0.43 ( ( B2 != zero_zero_int )
% 0.17/0.43 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
% 0.17/0.43 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_self2
% 0.17/0.43 thf(fact_1681_div__mult__self3,axiom,
% 0.17/0.43 ! [B2: nat,C: nat,A: nat] :
% 0.17/0.43 ( ( B2 != zero_zero_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
% 0.17/0.43 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_self3
% 0.17/0.43 thf(fact_1682_div__mult__self3,axiom,
% 0.17/0.43 ! [B2: int,C: int,A: int] :
% 0.17/0.43 ( ( B2 != zero_zero_int )
% 0.17/0.43 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
% 0.17/0.43 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_self3
% 0.17/0.43 thf(fact_1683_div__mult__self4,axiom,
% 0.17/0.43 ! [B2: nat,C: nat,A: nat] :
% 0.17/0.43 ( ( B2 != zero_zero_nat )
% 0.17/0.43 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
% 0.17/0.43 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_self4
% 0.17/0.43 thf(fact_1684_div__mult__self4,axiom,
% 0.17/0.43 ! [B2: int,C: int,A: int] :
% 0.17/0.43 ( ( B2 != zero_zero_int )
% 0.17/0.43 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
% 0.17/0.43 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mult_self4
% 0.17/0.43 thf(fact_1685_power__mono__iff,axiom,
% 0.17/0.43 ! [A: real,B2: real,N: nat] :
% 0.17/0.43 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.43 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_eq_real @ A @ B2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_mono_iff
% 0.17/0.43 thf(fact_1686_power__mono__iff,axiom,
% 0.17/0.43 ! [A: rat,B2: rat,N: nat] :
% 0.17/0.43 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.43 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_eq_rat @ A @ B2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_mono_iff
% 0.17/0.43 thf(fact_1687_power__mono__iff,axiom,
% 0.17/0.43 ! [A: nat,B2: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.43 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_eq_nat @ A @ B2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_mono_iff
% 0.17/0.43 thf(fact_1688_power__mono__iff,axiom,
% 0.17/0.43 ! [A: int,B2: int,N: nat] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.43 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_eq_int @ A @ B2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_mono_iff
% 0.17/0.43 thf(fact_1689_unit__mult__div__div,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.43 => ( ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.17/0.43 = ( divide6298287555418463151nteger @ B2 @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_mult_div_div
% 0.17/0.43 thf(fact_1690_unit__mult__div__div,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.43 => ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.17/0.43 = ( divide_divide_nat @ B2 @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_mult_div_div
% 0.17/0.43 thf(fact_1691_unit__mult__div__div,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.43 => ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A ) )
% 0.17/0.43 = ( divide_divide_int @ B2 @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_mult_div_div
% 0.17/0.43 thf(fact_1692_unit__div__mult__self,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.43 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_mult_self
% 0.17/0.43 thf(fact_1693_unit__div__mult__self,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.43 => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_mult_self
% 0.17/0.43 thf(fact_1694_unit__div__mult__self,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.43 => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
% 0.17/0.43 = B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_mult_self
% 0.17/0.43 thf(fact_1695_signed__take__bit__Suc__bit0,axiom,
% 0.17/0.43 ! [N: nat,K: num] :
% 0.17/0.43 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.17/0.43 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % signed_take_bit_Suc_bit0
% 0.17/0.43 thf(fact_1696_Suc__diff__1,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.17/0.43 = N ) ) ).
% 0.17/0.43
% 0.17/0.43 % Suc_diff_1
% 0.17/0.43 thf(fact_1697_half__negative__int__iff,axiom,
% 0.17/0.43 ! [K: int] :
% 0.17/0.43 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.17/0.43 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % half_negative_int_iff
% 0.17/0.43 thf(fact_1698_half__nonnegative__int__iff,axiom,
% 0.17/0.43 ! [K: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.17/0.43
% 0.17/0.43 % half_nonnegative_int_iff
% 0.17/0.43 thf(fact_1699_le__divide__eq__1__pos,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.43 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % le_divide_eq_1_pos
% 0.17/0.43 thf(fact_1700_le__divide__eq__1__pos,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.43 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % le_divide_eq_1_pos
% 0.17/0.43 thf(fact_1701_le__divide__eq__1__neg,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.43 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % le_divide_eq_1_neg
% 0.17/0.43 thf(fact_1702_le__divide__eq__1__neg,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.43 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.17/0.43 = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % le_divide_eq_1_neg
% 0.17/0.43 thf(fact_1703_divide__le__eq__1__pos,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.43 => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.17/0.43 = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_le_eq_1_pos
% 0.17/0.43 thf(fact_1704_divide__le__eq__1__pos,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.43 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.17/0.43 = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_le_eq_1_pos
% 0.17/0.43 thf(fact_1705_divide__le__eq__1__neg,axiom,
% 0.17/0.43 ! [A: real,B2: real] :
% 0.17/0.43 ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.43 => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.17/0.43 = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_le_eq_1_neg
% 0.17/0.43 thf(fact_1706_divide__le__eq__1__neg,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] :
% 0.17/0.43 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.43 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.17/0.43 = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % divide_le_eq_1_neg
% 0.17/0.43 thf(fact_1707_power__strict__decreasing__iff,axiom,
% 0.17/0.43 ! [B2: real,M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.17/0.43 => ( ( ord_less_real @ B2 @ one_one_real )
% 0.17/0.43 => ( ( ord_less_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_strict_decreasing_iff
% 0.17/0.43 thf(fact_1708_power__strict__decreasing__iff,axiom,
% 0.17/0.43 ! [B2: rat,M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.17/0.43 => ( ( ord_less_rat @ B2 @ one_one_rat )
% 0.17/0.43 => ( ( ord_less_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_strict_decreasing_iff
% 0.17/0.43 thf(fact_1709_power__strict__decreasing__iff,axiom,
% 0.17/0.43 ! [B2: nat,M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.17/0.43 => ( ( ord_less_nat @ B2 @ one_one_nat )
% 0.17/0.43 => ( ( ord_less_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_strict_decreasing_iff
% 0.17/0.43 thf(fact_1710_power__strict__decreasing__iff,axiom,
% 0.17/0.43 ! [B2: int,M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.43 => ( ( ord_less_int @ B2 @ one_one_int )
% 0.17/0.43 => ( ( ord_less_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_strict_decreasing_iff
% 0.17/0.43 thf(fact_1711_even__mult__iff,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 0.17/0.43 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_mult_iff
% 0.17/0.43 thf(fact_1712_even__mult__iff,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B2 ) )
% 0.17/0.43 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_mult_iff
% 0.17/0.43 thf(fact_1713_even__mult__iff,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B2 ) )
% 0.17/0.43 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_mult_iff
% 0.17/0.43 thf(fact_1714_zero__eq__power2,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_rat )
% 0.17/0.43 = ( A = zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_eq_power2
% 0.17/0.43 thf(fact_1715_zero__eq__power2,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_nat )
% 0.17/0.43 = ( A = zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_eq_power2
% 0.17/0.43 thf(fact_1716_zero__eq__power2,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_real )
% 0.17/0.43 = ( A = zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_eq_power2
% 0.17/0.43 thf(fact_1717_zero__eq__power2,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 = ( A = zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_eq_power2
% 0.17/0.43 thf(fact_1718_zero__eq__power2,axiom,
% 0.17/0.43 ! [A: complex] :
% 0.17/0.43 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_complex )
% 0.17/0.43 = ( A = zero_zero_complex ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_eq_power2
% 0.17/0.43 thf(fact_1719_odd__add,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) )
% 0.17/0.43 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.17/0.43 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % odd_add
% 0.17/0.43 thf(fact_1720_odd__add,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) )
% 0.17/0.43 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.17/0.43 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % odd_add
% 0.17/0.43 thf(fact_1721_odd__add,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) )
% 0.17/0.43 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.17/0.43 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % odd_add
% 0.17/0.43 thf(fact_1722_even__add,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.17/0.43 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_add
% 0.17/0.43 thf(fact_1723_even__add,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) )
% 0.17/0.43 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_add
% 0.17/0.43 thf(fact_1724_even__add,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) )
% 0.17/0.43 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_add
% 0.17/0.43 thf(fact_1725_even__mod__2__iff,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_mod_2_iff
% 0.17/0.43 thf(fact_1726_even__mod__2__iff,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_mod_2_iff
% 0.17/0.43 thf(fact_1727_even__mod__2__iff,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_mod_2_iff
% 0.17/0.43 thf(fact_1728_even__Suc,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 0.17/0.43 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_Suc
% 0.17/0.43 thf(fact_1729_even__Suc__Suc__iff,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 0.17/0.43 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_Suc_Suc_iff
% 0.17/0.43 thf(fact_1730_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 != ( suc @ zero_zero_nat ) )
% 0.17/0.43 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % not_mod2_eq_Suc_0_eq_0
% 0.17/0.43 thf(fact_1731_add__self__mod__2,axiom,
% 0.17/0.43 ! [M: nat] :
% 0.17/0.43 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % add_self_mod_2
% 0.17/0.43 thf(fact_1732_one__div__two__eq__zero,axiom,
% 0.17/0.43 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % one_div_two_eq_zero
% 0.17/0.43 thf(fact_1733_one__div__two__eq__zero,axiom,
% 0.17/0.43 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_int ) ).
% 0.17/0.43
% 0.17/0.43 % one_div_two_eq_zero
% 0.17/0.43 thf(fact_1734_bits__1__div__2,axiom,
% 0.17/0.43 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_nat ) ).
% 0.17/0.43
% 0.17/0.43 % bits_1_div_2
% 0.17/0.43 thf(fact_1735_bits__1__div__2,axiom,
% 0.17/0.43 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_int ) ).
% 0.17/0.43
% 0.17/0.43 % bits_1_div_2
% 0.17/0.43 thf(fact_1736_power__decreasing__iff,axiom,
% 0.17/0.43 ! [B2: real,M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.17/0.43 => ( ( ord_less_real @ B2 @ one_one_real )
% 0.17/0.43 => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_decreasing_iff
% 0.17/0.43 thf(fact_1737_power__decreasing__iff,axiom,
% 0.17/0.43 ! [B2: rat,M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.17/0.43 => ( ( ord_less_rat @ B2 @ one_one_rat )
% 0.17/0.43 => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_decreasing_iff
% 0.17/0.43 thf(fact_1738_power__decreasing__iff,axiom,
% 0.17/0.43 ! [B2: nat,M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.17/0.43 => ( ( ord_less_nat @ B2 @ one_one_nat )
% 0.17/0.43 => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_decreasing_iff
% 0.17/0.43 thf(fact_1739_power__decreasing__iff,axiom,
% 0.17/0.43 ! [B2: int,M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.43 => ( ( ord_less_int @ B2 @ one_one_int )
% 0.17/0.43 => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
% 0.17/0.43 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_decreasing_iff
% 0.17/0.43 thf(fact_1740_power2__less__eq__zero__iff,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.17/0.43 = ( A = zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % power2_less_eq_zero_iff
% 0.17/0.43 thf(fact_1741_power2__less__eq__zero__iff,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 0.17/0.43 = ( A = zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % power2_less_eq_zero_iff
% 0.17/0.43 thf(fact_1742_power2__less__eq__zero__iff,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.17/0.43 = ( A = zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % power2_less_eq_zero_iff
% 0.17/0.43 thf(fact_1743_power2__eq__iff__nonneg,axiom,
% 0.17/0.43 ! [X4: real,Y3: real] :
% 0.17/0.43 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.17/0.43 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.17/0.43 => ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( X4 = Y3 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power2_eq_iff_nonneg
% 0.17/0.43 thf(fact_1744_power2__eq__iff__nonneg,axiom,
% 0.17/0.43 ! [X4: rat,Y3: rat] :
% 0.17/0.43 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.17/0.43 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.17/0.43 => ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( X4 = Y3 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power2_eq_iff_nonneg
% 0.17/0.43 thf(fact_1745_power2__eq__iff__nonneg,axiom,
% 0.17/0.43 ! [X4: nat,Y3: nat] :
% 0.17/0.43 ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
% 0.17/0.43 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.17/0.43 => ( ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( X4 = Y3 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power2_eq_iff_nonneg
% 0.17/0.43 thf(fact_1746_power2__eq__iff__nonneg,axiom,
% 0.17/0.43 ! [X4: int,Y3: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.17/0.43 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.17/0.43 => ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( X4 = Y3 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power2_eq_iff_nonneg
% 0.17/0.43 thf(fact_1747_zero__less__power2,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( A != zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_less_power2
% 0.17/0.43 thf(fact_1748_zero__less__power2,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( A != zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_less_power2
% 0.17/0.43 thf(fact_1749_zero__less__power2,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( A != zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_less_power2
% 0.17/0.43 thf(fact_1750_sum__power2__eq__zero__iff,axiom,
% 0.17/0.43 ! [X4: rat,Y3: rat] :
% 0.17/0.43 ( ( ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = zero_zero_rat )
% 0.17/0.43 = ( ( X4 = zero_zero_rat )
% 0.17/0.43 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % sum_power2_eq_zero_iff
% 0.17/0.43 thf(fact_1751_sum__power2__eq__zero__iff,axiom,
% 0.17/0.43 ! [X4: real,Y3: real] :
% 0.17/0.43 ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = zero_zero_real )
% 0.17/0.43 = ( ( X4 = zero_zero_real )
% 0.17/0.43 & ( Y3 = zero_zero_real ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % sum_power2_eq_zero_iff
% 0.17/0.43 thf(fact_1752_sum__power2__eq__zero__iff,axiom,
% 0.17/0.43 ! [X4: int,Y3: int] :
% 0.17/0.43 ( ( ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 = ( ( X4 = zero_zero_int )
% 0.17/0.43 & ( Y3 = zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % sum_power2_eq_zero_iff
% 0.17/0.43 thf(fact_1753_even__plus__one__iff,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 0.17/0.43 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_plus_one_iff
% 0.17/0.43 thf(fact_1754_even__plus__one__iff,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 0.17/0.43 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_plus_one_iff
% 0.17/0.43 thf(fact_1755_even__plus__one__iff,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 0.17/0.43 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_plus_one_iff
% 0.17/0.43 thf(fact_1756_even__diff,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 0.17/0.43 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_diff
% 0.17/0.43 thf(fact_1757_even__diff,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B2 ) )
% 0.17/0.43 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_diff
% 0.17/0.43 thf(fact_1758_not__mod__2__eq__0__eq__1,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 != zero_zero_nat )
% 0.17/0.43 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = one_one_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % not_mod_2_eq_0_eq_1
% 0.17/0.43 thf(fact_1759_not__mod__2__eq__0__eq__1,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.43 != zero_zero_int )
% 0.17/0.43 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.43 = one_one_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % not_mod_2_eq_0_eq_1
% 0.17/0.43 thf(fact_1760_not__mod__2__eq__0__eq__1,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.43 != zero_z3403309356797280102nteger )
% 0.17/0.43 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.43 = one_one_Code_integer ) ) ).
% 0.17/0.43
% 0.17/0.43 % not_mod_2_eq_0_eq_1
% 0.17/0.43 thf(fact_1761_not__mod__2__eq__1__eq__0,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 != one_one_nat )
% 0.17/0.43 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % not_mod_2_eq_1_eq_0
% 0.17/0.43 thf(fact_1762_not__mod__2__eq__1__eq__0,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.43 != one_one_int )
% 0.17/0.43 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % not_mod_2_eq_1_eq_0
% 0.17/0.43 thf(fact_1763_not__mod__2__eq__1__eq__0,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.43 != one_one_Code_integer )
% 0.17/0.43 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.43 = zero_z3403309356797280102nteger ) ) ).
% 0.17/0.43
% 0.17/0.43 % not_mod_2_eq_1_eq_0
% 0.17/0.43 thf(fact_1764_even__power,axiom,
% 0.17/0.43 ! [A: code_integer,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.17/0.43 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_power
% 0.17/0.43 thf(fact_1765_even__power,axiom,
% 0.17/0.43 ! [A: nat,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 0.17/0.43 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_power
% 0.17/0.43 thf(fact_1766_even__power,axiom,
% 0.17/0.43 ! [A: int,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 0.17/0.43 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_power
% 0.17/0.43 thf(fact_1767_odd__Suc__minus__one,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.43 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.17/0.43 = N ) ) ).
% 0.17/0.43
% 0.17/0.43 % odd_Suc_minus_one
% 0.17/0.43 thf(fact_1768_odd__Suc__div__two,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.43 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % odd_Suc_div_two
% 0.17/0.43 thf(fact_1769_even__Suc__div__two,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.43 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_Suc_div_two
% 0.17/0.43 thf(fact_1770_mod2__gr__0,axiom,
% 0.17/0.43 ! [M: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.43 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = one_one_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % mod2_gr_0
% 0.17/0.43 thf(fact_1771_unset__bit__0,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 0.17/0.43 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unset_bit_0
% 0.17/0.43 thf(fact_1772_unset__bit__0,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 0.17/0.43 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unset_bit_0
% 0.17/0.43 thf(fact_1773_odd__succ__div__two,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % odd_succ_div_two
% 0.17/0.43 thf(fact_1774_odd__succ__div__two,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % odd_succ_div_two
% 0.17/0.43 thf(fact_1775_odd__succ__div__two,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % odd_succ_div_two
% 0.17/0.43 thf(fact_1776_even__succ__div__two,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_div_two
% 0.17/0.43 thf(fact_1777_even__succ__div__two,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_div_two
% 0.17/0.43 thf(fact_1778_even__succ__div__two,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_div_two
% 0.17/0.43 thf(fact_1779_even__succ__div__2,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_div_2
% 0.17/0.43 thf(fact_1780_even__succ__div__2,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_div_2
% 0.17/0.43 thf(fact_1781_even__succ__div__2,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.43 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_div_2
% 0.17/0.43 thf(fact_1782_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 0.17/0.43 = ( N = zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % semiring_parity_class.even_mask_iff
% 0.17/0.43 thf(fact_1783_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 0.17/0.43 = ( N = zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % semiring_parity_class.even_mask_iff
% 0.17/0.43 thf(fact_1784_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.17/0.43 = ( N = zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % semiring_parity_class.even_mask_iff
% 0.17/0.43 thf(fact_1785_zero__le__power__eq__numeral,axiom,
% 0.17/0.43 ! [A: real,W: num] :
% 0.17/0.43 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.17/0.43 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_le_power_eq_numeral
% 0.17/0.43 thf(fact_1786_zero__le__power__eq__numeral,axiom,
% 0.17/0.43 ! [A: rat,W: num] :
% 0.17/0.43 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.17/0.43 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_le_power_eq_numeral
% 0.17/0.43 thf(fact_1787_zero__le__power__eq__numeral,axiom,
% 0.17/0.43 ! [A: int,W: num] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.17/0.43 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_le_power_eq_numeral
% 0.17/0.43 thf(fact_1788_zero__less__power__eq__numeral,axiom,
% 0.17/0.43 ! [A: real,W: num] :
% 0.17/0.43 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.17/0.43 = ( ( ( numeral_numeral_nat @ W )
% 0.17/0.43 = zero_zero_nat )
% 0.17/0.43 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( A != zero_zero_real ) )
% 0.17/0.43 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_less_power_eq_numeral
% 0.17/0.43 thf(fact_1789_zero__less__power__eq__numeral,axiom,
% 0.17/0.43 ! [A: rat,W: num] :
% 0.17/0.43 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.17/0.43 = ( ( ( numeral_numeral_nat @ W )
% 0.17/0.43 = zero_zero_nat )
% 0.17/0.43 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( A != zero_zero_rat ) )
% 0.17/0.43 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_less_power_eq_numeral
% 0.17/0.43 thf(fact_1790_zero__less__power__eq__numeral,axiom,
% 0.17/0.43 ! [A: int,W: num] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.17/0.43 = ( ( ( numeral_numeral_nat @ W )
% 0.17/0.43 = zero_zero_nat )
% 0.17/0.43 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( A != zero_zero_int ) )
% 0.17/0.43 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_less_power_eq_numeral
% 0.17/0.43 thf(fact_1791_power__less__zero__eq,axiom,
% 0.17/0.43 ! [A: real,N: nat] :
% 0.17/0.43 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.17/0.43 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.43 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_less_zero_eq
% 0.17/0.43 thf(fact_1792_power__less__zero__eq,axiom,
% 0.17/0.43 ! [A: rat,N: nat] :
% 0.17/0.43 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.17/0.43 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.43 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_less_zero_eq
% 0.17/0.43 thf(fact_1793_power__less__zero__eq,axiom,
% 0.17/0.43 ! [A: int,N: nat] :
% 0.17/0.43 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.17/0.43 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.43 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_less_zero_eq
% 0.17/0.43 thf(fact_1794_power__less__zero__eq__numeral,axiom,
% 0.17/0.43 ! [A: real,W: num] :
% 0.17/0.43 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.17/0.43 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_less_zero_eq_numeral
% 0.17/0.43 thf(fact_1795_power__less__zero__eq__numeral,axiom,
% 0.17/0.43 ! [A: rat,W: num] :
% 0.17/0.43 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.17/0.43 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_less_zero_eq_numeral
% 0.17/0.43 thf(fact_1796_power__less__zero__eq__numeral,axiom,
% 0.17/0.43 ! [A: int,W: num] :
% 0.17/0.43 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.17/0.43 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_less_zero_eq_numeral
% 0.17/0.43 thf(fact_1797_even__diff__nat,axiom,
% 0.17/0.43 ! [M: nat,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 0.17/0.43 = ( ( ord_less_nat @ M @ N )
% 0.17/0.43 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_diff_nat
% 0.17/0.43 thf(fact_1798_odd__two__times__div__two__succ,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 0.17/0.43 = A ) ) ).
% 0.17/0.43
% 0.17/0.43 % odd_two_times_div_two_succ
% 0.17/0.43 thf(fact_1799_odd__two__times__div__two__succ,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 0.17/0.43 = A ) ) ).
% 0.17/0.43
% 0.17/0.43 % odd_two_times_div_two_succ
% 0.17/0.43 thf(fact_1800_odd__two__times__div__two__succ,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 0.17/0.43 = A ) ) ).
% 0.17/0.43
% 0.17/0.43 % odd_two_times_div_two_succ
% 0.17/0.43 thf(fact_1801_power__le__zero__eq__numeral,axiom,
% 0.17/0.43 ! [A: real,W: num] :
% 0.17/0.43 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.17/0.43 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.17/0.43 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_le_zero_eq_numeral
% 0.17/0.43 thf(fact_1802_power__le__zero__eq__numeral,axiom,
% 0.17/0.43 ! [A: rat,W: num] :
% 0.17/0.43 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.17/0.43 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.17/0.43 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_le_zero_eq_numeral
% 0.17/0.43 thf(fact_1803_power__le__zero__eq__numeral,axiom,
% 0.17/0.43 ! [A: int,W: num] :
% 0.17/0.43 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.17/0.43 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.17/0.43 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.17/0.43 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_le_zero_eq_numeral
% 0.17/0.43 thf(fact_1804_set__bit__0,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 0.17/0.43 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % set_bit_0
% 0.17/0.43 thf(fact_1805_set__bit__0,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 0.17/0.43 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % set_bit_0
% 0.17/0.43 thf(fact_1806_even__succ__div__exp,axiom,
% 0.17/0.43 ! [A: code_integer,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.43 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_div_exp
% 0.17/0.43 thf(fact_1807_even__succ__div__exp,axiom,
% 0.17/0.43 ! [A: nat,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.43 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_div_exp
% 0.17/0.43 thf(fact_1808_even__succ__div__exp,axiom,
% 0.17/0.43 ! [A: int,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.43 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_div_exp
% 0.17/0.43 thf(fact_1809_even__succ__mod__exp,axiom,
% 0.17/0.43 ! [A: nat,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.43 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_mod_exp
% 0.17/0.43 thf(fact_1810_even__succ__mod__exp,axiom,
% 0.17/0.43 ! [A: int,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.43 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_mod_exp
% 0.17/0.43 thf(fact_1811_even__succ__mod__exp,axiom,
% 0.17/0.43 ! [A: code_integer,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.17/0.43 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_succ_mod_exp
% 0.17/0.43 thf(fact_1812_split__zdiv,axiom,
% 0.17/0.43 ! [P: int > $o,N: int,K: int] :
% 0.17/0.43 ( ( P @ ( divide_divide_int @ N @ K ) )
% 0.17/0.43 = ( ( ( K = zero_zero_int )
% 0.17/0.43 => ( P @ zero_zero_int ) )
% 0.17/0.43 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.17/0.43 => ! [I2: int,J3: int] :
% 0.17/0.43 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.17/0.43 & ( ord_less_int @ J3 @ K )
% 0.17/0.43 & ( N
% 0.17/0.43 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.17/0.43 => ( P @ I2 ) ) )
% 0.17/0.43 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.17/0.43 => ! [I2: int,J3: int] :
% 0.17/0.43 ( ( ( ord_less_int @ K @ J3 )
% 0.17/0.43 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.17/0.43 & ( N
% 0.17/0.43 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.17/0.43 => ( P @ I2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % split_zdiv
% 0.17/0.43 thf(fact_1813_div__pos__geq,axiom,
% 0.17/0.43 ! [L2: int,K: int] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.17/0.43 => ( ( ord_less_eq_int @ L2 @ K )
% 0.17/0.43 => ( ( divide_divide_int @ K @ L2 )
% 0.17/0.43 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_pos_geq
% 0.17/0.43 thf(fact_1814_int__div__neg__eq,axiom,
% 0.17/0.43 ! [A: int,B2: int,Q2: int,R2: int] :
% 0.17/0.43 ( ( A
% 0.17/0.43 = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 0.17/0.43 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.17/0.43 => ( ( ord_less_int @ B2 @ R2 )
% 0.17/0.43 => ( ( divide_divide_int @ A @ B2 )
% 0.17/0.43 = Q2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % int_div_neg_eq
% 0.17/0.43 thf(fact_1815_int__div__pos__eq,axiom,
% 0.17/0.43 ! [A: int,B2: int,Q2: int,R2: int] :
% 0.17/0.43 ( ( A
% 0.17/0.43 = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 0.17/0.43 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.17/0.43 => ( ( ord_less_int @ R2 @ B2 )
% 0.17/0.43 => ( ( divide_divide_int @ A @ B2 )
% 0.17/0.43 = Q2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % int_div_pos_eq
% 0.17/0.43 thf(fact_1816_div__mod__decomp__int,axiom,
% 0.17/0.43 ! [A3: int,N: int] :
% 0.17/0.43 ( A3
% 0.17/0.43 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ N ) @ N ) @ ( modulo_modulo_int @ A3 @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_mod_decomp_int
% 0.17/0.43 thf(fact_1817_split__neg__lemma,axiom,
% 0.17/0.43 ! [K: int,P: int > int > $o,N: int] :
% 0.17/0.43 ( ( ord_less_int @ K @ zero_zero_int )
% 0.17/0.43 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.17/0.43 = ( ! [I2: int,J3: int] :
% 0.17/0.43 ( ( ( ord_less_int @ K @ J3 )
% 0.17/0.43 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.17/0.43 & ( N
% 0.17/0.43 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.17/0.43 => ( P @ I2 @ J3 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % split_neg_lemma
% 0.17/0.43 thf(fact_1818_split__pos__lemma,axiom,
% 0.17/0.43 ! [K: int,P: int > int > $o,N: int] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ K )
% 0.17/0.43 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.17/0.43 = ( ! [I2: int,J3: int] :
% 0.17/0.43 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.17/0.43 & ( ord_less_int @ J3 @ K )
% 0.17/0.43 & ( N
% 0.17/0.43 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.17/0.43 => ( P @ I2 @ J3 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % split_pos_lemma
% 0.17/0.43 thf(fact_1819_int__div__less__self,axiom,
% 0.17/0.43 ! [X4: int,K: int] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ X4 )
% 0.17/0.43 => ( ( ord_less_int @ one_one_int @ K )
% 0.17/0.43 => ( ord_less_int @ ( divide_divide_int @ X4 @ K ) @ X4 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % int_div_less_self
% 0.17/0.43 thf(fact_1820_verit__le__mono__div__int,axiom,
% 0.17/0.43 ! [A3: int,B4: int,N: int] :
% 0.17/0.43 ( ( ord_less_int @ A3 @ B4 )
% 0.17/0.43 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.17/0.43 => ( ord_less_eq_int
% 0.17/0.43 @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N )
% 0.17/0.43 @ ( if_int
% 0.17/0.43 @ ( ( modulo_modulo_int @ B4 @ N )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 @ one_one_int
% 0.17/0.43 @ zero_zero_int ) )
% 0.17/0.43 @ ( divide_divide_int @ B4 @ N ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % verit_le_mono_div_int
% 0.17/0.43 thf(fact_1821_zdiv__zmult2__eq,axiom,
% 0.17/0.43 ! [C: int,A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.43 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.17/0.43 = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zdiv_zmult2_eq
% 0.17/0.43 thf(fact_1822_zmod__zmult2__eq,axiom,
% 0.17/0.43 ! [C: int,A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.43 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.17/0.43 = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zmod_zmult2_eq
% 0.17/0.43 thf(fact_1823_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.43 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 0.17/0.43 = ( ( ord_less_eq_int @ B2 @ A )
% 0.17/0.43 & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nonneg1_imp_zdiv_pos_iff
% 0.17/0.43 thf(fact_1824_pos__imp__zdiv__nonneg__iff,axiom,
% 0.17/0.43 ! [B2: int,A: int] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.43 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % pos_imp_zdiv_nonneg_iff
% 0.17/0.43 thf(fact_1825_neg__imp__zdiv__nonneg__iff,axiom,
% 0.17/0.43 ! [B2: int,A: int] :
% 0.17/0.43 ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.17/0.43 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 0.17/0.43 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % neg_imp_zdiv_nonneg_iff
% 0.17/0.43 thf(fact_1826_pos__imp__zdiv__pos__iff,axiom,
% 0.17/0.43 ! [K: int,I: int] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ K )
% 0.17/0.43 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 0.17/0.43 = ( ord_less_eq_int @ K @ I ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % pos_imp_zdiv_pos_iff
% 0.17/0.43 thf(fact_1827_pos__imp__zdiv__neg__iff,axiom,
% 0.17/0.43 ! [B2: int,A: int] :
% 0.17/0.43 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.43 => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
% 0.17/0.43 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % pos_imp_zdiv_neg_iff
% 0.17/0.43 thf(fact_1828_neg__imp__zdiv__neg__iff,axiom,
% 0.17/0.43 ! [B2: int,A: int] :
% 0.17/0.43 ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.17/0.43 => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
% 0.17/0.43 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % neg_imp_zdiv_neg_iff
% 0.17/0.43 thf(fact_1829_div__nonpos__pos__le0,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.43 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.43 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_nonpos_pos_le0
% 0.17/0.43 thf(fact_1830_div__nonneg__neg__le0,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.43 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.17/0.43 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_nonneg_neg_le0
% 0.17/0.43 thf(fact_1831_div__neg__pos__less0,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_int @ A @ zero_zero_int )
% 0.17/0.43 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.43 => ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_neg_pos_less0
% 0.17/0.43 thf(fact_1832_div__positive__int,axiom,
% 0.17/0.43 ! [L2: int,K: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ L2 @ K )
% 0.17/0.43 => ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.17/0.43 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_positive_int
% 0.17/0.43 thf(fact_1833_div__int__pos__iff,axiom,
% 0.17/0.43 ! [K: int,L2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
% 0.17/0.43 = ( ( K = zero_zero_int )
% 0.17/0.43 | ( L2 = zero_zero_int )
% 0.17/0.43 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.17/0.43 & ( ord_less_eq_int @ zero_zero_int @ L2 ) )
% 0.17/0.43 | ( ( ord_less_int @ K @ zero_zero_int )
% 0.17/0.43 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_int_pos_iff
% 0.17/0.43 thf(fact_1834_zdiv__mono2__neg,axiom,
% 0.17/0.43 ! [A: int,B: int,B2: int] :
% 0.17/0.43 ( ( ord_less_int @ A @ zero_zero_int )
% 0.17/0.43 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.17/0.43 => ( ( ord_less_eq_int @ B @ B2 )
% 0.17/0.43 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zdiv_mono2_neg
% 0.17/0.43 thf(fact_1835_zdiv__mono1__neg,axiom,
% 0.17/0.43 ! [A: int,A2: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ A @ A2 )
% 0.17/0.43 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.17/0.43 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zdiv_mono1_neg
% 0.17/0.43 thf(fact_1836_zdiv__eq__0__iff,axiom,
% 0.17/0.43 ! [I: int,K: int] :
% 0.17/0.43 ( ( ( divide_divide_int @ I @ K )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 = ( ( K = zero_zero_int )
% 0.17/0.43 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.17/0.43 & ( ord_less_int @ I @ K ) )
% 0.17/0.43 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.17/0.43 & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zdiv_eq_0_iff
% 0.17/0.43 thf(fact_1837_zdiv__mono__strict,axiom,
% 0.17/0.43 ! [A3: int,B4: int,N: int] :
% 0.17/0.43 ( ( ord_less_int @ A3 @ B4 )
% 0.17/0.43 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.17/0.43 => ( ( ( modulo_modulo_int @ A3 @ N )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 => ( ( ( modulo_modulo_int @ B4 @ N )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 => ( ord_less_int @ ( divide_divide_int @ A3 @ N ) @ ( divide_divide_int @ B4 @ N ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zdiv_mono_strict
% 0.17/0.43 thf(fact_1838_zdiv__mono2,axiom,
% 0.17/0.43 ! [A: int,B: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.43 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.17/0.43 => ( ( ord_less_eq_int @ B @ B2 )
% 0.17/0.43 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zdiv_mono2
% 0.17/0.43 thf(fact_1839_zdiv__mono1,axiom,
% 0.17/0.43 ! [A: int,A2: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ A @ A2 )
% 0.17/0.43 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.43 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % zdiv_mono1
% 0.17/0.43 thf(fact_1840_dvd__field__iff,axiom,
% 0.17/0.43 ( dvd_dvd_real
% 0.17/0.43 = ( ^ [A4: real,B3: real] :
% 0.17/0.43 ( ( A4 = zero_zero_real )
% 0.17/0.43 => ( B3 = zero_zero_real ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_field_iff
% 0.17/0.43 thf(fact_1841_dvd__field__iff,axiom,
% 0.17/0.43 ( dvd_dvd_rat
% 0.17/0.43 = ( ^ [A4: rat,B3: rat] :
% 0.17/0.43 ( ( A4 = zero_zero_rat )
% 0.17/0.43 => ( B3 = zero_zero_rat ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_field_iff
% 0.17/0.43 thf(fact_1842_dvd__refl,axiom,
% 0.17/0.43 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_refl
% 0.17/0.43 thf(fact_1843_dvd__refl,axiom,
% 0.17/0.43 ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_refl
% 0.17/0.43 thf(fact_1844_dvd__refl,axiom,
% 0.17/0.43 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_refl
% 0.17/0.43 thf(fact_1845_dvd__trans,axiom,
% 0.17/0.43 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_nat @ B2 @ C )
% 0.17/0.43 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_trans
% 0.17/0.43 thf(fact_1846_dvd__trans,axiom,
% 0.17/0.43 ! [A: int,B2: int,C: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_int @ B2 @ C )
% 0.17/0.43 => ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_trans
% 0.17/0.43 thf(fact_1847_dvd__trans,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ B2 @ C )
% 0.17/0.43 => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_trans
% 0.17/0.43 thf(fact_1848_dvd__0__left,axiom,
% 0.17/0.43 ! [A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.17/0.43 => ( A = zero_z3403309356797280102nteger ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_0_left
% 0.17/0.43 thf(fact_1849_dvd__0__left,axiom,
% 0.17/0.43 ! [A: real] :
% 0.17/0.43 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.17/0.43 => ( A = zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_0_left
% 0.17/0.43 thf(fact_1850_dvd__0__left,axiom,
% 0.17/0.43 ! [A: rat] :
% 0.17/0.43 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.17/0.43 => ( A = zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_0_left
% 0.17/0.43 thf(fact_1851_dvd__0__left,axiom,
% 0.17/0.43 ! [A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.17/0.43 => ( A = zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_0_left
% 0.17/0.43 thf(fact_1852_dvd__0__left,axiom,
% 0.17/0.43 ! [A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.17/0.43 => ( A = zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_0_left
% 0.17/0.43 thf(fact_1853_dvd__antisym,axiom,
% 0.17/0.43 ! [M: nat,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ M @ N )
% 0.17/0.43 => ( ( dvd_dvd_nat @ N @ M )
% 0.17/0.43 => ( M = N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_antisym
% 0.17/0.43 thf(fact_1854_zero__reorient,axiom,
% 0.17/0.43 ! [X4: literal] :
% 0.17/0.43 ( ( zero_zero_literal = X4 )
% 0.17/0.43 = ( X4 = zero_zero_literal ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_reorient
% 0.17/0.43 thf(fact_1855_zero__reorient,axiom,
% 0.17/0.43 ! [X4: real] :
% 0.17/0.43 ( ( zero_zero_real = X4 )
% 0.17/0.43 = ( X4 = zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_reorient
% 0.17/0.43 thf(fact_1856_zero__reorient,axiom,
% 0.17/0.43 ! [X4: rat] :
% 0.17/0.43 ( ( zero_zero_rat = X4 )
% 0.17/0.43 = ( X4 = zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_reorient
% 0.17/0.43 thf(fact_1857_zero__reorient,axiom,
% 0.17/0.43 ! [X4: nat] :
% 0.17/0.43 ( ( zero_zero_nat = X4 )
% 0.17/0.43 = ( X4 = zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_reorient
% 0.17/0.43 thf(fact_1858_zero__reorient,axiom,
% 0.17/0.43 ! [X4: int] :
% 0.17/0.43 ( ( zero_zero_int = X4 )
% 0.17/0.43 = ( X4 = zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_reorient
% 0.17/0.43 thf(fact_1859_not__is__unit__0,axiom,
% 0.17/0.43 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 0.17/0.43
% 0.17/0.43 % not_is_unit_0
% 0.17/0.43 thf(fact_1860_not__is__unit__0,axiom,
% 0.17/0.43 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.17/0.43
% 0.17/0.43 % not_is_unit_0
% 0.17/0.43 thf(fact_1861_not__is__unit__0,axiom,
% 0.17/0.43 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 0.17/0.43
% 0.17/0.43 % not_is_unit_0
% 0.17/0.43 thf(fact_1862_dvd__div__eq__0__iff,axiom,
% 0.17/0.43 ! [B2: code_integer,A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.17/0.43 => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 0.17/0.43 = zero_z3403309356797280102nteger )
% 0.17/0.43 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_eq_0_iff
% 0.17/0.43 thf(fact_1863_dvd__div__eq__0__iff,axiom,
% 0.17/0.43 ! [B2: complex,A: complex] :
% 0.17/0.43 ( ( dvd_dvd_complex @ B2 @ A )
% 0.17/0.43 => ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 0.17/0.43 = zero_zero_complex )
% 0.17/0.43 = ( A = zero_zero_complex ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_eq_0_iff
% 0.17/0.43 thf(fact_1864_dvd__div__eq__0__iff,axiom,
% 0.17/0.43 ! [B2: real,A: real] :
% 0.17/0.43 ( ( dvd_dvd_real @ B2 @ A )
% 0.17/0.43 => ( ( ( divide_divide_real @ A @ B2 )
% 0.17/0.43 = zero_zero_real )
% 0.17/0.43 = ( A = zero_zero_real ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_eq_0_iff
% 0.17/0.43 thf(fact_1865_dvd__div__eq__0__iff,axiom,
% 0.17/0.43 ! [B2: rat,A: rat] :
% 0.17/0.43 ( ( dvd_dvd_rat @ B2 @ A )
% 0.17/0.43 => ( ( ( divide_divide_rat @ A @ B2 )
% 0.17/0.43 = zero_zero_rat )
% 0.17/0.43 = ( A = zero_zero_rat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_eq_0_iff
% 0.17/0.43 thf(fact_1866_dvd__div__eq__0__iff,axiom,
% 0.17/0.43 ! [B2: nat,A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ B2 @ A )
% 0.17/0.43 => ( ( ( divide_divide_nat @ A @ B2 )
% 0.17/0.43 = zero_zero_nat )
% 0.17/0.43 = ( A = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_eq_0_iff
% 0.17/0.43 thf(fact_1867_dvd__div__eq__0__iff,axiom,
% 0.17/0.43 ! [B2: int,A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ B2 @ A )
% 0.17/0.43 => ( ( ( divide_divide_int @ A @ B2 )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 = ( A = zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_eq_0_iff
% 0.17/0.43 thf(fact_1868_nat__dvd__not__less,axiom,
% 0.17/0.43 ! [M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.17/0.43 => ( ( ord_less_nat @ M @ N )
% 0.17/0.43 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nat_dvd_not_less
% 0.17/0.43 thf(fact_1869_mod__0__imp__dvd,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( ( modulo_modulo_nat @ A @ B2 )
% 0.17/0.43 = zero_zero_nat )
% 0.17/0.43 => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % mod_0_imp_dvd
% 0.17/0.43 thf(fact_1870_mod__0__imp__dvd,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ( modulo_modulo_int @ A @ B2 )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 => ( dvd_dvd_int @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % mod_0_imp_dvd
% 0.17/0.43 thf(fact_1871_mod__0__imp__dvd,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.17/0.43 = zero_z3403309356797280102nteger )
% 0.17/0.43 => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % mod_0_imp_dvd
% 0.17/0.43 thf(fact_1872_dvd__eq__mod__eq__0,axiom,
% 0.17/0.43 ( dvd_dvd_nat
% 0.17/0.43 = ( ^ [A4: nat,B3: nat] :
% 0.17/0.43 ( ( modulo_modulo_nat @ B3 @ A4 )
% 0.17/0.43 = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_eq_mod_eq_0
% 0.17/0.43 thf(fact_1873_dvd__eq__mod__eq__0,axiom,
% 0.17/0.43 ( dvd_dvd_int
% 0.17/0.43 = ( ^ [A4: int,B3: int] :
% 0.17/0.43 ( ( modulo_modulo_int @ B3 @ A4 )
% 0.17/0.43 = zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_eq_mod_eq_0
% 0.17/0.43 thf(fact_1874_dvd__eq__mod__eq__0,axiom,
% 0.17/0.43 ( dvd_dvd_Code_integer
% 0.17/0.43 = ( ^ [A4: code_integer,B3: code_integer] :
% 0.17/0.43 ( ( modulo364778990260209775nteger @ B3 @ A4 )
% 0.17/0.43 = zero_z3403309356797280102nteger ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_eq_mod_eq_0
% 0.17/0.43 thf(fact_1875_mod__eq__0__iff__dvd,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( ( modulo_modulo_nat @ A @ B2 )
% 0.17/0.43 = zero_zero_nat )
% 0.17/0.43 = ( dvd_dvd_nat @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % mod_eq_0_iff_dvd
% 0.17/0.43 thf(fact_1876_mod__eq__0__iff__dvd,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ( modulo_modulo_int @ A @ B2 )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 = ( dvd_dvd_int @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % mod_eq_0_iff_dvd
% 0.17/0.43 thf(fact_1877_mod__eq__0__iff__dvd,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.17/0.43 = zero_z3403309356797280102nteger )
% 0.17/0.43 = ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % mod_eq_0_iff_dvd
% 0.17/0.43 thf(fact_1878_list__decode_Ocases,axiom,
% 0.17/0.43 ! [X4: nat] :
% 0.17/0.43 ( ( X4 != zero_zero_nat )
% 0.17/0.43 => ~ ! [N4: nat] :
% 0.17/0.43 ( X4
% 0.17/0.43 != ( suc @ N4 ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % list_decode.cases
% 0.17/0.43 thf(fact_1879_pos__zmod__mult__2,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.43 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.17/0.43 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ A ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % pos_zmod_mult_2
% 0.17/0.43 thf(fact_1880_neg__zmod__mult__2,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.43 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.17/0.43 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % neg_zmod_mult_2
% 0.17/0.43 thf(fact_1881_unit__dvdE,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.43 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.17/0.43 => ! [C3: code_integer] :
% 0.17/0.43 ( B2
% 0.17/0.43 != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_dvdE
% 0.17/0.43 thf(fact_1882_unit__dvdE,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.43 => ~ ( ( A != zero_zero_nat )
% 0.17/0.43 => ! [C3: nat] :
% 0.17/0.43 ( B2
% 0.17/0.43 != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_dvdE
% 0.17/0.43 thf(fact_1883_unit__dvdE,axiom,
% 0.17/0.43 ! [A: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.43 => ~ ( ( A != zero_zero_int )
% 0.17/0.43 => ! [C3: int] :
% 0.17/0.43 ( B2
% 0.17/0.43 != ( times_times_int @ A @ C3 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_dvdE
% 0.17/0.43 thf(fact_1884_dvd__div__eq__mult,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.43 ( ( A != zero_z3403309356797280102nteger )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.43 => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 0.17/0.43 = C )
% 0.17/0.43 = ( B2
% 0.17/0.43 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_eq_mult
% 0.17/0.43 thf(fact_1885_dvd__div__eq__mult,axiom,
% 0.17/0.43 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.43 ( ( A != zero_zero_nat )
% 0.17/0.43 => ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.43 => ( ( ( divide_divide_nat @ B2 @ A )
% 0.17/0.43 = C )
% 0.17/0.43 = ( B2
% 0.17/0.43 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_eq_mult
% 0.17/0.43 thf(fact_1886_dvd__div__eq__mult,axiom,
% 0.17/0.43 ! [A: int,B2: int,C: int] :
% 0.17/0.43 ( ( A != zero_zero_int )
% 0.17/0.43 => ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.43 => ( ( ( divide_divide_int @ B2 @ A )
% 0.17/0.43 = C )
% 0.17/0.43 = ( B2
% 0.17/0.43 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_eq_mult
% 0.17/0.43 thf(fact_1887_div__dvd__iff__mult,axiom,
% 0.17/0.43 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.17/0.43 ( ( B2 != zero_z3403309356797280102nteger )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 0.17/0.43 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_dvd_iff_mult
% 0.17/0.43 thf(fact_1888_div__dvd__iff__mult,axiom,
% 0.17/0.43 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.43 ( ( B2 != zero_zero_nat )
% 0.17/0.43 => ( ( dvd_dvd_nat @ B2 @ A )
% 0.17/0.43 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 0.17/0.43 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_dvd_iff_mult
% 0.17/0.43 thf(fact_1889_div__dvd__iff__mult,axiom,
% 0.17/0.43 ! [B2: int,A: int,C: int] :
% 0.17/0.43 ( ( B2 != zero_zero_int )
% 0.17/0.43 => ( ( dvd_dvd_int @ B2 @ A )
% 0.17/0.43 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 0.17/0.43 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % div_dvd_iff_mult
% 0.17/0.43 thf(fact_1890_dvd__div__iff__mult,axiom,
% 0.17/0.43 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.17/0.43 ( ( C != zero_z3403309356797280102nteger )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.17/0.43 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_iff_mult
% 0.17/0.43 thf(fact_1891_dvd__div__iff__mult,axiom,
% 0.17/0.43 ! [C: nat,B2: nat,A: nat] :
% 0.17/0.43 ( ( C != zero_zero_nat )
% 0.17/0.43 => ( ( dvd_dvd_nat @ C @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 0.17/0.43 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_iff_mult
% 0.17/0.43 thf(fact_1892_dvd__div__iff__mult,axiom,
% 0.17/0.43 ! [C: int,B2: int,A: int] :
% 0.17/0.43 ( ( C != zero_zero_int )
% 0.17/0.43 => ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 0.17/0.43 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_iff_mult
% 0.17/0.43 thf(fact_1893_dvd__div__div__eq__mult,axiom,
% 0.17/0.43 ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
% 0.17/0.43 ( ( A != zero_z3403309356797280102nteger )
% 0.17/0.43 => ( ( C != zero_z3403309356797280102nteger )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.17/0.43 => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 0.17/0.43 = ( divide6298287555418463151nteger @ D @ C ) )
% 0.17/0.43 = ( ( times_3573771949741848930nteger @ B2 @ C )
% 0.17/0.43 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_div_eq_mult
% 0.17/0.43 thf(fact_1894_dvd__div__div__eq__mult,axiom,
% 0.17/0.43 ! [A: nat,C: nat,B2: nat,D: nat] :
% 0.17/0.43 ( ( A != zero_zero_nat )
% 0.17/0.43 => ( ( C != zero_zero_nat )
% 0.17/0.43 => ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_nat @ C @ D )
% 0.17/0.43 => ( ( ( divide_divide_nat @ B2 @ A )
% 0.17/0.43 = ( divide_divide_nat @ D @ C ) )
% 0.17/0.43 = ( ( times_times_nat @ B2 @ C )
% 0.17/0.43 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_div_eq_mult
% 0.17/0.43 thf(fact_1895_dvd__div__div__eq__mult,axiom,
% 0.17/0.43 ! [A: int,C: int,B2: int,D: int] :
% 0.17/0.43 ( ( A != zero_zero_int )
% 0.17/0.43 => ( ( C != zero_zero_int )
% 0.17/0.43 => ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_int @ C @ D )
% 0.17/0.43 => ( ( ( divide_divide_int @ B2 @ A )
% 0.17/0.43 = ( divide_divide_int @ D @ C ) )
% 0.17/0.43 = ( ( times_times_int @ B2 @ C )
% 0.17/0.43 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_div_div_eq_mult
% 0.17/0.43 thf(fact_1896_unit__div__eq__0__iff,axiom,
% 0.17/0.43 ! [B2: code_integer,A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.43 => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 0.17/0.43 = zero_z3403309356797280102nteger )
% 0.17/0.43 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_eq_0_iff
% 0.17/0.43 thf(fact_1897_unit__div__eq__0__iff,axiom,
% 0.17/0.43 ! [B2: nat,A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.43 => ( ( ( divide_divide_nat @ A @ B2 )
% 0.17/0.43 = zero_zero_nat )
% 0.17/0.43 = ( A = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_eq_0_iff
% 0.17/0.43 thf(fact_1898_unit__div__eq__0__iff,axiom,
% 0.17/0.43 ! [B2: int,A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.43 => ( ( ( divide_divide_int @ A @ B2 )
% 0.17/0.43 = zero_zero_int )
% 0.17/0.43 = ( A = zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_div_eq_0_iff
% 0.17/0.43 thf(fact_1899_is__unit__power__iff,axiom,
% 0.17/0.43 ! [A: code_integer,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 0.17/0.43 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.43 | ( N = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % is_unit_power_iff
% 0.17/0.43 thf(fact_1900_is__unit__power__iff,axiom,
% 0.17/0.43 ! [A: nat,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 0.17/0.43 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.43 | ( N = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % is_unit_power_iff
% 0.17/0.43 thf(fact_1901_is__unit__power__iff,axiom,
% 0.17/0.43 ! [A: int,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 0.17/0.43 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.43 | ( N = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % is_unit_power_iff
% 0.17/0.43 thf(fact_1902_unit__imp__mod__eq__0,axiom,
% 0.17/0.43 ! [B2: nat,A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.43 => ( ( modulo_modulo_nat @ A @ B2 )
% 0.17/0.43 = zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_imp_mod_eq_0
% 0.17/0.43 thf(fact_1903_unit__imp__mod__eq__0,axiom,
% 0.17/0.43 ! [B2: int,A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.43 => ( ( modulo_modulo_int @ A @ B2 )
% 0.17/0.43 = zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_imp_mod_eq_0
% 0.17/0.43 thf(fact_1904_unit__imp__mod__eq__0,axiom,
% 0.17/0.43 ! [B2: code_integer,A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.43 => ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.17/0.43 = zero_z3403309356797280102nteger ) ) ).
% 0.17/0.43
% 0.17/0.43 % unit_imp_mod_eq_0
% 0.17/0.43 thf(fact_1905_dvd__imp__le,axiom,
% 0.17/0.43 ! [K: nat,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ K @ N )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_imp_le
% 0.17/0.43 thf(fact_1906_nat__mult__dvd__cancel1,axiom,
% 0.17/0.43 ! [K: nat,M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.43 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.17/0.43 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % nat_mult_dvd_cancel1
% 0.17/0.43 thf(fact_1907_dvd__mult__cancel,axiom,
% 0.17/0.43 ! [K: nat,M: nat,N: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.17/0.43 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.43 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_cancel
% 0.17/0.43 thf(fact_1908_power__0__left,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ( N = zero_zero_nat )
% 0.17/0.43 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.17/0.43 = one_one_rat ) )
% 0.17/0.43 & ( ( N != zero_zero_nat )
% 0.17/0.43 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.17/0.43 = zero_zero_rat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_0_left
% 0.17/0.43 thf(fact_1909_power__0__left,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ( N = zero_zero_nat )
% 0.17/0.43 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.17/0.43 = one_one_nat ) )
% 0.17/0.43 & ( ( N != zero_zero_nat )
% 0.17/0.43 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.17/0.43 = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_0_left
% 0.17/0.43 thf(fact_1910_power__0__left,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ( N = zero_zero_nat )
% 0.17/0.43 => ( ( power_power_real @ zero_zero_real @ N )
% 0.17/0.43 = one_one_real ) )
% 0.17/0.43 & ( ( N != zero_zero_nat )
% 0.17/0.43 => ( ( power_power_real @ zero_zero_real @ N )
% 0.17/0.43 = zero_zero_real ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_0_left
% 0.17/0.43 thf(fact_1911_power__0__left,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ( N = zero_zero_nat )
% 0.17/0.43 => ( ( power_power_int @ zero_zero_int @ N )
% 0.17/0.43 = one_one_int ) )
% 0.17/0.43 & ( ( N != zero_zero_nat )
% 0.17/0.43 => ( ( power_power_int @ zero_zero_int @ N )
% 0.17/0.43 = zero_zero_int ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_0_left
% 0.17/0.43 thf(fact_1912_power__0__left,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ( N = zero_zero_nat )
% 0.17/0.43 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.17/0.43 = one_one_complex ) )
% 0.17/0.43 & ( ( N != zero_zero_nat )
% 0.17/0.43 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.17/0.43 = zero_zero_complex ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % power_0_left
% 0.17/0.43 thf(fact_1913_zero__power,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.17/0.43 = zero_zero_rat ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_power
% 0.17/0.43 thf(fact_1914_zero__power,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.17/0.43 = zero_zero_nat ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_power
% 0.17/0.43 thf(fact_1915_zero__power,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( power_power_real @ zero_zero_real @ N )
% 0.17/0.43 = zero_zero_real ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_power
% 0.17/0.43 thf(fact_1916_zero__power,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( power_power_int @ zero_zero_int @ N )
% 0.17/0.43 = zero_zero_int ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_power
% 0.17/0.43 thf(fact_1917_zero__power,axiom,
% 0.17/0.43 ! [N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.43 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.17/0.43 = zero_zero_complex ) ) ).
% 0.17/0.43
% 0.17/0.43 % zero_power
% 0.17/0.43 thf(fact_1918_mod__greater__zero__iff__not__dvd,axiom,
% 0.17/0.43 ! [M: nat,N: nat] :
% 0.17/0.43 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.17/0.43 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mod_greater_zero_iff_not_dvd
% 0.17/0.43 thf(fact_1919_even__set__bit__iff,axiom,
% 0.17/0.43 ! [M: nat,A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.17/0.43 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 & ( M != zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_set_bit_iff
% 0.17/0.43 thf(fact_1920_even__set__bit__iff,axiom,
% 0.17/0.43 ! [M: nat,A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.17/0.43 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 & ( M != zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_set_bit_iff
% 0.17/0.43 thf(fact_1921_even__set__bit__iff,axiom,
% 0.17/0.43 ! [M: nat,A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.17/0.43 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 & ( M != zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_set_bit_iff
% 0.17/0.43 thf(fact_1922_even__flip__bit__iff,axiom,
% 0.17/0.43 ! [M: nat,A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.17/0.43 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 != ( M = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_flip_bit_iff
% 0.17/0.43 thf(fact_1923_even__flip__bit__iff,axiom,
% 0.17/0.43 ! [M: nat,A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.17/0.43 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 != ( M = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_flip_bit_iff
% 0.17/0.43 thf(fact_1924_even__flip__bit__iff,axiom,
% 0.17/0.43 ! [M: nat,A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.17/0.43 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 != ( M = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_flip_bit_iff
% 0.17/0.43 thf(fact_1925_even__unset__bit__iff,axiom,
% 0.17/0.43 ! [M: nat,A: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.17/0.43 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 | ( M = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_unset_bit_iff
% 0.17/0.43 thf(fact_1926_even__unset__bit__iff,axiom,
% 0.17/0.43 ! [M: nat,A: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.17/0.43 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 | ( M = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_unset_bit_iff
% 0.17/0.43 thf(fact_1927_even__unset__bit__iff,axiom,
% 0.17/0.43 ! [M: nat,A: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.17/0.43 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.43 | ( M = zero_zero_nat ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % even_unset_bit_iff
% 0.17/0.43 thf(fact_1928_dvd__triv__right,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_triv_right
% 0.17/0.43 thf(fact_1929_dvd__triv__right,axiom,
% 0.17/0.43 ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_triv_right
% 0.17/0.43 thf(fact_1930_dvd__triv__right,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_triv_right
% 0.17/0.43 thf(fact_1931_dvd__triv__right,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_triv_right
% 0.17/0.43 thf(fact_1932_dvd__triv__right,axiom,
% 0.17/0.43 ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ A ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_triv_right
% 0.17/0.43 thf(fact_1933_dvd__mult__right,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.17/0.43 => ( dvd_dvd_Code_integer @ B2 @ C ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_right
% 0.17/0.43 thf(fact_1934_dvd__mult__right,axiom,
% 0.17/0.43 ! [A: real,B2: real,C: real] :
% 0.17/0.43 ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
% 0.17/0.43 => ( dvd_dvd_real @ B2 @ C ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_right
% 0.17/0.43 thf(fact_1935_dvd__mult__right,axiom,
% 0.17/0.43 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.43 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 0.17/0.43 => ( dvd_dvd_rat @ B2 @ C ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_right
% 0.17/0.43 thf(fact_1936_dvd__mult__right,axiom,
% 0.17/0.43 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.17/0.43 => ( dvd_dvd_nat @ B2 @ C ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_right
% 0.17/0.43 thf(fact_1937_dvd__mult__right,axiom,
% 0.17/0.43 ! [A: int,B2: int,C: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.17/0.43 => ( dvd_dvd_int @ B2 @ C ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_right
% 0.17/0.43 thf(fact_1938_mult__dvd__mono,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.17/0.43 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_dvd_mono
% 0.17/0.43 thf(fact_1939_mult__dvd__mono,axiom,
% 0.17/0.43 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.43 ( ( dvd_dvd_real @ A @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_real @ C @ D )
% 0.17/0.43 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_dvd_mono
% 0.17/0.43 thf(fact_1940_mult__dvd__mono,axiom,
% 0.17/0.43 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.43 ( ( dvd_dvd_rat @ A @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_rat @ C @ D )
% 0.17/0.43 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_dvd_mono
% 0.17/0.43 thf(fact_1941_mult__dvd__mono,axiom,
% 0.17/0.43 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_nat @ C @ D )
% 0.17/0.43 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_dvd_mono
% 0.17/0.43 thf(fact_1942_mult__dvd__mono,axiom,
% 0.17/0.43 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.43 => ( ( dvd_dvd_int @ C @ D )
% 0.17/0.43 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % mult_dvd_mono
% 0.17/0.43 thf(fact_1943_dvd__triv__left,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_triv_left
% 0.17/0.43 thf(fact_1944_dvd__triv__left,axiom,
% 0.17/0.43 ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_triv_left
% 0.17/0.43 thf(fact_1945_dvd__triv__left,axiom,
% 0.17/0.43 ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_triv_left
% 0.17/0.43 thf(fact_1946_dvd__triv__left,axiom,
% 0.17/0.43 ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_triv_left
% 0.17/0.43 thf(fact_1947_dvd__triv__left,axiom,
% 0.17/0.43 ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B2 ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_triv_left
% 0.17/0.43 thf(fact_1948_dvd__mult__left,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.17/0.43 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_left
% 0.17/0.43 thf(fact_1949_dvd__mult__left,axiom,
% 0.17/0.43 ! [A: real,B2: real,C: real] :
% 0.17/0.43 ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
% 0.17/0.43 => ( dvd_dvd_real @ A @ C ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_left
% 0.17/0.43 thf(fact_1950_dvd__mult__left,axiom,
% 0.17/0.43 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.43 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 0.17/0.43 => ( dvd_dvd_rat @ A @ C ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_left
% 0.17/0.43 thf(fact_1951_dvd__mult__left,axiom,
% 0.17/0.43 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.17/0.43 => ( dvd_dvd_nat @ A @ C ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_left
% 0.17/0.43 thf(fact_1952_dvd__mult__left,axiom,
% 0.17/0.43 ! [A: int,B2: int,C: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.17/0.43 => ( dvd_dvd_int @ A @ C ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult_left
% 0.17/0.43 thf(fact_1953_dvd__mult2,axiom,
% 0.17/0.43 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.43 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult2
% 0.17/0.43 thf(fact_1954_dvd__mult2,axiom,
% 0.17/0.43 ! [A: real,B2: real,C: real] :
% 0.17/0.43 ( ( dvd_dvd_real @ A @ B2 )
% 0.17/0.43 => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult2
% 0.17/0.43 thf(fact_1955_dvd__mult2,axiom,
% 0.17/0.43 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.43 ( ( dvd_dvd_rat @ A @ B2 )
% 0.17/0.43 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult2
% 0.17/0.43 thf(fact_1956_dvd__mult2,axiom,
% 0.17/0.43 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.43 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult2
% 0.17/0.43 thf(fact_1957_dvd__mult2,axiom,
% 0.17/0.43 ! [A: int,B2: int,C: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.43 => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult2
% 0.17/0.43 thf(fact_1958_dvd__mult,axiom,
% 0.17/0.43 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.43 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.17/0.43 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult
% 0.17/0.43 thf(fact_1959_dvd__mult,axiom,
% 0.17/0.43 ! [A: real,C: real,B2: real] :
% 0.17/0.43 ( ( dvd_dvd_real @ A @ C )
% 0.17/0.43 => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult
% 0.17/0.43 thf(fact_1960_dvd__mult,axiom,
% 0.17/0.43 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.43 ( ( dvd_dvd_rat @ A @ C )
% 0.17/0.43 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult
% 0.17/0.43 thf(fact_1961_dvd__mult,axiom,
% 0.17/0.43 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.43 ( ( dvd_dvd_nat @ A @ C )
% 0.17/0.43 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult
% 0.17/0.43 thf(fact_1962_dvd__mult,axiom,
% 0.17/0.43 ! [A: int,C: int,B2: int] :
% 0.17/0.43 ( ( dvd_dvd_int @ A @ C )
% 0.17/0.43 => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.17/0.43
% 0.17/0.43 % dvd_mult
% 0.17/0.43 thf(fact_1963_dvd__def,axiom,
% 0.17/0.43 ( dvd_dvd_Code_integer
% 0.17/0.43 = ( ^ [B3: code_integer,A4: code_integer] :
% 0.17/0.43 ? [K3: code_integer] :
% 0.17/0.44 ( A4
% 0.17/0.44 = ( times_3573771949741848930nteger @ B3 @ K3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_def
% 0.17/0.44 thf(fact_1964_dvd__def,axiom,
% 0.17/0.44 ( dvd_dvd_real
% 0.17/0.44 = ( ^ [B3: real,A4: real] :
% 0.17/0.44 ? [K3: real] :
% 0.17/0.44 ( A4
% 0.17/0.44 = ( times_times_real @ B3 @ K3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_def
% 0.17/0.44 thf(fact_1965_dvd__def,axiom,
% 0.17/0.44 ( dvd_dvd_rat
% 0.17/0.44 = ( ^ [B3: rat,A4: rat] :
% 0.17/0.44 ? [K3: rat] :
% 0.17/0.44 ( A4
% 0.17/0.44 = ( times_times_rat @ B3 @ K3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_def
% 0.17/0.44 thf(fact_1966_dvd__def,axiom,
% 0.17/0.44 ( dvd_dvd_nat
% 0.17/0.44 = ( ^ [B3: nat,A4: nat] :
% 0.17/0.44 ? [K3: nat] :
% 0.17/0.44 ( A4
% 0.17/0.44 = ( times_times_nat @ B3 @ K3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_def
% 0.17/0.44 thf(fact_1967_dvd__def,axiom,
% 0.17/0.44 ( dvd_dvd_int
% 0.17/0.44 = ( ^ [B3: int,A4: int] :
% 0.17/0.44 ? [K3: int] :
% 0.17/0.44 ( A4
% 0.17/0.44 = ( times_times_int @ B3 @ K3 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_def
% 0.17/0.44 thf(fact_1968_dvdI,axiom,
% 0.17/0.44 ! [A: code_integer,B2: code_integer,K: code_integer] :
% 0.17/0.44 ( ( A
% 0.17/0.44 = ( times_3573771949741848930nteger @ B2 @ K ) )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvdI
% 0.17/0.44 thf(fact_1969_dvdI,axiom,
% 0.17/0.44 ! [A: real,B2: real,K: real] :
% 0.17/0.44 ( ( A
% 0.17/0.44 = ( times_times_real @ B2 @ K ) )
% 0.17/0.44 => ( dvd_dvd_real @ B2 @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvdI
% 0.17/0.44 thf(fact_1970_dvdI,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,K: rat] :
% 0.17/0.44 ( ( A
% 0.17/0.44 = ( times_times_rat @ B2 @ K ) )
% 0.17/0.44 => ( dvd_dvd_rat @ B2 @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvdI
% 0.17/0.44 thf(fact_1971_dvdI,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,K: nat] :
% 0.17/0.44 ( ( A
% 0.17/0.44 = ( times_times_nat @ B2 @ K ) )
% 0.17/0.44 => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvdI
% 0.17/0.44 thf(fact_1972_dvdI,axiom,
% 0.17/0.44 ! [A: int,B2: int,K: int] :
% 0.17/0.44 ( ( A
% 0.17/0.44 = ( times_times_int @ B2 @ K ) )
% 0.17/0.44 => ( dvd_dvd_int @ B2 @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvdI
% 0.17/0.44 thf(fact_1973_dvdE,axiom,
% 0.17/0.44 ! [B2: code_integer,A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.17/0.44 => ~ ! [K2: code_integer] :
% 0.17/0.44 ( A
% 0.17/0.44 != ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvdE
% 0.17/0.44 thf(fact_1974_dvdE,axiom,
% 0.17/0.44 ! [B2: real,A: real] :
% 0.17/0.44 ( ( dvd_dvd_real @ B2 @ A )
% 0.17/0.44 => ~ ! [K2: real] :
% 0.17/0.44 ( A
% 0.17/0.44 != ( times_times_real @ B2 @ K2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvdE
% 0.17/0.44 thf(fact_1975_dvdE,axiom,
% 0.17/0.44 ! [B2: rat,A: rat] :
% 0.17/0.44 ( ( dvd_dvd_rat @ B2 @ A )
% 0.17/0.44 => ~ ! [K2: rat] :
% 0.17/0.44 ( A
% 0.17/0.44 != ( times_times_rat @ B2 @ K2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvdE
% 0.17/0.44 thf(fact_1976_dvdE,axiom,
% 0.17/0.44 ! [B2: nat,A: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ B2 @ A )
% 0.17/0.44 => ~ ! [K2: nat] :
% 0.17/0.44 ( A
% 0.17/0.44 != ( times_times_nat @ B2 @ K2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvdE
% 0.17/0.44 thf(fact_1977_dvdE,axiom,
% 0.17/0.44 ! [B2: int,A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ B2 @ A )
% 0.17/0.44 => ~ ! [K2: int] :
% 0.17/0.44 ( A
% 0.17/0.44 != ( times_times_int @ B2 @ K2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvdE
% 0.17/0.44 thf(fact_1978_dvd__unit__imp__unit,axiom,
% 0.17/0.44 ! [A: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_unit_imp_unit
% 0.17/0.44 thf(fact_1979_dvd__unit__imp__unit,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.44 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_unit_imp_unit
% 0.17/0.44 thf(fact_1980_dvd__unit__imp__unit,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.44 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_unit_imp_unit
% 0.17/0.44 thf(fact_1981_unit__imp__dvd,axiom,
% 0.17/0.44 ! [B2: code_integer,A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_imp_dvd
% 0.17/0.44 thf(fact_1982_unit__imp__dvd,axiom,
% 0.17/0.44 ! [B2: nat,A: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.44 => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_imp_dvd
% 0.17/0.44 thf(fact_1983_unit__imp__dvd,axiom,
% 0.17/0.44 ! [B2: int,A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.44 => ( dvd_dvd_int @ B2 @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_imp_dvd
% 0.17/0.44 thf(fact_1984_one__dvd,axiom,
% 0.17/0.44 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 0.17/0.44
% 0.17/0.44 % one_dvd
% 0.17/0.44 thf(fact_1985_one__dvd,axiom,
% 0.17/0.44 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 0.17/0.44
% 0.17/0.44 % one_dvd
% 0.17/0.44 thf(fact_1986_one__dvd,axiom,
% 0.17/0.44 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 0.17/0.44
% 0.17/0.44 % one_dvd
% 0.17/0.44 thf(fact_1987_one__dvd,axiom,
% 0.17/0.44 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 0.17/0.44
% 0.17/0.44 % one_dvd
% 0.17/0.44 thf(fact_1988_one__dvd,axiom,
% 0.17/0.44 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 0.17/0.44
% 0.17/0.44 % one_dvd
% 0.17/0.44 thf(fact_1989_one__dvd,axiom,
% 0.17/0.44 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 0.17/0.44
% 0.17/0.44 % one_dvd
% 0.17/0.44 thf(fact_1990_dvd__add__right__iff,axiom,
% 0.17/0.44 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add_right_iff
% 0.17/0.44 thf(fact_1991_dvd__add__right__iff,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real] :
% 0.17/0.44 ( ( dvd_dvd_real @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add_right_iff
% 0.17/0.44 thf(fact_1992_dvd__add__right__iff,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.44 ( ( dvd_dvd_rat @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add_right_iff
% 0.17/0.44 thf(fact_1993_dvd__add__right__iff,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add_right_iff
% 0.17/0.44 thf(fact_1994_dvd__add__right__iff,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add_right_iff
% 0.17/0.44 thf(fact_1995_dvd__add__left__iff,axiom,
% 0.17/0.44 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add_left_iff
% 0.17/0.44 thf(fact_1996_dvd__add__left__iff,axiom,
% 0.17/0.44 ! [A: real,C: real,B2: real] :
% 0.17/0.44 ( ( dvd_dvd_real @ A @ C )
% 0.17/0.44 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add_left_iff
% 0.17/0.44 thf(fact_1997_dvd__add__left__iff,axiom,
% 0.17/0.44 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.44 ( ( dvd_dvd_rat @ A @ C )
% 0.17/0.44 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add_left_iff
% 0.17/0.44 thf(fact_1998_dvd__add__left__iff,axiom,
% 0.17/0.44 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ A @ C )
% 0.17/0.44 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_nat @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add_left_iff
% 0.17/0.44 thf(fact_1999_dvd__add__left__iff,axiom,
% 0.17/0.44 ! [A: int,C: int,B2: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ A @ C )
% 0.17/0.44 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add_left_iff
% 0.17/0.44 thf(fact_2000_dvd__add,axiom,
% 0.17/0.44 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add
% 0.17/0.44 thf(fact_2001_dvd__add,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real] :
% 0.17/0.44 ( ( dvd_dvd_real @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_real @ A @ C )
% 0.17/0.44 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add
% 0.17/0.44 thf(fact_2002_dvd__add,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.44 ( ( dvd_dvd_rat @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_rat @ A @ C )
% 0.17/0.44 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add
% 0.17/0.44 thf(fact_2003_dvd__add,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_nat @ A @ C )
% 0.17/0.44 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add
% 0.17/0.44 thf(fact_2004_dvd__add,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ A @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_int @ A @ C )
% 0.17/0.44 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_add
% 0.17/0.44 thf(fact_2005_dvd__diff__commute,axiom,
% 0.17/0.44 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B2 ) )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B2 @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_diff_commute
% 0.17/0.44 thf(fact_2006_dvd__diff__commute,axiom,
% 0.17/0.44 ! [A: int,C: int,B2: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 0.17/0.44 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_diff_commute
% 0.17/0.44 thf(fact_2007_dvd__diff,axiom,
% 0.17/0.44 ! [X4: code_integer,Y3: code_integer,Z: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ X4 @ Y3 )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ X4 @ Z )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ X4 @ ( minus_8373710615458151222nteger @ Y3 @ Z ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_diff
% 0.17/0.44 thf(fact_2008_dvd__diff,axiom,
% 0.17/0.44 ! [X4: real,Y3: real,Z: real] :
% 0.17/0.44 ( ( dvd_dvd_real @ X4 @ Y3 )
% 0.17/0.44 => ( ( dvd_dvd_real @ X4 @ Z )
% 0.17/0.44 => ( dvd_dvd_real @ X4 @ ( minus_minus_real @ Y3 @ Z ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_diff
% 0.17/0.44 thf(fact_2009_dvd__diff,axiom,
% 0.17/0.44 ! [X4: rat,Y3: rat,Z: rat] :
% 0.17/0.44 ( ( dvd_dvd_rat @ X4 @ Y3 )
% 0.17/0.44 => ( ( dvd_dvd_rat @ X4 @ Z )
% 0.17/0.44 => ( dvd_dvd_rat @ X4 @ ( minus_minus_rat @ Y3 @ Z ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_diff
% 0.17/0.44 thf(fact_2010_dvd__diff,axiom,
% 0.17/0.44 ! [X4: int,Y3: int,Z: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ X4 @ Y3 )
% 0.17/0.44 => ( ( dvd_dvd_int @ X4 @ Z )
% 0.17/0.44 => ( dvd_dvd_int @ X4 @ ( minus_minus_int @ Y3 @ Z ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_diff
% 0.17/0.44 thf(fact_2011_dvd__div__eq__iff,axiom,
% 0.17/0.44 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.44 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.17/0.44 = ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.17/0.44 = ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_iff
% 0.17/0.44 thf(fact_2012_dvd__div__eq__iff,axiom,
% 0.17/0.44 ! [C: complex,A: complex,B2: complex] :
% 0.17/0.44 ( ( dvd_dvd_complex @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_complex @ C @ B2 )
% 0.17/0.44 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.17/0.44 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.17/0.44 = ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_iff
% 0.17/0.44 thf(fact_2013_dvd__div__eq__iff,axiom,
% 0.17/0.44 ! [C: real,A: real,B2: real] :
% 0.17/0.44 ( ( dvd_dvd_real @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_real @ C @ B2 )
% 0.17/0.44 => ( ( ( divide_divide_real @ A @ C )
% 0.17/0.44 = ( divide_divide_real @ B2 @ C ) )
% 0.17/0.44 = ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_iff
% 0.17/0.44 thf(fact_2014_dvd__div__eq__iff,axiom,
% 0.17/0.44 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.44 ( ( dvd_dvd_rat @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_rat @ C @ B2 )
% 0.17/0.44 => ( ( ( divide_divide_rat @ A @ C )
% 0.17/0.44 = ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.44 = ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_iff
% 0.17/0.44 thf(fact_2015_dvd__div__eq__iff,axiom,
% 0.17/0.44 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_nat @ C @ B2 )
% 0.17/0.44 => ( ( ( divide_divide_nat @ A @ C )
% 0.17/0.44 = ( divide_divide_nat @ B2 @ C ) )
% 0.17/0.44 = ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_iff
% 0.17/0.44 thf(fact_2016_dvd__div__eq__iff,axiom,
% 0.17/0.44 ! [C: int,A: int,B2: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.44 => ( ( ( divide_divide_int @ A @ C )
% 0.17/0.44 = ( divide_divide_int @ B2 @ C ) )
% 0.17/0.44 = ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_iff
% 0.17/0.44 thf(fact_2017_dvd__div__eq__cancel,axiom,
% 0.17/0.44 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.17/0.44 = ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.44 => ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_cancel
% 0.17/0.44 thf(fact_2018_dvd__div__eq__cancel,axiom,
% 0.17/0.44 ! [A: complex,C: complex,B2: complex] :
% 0.17/0.44 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.17/0.44 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.17/0.44 => ( ( dvd_dvd_complex @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_complex @ C @ B2 )
% 0.17/0.44 => ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_cancel
% 0.17/0.44 thf(fact_2019_dvd__div__eq__cancel,axiom,
% 0.17/0.44 ! [A: real,C: real,B2: real] :
% 0.17/0.44 ( ( ( divide_divide_real @ A @ C )
% 0.17/0.44 = ( divide_divide_real @ B2 @ C ) )
% 0.17/0.44 => ( ( dvd_dvd_real @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_real @ C @ B2 )
% 0.17/0.44 => ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_cancel
% 0.17/0.44 thf(fact_2020_dvd__div__eq__cancel,axiom,
% 0.17/0.44 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.44 ( ( ( divide_divide_rat @ A @ C )
% 0.17/0.44 = ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.44 => ( ( dvd_dvd_rat @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_rat @ C @ B2 )
% 0.17/0.44 => ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_cancel
% 0.17/0.44 thf(fact_2021_dvd__div__eq__cancel,axiom,
% 0.17/0.44 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.44 ( ( ( divide_divide_nat @ A @ C )
% 0.17/0.44 = ( divide_divide_nat @ B2 @ C ) )
% 0.17/0.44 => ( ( dvd_dvd_nat @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_nat @ C @ B2 )
% 0.17/0.44 => ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_cancel
% 0.17/0.44 thf(fact_2022_dvd__div__eq__cancel,axiom,
% 0.17/0.44 ! [A: int,C: int,B2: int] :
% 0.17/0.44 ( ( ( divide_divide_int @ A @ C )
% 0.17/0.44 = ( divide_divide_int @ B2 @ C ) )
% 0.17/0.44 => ( ( dvd_dvd_int @ C @ A )
% 0.17/0.44 => ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.44 => ( A = B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_eq_cancel
% 0.17/0.44 thf(fact_2023_div__div__div__same,axiom,
% 0.17/0.44 ! [D: code_integer,B2: code_integer,A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ D @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.17/0.44 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B2 @ D ) )
% 0.17/0.44 = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_div_div_same
% 0.17/0.44 thf(fact_2024_div__div__div__same,axiom,
% 0.17/0.44 ! [D: nat,B2: nat,A: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ D @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_nat @ B2 @ A )
% 0.17/0.44 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B2 @ D ) )
% 0.17/0.44 = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_div_div_same
% 0.17/0.44 thf(fact_2025_div__div__div__same,axiom,
% 0.17/0.44 ! [D: int,B2: int,A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ D @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_int @ B2 @ A )
% 0.17/0.44 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B2 @ D ) )
% 0.17/0.44 = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_div_div_same
% 0.17/0.44 thf(fact_2026_dvd__power__same,axiom,
% 0.17/0.44 ! [X4: code_integer,Y3: code_integer,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ X4 @ Y3 )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_same
% 0.17/0.44 thf(fact_2027_dvd__power__same,axiom,
% 0.17/0.44 ! [X4: nat,Y3: nat,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ X4 @ Y3 )
% 0.17/0.44 => ( dvd_dvd_nat @ ( power_power_nat @ X4 @ N ) @ ( power_power_nat @ Y3 @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_same
% 0.17/0.44 thf(fact_2028_dvd__power__same,axiom,
% 0.17/0.44 ! [X4: real,Y3: real,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_real @ X4 @ Y3 )
% 0.17/0.44 => ( dvd_dvd_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ Y3 @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_same
% 0.17/0.44 thf(fact_2029_dvd__power__same,axiom,
% 0.17/0.44 ! [X4: int,Y3: int,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_int @ X4 @ Y3 )
% 0.17/0.44 => ( dvd_dvd_int @ ( power_power_int @ X4 @ N ) @ ( power_power_int @ Y3 @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_same
% 0.17/0.44 thf(fact_2030_dvd__power__same,axiom,
% 0.17/0.44 ! [X4: complex,Y3: complex,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_complex @ X4 @ Y3 )
% 0.17/0.44 => ( dvd_dvd_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ Y3 @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_same
% 0.17/0.44 thf(fact_2031_dvd__mod__iff,axiom,
% 0.17/0.44 ! [C: nat,B2: nat,A: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ C @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.17/0.44 = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mod_iff
% 0.17/0.44 thf(fact_2032_dvd__mod__iff,axiom,
% 0.17/0.44 ! [C: int,B2: int,A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 0.17/0.44 = ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mod_iff
% 0.17/0.44 thf(fact_2033_dvd__mod__iff,axiom,
% 0.17/0.44 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mod_iff
% 0.17/0.44 thf(fact_2034_dvd__mod__imp__dvd,axiom,
% 0.17/0.44 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.17/0.44 => ( ( dvd_dvd_nat @ C @ B2 )
% 0.17/0.44 => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mod_imp_dvd
% 0.17/0.44 thf(fact_2035_dvd__mod__imp__dvd,axiom,
% 0.17/0.44 ! [C: int,A: int,B2: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 0.17/0.44 => ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.44 => ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mod_imp_dvd
% 0.17/0.44 thf(fact_2036_dvd__mod__imp__dvd,axiom,
% 0.17/0.44 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mod_imp_dvd
% 0.17/0.44 thf(fact_2037_dvd__mod,axiom,
% 0.17/0.44 ! [K: nat,M: nat,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ K @ M )
% 0.17/0.44 => ( ( dvd_dvd_nat @ K @ N )
% 0.17/0.44 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mod
% 0.17/0.44 thf(fact_2038_dvd__mod,axiom,
% 0.17/0.44 ! [K: int,M: int,N: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ K @ M )
% 0.17/0.44 => ( ( dvd_dvd_int @ K @ N )
% 0.17/0.44 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mod
% 0.17/0.44 thf(fact_2039_dvd__mod,axiom,
% 0.17/0.44 ! [K: code_integer,M: code_integer,N: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ K @ M )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ K @ N )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mod
% 0.17/0.44 thf(fact_2040_mod__mod__cancel,axiom,
% 0.17/0.44 ! [C: nat,B2: nat,A: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ C @ B2 )
% 0.17/0.44 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C )
% 0.17/0.44 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mod_mod_cancel
% 0.17/0.44 thf(fact_2041_mod__mod__cancel,axiom,
% 0.17/0.44 ! [C: int,B2: int,A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.44 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ C )
% 0.17/0.44 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mod_mod_cancel
% 0.17/0.44 thf(fact_2042_mod__mod__cancel,axiom,
% 0.17/0.44 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.44 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C )
% 0.17/0.44 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mod_mod_cancel
% 0.17/0.44 thf(fact_2043_dvd__diff__nat,axiom,
% 0.17/0.44 ! [K: nat,M: nat,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ K @ M )
% 0.17/0.44 => ( ( dvd_dvd_nat @ K @ N )
% 0.17/0.44 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_diff_nat
% 0.17/0.44 thf(fact_2044_zero__le,axiom,
% 0.17/0.44 ! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% 0.17/0.44
% 0.17/0.44 % zero_le
% 0.17/0.44 thf(fact_2045_le__numeral__extra_I3_J,axiom,
% 0.17/0.44 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.17/0.44
% 0.17/0.44 % le_numeral_extra(3)
% 0.17/0.44 thf(fact_2046_le__numeral__extra_I3_J,axiom,
% 0.17/0.44 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 0.17/0.44
% 0.17/0.44 % le_numeral_extra(3)
% 0.17/0.44 thf(fact_2047_le__numeral__extra_I3_J,axiom,
% 0.17/0.44 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.17/0.44
% 0.17/0.44 % le_numeral_extra(3)
% 0.17/0.44 thf(fact_2048_le__numeral__extra_I3_J,axiom,
% 0.17/0.44 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.17/0.44
% 0.17/0.44 % le_numeral_extra(3)
% 0.17/0.44 thf(fact_2049_zero__less__iff__neq__zero,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 = ( N != zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_iff_neq_zero
% 0.17/0.44 thf(fact_2050_gr__implies__not__zero,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_nat @ M @ N )
% 0.17/0.44 => ( N != zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % gr_implies_not_zero
% 0.17/0.44 thf(fact_2051_not__less__zero,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % not_less_zero
% 0.17/0.44 thf(fact_2052_gr__zeroI,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ( ( N != zero_zero_nat )
% 0.17/0.44 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % gr_zeroI
% 0.17/0.44 thf(fact_2053_less__numeral__extra_I3_J,axiom,
% 0.17/0.44 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.17/0.44
% 0.17/0.44 % less_numeral_extra(3)
% 0.17/0.44 thf(fact_2054_less__numeral__extra_I3_J,axiom,
% 0.17/0.44 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 0.17/0.44
% 0.17/0.44 % less_numeral_extra(3)
% 0.17/0.44 thf(fact_2055_less__numeral__extra_I3_J,axiom,
% 0.17/0.44 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % less_numeral_extra(3)
% 0.17/0.44 thf(fact_2056_less__numeral__extra_I3_J,axiom,
% 0.17/0.44 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.17/0.44
% 0.17/0.44 % less_numeral_extra(3)
% 0.17/0.44 thf(fact_2057_field__lbound__gt__zero,axiom,
% 0.17/0.44 ! [D1: real,D22: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.17/0.44 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 0.17/0.44 => ? [E2: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.17/0.44 & ( ord_less_real @ E2 @ D1 )
% 0.17/0.44 & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % field_lbound_gt_zero
% 0.17/0.44 thf(fact_2058_field__lbound__gt__zero,axiom,
% 0.17/0.44 ! [D1: rat,D22: rat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 0.17/0.44 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 0.17/0.44 => ? [E2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.17/0.44 & ( ord_less_rat @ E2 @ D1 )
% 0.17/0.44 & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % field_lbound_gt_zero
% 0.17/0.44 thf(fact_2059_zero__neq__numeral,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ( zero_zero_complex
% 0.17/0.44 != ( numera6690914467698888265omplex @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_neq_numeral
% 0.17/0.44 thf(fact_2060_zero__neq__numeral,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ( zero_zero_real
% 0.17/0.44 != ( numeral_numeral_real @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_neq_numeral
% 0.17/0.44 thf(fact_2061_zero__neq__numeral,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ( zero_zero_rat
% 0.17/0.44 != ( numeral_numeral_rat @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_neq_numeral
% 0.17/0.44 thf(fact_2062_zero__neq__numeral,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ( zero_zero_nat
% 0.17/0.44 != ( numeral_numeral_nat @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_neq_numeral
% 0.17/0.44 thf(fact_2063_zero__neq__numeral,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ( zero_zero_int
% 0.17/0.44 != ( numeral_numeral_int @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_neq_numeral
% 0.17/0.44 thf(fact_2064_mult__right__cancel,axiom,
% 0.17/0.44 ! [C: real,A: real,B2: real] :
% 0.17/0.44 ( ( C != zero_zero_real )
% 0.17/0.44 => ( ( ( times_times_real @ A @ C )
% 0.17/0.44 = ( times_times_real @ B2 @ C ) )
% 0.17/0.44 = ( A = B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_right_cancel
% 0.17/0.44 thf(fact_2065_mult__right__cancel,axiom,
% 0.17/0.44 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.44 ( ( C != zero_zero_rat )
% 0.17/0.44 => ( ( ( times_times_rat @ A @ C )
% 0.17/0.44 = ( times_times_rat @ B2 @ C ) )
% 0.17/0.44 = ( A = B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_right_cancel
% 0.17/0.44 thf(fact_2066_mult__right__cancel,axiom,
% 0.17/0.44 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.44 ( ( C != zero_zero_nat )
% 0.17/0.44 => ( ( ( times_times_nat @ A @ C )
% 0.17/0.44 = ( times_times_nat @ B2 @ C ) )
% 0.17/0.44 = ( A = B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_right_cancel
% 0.17/0.44 thf(fact_2067_mult__right__cancel,axiom,
% 0.17/0.44 ! [C: int,A: int,B2: int] :
% 0.17/0.44 ( ( C != zero_zero_int )
% 0.17/0.44 => ( ( ( times_times_int @ A @ C )
% 0.17/0.44 = ( times_times_int @ B2 @ C ) )
% 0.17/0.44 = ( A = B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_right_cancel
% 0.17/0.44 thf(fact_2068_mult__left__cancel,axiom,
% 0.17/0.44 ! [C: real,A: real,B2: real] :
% 0.17/0.44 ( ( C != zero_zero_real )
% 0.17/0.44 => ( ( ( times_times_real @ C @ A )
% 0.17/0.44 = ( times_times_real @ C @ B2 ) )
% 0.17/0.44 = ( A = B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_left_cancel
% 0.17/0.44 thf(fact_2069_mult__left__cancel,axiom,
% 0.17/0.44 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.44 ( ( C != zero_zero_rat )
% 0.17/0.44 => ( ( ( times_times_rat @ C @ A )
% 0.17/0.44 = ( times_times_rat @ C @ B2 ) )
% 0.17/0.44 = ( A = B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_left_cancel
% 0.17/0.44 thf(fact_2070_mult__left__cancel,axiom,
% 0.17/0.44 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.44 ( ( C != zero_zero_nat )
% 0.17/0.44 => ( ( ( times_times_nat @ C @ A )
% 0.17/0.44 = ( times_times_nat @ C @ B2 ) )
% 0.17/0.44 = ( A = B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_left_cancel
% 0.17/0.44 thf(fact_2071_mult__left__cancel,axiom,
% 0.17/0.44 ! [C: int,A: int,B2: int] :
% 0.17/0.44 ( ( C != zero_zero_int )
% 0.17/0.44 => ( ( ( times_times_int @ C @ A )
% 0.17/0.44 = ( times_times_int @ C @ B2 ) )
% 0.17/0.44 = ( A = B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_left_cancel
% 0.17/0.44 thf(fact_2072_no__zero__divisors,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( A != zero_zero_real )
% 0.17/0.44 => ( ( B2 != zero_zero_real )
% 0.17/0.44 => ( ( times_times_real @ A @ B2 )
% 0.17/0.44 != zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % no_zero_divisors
% 0.17/0.44 thf(fact_2073_no__zero__divisors,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( A != zero_zero_rat )
% 0.17/0.44 => ( ( B2 != zero_zero_rat )
% 0.17/0.44 => ( ( times_times_rat @ A @ B2 )
% 0.17/0.44 != zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % no_zero_divisors
% 0.17/0.44 thf(fact_2074_no__zero__divisors,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( A != zero_zero_nat )
% 0.17/0.44 => ( ( B2 != zero_zero_nat )
% 0.17/0.44 => ( ( times_times_nat @ A @ B2 )
% 0.17/0.44 != zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % no_zero_divisors
% 0.17/0.44 thf(fact_2075_no__zero__divisors,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( A != zero_zero_int )
% 0.17/0.44 => ( ( B2 != zero_zero_int )
% 0.17/0.44 => ( ( times_times_int @ A @ B2 )
% 0.17/0.44 != zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % no_zero_divisors
% 0.17/0.44 thf(fact_2076_divisors__zero,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ( times_times_real @ A @ B2 )
% 0.17/0.44 = zero_zero_real )
% 0.17/0.44 => ( ( A = zero_zero_real )
% 0.17/0.44 | ( B2 = zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % divisors_zero
% 0.17/0.44 thf(fact_2077_divisors__zero,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ( times_times_rat @ A @ B2 )
% 0.17/0.44 = zero_zero_rat )
% 0.17/0.44 => ( ( A = zero_zero_rat )
% 0.17/0.44 | ( B2 = zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % divisors_zero
% 0.17/0.44 thf(fact_2078_divisors__zero,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ( times_times_nat @ A @ B2 )
% 0.17/0.44 = zero_zero_nat )
% 0.17/0.44 => ( ( A = zero_zero_nat )
% 0.17/0.44 | ( B2 = zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % divisors_zero
% 0.17/0.44 thf(fact_2079_divisors__zero,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ( times_times_int @ A @ B2 )
% 0.17/0.44 = zero_zero_int )
% 0.17/0.44 => ( ( A = zero_zero_int )
% 0.17/0.44 | ( B2 = zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % divisors_zero
% 0.17/0.44 thf(fact_2080_mult__not__zero,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ( times_times_real @ A @ B2 )
% 0.17/0.44 != zero_zero_real )
% 0.17/0.44 => ( ( A != zero_zero_real )
% 0.17/0.44 & ( B2 != zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_not_zero
% 0.17/0.44 thf(fact_2081_mult__not__zero,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ( times_times_rat @ A @ B2 )
% 0.17/0.44 != zero_zero_rat )
% 0.17/0.44 => ( ( A != zero_zero_rat )
% 0.17/0.44 & ( B2 != zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_not_zero
% 0.17/0.44 thf(fact_2082_mult__not__zero,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ( times_times_nat @ A @ B2 )
% 0.17/0.44 != zero_zero_nat )
% 0.17/0.44 => ( ( A != zero_zero_nat )
% 0.17/0.44 & ( B2 != zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_not_zero
% 0.17/0.44 thf(fact_2083_mult__not__zero,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ( times_times_int @ A @ B2 )
% 0.17/0.44 != zero_zero_int )
% 0.17/0.44 => ( ( A != zero_zero_int )
% 0.17/0.44 & ( B2 != zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_not_zero
% 0.17/0.44 thf(fact_2084_zero__neq__one,axiom,
% 0.17/0.44 zero_zero_complex != one_one_complex ).
% 0.17/0.44
% 0.17/0.44 % zero_neq_one
% 0.17/0.44 thf(fact_2085_zero__neq__one,axiom,
% 0.17/0.44 zero_zero_real != one_one_real ).
% 0.17/0.44
% 0.17/0.44 % zero_neq_one
% 0.17/0.44 thf(fact_2086_zero__neq__one,axiom,
% 0.17/0.44 zero_zero_rat != one_one_rat ).
% 0.17/0.44
% 0.17/0.44 % zero_neq_one
% 0.17/0.44 thf(fact_2087_zero__neq__one,axiom,
% 0.17/0.44 zero_zero_nat != one_one_nat ).
% 0.17/0.44
% 0.17/0.44 % zero_neq_one
% 0.17/0.44 thf(fact_2088_zero__neq__one,axiom,
% 0.17/0.44 zero_zero_int != one_one_int ).
% 0.17/0.44
% 0.17/0.44 % zero_neq_one
% 0.17/0.44 thf(fact_2089_add_Ogroup__left__neutral,axiom,
% 0.17/0.44 ! [A: real] :
% 0.17/0.44 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % add.group_left_neutral
% 0.17/0.44 thf(fact_2090_add_Ogroup__left__neutral,axiom,
% 0.17/0.44 ! [A: rat] :
% 0.17/0.44 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % add.group_left_neutral
% 0.17/0.44 thf(fact_2091_add_Ogroup__left__neutral,axiom,
% 0.17/0.44 ! [A: int] :
% 0.17/0.44 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % add.group_left_neutral
% 0.17/0.44 thf(fact_2092_add_Ocomm__neutral,axiom,
% 0.17/0.44 ! [A: real] :
% 0.17/0.44 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % add.comm_neutral
% 0.17/0.44 thf(fact_2093_add_Ocomm__neutral,axiom,
% 0.17/0.44 ! [A: rat] :
% 0.17/0.44 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % add.comm_neutral
% 0.17/0.44 thf(fact_2094_add_Ocomm__neutral,axiom,
% 0.17/0.44 ! [A: nat] :
% 0.17/0.44 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % add.comm_neutral
% 0.17/0.44 thf(fact_2095_add_Ocomm__neutral,axiom,
% 0.17/0.44 ! [A: int] :
% 0.17/0.44 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % add.comm_neutral
% 0.17/0.44 thf(fact_2096_comm__monoid__add__class_Oadd__0,axiom,
% 0.17/0.44 ! [A: real] :
% 0.17/0.44 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % comm_monoid_add_class.add_0
% 0.17/0.44 thf(fact_2097_comm__monoid__add__class_Oadd__0,axiom,
% 0.17/0.44 ! [A: rat] :
% 0.17/0.44 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % comm_monoid_add_class.add_0
% 0.17/0.44 thf(fact_2098_comm__monoid__add__class_Oadd__0,axiom,
% 0.17/0.44 ! [A: nat] :
% 0.17/0.44 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % comm_monoid_add_class.add_0
% 0.17/0.44 thf(fact_2099_comm__monoid__add__class_Oadd__0,axiom,
% 0.17/0.44 ! [A: int] :
% 0.17/0.44 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % comm_monoid_add_class.add_0
% 0.17/0.44 thf(fact_2100_verit__sum__simplify,axiom,
% 0.17/0.44 ! [A: real] :
% 0.17/0.44 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % verit_sum_simplify
% 0.17/0.44 thf(fact_2101_verit__sum__simplify,axiom,
% 0.17/0.44 ! [A: rat] :
% 0.17/0.44 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % verit_sum_simplify
% 0.17/0.44 thf(fact_2102_verit__sum__simplify,axiom,
% 0.17/0.44 ! [A: nat] :
% 0.17/0.44 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % verit_sum_simplify
% 0.17/0.44 thf(fact_2103_verit__sum__simplify,axiom,
% 0.17/0.44 ! [A: int] :
% 0.17/0.44 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.17/0.44 = A ) ).
% 0.17/0.44
% 0.17/0.44 % verit_sum_simplify
% 0.17/0.44 thf(fact_2104_eq__iff__diff__eq__0,axiom,
% 0.17/0.44 ( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
% 0.17/0.44 = ( ^ [A4: real,B3: real] :
% 0.17/0.44 ( ( minus_minus_real @ A4 @ B3 )
% 0.17/0.44 = zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % eq_iff_diff_eq_0
% 0.17/0.44 thf(fact_2105_eq__iff__diff__eq__0,axiom,
% 0.17/0.44 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 0.17/0.44 = ( ^ [A4: rat,B3: rat] :
% 0.17/0.44 ( ( minus_minus_rat @ A4 @ B3 )
% 0.17/0.44 = zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % eq_iff_diff_eq_0
% 0.17/0.44 thf(fact_2106_eq__iff__diff__eq__0,axiom,
% 0.17/0.44 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.17/0.44 = ( ^ [A4: int,B3: int] :
% 0.17/0.44 ( ( minus_minus_int @ A4 @ B3 )
% 0.17/0.44 = zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % eq_iff_diff_eq_0
% 0.17/0.44 thf(fact_2107_power__not__zero,axiom,
% 0.17/0.44 ! [A: rat,N: nat] :
% 0.17/0.44 ( ( A != zero_zero_rat )
% 0.17/0.44 => ( ( power_power_rat @ A @ N )
% 0.17/0.44 != zero_zero_rat ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_not_zero
% 0.17/0.44 thf(fact_2108_power__not__zero,axiom,
% 0.17/0.44 ! [A: nat,N: nat] :
% 0.17/0.44 ( ( A != zero_zero_nat )
% 0.17/0.44 => ( ( power_power_nat @ A @ N )
% 0.17/0.44 != zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_not_zero
% 0.17/0.44 thf(fact_2109_power__not__zero,axiom,
% 0.17/0.44 ! [A: real,N: nat] :
% 0.17/0.44 ( ( A != zero_zero_real )
% 0.17/0.44 => ( ( power_power_real @ A @ N )
% 0.17/0.44 != zero_zero_real ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_not_zero
% 0.17/0.44 thf(fact_2110_power__not__zero,axiom,
% 0.17/0.44 ! [A: int,N: nat] :
% 0.17/0.44 ( ( A != zero_zero_int )
% 0.17/0.44 => ( ( power_power_int @ A @ N )
% 0.17/0.44 != zero_zero_int ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_not_zero
% 0.17/0.44 thf(fact_2111_power__not__zero,axiom,
% 0.17/0.44 ! [A: complex,N: nat] :
% 0.17/0.44 ( ( A != zero_zero_complex )
% 0.17/0.44 => ( ( power_power_complex @ A @ N )
% 0.17/0.44 != zero_zero_complex ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_not_zero
% 0.17/0.44 thf(fact_2112_not0__implies__Suc,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ( ( N != zero_zero_nat )
% 0.17/0.44 => ? [M4: nat] :
% 0.17/0.44 ( N
% 0.17/0.44 = ( suc @ M4 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % not0_implies_Suc
% 0.17/0.44 thf(fact_2113_Zero__not__Suc,axiom,
% 0.17/0.44 ! [M: nat] :
% 0.17/0.44 ( zero_zero_nat
% 0.17/0.44 != ( suc @ M ) ) ).
% 0.17/0.44
% 0.17/0.44 % Zero_not_Suc
% 0.17/0.44 thf(fact_2114_Zero__neq__Suc,axiom,
% 0.17/0.44 ! [M: nat] :
% 0.17/0.44 ( zero_zero_nat
% 0.17/0.44 != ( suc @ M ) ) ).
% 0.17/0.44
% 0.17/0.44 % Zero_neq_Suc
% 0.17/0.44 thf(fact_2115_Suc__neq__Zero,axiom,
% 0.17/0.44 ! [M: nat] :
% 0.17/0.44 ( ( suc @ M )
% 0.17/0.44 != zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % Suc_neq_Zero
% 0.17/0.44 thf(fact_2116_zero__induct,axiom,
% 0.17/0.44 ! [P: nat > $o,K: nat] :
% 0.17/0.44 ( ( P @ K )
% 0.17/0.44 => ( ! [N4: nat] :
% 0.17/0.44 ( ( P @ ( suc @ N4 ) )
% 0.17/0.44 => ( P @ N4 ) )
% 0.17/0.44 => ( P @ zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_induct
% 0.17/0.44 thf(fact_2117_diff__induct,axiom,
% 0.17/0.44 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.17/0.44 ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 0.17/0.44 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 0.17/0.44 => ( ! [X3: nat,Y4: nat] :
% 0.17/0.44 ( ( P @ X3 @ Y4 )
% 0.17/0.44 => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 0.17/0.44 => ( P @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diff_induct
% 0.17/0.44 thf(fact_2118_nat__induct,axiom,
% 0.17/0.44 ! [P: nat > $o,N: nat] :
% 0.17/0.44 ( ( P @ zero_zero_nat )
% 0.17/0.44 => ( ! [N4: nat] :
% 0.17/0.44 ( ( P @ N4 )
% 0.17/0.44 => ( P @ ( suc @ N4 ) ) )
% 0.17/0.44 => ( P @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % nat_induct
% 0.17/0.44 thf(fact_2119_old_Onat_Oexhaust,axiom,
% 0.17/0.44 ! [Y3: nat] :
% 0.17/0.44 ( ( Y3 != zero_zero_nat )
% 0.17/0.44 => ~ ! [Nat3: nat] :
% 0.17/0.44 ( Y3
% 0.17/0.44 != ( suc @ Nat3 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % old.nat.exhaust
% 0.17/0.44 thf(fact_2120_nat_OdiscI,axiom,
% 0.17/0.44 ! [Nat: nat,X2: nat] :
% 0.17/0.44 ( ( Nat
% 0.17/0.44 = ( suc @ X2 ) )
% 0.17/0.44 => ( Nat != zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % nat.discI
% 0.17/0.44 thf(fact_2121_old_Onat_Odistinct_I1_J,axiom,
% 0.17/0.44 ! [Nat2: nat] :
% 0.17/0.44 ( zero_zero_nat
% 0.17/0.44 != ( suc @ Nat2 ) ) ).
% 0.17/0.44
% 0.17/0.44 % old.nat.distinct(1)
% 0.17/0.44 thf(fact_2122_old_Onat_Odistinct_I2_J,axiom,
% 0.17/0.44 ! [Nat2: nat] :
% 0.17/0.44 ( ( suc @ Nat2 )
% 0.17/0.44 != zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % old.nat.distinct(2)
% 0.17/0.44 thf(fact_2123_nat_Odistinct_I1_J,axiom,
% 0.17/0.44 ! [X2: nat] :
% 0.17/0.44 ( zero_zero_nat
% 0.17/0.44 != ( suc @ X2 ) ) ).
% 0.17/0.44
% 0.17/0.44 % nat.distinct(1)
% 0.17/0.44 thf(fact_2124_infinite__descent0,axiom,
% 0.17/0.44 ! [P: nat > $o,N: nat] :
% 0.17/0.44 ( ( P @ zero_zero_nat )
% 0.17/0.44 => ( ! [N4: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.17/0.44 => ( ~ ( P @ N4 )
% 0.17/0.44 => ? [M5: nat] :
% 0.17/0.44 ( ( ord_less_nat @ M5 @ N4 )
% 0.17/0.44 & ~ ( P @ M5 ) ) ) )
% 0.17/0.44 => ( P @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % infinite_descent0
% 0.17/0.44 thf(fact_2125_gr__implies__not0,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_nat @ M @ N )
% 0.17/0.44 => ( N != zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % gr_implies_not0
% 0.17/0.44 thf(fact_2126_less__zeroE,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % less_zeroE
% 0.17/0.44 thf(fact_2127_not__less0,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % not_less0
% 0.17/0.44 thf(fact_2128_not__gr0,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.17/0.44 = ( N = zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % not_gr0
% 0.17/0.44 thf(fact_2129_gr0I,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ( ( N != zero_zero_nat )
% 0.17/0.44 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % gr0I
% 0.17/0.44 thf(fact_2130_bot__nat__0_Oextremum__strict,axiom,
% 0.17/0.44 ! [A: nat] :
% 0.17/0.44 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % bot_nat_0.extremum_strict
% 0.17/0.44 thf(fact_2131_le__0__eq,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.17/0.44 = ( N = zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_0_eq
% 0.17/0.44 thf(fact_2132_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.17/0.44 ! [A: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.44 => ( A = zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % bot_nat_0.extremum_uniqueI
% 0.17/0.44 thf(fact_2133_bot__nat__0_Oextremum__unique,axiom,
% 0.17/0.44 ! [A: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.44 = ( A = zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % bot_nat_0.extremum_unique
% 0.17/0.44 thf(fact_2134_less__eq__nat_Osimps_I1_J,axiom,
% 0.17/0.44 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.17/0.44
% 0.17/0.44 % less_eq_nat.simps(1)
% 0.17/0.44 thf(fact_2135_add__eq__self__zero,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ( plus_plus_nat @ M @ N )
% 0.17/0.44 = M )
% 0.17/0.44 => ( N = zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_eq_self_zero
% 0.17/0.44 thf(fact_2136_plus__nat_Oadd__0,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.17/0.44 = N ) ).
% 0.17/0.44
% 0.17/0.44 % plus_nat.add_0
% 0.17/0.44 thf(fact_2137_diffs0__imp__equal,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ( minus_minus_nat @ M @ N )
% 0.17/0.44 = zero_zero_nat )
% 0.17/0.44 => ( ( ( minus_minus_nat @ N @ M )
% 0.17/0.44 = zero_zero_nat )
% 0.17/0.44 => ( M = N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % diffs0_imp_equal
% 0.17/0.44 thf(fact_2138_minus__nat_Odiff__0,axiom,
% 0.17/0.44 ! [M: nat] :
% 0.17/0.44 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.17/0.44 = M ) ).
% 0.17/0.44
% 0.17/0.44 % minus_nat.diff_0
% 0.17/0.44 thf(fact_2139_nat__mult__eq__cancel__disj,axiom,
% 0.17/0.44 ! [K: nat,M: nat,N: nat] :
% 0.17/0.44 ( ( ( times_times_nat @ K @ M )
% 0.17/0.44 = ( times_times_nat @ K @ N ) )
% 0.17/0.44 = ( ( K = zero_zero_nat )
% 0.17/0.44 | ( M = N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % nat_mult_eq_cancel_disj
% 0.17/0.44 thf(fact_2140_mult__0,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ( ( times_times_nat @ zero_zero_nat @ N )
% 0.17/0.44 = zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % mult_0
% 0.17/0.44 thf(fact_2141_even__zero,axiom,
% 0.17/0.44 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 0.17/0.44
% 0.17/0.44 % even_zero
% 0.17/0.44 thf(fact_2142_even__zero,axiom,
% 0.17/0.44 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 0.17/0.44
% 0.17/0.44 % even_zero
% 0.17/0.44 thf(fact_2143_even__zero,axiom,
% 0.17/0.44 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 0.17/0.44
% 0.17/0.44 % even_zero
% 0.17/0.44 thf(fact_2144_is__unitE,axiom,
% 0.17/0.44 ! [A: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.44 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.17/0.44 => ! [B5: code_integer] :
% 0.17/0.44 ( ( B5 != zero_z3403309356797280102nteger )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ B5 @ one_one_Code_integer )
% 0.17/0.44 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 0.17/0.44 = B5 )
% 0.17/0.44 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B5 )
% 0.17/0.44 = A )
% 0.17/0.44 => ( ( ( times_3573771949741848930nteger @ A @ B5 )
% 0.17/0.44 = one_one_Code_integer )
% 0.17/0.44 => ( ( divide6298287555418463151nteger @ C @ A )
% 0.17/0.44 != ( times_3573771949741848930nteger @ C @ B5 ) ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unitE
% 0.17/0.44 thf(fact_2145_is__unitE,axiom,
% 0.17/0.44 ! [A: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.44 => ~ ( ( A != zero_zero_nat )
% 0.17/0.44 => ! [B5: nat] :
% 0.17/0.44 ( ( B5 != zero_zero_nat )
% 0.17/0.44 => ( ( dvd_dvd_nat @ B5 @ one_one_nat )
% 0.17/0.44 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 0.17/0.44 = B5 )
% 0.17/0.44 => ( ( ( divide_divide_nat @ one_one_nat @ B5 )
% 0.17/0.44 = A )
% 0.17/0.44 => ( ( ( times_times_nat @ A @ B5 )
% 0.17/0.44 = one_one_nat )
% 0.17/0.44 => ( ( divide_divide_nat @ C @ A )
% 0.17/0.44 != ( times_times_nat @ C @ B5 ) ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unitE
% 0.17/0.44 thf(fact_2146_is__unitE,axiom,
% 0.17/0.44 ! [A: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.44 => ~ ( ( A != zero_zero_int )
% 0.17/0.44 => ! [B5: int] :
% 0.17/0.44 ( ( B5 != zero_zero_int )
% 0.17/0.44 => ( ( dvd_dvd_int @ B5 @ one_one_int )
% 0.17/0.44 => ( ( ( divide_divide_int @ one_one_int @ A )
% 0.17/0.44 = B5 )
% 0.17/0.44 => ( ( ( divide_divide_int @ one_one_int @ B5 )
% 0.17/0.44 = A )
% 0.17/0.44 => ( ( ( times_times_int @ A @ B5 )
% 0.17/0.44 = one_one_int )
% 0.17/0.44 => ( ( divide_divide_int @ C @ A )
% 0.17/0.44 != ( times_times_int @ C @ B5 ) ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unitE
% 0.17/0.44 thf(fact_2147_is__unit__div__mult__cancel__left,axiom,
% 0.17/0.44 ! [A: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( A != zero_z3403309356797280102nteger )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.44 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 0.17/0.44 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unit_div_mult_cancel_left
% 0.17/0.44 thf(fact_2148_is__unit__div__mult__cancel__left,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( A != zero_zero_nat )
% 0.17/0.44 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.44 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B2 ) )
% 0.17/0.44 = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unit_div_mult_cancel_left
% 0.17/0.44 thf(fact_2149_is__unit__div__mult__cancel__left,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( A != zero_zero_int )
% 0.17/0.44 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.44 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B2 ) )
% 0.17/0.44 = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unit_div_mult_cancel_left
% 0.17/0.44 thf(fact_2150_is__unit__div__mult__cancel__right,axiom,
% 0.17/0.44 ! [A: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( A != zero_z3403309356797280102nteger )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.44 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ A ) )
% 0.17/0.44 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unit_div_mult_cancel_right
% 0.17/0.44 thf(fact_2151_is__unit__div__mult__cancel__right,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( A != zero_zero_nat )
% 0.17/0.44 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.44 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ A ) )
% 0.17/0.44 = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unit_div_mult_cancel_right
% 0.17/0.44 thf(fact_2152_is__unit__div__mult__cancel__right,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( A != zero_zero_int )
% 0.17/0.44 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.44 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ A ) )
% 0.17/0.44 = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unit_div_mult_cancel_right
% 0.17/0.44 thf(fact_2153_dvd__power__iff,axiom,
% 0.17/0.44 ! [X4: code_integer,M: nat,N: nat] :
% 0.17/0.44 ( ( X4 != zero_z3403309356797280102nteger )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ M ) @ ( power_8256067586552552935nteger @ X4 @ N ) )
% 0.17/0.44 = ( ( dvd_dvd_Code_integer @ X4 @ one_one_Code_integer )
% 0.17/0.44 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_iff
% 0.17/0.44 thf(fact_2154_dvd__power__iff,axiom,
% 0.17/0.44 ! [X4: nat,M: nat,N: nat] :
% 0.17/0.44 ( ( X4 != zero_zero_nat )
% 0.17/0.44 => ( ( dvd_dvd_nat @ ( power_power_nat @ X4 @ M ) @ ( power_power_nat @ X4 @ N ) )
% 0.17/0.44 = ( ( dvd_dvd_nat @ X4 @ one_one_nat )
% 0.17/0.44 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_iff
% 0.17/0.44 thf(fact_2155_dvd__power__iff,axiom,
% 0.17/0.44 ! [X4: int,M: nat,N: nat] :
% 0.17/0.44 ( ( X4 != zero_zero_int )
% 0.17/0.44 => ( ( dvd_dvd_int @ ( power_power_int @ X4 @ M ) @ ( power_power_int @ X4 @ N ) )
% 0.17/0.44 = ( ( dvd_dvd_int @ X4 @ one_one_int )
% 0.17/0.44 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_iff
% 0.17/0.44 thf(fact_2156_dvd__power,axiom,
% 0.17/0.44 ! [N: nat,X4: code_integer] :
% 0.17/0.44 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 | ( X4 = one_one_Code_integer ) )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ X4 @ ( power_8256067586552552935nteger @ X4 @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power
% 0.17/0.44 thf(fact_2157_dvd__power,axiom,
% 0.17/0.44 ! [N: nat,X4: rat] :
% 0.17/0.44 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 | ( X4 = one_one_rat ) )
% 0.17/0.44 => ( dvd_dvd_rat @ X4 @ ( power_power_rat @ X4 @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power
% 0.17/0.44 thf(fact_2158_dvd__power,axiom,
% 0.17/0.44 ! [N: nat,X4: nat] :
% 0.17/0.44 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 | ( X4 = one_one_nat ) )
% 0.17/0.44 => ( dvd_dvd_nat @ X4 @ ( power_power_nat @ X4 @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power
% 0.17/0.44 thf(fact_2159_dvd__power,axiom,
% 0.17/0.44 ! [N: nat,X4: real] :
% 0.17/0.44 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 | ( X4 = one_one_real ) )
% 0.17/0.44 => ( dvd_dvd_real @ X4 @ ( power_power_real @ X4 @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power
% 0.17/0.44 thf(fact_2160_dvd__power,axiom,
% 0.17/0.44 ! [N: nat,X4: int] :
% 0.17/0.44 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 | ( X4 = one_one_int ) )
% 0.17/0.44 => ( dvd_dvd_int @ X4 @ ( power_power_int @ X4 @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power
% 0.17/0.44 thf(fact_2161_dvd__power,axiom,
% 0.17/0.44 ! [N: nat,X4: complex] :
% 0.17/0.44 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 | ( X4 = one_one_complex ) )
% 0.17/0.44 => ( dvd_dvd_complex @ X4 @ ( power_power_complex @ X4 @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power
% 0.17/0.44 thf(fact_2162_enat__0__less__mult__iff,axiom,
% 0.17/0.44 ! [M: extended_enat,N: extended_enat] :
% 0.17/0.44 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 0.17/0.44 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 0.17/0.44 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % enat_0_less_mult_iff
% 0.17/0.44 thf(fact_2163_not__iless0,axiom,
% 0.17/0.44 ! [N: extended_enat] :
% 0.17/0.44 ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 0.17/0.44
% 0.17/0.44 % not_iless0
% 0.17/0.44 thf(fact_2164_iadd__is__0,axiom,
% 0.17/0.44 ! [M: extended_enat,N: extended_enat] :
% 0.17/0.44 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 0.17/0.44 = zero_z5237406670263579293d_enat )
% 0.17/0.44 = ( ( M = zero_z5237406670263579293d_enat )
% 0.17/0.44 & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % iadd_is_0
% 0.17/0.44 thf(fact_2165_ile0__eq,axiom,
% 0.17/0.44 ! [N: extended_enat] :
% 0.17/0.44 ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.17/0.44 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 0.17/0.44
% 0.17/0.44 % ile0_eq
% 0.17/0.44 thf(fact_2166_i0__lb,axiom,
% 0.17/0.44 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 0.17/0.44
% 0.17/0.44 % i0_lb
% 0.17/0.44 thf(fact_2167_dvd__mult__cancel1,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.17/0.44 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 0.17/0.44 = ( N = one_one_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mult_cancel1
% 0.17/0.44 thf(fact_2168_dvd__mult__cancel2,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.17/0.44 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 0.17/0.44 = ( N = one_one_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mult_cancel2
% 0.17/0.44 thf(fact_2169_power__eq__imp__eq__base,axiom,
% 0.17/0.44 ! [A: real,N: nat,B2: real] :
% 0.17/0.44 ( ( ( power_power_real @ A @ N )
% 0.17/0.44 = ( power_power_real @ B2 @ N ) )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( A = B2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_eq_imp_eq_base
% 0.17/0.44 thf(fact_2170_power__eq__imp__eq__base,axiom,
% 0.17/0.44 ! [A: rat,N: nat,B2: rat] :
% 0.17/0.44 ( ( ( power_power_rat @ A @ N )
% 0.17/0.44 = ( power_power_rat @ B2 @ N ) )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( A = B2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_eq_imp_eq_base
% 0.17/0.44 thf(fact_2171_power__eq__imp__eq__base,axiom,
% 0.17/0.44 ! [A: nat,N: nat,B2: nat] :
% 0.17/0.44 ( ( ( power_power_nat @ A @ N )
% 0.17/0.44 = ( power_power_nat @ B2 @ N ) )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( A = B2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_eq_imp_eq_base
% 0.17/0.44 thf(fact_2172_power__eq__imp__eq__base,axiom,
% 0.17/0.44 ! [A: int,N: nat,B2: int] :
% 0.17/0.44 ( ( ( power_power_int @ A @ N )
% 0.17/0.44 = ( power_power_int @ B2 @ N ) )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( A = B2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_eq_imp_eq_base
% 0.17/0.44 thf(fact_2173_power__eq__iff__eq__base,axiom,
% 0.17/0.44 ! [N: nat,A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.44 => ( ( ( power_power_real @ A @ N )
% 0.17/0.44 = ( power_power_real @ B2 @ N ) )
% 0.17/0.44 = ( A = B2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_eq_iff_eq_base
% 0.17/0.44 thf(fact_2174_power__eq__iff__eq__base,axiom,
% 0.17/0.44 ! [N: nat,A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.44 => ( ( ( power_power_rat @ A @ N )
% 0.17/0.44 = ( power_power_rat @ B2 @ N ) )
% 0.17/0.44 = ( A = B2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_eq_iff_eq_base
% 0.17/0.44 thf(fact_2175_power__eq__iff__eq__base,axiom,
% 0.17/0.44 ! [N: nat,A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.44 => ( ( ( power_power_nat @ A @ N )
% 0.17/0.44 = ( power_power_nat @ B2 @ N ) )
% 0.17/0.44 = ( A = B2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_eq_iff_eq_base
% 0.17/0.44 thf(fact_2176_power__eq__iff__eq__base,axiom,
% 0.17/0.44 ! [N: nat,A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.44 => ( ( ( power_power_int @ A @ N )
% 0.17/0.44 = ( power_power_int @ B2 @ N ) )
% 0.17/0.44 = ( A = B2 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_eq_iff_eq_base
% 0.17/0.44 thf(fact_2177_int__power__div__base,axiom,
% 0.17/0.44 ! [M: nat,K: int] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.17/0.44 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.17/0.44 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 0.17/0.44 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % int_power_div_base
% 0.17/0.44 thf(fact_2178_zero__less__power__eq,axiom,
% 0.17/0.44 ! [A: real,N: nat] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.17/0.44 = ( ( N = zero_zero_nat )
% 0.17/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( A != zero_zero_real ) )
% 0.17/0.44 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_power_eq
% 0.17/0.44 thf(fact_2179_zero__less__power__eq,axiom,
% 0.17/0.44 ! [A: rat,N: nat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.17/0.44 = ( ( N = zero_zero_nat )
% 0.17/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( A != zero_zero_rat ) )
% 0.17/0.44 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_power_eq
% 0.17/0.44 thf(fact_2180_zero__less__power__eq,axiom,
% 0.17/0.44 ! [A: int,N: nat] :
% 0.17/0.44 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.17/0.44 = ( ( N = zero_zero_nat )
% 0.17/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( A != zero_zero_int ) )
% 0.17/0.44 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_power_eq
% 0.17/0.44 thf(fact_2181_even__signed__take__bit__iff,axiom,
% 0.17/0.44 ! [M: nat,A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % even_signed_take_bit_iff
% 0.17/0.44 thf(fact_2182_even__signed__take__bit__iff,axiom,
% 0.17/0.44 ! [M: nat,A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 0.17/0.44 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % even_signed_take_bit_iff
% 0.17/0.44 thf(fact_2183_realpow__pos__nth,axiom,
% 0.17/0.44 ! [N: nat,A: real] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.44 => ? [R3: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.17/0.44 & ( ( power_power_real @ R3 @ N )
% 0.17/0.44 = A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % realpow_pos_nth
% 0.17/0.44 thf(fact_2184_realpow__pos__nth__unique,axiom,
% 0.17/0.44 ! [N: nat,A: real] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.44 => ? [X3: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.17/0.44 & ( ( power_power_real @ X3 @ N )
% 0.17/0.44 = A )
% 0.17/0.44 & ! [Y5: real] :
% 0.17/0.44 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 0.17/0.44 & ( ( power_power_real @ Y5 @ N )
% 0.17/0.44 = A ) )
% 0.17/0.44 => ( Y5 = X3 ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % realpow_pos_nth_unique
% 0.17/0.44 thf(fact_2185_power__le__zero__eq,axiom,
% 0.17/0.44 ! [A: real,N: nat] :
% 0.17/0.44 ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.17/0.44 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.17/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_le_zero_eq
% 0.17/0.44 thf(fact_2186_power__le__zero__eq,axiom,
% 0.17/0.44 ! [A: rat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.17/0.44 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.17/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_le_zero_eq
% 0.17/0.44 thf(fact_2187_power__le__zero__eq,axiom,
% 0.17/0.44 ! [A: int,N: nat] :
% 0.17/0.44 ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.17/0.44 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.17/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_le_zero_eq
% 0.17/0.44 thf(fact_2188_even__iff__mod__2__eq__zero,axiom,
% 0.17/0.44 ! [A: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.44 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.44 = zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % even_iff_mod_2_eq_zero
% 0.17/0.44 thf(fact_2189_even__iff__mod__2__eq__zero,axiom,
% 0.17/0.44 ! [A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.44 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.44 = zero_zero_int ) ) ).
% 0.17/0.44
% 0.17/0.44 % even_iff_mod_2_eq_zero
% 0.17/0.44 thf(fact_2190_even__iff__mod__2__eq__zero,axiom,
% 0.17/0.44 ! [A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.44 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.44 = zero_z3403309356797280102nteger ) ) ).
% 0.17/0.44
% 0.17/0.44 % even_iff_mod_2_eq_zero
% 0.17/0.44 thf(fact_2191_odd__pos,axiom,
% 0.17/0.44 ! [N: nat] :
% 0.17/0.44 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.44 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % odd_pos
% 0.17/0.44 thf(fact_2192_signed__take__bit__int__less__self__iff,axiom,
% 0.17/0.44 ! [N: nat,K: int] :
% 0.17/0.44 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.17/0.44 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.17/0.44
% 0.17/0.44 % signed_take_bit_int_less_self_iff
% 0.17/0.44 thf(fact_2193_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.17/0.44 ! [K: int,N: nat] :
% 0.17/0.44 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.17/0.44 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % signed_take_bit_int_greater_eq_self_iff
% 0.17/0.44 thf(fact_2194_signed__take__bit__int__less__exp,axiom,
% 0.17/0.44 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % signed_take_bit_int_less_exp
% 0.17/0.44 thf(fact_2195_is__unit__mult__iff,axiom,
% 0.17/0.44 ! [A: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer )
% 0.17/0.44 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.44 & ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unit_mult_iff
% 0.17/0.44 thf(fact_2196_is__unit__mult__iff,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat )
% 0.17/0.44 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.44 & ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unit_mult_iff
% 0.17/0.44 thf(fact_2197_is__unit__mult__iff,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int )
% 0.17/0.44 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.44 & ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % is_unit_mult_iff
% 0.17/0.44 thf(fact_2198_dvd__mult__unit__iff,axiom,
% 0.17/0.44 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mult_unit_iff
% 0.17/0.44 thf(fact_2199_dvd__mult__unit__iff,axiom,
% 0.17/0.44 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.44 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) )
% 0.17/0.44 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mult_unit_iff
% 0.17/0.44 thf(fact_2200_dvd__mult__unit__iff,axiom,
% 0.17/0.44 ! [B2: int,A: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.44 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) )
% 0.17/0.44 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mult_unit_iff
% 0.17/0.44 thf(fact_2201_mult__unit__dvd__iff,axiom,
% 0.17/0.44 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_unit_dvd_iff
% 0.17/0.44 thf(fact_2202_mult__unit__dvd__iff,axiom,
% 0.17/0.44 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.44 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.17/0.44 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_unit_dvd_iff
% 0.17/0.44 thf(fact_2203_mult__unit__dvd__iff,axiom,
% 0.17/0.44 ! [B2: int,A: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.44 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.17/0.44 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_unit_dvd_iff
% 0.17/0.44 thf(fact_2204_dvd__mult__unit__iff_H,axiom,
% 0.17/0.44 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mult_unit_iff'
% 0.17/0.44 thf(fact_2205_dvd__mult__unit__iff_H,axiom,
% 0.17/0.44 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.44 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mult_unit_iff'
% 0.17/0.44 thf(fact_2206_dvd__mult__unit__iff_H,axiom,
% 0.17/0.44 ! [B2: int,A: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.44 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mult_unit_iff'
% 0.17/0.44 thf(fact_2207_mult__unit__dvd__iff_H,axiom,
% 0.17/0.44 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_unit_dvd_iff'
% 0.17/0.44 thf(fact_2208_mult__unit__dvd__iff_H,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.44 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.17/0.44 = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_unit_dvd_iff'
% 0.17/0.44 thf(fact_2209_mult__unit__dvd__iff_H,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.44 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.17/0.44 = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_unit_dvd_iff'
% 0.17/0.44 thf(fact_2210_unit__mult__left__cancel,axiom,
% 0.17/0.44 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.44 => ( ( ( times_3573771949741848930nteger @ A @ B2 )
% 0.17/0.44 = ( times_3573771949741848930nteger @ A @ C ) )
% 0.17/0.44 = ( B2 = C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_mult_left_cancel
% 0.17/0.44 thf(fact_2211_unit__mult__left__cancel,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.44 => ( ( ( times_times_nat @ A @ B2 )
% 0.17/0.44 = ( times_times_nat @ A @ C ) )
% 0.17/0.44 = ( B2 = C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_mult_left_cancel
% 0.17/0.44 thf(fact_2212_unit__mult__left__cancel,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.44 => ( ( ( times_times_int @ A @ B2 )
% 0.17/0.44 = ( times_times_int @ A @ C ) )
% 0.17/0.44 = ( B2 = C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_mult_left_cancel
% 0.17/0.44 thf(fact_2213_unit__mult__right__cancel,axiom,
% 0.17/0.44 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.44 => ( ( ( times_3573771949741848930nteger @ B2 @ A )
% 0.17/0.44 = ( times_3573771949741848930nteger @ C @ A ) )
% 0.17/0.44 = ( B2 = C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_mult_right_cancel
% 0.17/0.44 thf(fact_2214_unit__mult__right__cancel,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.44 => ( ( ( times_times_nat @ B2 @ A )
% 0.17/0.44 = ( times_times_nat @ C @ A ) )
% 0.17/0.44 = ( B2 = C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_mult_right_cancel
% 0.17/0.44 thf(fact_2215_unit__mult__right__cancel,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.44 => ( ( ( times_times_int @ B2 @ A )
% 0.17/0.44 = ( times_times_int @ C @ A ) )
% 0.17/0.44 = ( B2 = C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_mult_right_cancel
% 0.17/0.44 thf(fact_2216_dvd__div__mult,axiom,
% 0.17/0.44 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.44 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ C ) @ A )
% 0.17/0.44 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_mult
% 0.17/0.44 thf(fact_2217_dvd__div__mult,axiom,
% 0.17/0.44 ! [C: nat,B2: nat,A: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ C @ B2 )
% 0.17/0.44 => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A )
% 0.17/0.44 = ( divide_divide_nat @ ( times_times_nat @ B2 @ A ) @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_mult
% 0.17/0.44 thf(fact_2218_dvd__div__mult,axiom,
% 0.17/0.44 ! [C: int,B2: int,A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.44 => ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A )
% 0.17/0.44 = ( divide_divide_int @ ( times_times_int @ B2 @ A ) @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_mult
% 0.17/0.44 thf(fact_2219_div__mult__swap,axiom,
% 0.17/0.44 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.44 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.17/0.44 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_mult_swap
% 0.17/0.44 thf(fact_2220_div__mult__swap,axiom,
% 0.17/0.44 ! [C: nat,B2: nat,A: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ C @ B2 )
% 0.17/0.44 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 0.17/0.44 = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_mult_swap
% 0.17/0.44 thf(fact_2221_div__mult__swap,axiom,
% 0.17/0.44 ! [C: int,B2: int,A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.44 => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 0.17/0.44 = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_mult_swap
% 0.17/0.44 thf(fact_2222_div__div__eq__right,axiom,
% 0.17/0.44 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.17/0.44 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.17/0.44 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_div_eq_right
% 0.17/0.44 thf(fact_2223_div__div__eq__right,axiom,
% 0.17/0.44 ! [C: nat,B2: nat,A: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ C @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_nat @ B2 @ A )
% 0.17/0.44 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 0.17/0.44 = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_div_eq_right
% 0.17/0.44 thf(fact_2224_div__div__eq__right,axiom,
% 0.17/0.44 ! [C: int,B2: int,A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.44 => ( ( dvd_dvd_int @ B2 @ A )
% 0.17/0.44 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 0.17/0.44 = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_div_eq_right
% 0.17/0.44 thf(fact_2225_dvd__div__mult2__eq,axiom,
% 0.17/0.44 ! [B2: code_integer,C: code_integer,A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ C ) @ A )
% 0.17/0.44 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.17/0.44 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_mult2_eq
% 0.17/0.44 thf(fact_2226_dvd__div__mult2__eq,axiom,
% 0.17/0.44 ! [B2: nat,C: nat,A: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A )
% 0.17/0.44 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.17/0.44 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_mult2_eq
% 0.17/0.44 thf(fact_2227_dvd__div__mult2__eq,axiom,
% 0.17/0.44 ! [B2: int,C: int,A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A )
% 0.17/0.44 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.17/0.44 = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_mult2_eq
% 0.17/0.44 thf(fact_2228_dvd__mult__imp__div,axiom,
% 0.17/0.44 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mult_imp_div
% 0.17/0.44 thf(fact_2229_dvd__mult__imp__div,axiom,
% 0.17/0.44 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 )
% 0.17/0.44 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mult_imp_div
% 0.17/0.44 thf(fact_2230_dvd__mult__imp__div,axiom,
% 0.17/0.44 ! [A: int,C: int,B2: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 )
% 0.17/0.44 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_mult_imp_div
% 0.17/0.44 thf(fact_2231_div__mult__div__if__dvd,axiom,
% 0.17/0.44 ! [B2: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ D @ C )
% 0.17/0.44 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 0.17/0.44 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_mult_div_if_dvd
% 0.17/0.44 thf(fact_2232_div__mult__div__if__dvd,axiom,
% 0.17/0.44 ! [B2: nat,A: nat,D: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ B2 @ A )
% 0.17/0.44 => ( ( dvd_dvd_nat @ D @ C )
% 0.17/0.44 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ ( divide_divide_nat @ C @ D ) )
% 0.17/0.44 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_mult_div_if_dvd
% 0.17/0.44 thf(fact_2233_div__mult__div__if__dvd,axiom,
% 0.17/0.44 ! [B2: int,A: int,D: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ B2 @ A )
% 0.17/0.44 => ( ( dvd_dvd_int @ D @ C )
% 0.17/0.44 => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ C @ D ) )
% 0.17/0.44 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_mult_div_if_dvd
% 0.17/0.44 thf(fact_2234_unit__div__cancel,axiom,
% 0.17/0.44 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.17/0.44 => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 0.17/0.44 = ( divide6298287555418463151nteger @ C @ A ) )
% 0.17/0.44 = ( B2 = C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_div_cancel
% 0.17/0.44 thf(fact_2235_unit__div__cancel,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.17/0.44 => ( ( ( divide_divide_nat @ B2 @ A )
% 0.17/0.44 = ( divide_divide_nat @ C @ A ) )
% 0.17/0.44 = ( B2 = C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_div_cancel
% 0.17/0.44 thf(fact_2236_unit__div__cancel,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.17/0.44 => ( ( ( divide_divide_int @ B2 @ A )
% 0.17/0.44 = ( divide_divide_int @ C @ A ) )
% 0.17/0.44 = ( B2 = C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % unit_div_cancel
% 0.17/0.44 thf(fact_2237_div__unit__dvd__iff,axiom,
% 0.17/0.44 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_unit_dvd_iff
% 0.17/0.44 thf(fact_2238_div__unit__dvd__iff,axiom,
% 0.17/0.44 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.44 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 0.17/0.44 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_unit_dvd_iff
% 0.17/0.44 thf(fact_2239_div__unit__dvd__iff,axiom,
% 0.17/0.44 ! [B2: int,A: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.44 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 0.17/0.44 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_unit_dvd_iff
% 0.17/0.44 thf(fact_2240_dvd__div__unit__iff,axiom,
% 0.17/0.44 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.44 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B2 ) )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_unit_iff
% 0.17/0.44 thf(fact_2241_dvd__div__unit__iff,axiom,
% 0.17/0.44 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.44 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B2 ) )
% 0.17/0.44 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_unit_iff
% 0.17/0.44 thf(fact_2242_dvd__div__unit__iff,axiom,
% 0.17/0.44 ! [B2: int,A: int,C: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.44 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B2 ) )
% 0.17/0.44 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_div_unit_iff
% 0.17/0.44 thf(fact_2243_div__plus__div__distrib__dvd__right,axiom,
% 0.17/0.44 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.17/0.44 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 0.17/0.44 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_plus_div_distrib_dvd_right
% 0.17/0.44 thf(fact_2244_div__plus__div__distrib__dvd__right,axiom,
% 0.17/0.44 ! [C: nat,B2: nat,A: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ C @ B2 )
% 0.17/0.44 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.17/0.44 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_plus_div_distrib_dvd_right
% 0.17/0.44 thf(fact_2245_div__plus__div__distrib__dvd__right,axiom,
% 0.17/0.44 ! [C: int,B2: int,A: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ C @ B2 )
% 0.17/0.44 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.17/0.44 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_plus_div_distrib_dvd_right
% 0.17/0.44 thf(fact_2246_div__plus__div__distrib__dvd__left,axiom,
% 0.17/0.44 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.17/0.44 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 0.17/0.44 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_plus_div_distrib_dvd_left
% 0.17/0.44 thf(fact_2247_div__plus__div__distrib__dvd__left,axiom,
% 0.17/0.44 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ C @ A )
% 0.17/0.44 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.17/0.44 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_plus_div_distrib_dvd_left
% 0.17/0.44 thf(fact_2248_div__plus__div__distrib__dvd__left,axiom,
% 0.17/0.44 ! [C: int,A: int,B2: int] :
% 0.17/0.44 ( ( dvd_dvd_int @ C @ A )
% 0.17/0.44 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.17/0.44 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_plus_div_distrib_dvd_left
% 0.17/0.44 thf(fact_2249_div__power,axiom,
% 0.17/0.44 ! [B2: code_integer,A: code_integer,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.17/0.44 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ N )
% 0.17/0.44 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_power
% 0.17/0.44 thf(fact_2250_div__power,axiom,
% 0.17/0.44 ! [B2: nat,A: nat,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ B2 @ A )
% 0.17/0.44 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B2 ) @ N )
% 0.17/0.44 = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_power
% 0.17/0.44 thf(fact_2251_div__power,axiom,
% 0.17/0.44 ! [B2: int,A: int,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_int @ B2 @ A )
% 0.17/0.44 => ( ( power_power_int @ ( divide_divide_int @ A @ B2 ) @ N )
% 0.17/0.44 = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % div_power
% 0.17/0.44 thf(fact_2252_power__strict__mono,axiom,
% 0.17/0.44 ! [A: real,B2: real,N: nat] :
% 0.17/0.44 ( ( ord_less_real @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_strict_mono
% 0.17/0.44 thf(fact_2253_power__strict__mono,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,N: nat] :
% 0.17/0.44 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_strict_mono
% 0.17/0.44 thf(fact_2254_power__strict__mono,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_strict_mono
% 0.17/0.44 thf(fact_2255_power__strict__mono,axiom,
% 0.17/0.44 ! [A: int,B2: int,N: nat] :
% 0.17/0.44 ( ( ord_less_int @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.44 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_strict_mono
% 0.17/0.44 thf(fact_2256_le__imp__power__dvd,axiom,
% 0.17/0.44 ! [M: nat,N: nat,A: code_integer] :
% 0.17/0.44 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_imp_power_dvd
% 0.17/0.44 thf(fact_2257_le__imp__power__dvd,axiom,
% 0.17/0.44 ! [M: nat,N: nat,A: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.44 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_imp_power_dvd
% 0.17/0.44 thf(fact_2258_le__imp__power__dvd,axiom,
% 0.17/0.44 ! [M: nat,N: nat,A: real] :
% 0.17/0.44 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.44 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_imp_power_dvd
% 0.17/0.44 thf(fact_2259_le__imp__power__dvd,axiom,
% 0.17/0.44 ! [M: nat,N: nat,A: int] :
% 0.17/0.44 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.44 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_imp_power_dvd
% 0.17/0.44 thf(fact_2260_le__imp__power__dvd,axiom,
% 0.17/0.44 ! [M: nat,N: nat,A: complex] :
% 0.17/0.44 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.44 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_imp_power_dvd
% 0.17/0.44 thf(fact_2261_power__le__dvd,axiom,
% 0.17/0.44 ! [A: code_integer,N: nat,B2: code_integer,M: nat] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_le_dvd
% 0.17/0.44 thf(fact_2262_power__le__dvd,axiom,
% 0.17/0.44 ! [A: nat,N: nat,B2: nat,M: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.44 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_le_dvd
% 0.17/0.44 thf(fact_2263_power__le__dvd,axiom,
% 0.17/0.44 ! [A: real,N: nat,B2: real,M: nat] :
% 0.17/0.44 ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.44 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_le_dvd
% 0.17/0.44 thf(fact_2264_power__le__dvd,axiom,
% 0.17/0.44 ! [A: int,N: nat,B2: int,M: nat] :
% 0.17/0.44 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.44 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_le_dvd
% 0.17/0.44 thf(fact_2265_power__le__dvd,axiom,
% 0.17/0.44 ! [A: complex,N: nat,B2: complex,M: nat] :
% 0.17/0.44 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.44 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % power_le_dvd
% 0.17/0.44 thf(fact_2266_dvd__power__le,axiom,
% 0.17/0.44 ! [X4: code_integer,Y3: code_integer,N: nat,M: nat] :
% 0.17/0.44 ( ( dvd_dvd_Code_integer @ X4 @ Y3 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.44 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_le
% 0.17/0.44 thf(fact_2267_dvd__power__le,axiom,
% 0.17/0.44 ! [X4: nat,Y3: nat,N: nat,M: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ X4 @ Y3 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.44 => ( dvd_dvd_nat @ ( power_power_nat @ X4 @ N ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_le
% 0.17/0.44 thf(fact_2268_dvd__power__le,axiom,
% 0.17/0.44 ! [X4: real,Y3: real,N: nat,M: nat] :
% 0.17/0.44 ( ( dvd_dvd_real @ X4 @ Y3 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.44 => ( dvd_dvd_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_le
% 0.17/0.44 thf(fact_2269_dvd__power__le,axiom,
% 0.17/0.44 ! [X4: int,Y3: int,N: nat,M: nat] :
% 0.17/0.44 ( ( dvd_dvd_int @ X4 @ Y3 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.44 => ( dvd_dvd_int @ ( power_power_int @ X4 @ N ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_le
% 0.17/0.44 thf(fact_2270_dvd__power__le,axiom,
% 0.17/0.44 ! [X4: complex,Y3: complex,N: nat,M: nat] :
% 0.17/0.44 ( ( dvd_dvd_complex @ X4 @ Y3 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.44 => ( dvd_dvd_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_power_le
% 0.17/0.44 thf(fact_2271_mod__eq__dvd__iff,axiom,
% 0.17/0.44 ! [A: int,C: int,B2: int] :
% 0.17/0.44 ( ( ( modulo_modulo_int @ A @ C )
% 0.17/0.44 = ( modulo_modulo_int @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mod_eq_dvd_iff
% 0.17/0.44 thf(fact_2272_mod__eq__dvd__iff,axiom,
% 0.17/0.44 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.17/0.44 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.17/0.44 = ( modulo364778990260209775nteger @ B2 @ C ) )
% 0.17/0.44 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mod_eq_dvd_iff
% 0.17/0.44 thf(fact_2273_dvd__minus__mod,axiom,
% 0.17/0.44 ! [B2: nat,A: nat] : ( dvd_dvd_nat @ B2 @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_minus_mod
% 0.17/0.44 thf(fact_2274_dvd__minus__mod,axiom,
% 0.17/0.44 ! [B2: int,A: int] : ( dvd_dvd_int @ B2 @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_minus_mod
% 0.17/0.44 thf(fact_2275_dvd__minus__mod,axiom,
% 0.17/0.44 ! [B2: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B2 @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_minus_mod
% 0.17/0.44 thf(fact_2276_dvd__minus__self,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 0.17/0.44 = ( ( ord_less_nat @ N @ M )
% 0.17/0.44 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_minus_self
% 0.17/0.44 thf(fact_2277_dvd__diffD,axiom,
% 0.17/0.44 ! [K: nat,M: nat,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.17/0.44 => ( ( dvd_dvd_nat @ K @ N )
% 0.17/0.44 => ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.44 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_diffD
% 0.17/0.44 thf(fact_2278_dvd__diffD1,axiom,
% 0.17/0.44 ! [K: nat,M: nat,N: nat] :
% 0.17/0.44 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.17/0.44 => ( ( dvd_dvd_nat @ K @ M )
% 0.17/0.44 => ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.44 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dvd_diffD1
% 0.17/0.44 thf(fact_2279_less__eq__dvd__minus,axiom,
% 0.17/0.44 ! [M: nat,N: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.44 => ( ( dvd_dvd_nat @ M @ N )
% 0.17/0.44 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % less_eq_dvd_minus
% 0.17/0.44 thf(fact_2280_dbl__def,axiom,
% 0.17/0.44 ( neg_numeral_dbl_real
% 0.17/0.44 = ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dbl_def
% 0.17/0.44 thf(fact_2281_dbl__def,axiom,
% 0.17/0.44 ( neg_numeral_dbl_rat
% 0.17/0.44 = ( ^ [X: rat] : ( plus_plus_rat @ X @ X ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dbl_def
% 0.17/0.44 thf(fact_2282_dbl__def,axiom,
% 0.17/0.44 ( neg_numeral_dbl_int
% 0.17/0.44 = ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % dbl_def
% 0.17/0.44 thf(fact_2283_zero__le__numeral,axiom,
% 0.17/0.44 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_le_numeral
% 0.17/0.44 thf(fact_2284_zero__le__numeral,axiom,
% 0.17/0.44 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_le_numeral
% 0.17/0.44 thf(fact_2285_zero__le__numeral,axiom,
% 0.17/0.44 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_le_numeral
% 0.17/0.44 thf(fact_2286_zero__le__numeral,axiom,
% 0.17/0.44 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_le_numeral
% 0.17/0.44 thf(fact_2287_not__numeral__le__zero,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.17/0.44
% 0.17/0.44 % not_numeral_le_zero
% 0.17/0.44 thf(fact_2288_not__numeral__le__zero,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.17/0.44
% 0.17/0.44 % not_numeral_le_zero
% 0.17/0.44 thf(fact_2289_not__numeral__le__zero,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % not_numeral_le_zero
% 0.17/0.44 thf(fact_2290_not__numeral__le__zero,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.17/0.44
% 0.17/0.44 % not_numeral_le_zero
% 0.17/0.44 thf(fact_2291_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.44 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.17/0.44 thf(fact_2292_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.17/0.44 thf(fact_2293_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.44 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.17/0.44 thf(fact_2294_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.44 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.17/0.44 thf(fact_2295_zero__le__mult__iff,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.17/0.44 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 0.17/0.44 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.44 & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_le_mult_iff
% 0.17/0.44 thf(fact_2296_zero__le__mult__iff,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.17/0.44 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 0.17/0.44 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.17/0.44 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_le_mult_iff
% 0.17/0.44 thf(fact_2297_zero__le__mult__iff,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 0.17/0.44 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
% 0.17/0.44 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.44 & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_le_mult_iff
% 0.17/0.44 thf(fact_2298_mult__nonneg__nonpos2,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 0.17/0.44 => ( ord_less_eq_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonpos2
% 0.17/0.44 thf(fact_2299_mult__nonneg__nonpos2,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonpos2
% 0.17/0.44 thf(fact_2300_mult__nonneg__nonpos2,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 0.17/0.44 => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonpos2
% 0.17/0.44 thf(fact_2301_mult__nonneg__nonpos2,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.17/0.44 => ( ord_less_eq_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonpos2
% 0.17/0.44 thf(fact_2302_mult__nonpos__nonneg,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.44 => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonpos_nonneg
% 0.17/0.44 thf(fact_2303_mult__nonpos__nonneg,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonpos_nonneg
% 0.17/0.44 thf(fact_2304_mult__nonpos__nonneg,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.44 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonpos_nonneg
% 0.17/0.44 thf(fact_2305_mult__nonpos__nonneg,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.44 => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonpos_nonneg
% 0.17/0.44 thf(fact_2306_mult__nonneg__nonpos,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 0.17/0.44 => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonpos
% 0.17/0.44 thf(fact_2307_mult__nonneg__nonpos,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonpos
% 0.17/0.44 thf(fact_2308_mult__nonneg__nonpos,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 0.17/0.44 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonpos
% 0.17/0.44 thf(fact_2309_mult__nonneg__nonpos,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.17/0.44 => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonpos
% 0.17/0.44 thf(fact_2310_mult__nonneg__nonneg,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.44 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonneg
% 0.17/0.44 thf(fact_2311_mult__nonneg__nonneg,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.44 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonneg
% 0.17/0.44 thf(fact_2312_mult__nonneg__nonneg,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.44 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonneg
% 0.17/0.44 thf(fact_2313_mult__nonneg__nonneg,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.44 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonneg_nonneg
% 0.17/0.44 thf(fact_2314_split__mult__neg__le,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 0.17/0.44 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.44 & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
% 0.17/0.44 => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ).
% 0.17/0.44
% 0.17/0.44 % split_mult_neg_le
% 0.17/0.44 thf(fact_2315_split__mult__neg__le,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 0.17/0.44 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.17/0.44 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) )
% 0.17/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ).
% 0.17/0.44
% 0.17/0.44 % split_mult_neg_le
% 0.17/0.44 thf(fact_2316_split__mult__neg__le,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.44 & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
% 0.17/0.44 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.44 & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
% 0.17/0.44 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ).
% 0.17/0.44
% 0.17/0.44 % split_mult_neg_le
% 0.17/0.44 thf(fact_2317_split__mult__neg__le,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
% 0.17/0.44 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.44 & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
% 0.17/0.44 => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ).
% 0.17/0.44
% 0.17/0.44 % split_mult_neg_le
% 0.17/0.44 thf(fact_2318_mult__le__0__iff,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 0.17/0.44 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 0.17/0.44 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.44 & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_le_0_iff
% 0.17/0.44 thf(fact_2319_mult__le__0__iff,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 0.17/0.44 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 0.17/0.44 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.17/0.44 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_le_0_iff
% 0.17/0.44 thf(fact_2320_mult__le__0__iff,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
% 0.17/0.44 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
% 0.17/0.44 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.44 & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_le_0_iff
% 0.17/0.44 thf(fact_2321_mult__right__mono,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.44 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_right_mono
% 0.17/0.44 thf(fact_2322_mult__right__mono,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_right_mono
% 0.17/0.44 thf(fact_2323_mult__right__mono,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.44 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_right_mono
% 0.17/0.44 thf(fact_2324_mult__right__mono,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.44 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_right_mono
% 0.17/0.44 thf(fact_2325_mult__right__mono__neg,axiom,
% 0.17/0.44 ! [B2: real,A: real,C: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.17/0.44 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_right_mono_neg
% 0.17/0.44 thf(fact_2326_mult__right__mono__neg,axiom,
% 0.17/0.44 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_right_mono_neg
% 0.17/0.44 thf(fact_2327_mult__right__mono__neg,axiom,
% 0.17/0.44 ! [B2: int,A: int,C: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.17/0.44 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_right_mono_neg
% 0.17/0.44 thf(fact_2328_mult__left__mono,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.44 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_left_mono
% 0.17/0.44 thf(fact_2329_mult__left__mono,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_left_mono
% 0.17/0.44 thf(fact_2330_mult__left__mono,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.44 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_left_mono
% 0.17/0.44 thf(fact_2331_mult__left__mono,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.44 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_left_mono
% 0.17/0.44 thf(fact_2332_mult__nonpos__nonpos,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.44 => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 0.17/0.44 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonpos_nonpos
% 0.17/0.44 thf(fact_2333_mult__nonpos__nonpos,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.17/0.44 => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonpos_nonpos
% 0.17/0.44 thf(fact_2334_mult__nonpos__nonpos,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.44 => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.17/0.44 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_nonpos_nonpos
% 0.17/0.44 thf(fact_2335_mult__left__mono__neg,axiom,
% 0.17/0.44 ! [B2: real,A: real,C: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.17/0.44 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_left_mono_neg
% 0.17/0.44 thf(fact_2336_mult__left__mono__neg,axiom,
% 0.17/0.44 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_left_mono_neg
% 0.17/0.44 thf(fact_2337_mult__left__mono__neg,axiom,
% 0.17/0.44 ! [B2: int,A: int,C: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.17/0.44 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_left_mono_neg
% 0.17/0.44 thf(fact_2338_split__mult__pos__le,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 0.17/0.44 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.44 & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
% 0.17/0.44 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % split_mult_pos_le
% 0.17/0.44 thf(fact_2339_split__mult__pos__le,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 0.17/0.44 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.17/0.44 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
% 0.17/0.44 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % split_mult_pos_le
% 0.17/0.44 thf(fact_2340_split__mult__pos__le,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
% 0.17/0.44 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.44 & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
% 0.17/0.44 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % split_mult_pos_le
% 0.17/0.44 thf(fact_2341_zero__le__square,axiom,
% 0.17/0.44 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_le_square
% 0.17/0.44 thf(fact_2342_zero__le__square,axiom,
% 0.17/0.44 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_le_square
% 0.17/0.44 thf(fact_2343_zero__le__square,axiom,
% 0.17/0.44 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_le_square
% 0.17/0.44 thf(fact_2344_mult__mono_H,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_real @ C @ D )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.44 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_mono'
% 0.17/0.44 thf(fact_2345_mult__mono_H,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_rat @ C @ D )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_mono'
% 0.17/0.44 thf(fact_2346_mult__mono_H,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ C @ D )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.44 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_mono'
% 0.17/0.44 thf(fact_2347_mult__mono_H,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_int @ C @ D )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.44 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_mono'
% 0.17/0.44 thf(fact_2348_mult__mono,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_real @ C @ D )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.44 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_mono
% 0.17/0.44 thf(fact_2349_mult__mono,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_rat @ C @ D )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_mono
% 0.17/0.44 thf(fact_2350_mult__mono,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ C @ D )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.44 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_mono
% 0.17/0.44 thf(fact_2351_mult__mono,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_int @ C @ D )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.44 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_mono
% 0.17/0.44 thf(fact_2352_not__numeral__less__zero,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.17/0.44
% 0.17/0.44 % not_numeral_less_zero
% 0.17/0.44 thf(fact_2353_not__numeral__less__zero,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.17/0.44
% 0.17/0.44 % not_numeral_less_zero
% 0.17/0.44 thf(fact_2354_not__numeral__less__zero,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % not_numeral_less_zero
% 0.17/0.44 thf(fact_2355_not__numeral__less__zero,axiom,
% 0.17/0.44 ! [N: num] :
% 0.17/0.44 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.17/0.44
% 0.17/0.44 % not_numeral_less_zero
% 0.17/0.44 thf(fact_2356_zero__less__numeral,axiom,
% 0.17/0.44 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_numeral
% 0.17/0.44 thf(fact_2357_zero__less__numeral,axiom,
% 0.17/0.44 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_numeral
% 0.17/0.44 thf(fact_2358_zero__less__numeral,axiom,
% 0.17/0.44 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_numeral
% 0.17/0.44 thf(fact_2359_zero__less__numeral,axiom,
% 0.17/0.44 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_numeral
% 0.17/0.44 thf(fact_2360_zero__less__one__class_Ozero__le__one,axiom,
% 0.17/0.44 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.17/0.44
% 0.17/0.44 % zero_less_one_class.zero_le_one
% 0.17/0.44 thf(fact_2361_zero__less__one__class_Ozero__le__one,axiom,
% 0.17/0.44 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.17/0.44
% 0.17/0.44 % zero_less_one_class.zero_le_one
% 0.17/0.44 thf(fact_2362_zero__less__one__class_Ozero__le__one,axiom,
% 0.17/0.44 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.17/0.44
% 0.17/0.44 % zero_less_one_class.zero_le_one
% 0.17/0.44 thf(fact_2363_zero__less__one__class_Ozero__le__one,axiom,
% 0.17/0.44 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.17/0.44
% 0.17/0.44 % zero_less_one_class.zero_le_one
% 0.17/0.44 thf(fact_2364_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.17/0.44 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.17/0.44
% 0.17/0.44 % linordered_nonzero_semiring_class.zero_le_one
% 0.17/0.44 thf(fact_2365_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.17/0.44 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.17/0.44
% 0.17/0.44 % linordered_nonzero_semiring_class.zero_le_one
% 0.17/0.44 thf(fact_2366_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.17/0.44 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.17/0.44
% 0.17/0.44 % linordered_nonzero_semiring_class.zero_le_one
% 0.17/0.44 thf(fact_2367_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.17/0.44 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.17/0.44
% 0.17/0.44 % linordered_nonzero_semiring_class.zero_le_one
% 0.17/0.44 thf(fact_2368_not__one__le__zero,axiom,
% 0.17/0.44 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.17/0.44
% 0.17/0.44 % not_one_le_zero
% 0.17/0.44 thf(fact_2369_not__one__le__zero,axiom,
% 0.17/0.44 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.17/0.44
% 0.17/0.44 % not_one_le_zero
% 0.17/0.44 thf(fact_2370_not__one__le__zero,axiom,
% 0.17/0.44 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % not_one_le_zero
% 0.17/0.44 thf(fact_2371_not__one__le__zero,axiom,
% 0.17/0.44 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.17/0.44
% 0.17/0.44 % not_one_le_zero
% 0.17/0.44 thf(fact_2372_add__decreasing,axiom,
% 0.17/0.44 ! [A: real,C: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.44 => ( ( ord_less_eq_real @ C @ B2 )
% 0.17/0.44 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_decreasing
% 0.17/0.44 thf(fact_2373_add__decreasing,axiom,
% 0.17/0.44 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.17/0.44 => ( ( ord_less_eq_rat @ C @ B2 )
% 0.17/0.44 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_decreasing
% 0.17/0.44 thf(fact_2374_add__decreasing,axiom,
% 0.17/0.44 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.44 => ( ( ord_less_eq_nat @ C @ B2 )
% 0.17/0.44 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_decreasing
% 0.17/0.44 thf(fact_2375_add__decreasing,axiom,
% 0.17/0.44 ! [A: int,C: int,B2: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.44 => ( ( ord_less_eq_int @ C @ B2 )
% 0.17/0.44 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_decreasing
% 0.17/0.44 thf(fact_2376_add__increasing,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_eq_real @ B2 @ C )
% 0.17/0.44 => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_increasing
% 0.17/0.44 thf(fact_2377_add__increasing,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.17/0.44 => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_increasing
% 0.17/0.44 thf(fact_2378_add__increasing,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.17/0.44 => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_increasing
% 0.17/0.44 thf(fact_2379_add__increasing,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_eq_int @ B2 @ C )
% 0.17/0.44 => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_increasing
% 0.17/0.44 thf(fact_2380_add__decreasing2,axiom,
% 0.17/0.44 ! [C: real,A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.17/0.44 => ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.44 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_decreasing2
% 0.17/0.44 thf(fact_2381_add__decreasing2,axiom,
% 0.17/0.44 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.17/0.44 => ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.44 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_decreasing2
% 0.17/0.44 thf(fact_2382_add__decreasing2,axiom,
% 0.17/0.44 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.17/0.44 => ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.44 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_decreasing2
% 0.17/0.44 thf(fact_2383_add__decreasing2,axiom,
% 0.17/0.44 ! [C: int,A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.17/0.44 => ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.44 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_decreasing2
% 0.17/0.44 thf(fact_2384_add__increasing2,axiom,
% 0.17/0.44 ! [C: real,B2: real,A: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.44 => ( ( ord_less_eq_real @ B2 @ A )
% 0.17/0.44 => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_increasing2
% 0.17/0.44 thf(fact_2385_add__increasing2,axiom,
% 0.17/0.44 ! [C: rat,B2: rat,A: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.44 => ( ( ord_less_eq_rat @ B2 @ A )
% 0.17/0.44 => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_increasing2
% 0.17/0.44 thf(fact_2386_add__increasing2,axiom,
% 0.17/0.44 ! [C: nat,B2: nat,A: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.44 => ( ( ord_less_eq_nat @ B2 @ A )
% 0.17/0.44 => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_increasing2
% 0.17/0.44 thf(fact_2387_add__increasing2,axiom,
% 0.17/0.44 ! [C: int,B2: int,A: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.44 => ( ( ord_less_eq_int @ B2 @ A )
% 0.17/0.44 => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_increasing2
% 0.17/0.44 thf(fact_2388_add__nonneg__nonneg,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.44 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonneg_nonneg
% 0.17/0.44 thf(fact_2389_add__nonneg__nonneg,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.44 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonneg_nonneg
% 0.17/0.44 thf(fact_2390_add__nonneg__nonneg,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.44 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonneg_nonneg
% 0.17/0.44 thf(fact_2391_add__nonneg__nonneg,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.44 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonneg_nonneg
% 0.17/0.44 thf(fact_2392_add__nonpos__nonpos,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.44 => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 0.17/0.44 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonpos_nonpos
% 0.17/0.44 thf(fact_2393_add__nonpos__nonpos,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.17/0.44 => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonpos_nonpos
% 0.17/0.44 thf(fact_2394_add__nonpos__nonpos,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.44 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 0.17/0.44 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonpos_nonpos
% 0.17/0.44 thf(fact_2395_add__nonpos__nonpos,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.44 => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.17/0.44 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonpos_nonpos
% 0.17/0.44 thf(fact_2396_add__nonneg__eq__0__iff,axiom,
% 0.17/0.44 ! [X4: real,Y3: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.17/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.17/0.44 => ( ( ( plus_plus_real @ X4 @ Y3 )
% 0.17/0.44 = zero_zero_real )
% 0.17/0.44 = ( ( X4 = zero_zero_real )
% 0.17/0.44 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonneg_eq_0_iff
% 0.17/0.44 thf(fact_2397_add__nonneg__eq__0__iff,axiom,
% 0.17/0.44 ! [X4: rat,Y3: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.17/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.17/0.44 => ( ( ( plus_plus_rat @ X4 @ Y3 )
% 0.17/0.44 = zero_zero_rat )
% 0.17/0.44 = ( ( X4 = zero_zero_rat )
% 0.17/0.44 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonneg_eq_0_iff
% 0.17/0.44 thf(fact_2398_add__nonneg__eq__0__iff,axiom,
% 0.17/0.44 ! [X4: nat,Y3: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
% 0.17/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.17/0.44 => ( ( ( plus_plus_nat @ X4 @ Y3 )
% 0.17/0.44 = zero_zero_nat )
% 0.17/0.44 = ( ( X4 = zero_zero_nat )
% 0.17/0.44 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonneg_eq_0_iff
% 0.17/0.44 thf(fact_2399_add__nonneg__eq__0__iff,axiom,
% 0.17/0.44 ! [X4: int,Y3: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.17/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.17/0.44 => ( ( ( plus_plus_int @ X4 @ Y3 )
% 0.17/0.44 = zero_zero_int )
% 0.17/0.44 = ( ( X4 = zero_zero_int )
% 0.17/0.44 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonneg_eq_0_iff
% 0.17/0.44 thf(fact_2400_add__nonpos__eq__0__iff,axiom,
% 0.17/0.44 ! [X4: real,Y3: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.17/0.44 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.17/0.44 => ( ( ( plus_plus_real @ X4 @ Y3 )
% 0.17/0.44 = zero_zero_real )
% 0.17/0.44 = ( ( X4 = zero_zero_real )
% 0.17/0.44 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonpos_eq_0_iff
% 0.17/0.44 thf(fact_2401_add__nonpos__eq__0__iff,axiom,
% 0.17/0.44 ! [X4: rat,Y3: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 0.17/0.44 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.17/0.44 => ( ( ( plus_plus_rat @ X4 @ Y3 )
% 0.17/0.44 = zero_zero_rat )
% 0.17/0.44 = ( ( X4 = zero_zero_rat )
% 0.17/0.44 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonpos_eq_0_iff
% 0.17/0.44 thf(fact_2402_add__nonpos__eq__0__iff,axiom,
% 0.17/0.44 ! [X4: nat,Y3: nat] :
% 0.17/0.44 ( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
% 0.17/0.44 => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 0.17/0.44 => ( ( ( plus_plus_nat @ X4 @ Y3 )
% 0.17/0.44 = zero_zero_nat )
% 0.17/0.44 = ( ( X4 = zero_zero_nat )
% 0.17/0.44 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonpos_eq_0_iff
% 0.17/0.44 thf(fact_2403_add__nonpos__eq__0__iff,axiom,
% 0.17/0.44 ! [X4: int,Y3: int] :
% 0.17/0.44 ( ( ord_less_eq_int @ X4 @ zero_zero_int )
% 0.17/0.44 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.17/0.44 => ( ( ( plus_plus_int @ X4 @ Y3 )
% 0.17/0.44 = zero_zero_int )
% 0.17/0.44 = ( ( X4 = zero_zero_int )
% 0.17/0.44 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_nonpos_eq_0_iff
% 0.17/0.44 thf(fact_2404_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real] :
% 0.17/0.44 ( ( ord_less_real @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.44 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.17/0.44 thf(fact_2405_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.44 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.17/0.44 thf(fact_2406_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.17/0.44 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.17/0.44 thf(fact_2407_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( ord_less_int @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.44 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.17/0.44 thf(fact_2408_mult__less__cancel__right__disj,axiom,
% 0.17/0.44 ! [A: real,C: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.17/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.44 & ( ord_less_real @ A @ B2 ) )
% 0.17/0.44 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.44 & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_right_disj
% 0.17/0.44 thf(fact_2409_mult__less__cancel__right__disj,axiom,
% 0.17/0.44 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.17/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.44 & ( ord_less_rat @ A @ B2 ) )
% 0.17/0.44 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.44 & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_right_disj
% 0.17/0.44 thf(fact_2410_mult__less__cancel__right__disj,axiom,
% 0.17/0.44 ! [A: int,C: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.17/0.44 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.44 & ( ord_less_int @ A @ B2 ) )
% 0.17/0.44 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.17/0.44 & ( ord_less_int @ B2 @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_right_disj
% 0.17/0.44 thf(fact_2411_mult__strict__right__mono,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real] :
% 0.17/0.44 ( ( ord_less_real @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.44 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_right_mono
% 0.17/0.44 thf(fact_2412_mult__strict__right__mono,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_right_mono
% 0.17/0.44 thf(fact_2413_mult__strict__right__mono,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.17/0.44 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_right_mono
% 0.17/0.44 thf(fact_2414_mult__strict__right__mono,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( ord_less_int @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.44 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_right_mono
% 0.17/0.44 thf(fact_2415_mult__strict__right__mono__neg,axiom,
% 0.17/0.44 ! [B2: real,A: real,C: real] :
% 0.17/0.44 ( ( ord_less_real @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.44 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_right_mono_neg
% 0.17/0.44 thf(fact_2416_mult__strict__right__mono__neg,axiom,
% 0.17/0.44 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_rat @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_right_mono_neg
% 0.17/0.44 thf(fact_2417_mult__strict__right__mono__neg,axiom,
% 0.17/0.44 ! [B2: int,A: int,C: int] :
% 0.17/0.44 ( ( ord_less_int @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.17/0.44 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_right_mono_neg
% 0.17/0.44 thf(fact_2418_mult__less__cancel__left__disj,axiom,
% 0.17/0.44 ! [C: real,A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.44 & ( ord_less_real @ A @ B2 ) )
% 0.17/0.44 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.44 & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_left_disj
% 0.17/0.44 thf(fact_2419_mult__less__cancel__left__disj,axiom,
% 0.17/0.44 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.44 & ( ord_less_rat @ A @ B2 ) )
% 0.17/0.44 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.44 & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_left_disj
% 0.17/0.44 thf(fact_2420_mult__less__cancel__left__disj,axiom,
% 0.17/0.44 ! [C: int,A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.44 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.44 & ( ord_less_int @ A @ B2 ) )
% 0.17/0.44 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.17/0.44 & ( ord_less_int @ B2 @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_left_disj
% 0.17/0.44 thf(fact_2421_mult__strict__left__mono,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real] :
% 0.17/0.44 ( ( ord_less_real @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.44 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_left_mono
% 0.17/0.44 thf(fact_2422_mult__strict__left__mono,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.44 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_left_mono
% 0.17/0.44 thf(fact_2423_mult__strict__left__mono,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.17/0.44 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_left_mono
% 0.17/0.44 thf(fact_2424_mult__strict__left__mono,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( ord_less_int @ A @ B2 )
% 0.17/0.44 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.44 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_left_mono
% 0.17/0.44 thf(fact_2425_mult__strict__left__mono__neg,axiom,
% 0.17/0.44 ! [B2: real,A: real,C: real] :
% 0.17/0.44 ( ( ord_less_real @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.44 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_left_mono_neg
% 0.17/0.44 thf(fact_2426_mult__strict__left__mono__neg,axiom,
% 0.17/0.44 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_rat @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_left_mono_neg
% 0.17/0.44 thf(fact_2427_mult__strict__left__mono__neg,axiom,
% 0.17/0.44 ! [B2: int,A: int,C: int] :
% 0.17/0.44 ( ( ord_less_int @ B2 @ A )
% 0.17/0.44 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.17/0.44 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_strict_left_mono_neg
% 0.17/0.44 thf(fact_2428_mult__less__cancel__left__pos,axiom,
% 0.17/0.44 ! [C: real,A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.44 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.44 = ( ord_less_real @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_left_pos
% 0.17/0.44 thf(fact_2429_mult__less__cancel__left__pos,axiom,
% 0.17/0.44 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.44 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.44 = ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_left_pos
% 0.17/0.44 thf(fact_2430_mult__less__cancel__left__pos,axiom,
% 0.17/0.44 ! [C: int,A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.44 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.44 = ( ord_less_int @ A @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_left_pos
% 0.17/0.44 thf(fact_2431_mult__less__cancel__left__neg,axiom,
% 0.17/0.44 ! [C: real,A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.44 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.44 = ( ord_less_real @ B2 @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_left_neg
% 0.17/0.44 thf(fact_2432_mult__less__cancel__left__neg,axiom,
% 0.17/0.44 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.44 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.44 = ( ord_less_rat @ B2 @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_left_neg
% 0.17/0.44 thf(fact_2433_mult__less__cancel__left__neg,axiom,
% 0.17/0.44 ! [C: int,A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ C @ zero_zero_int )
% 0.17/0.44 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.44 = ( ord_less_int @ B2 @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_cancel_left_neg
% 0.17/0.44 thf(fact_2434_zero__less__mult__pos2,axiom,
% 0.17/0.44 ! [B2: real,A: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A ) )
% 0.17/0.44 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_mult_pos2
% 0.17/0.44 thf(fact_2435_zero__less__mult__pos2,axiom,
% 0.17/0.44 ! [B2: rat,A: rat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B2 @ A ) )
% 0.17/0.44 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_mult_pos2
% 0.17/0.44 thf(fact_2436_zero__less__mult__pos2,axiom,
% 0.17/0.44 ! [B2: nat,A: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A ) )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_mult_pos2
% 0.17/0.44 thf(fact_2437_zero__less__mult__pos2,axiom,
% 0.17/0.44 ! [B2: int,A: int] :
% 0.17/0.44 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A ) )
% 0.17/0.44 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_mult_pos2
% 0.17/0.44 thf(fact_2438_zero__less__mult__pos,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.17/0.44 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_mult_pos
% 0.17/0.44 thf(fact_2439_zero__less__mult__pos,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.17/0.44 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_mult_pos
% 0.17/0.44 thf(fact_2440_zero__less__mult__pos,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_mult_pos
% 0.17/0.44 thf(fact_2441_zero__less__mult__pos,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 0.17/0.44 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_mult_pos
% 0.17/0.44 thf(fact_2442_zero__less__mult__iff,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.17/0.44 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.44 & ( ord_less_real @ zero_zero_real @ B2 ) )
% 0.17/0.44 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.44 & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_mult_iff
% 0.17/0.44 thf(fact_2443_zero__less__mult__iff,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.17/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.44 & ( ord_less_rat @ zero_zero_rat @ B2 ) )
% 0.17/0.44 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.44 & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_mult_iff
% 0.17/0.44 thf(fact_2444_zero__less__mult__iff,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 0.17/0.44 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.44 & ( ord_less_int @ zero_zero_int @ B2 ) )
% 0.17/0.44 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.17/0.44 & ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % zero_less_mult_iff
% 0.17/0.44 thf(fact_2445_mult__pos__neg2,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.17/0.44 => ( ord_less_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_neg2
% 0.17/0.44 thf(fact_2446_mult__pos__neg2,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_neg2
% 0.17/0.44 thf(fact_2447_mult__pos__neg2,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 0.17/0.44 => ( ord_less_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_neg2
% 0.17/0.44 thf(fact_2448_mult__pos__neg2,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.17/0.44 => ( ord_less_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_neg2
% 0.17/0.44 thf(fact_2449_mult__pos__pos,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.17/0.44 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_pos
% 0.17/0.44 thf(fact_2450_mult__pos__pos,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.17/0.44 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_pos
% 0.17/0.44 thf(fact_2451_mult__pos__pos,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.17/0.44 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_pos
% 0.17/0.44 thf(fact_2452_mult__pos__pos,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.44 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_pos
% 0.17/0.44 thf(fact_2453_mult__pos__neg,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.17/0.44 => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_neg
% 0.17/0.44 thf(fact_2454_mult__pos__neg,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_neg
% 0.17/0.44 thf(fact_2455_mult__pos__neg,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 0.17/0.44 => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_neg
% 0.17/0.44 thf(fact_2456_mult__pos__neg,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.17/0.44 => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_pos_neg
% 0.17/0.44 thf(fact_2457_mult__neg__pos,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.44 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.17/0.44 => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_neg_pos
% 0.17/0.44 thf(fact_2458_mult__neg__pos,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.44 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.17/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_neg_pos
% 0.17/0.44 thf(fact_2459_mult__neg__pos,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.17/0.44 => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_neg_pos
% 0.17/0.44 thf(fact_2460_mult__neg__pos,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ A @ zero_zero_int )
% 0.17/0.44 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.44 => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_neg_pos
% 0.17/0.44 thf(fact_2461_mult__less__0__iff,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 0.17/0.44 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.44 & ( ord_less_real @ B2 @ zero_zero_real ) )
% 0.17/0.44 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.44 & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_0_iff
% 0.17/0.44 thf(fact_2462_mult__less__0__iff,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 0.17/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.44 & ( ord_less_rat @ B2 @ zero_zero_rat ) )
% 0.17/0.44 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.44 & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_0_iff
% 0.17/0.44 thf(fact_2463_mult__less__0__iff,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
% 0.17/0.44 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.44 & ( ord_less_int @ B2 @ zero_zero_int ) )
% 0.17/0.44 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.17/0.44 & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_less_0_iff
% 0.17/0.44 thf(fact_2464_not__square__less__zero,axiom,
% 0.17/0.44 ! [A: real] :
% 0.17/0.44 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 0.17/0.44
% 0.17/0.44 % not_square_less_zero
% 0.17/0.44 thf(fact_2465_not__square__less__zero,axiom,
% 0.17/0.44 ! [A: rat] :
% 0.17/0.44 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 0.17/0.44
% 0.17/0.44 % not_square_less_zero
% 0.17/0.44 thf(fact_2466_not__square__less__zero,axiom,
% 0.17/0.44 ! [A: int] :
% 0.17/0.44 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 0.17/0.44
% 0.17/0.44 % not_square_less_zero
% 0.17/0.44 thf(fact_2467_mult__neg__neg,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.44 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.17/0.44 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_neg_neg
% 0.17/0.44 thf(fact_2468_mult__neg__neg,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.44 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_neg_neg
% 0.17/0.44 thf(fact_2469_mult__neg__neg,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ A @ zero_zero_int )
% 0.17/0.44 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.17/0.44 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % mult_neg_neg
% 0.17/0.44 thf(fact_2470_le__iff__diff__le__0,axiom,
% 0.17/0.44 ( ord_less_eq_real
% 0.17/0.44 = ( ^ [A4: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_iff_diff_le_0
% 0.17/0.44 thf(fact_2471_le__iff__diff__le__0,axiom,
% 0.17/0.44 ( ord_less_eq_rat
% 0.17/0.44 = ( ^ [A4: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_iff_diff_le_0
% 0.17/0.44 thf(fact_2472_le__iff__diff__le__0,axiom,
% 0.17/0.44 ( ord_less_eq_int
% 0.17/0.44 = ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % le_iff_diff_le_0
% 0.17/0.44 thf(fact_2473_less__numeral__extra_I1_J,axiom,
% 0.17/0.44 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.17/0.44
% 0.17/0.44 % less_numeral_extra(1)
% 0.17/0.44 thf(fact_2474_less__numeral__extra_I1_J,axiom,
% 0.17/0.44 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.17/0.44
% 0.17/0.44 % less_numeral_extra(1)
% 0.17/0.44 thf(fact_2475_less__numeral__extra_I1_J,axiom,
% 0.17/0.44 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.17/0.44
% 0.17/0.44 % less_numeral_extra(1)
% 0.17/0.44 thf(fact_2476_less__numeral__extra_I1_J,axiom,
% 0.17/0.44 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.17/0.44
% 0.17/0.44 % less_numeral_extra(1)
% 0.17/0.44 thf(fact_2477_zero__less__one,axiom,
% 0.17/0.44 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.17/0.44
% 0.17/0.44 % zero_less_one
% 0.17/0.44 thf(fact_2478_zero__less__one,axiom,
% 0.17/0.44 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.17/0.44
% 0.17/0.44 % zero_less_one
% 0.17/0.44 thf(fact_2479_zero__less__one,axiom,
% 0.17/0.44 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.17/0.44
% 0.17/0.44 % zero_less_one
% 0.17/0.44 thf(fact_2480_zero__less__one,axiom,
% 0.17/0.44 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.17/0.44
% 0.17/0.44 % zero_less_one
% 0.17/0.44 thf(fact_2481_not__one__less__zero,axiom,
% 0.17/0.44 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.17/0.44
% 0.17/0.44 % not_one_less_zero
% 0.17/0.44 thf(fact_2482_not__one__less__zero,axiom,
% 0.17/0.44 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.17/0.44
% 0.17/0.44 % not_one_less_zero
% 0.17/0.44 thf(fact_2483_not__one__less__zero,axiom,
% 0.17/0.44 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.17/0.44
% 0.17/0.44 % not_one_less_zero
% 0.17/0.44 thf(fact_2484_not__one__less__zero,axiom,
% 0.17/0.44 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.17/0.44
% 0.17/0.44 % not_one_less_zero
% 0.17/0.44 thf(fact_2485_add__neg__neg,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.44 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.17/0.44 => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_neg_neg
% 0.17/0.44 thf(fact_2486_add__neg__neg,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.44 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.17/0.44 => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_neg_neg
% 0.17/0.44 thf(fact_2487_add__neg__neg,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.17/0.44 => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 0.17/0.44 => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_neg_neg
% 0.17/0.44 thf(fact_2488_add__neg__neg,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ A @ zero_zero_int )
% 0.17/0.44 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.17/0.44 => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_neg_neg
% 0.17/0.44 thf(fact_2489_add__pos__pos,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.17/0.44 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_pos_pos
% 0.17/0.44 thf(fact_2490_add__pos__pos,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.17/0.44 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_pos_pos
% 0.17/0.44 thf(fact_2491_add__pos__pos,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.17/0.44 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_pos_pos
% 0.17/0.44 thf(fact_2492_add__pos__pos,axiom,
% 0.17/0.44 ! [A: int,B2: int] :
% 0.17/0.44 ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.44 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_pos_pos
% 0.17/0.44 thf(fact_2493_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.17/0.44 ! [A: nat,B2: nat] :
% 0.17/0.44 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.44 => ~ ! [C3: nat] :
% 0.17/0.44 ( ( B2
% 0.17/0.44 = ( plus_plus_nat @ A @ C3 ) )
% 0.17/0.44 => ( C3 = zero_zero_nat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % canonically_ordered_monoid_add_class.lessE
% 0.17/0.44 thf(fact_2494_pos__add__strict,axiom,
% 0.17/0.44 ! [A: real,B2: real,C: real] :
% 0.17/0.44 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.44 => ( ( ord_less_real @ B2 @ C )
% 0.17/0.44 => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pos_add_strict
% 0.17/0.44 thf(fact_2495_pos__add__strict,axiom,
% 0.17/0.44 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.44 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.44 => ( ( ord_less_rat @ B2 @ C )
% 0.17/0.44 => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pos_add_strict
% 0.17/0.44 thf(fact_2496_pos__add__strict,axiom,
% 0.17/0.44 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.44 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.17/0.44 => ( ( ord_less_nat @ B2 @ C )
% 0.17/0.44 => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pos_add_strict
% 0.17/0.44 thf(fact_2497_pos__add__strict,axiom,
% 0.17/0.44 ! [A: int,B2: int,C: int] :
% 0.17/0.44 ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.44 => ( ( ord_less_int @ B2 @ C )
% 0.17/0.44 => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % pos_add_strict
% 0.17/0.44 thf(fact_2498_add__less__zeroD,axiom,
% 0.17/0.44 ! [X4: real,Y3: real] :
% 0.17/0.44 ( ( ord_less_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
% 0.17/0.44 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 0.17/0.44 | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_less_zeroD
% 0.17/0.44 thf(fact_2499_add__less__zeroD,axiom,
% 0.17/0.44 ! [X4: rat,Y3: rat] :
% 0.17/0.44 ( ( ord_less_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ zero_zero_rat )
% 0.17/0.44 => ( ( ord_less_rat @ X4 @ zero_zero_rat )
% 0.17/0.44 | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_less_zeroD
% 0.17/0.44 thf(fact_2500_add__less__zeroD,axiom,
% 0.17/0.44 ! [X4: int,Y3: int] :
% 0.17/0.44 ( ( ord_less_int @ ( plus_plus_int @ X4 @ Y3 ) @ zero_zero_int )
% 0.17/0.44 => ( ( ord_less_int @ X4 @ zero_zero_int )
% 0.17/0.44 | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % add_less_zeroD
% 0.17/0.44 thf(fact_2501_divide__le__0__iff,axiom,
% 0.17/0.44 ! [A: real,B2: real] :
% 0.17/0.44 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
% 0.17/0.44 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.44 & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 0.17/0.44 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.44 & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 % divide_le_0_iff
% 0.17/0.44 thf(fact_2502_divide__le__0__iff,axiom,
% 0.17/0.44 ! [A: rat,B2: rat] :
% 0.17/0.44 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
% 0.17/0.45 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.45 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 0.17/0.45 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.17/0.45 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_le_0_iff
% 0.17/0.45 thf(fact_2503_divide__right__mono,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_right_mono
% 0.17/0.45 thf(fact_2504_divide__right__mono,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_right_mono
% 0.17/0.45 thf(fact_2505_zero__le__divide__iff,axiom,
% 0.17/0.45 ! [A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
% 0.17/0.45 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 0.17/0.45 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.45 & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_divide_iff
% 0.17/0.45 thf(fact_2506_zero__le__divide__iff,axiom,
% 0.17/0.45 ! [A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.17/0.45 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.45 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 0.17/0.45 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.17/0.45 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_divide_iff
% 0.17/0.45 thf(fact_2507_divide__nonneg__nonneg,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonneg_nonneg
% 0.17/0.45 thf(fact_2508_divide__nonneg__nonneg,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.17/0.45 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonneg_nonneg
% 0.17/0.45 thf(fact_2509_divide__nonneg__nonpos,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.17/0.45 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonneg_nonpos
% 0.17/0.45 thf(fact_2510_divide__nonneg__nonpos,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonneg_nonpos
% 0.17/0.45 thf(fact_2511_divide__nonpos__nonneg,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonpos_nonneg
% 0.17/0.45 thf(fact_2512_divide__nonpos__nonneg,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.17/0.45 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonpos_nonneg
% 0.17/0.45 thf(fact_2513_divide__nonpos__nonpos,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.17/0.45 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonpos_nonpos
% 0.17/0.45 thf(fact_2514_divide__nonpos__nonpos,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonpos_nonpos
% 0.17/0.45 thf(fact_2515_divide__right__mono__neg,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_right_mono_neg
% 0.17/0.45 thf(fact_2516_divide__right__mono__neg,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_right_mono_neg
% 0.17/0.45 thf(fact_2517_less__iff__diff__less__0,axiom,
% 0.17/0.45 ( ord_less_real
% 0.17/0.45 = ( ^ [A4: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % less_iff_diff_less_0
% 0.17/0.45 thf(fact_2518_less__iff__diff__less__0,axiom,
% 0.17/0.45 ( ord_less_rat
% 0.17/0.45 = ( ^ [A4: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % less_iff_diff_less_0
% 0.17/0.45 thf(fact_2519_less__iff__diff__less__0,axiom,
% 0.17/0.45 ( ord_less_int
% 0.17/0.45 = ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % less_iff_diff_less_0
% 0.17/0.45 thf(fact_2520_divide__strict__right__mono__neg,axiom,
% 0.17/0.45 ! [B2: real,A: real,C: real] :
% 0.17/0.45 ( ( ord_less_real @ B2 @ A )
% 0.17/0.45 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_strict_right_mono_neg
% 0.17/0.45 thf(fact_2521_divide__strict__right__mono__neg,axiom,
% 0.17/0.45 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.45 ( ( ord_less_rat @ B2 @ A )
% 0.17/0.45 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_strict_right_mono_neg
% 0.17/0.45 thf(fact_2522_divide__strict__right__mono,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real] :
% 0.17/0.45 ( ( ord_less_real @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_strict_right_mono
% 0.17/0.45 thf(fact_2523_divide__strict__right__mono,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.45 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_strict_right_mono
% 0.17/0.45 thf(fact_2524_zero__less__divide__iff,axiom,
% 0.17/0.45 ! [A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
% 0.17/0.45 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.45 & ( ord_less_real @ zero_zero_real @ B2 ) )
% 0.17/0.45 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.45 & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_less_divide_iff
% 0.17/0.45 thf(fact_2525_zero__less__divide__iff,axiom,
% 0.17/0.45 ! [A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.17/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.45 & ( ord_less_rat @ zero_zero_rat @ B2 ) )
% 0.17/0.45 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.45 & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_less_divide_iff
% 0.17/0.45 thf(fact_2526_divide__less__cancel,axiom,
% 0.17/0.45 ! [A: real,C: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ B2 @ A ) )
% 0.17/0.45 & ( C != zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_less_cancel
% 0.17/0.45 thf(fact_2527_divide__less__cancel,axiom,
% 0.17/0.45 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ B2 @ A ) )
% 0.17/0.45 & ( C != zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_less_cancel
% 0.17/0.45 thf(fact_2528_divide__less__0__iff,axiom,
% 0.17/0.45 ! [A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
% 0.17/0.45 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.45 & ( ord_less_real @ B2 @ zero_zero_real ) )
% 0.17/0.45 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.45 & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_less_0_iff
% 0.17/0.45 thf(fact_2529_divide__less__0__iff,axiom,
% 0.17/0.45 ! [A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
% 0.17/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.45 & ( ord_less_rat @ B2 @ zero_zero_rat ) )
% 0.17/0.45 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.45 & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_less_0_iff
% 0.17/0.45 thf(fact_2530_divide__pos__pos,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_pos_pos
% 0.17/0.45 thf(fact_2531_divide__pos__pos,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.17/0.45 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_pos_pos
% 0.17/0.45 thf(fact_2532_divide__pos__neg,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.17/0.45 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_pos_neg
% 0.17/0.45 thf(fact_2533_divide__pos__neg,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 0.17/0.45 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_pos_neg
% 0.17/0.45 thf(fact_2534_divide__neg__pos,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_real @ X4 @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_neg_pos
% 0.17/0.45 thf(fact_2535_divide__neg__pos,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_rat @ X4 @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.17/0.45 => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_neg_pos
% 0.17/0.45 thf(fact_2536_divide__neg__neg,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_real @ X4 @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_neg_neg
% 0.17/0.45 thf(fact_2537_divide__neg__neg,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_rat @ X4 @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_neg_neg
% 0.17/0.45 thf(fact_2538_zero__le__power,axiom,
% 0.17/0.45 ! [A: real,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_power
% 0.17/0.45 thf(fact_2539_zero__le__power,axiom,
% 0.17/0.45 ! [A: rat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_power
% 0.17/0.45 thf(fact_2540_zero__le__power,axiom,
% 0.17/0.45 ! [A: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_power
% 0.17/0.45 thf(fact_2541_zero__le__power,axiom,
% 0.17/0.45 ! [A: int,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_power
% 0.17/0.45 thf(fact_2542_power__mono,axiom,
% 0.17/0.45 ! [A: real,B2: real,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_mono
% 0.17/0.45 thf(fact_2543_power__mono,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_mono
% 0.17/0.45 thf(fact_2544_power__mono,axiom,
% 0.17/0.45 ! [A: nat,B2: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_mono
% 0.17/0.45 thf(fact_2545_power__mono,axiom,
% 0.17/0.45 ! [A: int,B2: int,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_mono
% 0.17/0.45 thf(fact_2546_zero__less__power,axiom,
% 0.17/0.45 ! [A: real,N: nat] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_less_power
% 0.17/0.45 thf(fact_2547_zero__less__power,axiom,
% 0.17/0.45 ! [A: rat,N: nat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_less_power
% 0.17/0.45 thf(fact_2548_zero__less__power,axiom,
% 0.17/0.45 ! [A: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_less_power
% 0.17/0.45 thf(fact_2549_zero__less__power,axiom,
% 0.17/0.45 ! [A: int,N: nat] :
% 0.17/0.45 ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_less_power
% 0.17/0.45 thf(fact_2550_frac__eq__eq,axiom,
% 0.17/0.45 ! [Y3: complex,Z: complex,X4: complex,W: complex] :
% 0.17/0.45 ( ( Y3 != zero_zero_complex )
% 0.17/0.45 => ( ( Z != zero_zero_complex )
% 0.17/0.45 => ( ( ( divide1717551699836669952omplex @ X4 @ Y3 )
% 0.17/0.45 = ( divide1717551699836669952omplex @ W @ Z ) )
% 0.17/0.45 = ( ( times_times_complex @ X4 @ Z )
% 0.17/0.45 = ( times_times_complex @ W @ Y3 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % frac_eq_eq
% 0.17/0.45 thf(fact_2551_frac__eq__eq,axiom,
% 0.17/0.45 ! [Y3: real,Z: real,X4: real,W: real] :
% 0.17/0.45 ( ( Y3 != zero_zero_real )
% 0.17/0.45 => ( ( Z != zero_zero_real )
% 0.17/0.45 => ( ( ( divide_divide_real @ X4 @ Y3 )
% 0.17/0.45 = ( divide_divide_real @ W @ Z ) )
% 0.17/0.45 = ( ( times_times_real @ X4 @ Z )
% 0.17/0.45 = ( times_times_real @ W @ Y3 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % frac_eq_eq
% 0.17/0.45 thf(fact_2552_frac__eq__eq,axiom,
% 0.17/0.45 ! [Y3: rat,Z: rat,X4: rat,W: rat] :
% 0.17/0.45 ( ( Y3 != zero_zero_rat )
% 0.17/0.45 => ( ( Z != zero_zero_rat )
% 0.17/0.45 => ( ( ( divide_divide_rat @ X4 @ Y3 )
% 0.17/0.45 = ( divide_divide_rat @ W @ Z ) )
% 0.17/0.45 = ( ( times_times_rat @ X4 @ Z )
% 0.17/0.45 = ( times_times_rat @ W @ Y3 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % frac_eq_eq
% 0.17/0.45 thf(fact_2553_divide__eq__eq,axiom,
% 0.17/0.45 ! [B2: complex,C: complex,A: complex] :
% 0.17/0.45 ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 0.17/0.45 = A )
% 0.17/0.45 = ( ( ( C != zero_zero_complex )
% 0.17/0.45 => ( B2
% 0.17/0.45 = ( times_times_complex @ A @ C ) ) )
% 0.17/0.45 & ( ( C = zero_zero_complex )
% 0.17/0.45 => ( A = zero_zero_complex ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_eq_eq
% 0.17/0.45 thf(fact_2554_divide__eq__eq,axiom,
% 0.17/0.45 ! [B2: real,C: real,A: real] :
% 0.17/0.45 ( ( ( divide_divide_real @ B2 @ C )
% 0.17/0.45 = A )
% 0.17/0.45 = ( ( ( C != zero_zero_real )
% 0.17/0.45 => ( B2
% 0.17/0.45 = ( times_times_real @ A @ C ) ) )
% 0.17/0.45 & ( ( C = zero_zero_real )
% 0.17/0.45 => ( A = zero_zero_real ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_eq_eq
% 0.17/0.45 thf(fact_2555_divide__eq__eq,axiom,
% 0.17/0.45 ! [B2: rat,C: rat,A: rat] :
% 0.17/0.45 ( ( ( divide_divide_rat @ B2 @ C )
% 0.17/0.45 = A )
% 0.17/0.45 = ( ( ( C != zero_zero_rat )
% 0.17/0.45 => ( B2
% 0.17/0.45 = ( times_times_rat @ A @ C ) ) )
% 0.17/0.45 & ( ( C = zero_zero_rat )
% 0.17/0.45 => ( A = zero_zero_rat ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_eq_eq
% 0.17/0.45 thf(fact_2556_eq__divide__eq,axiom,
% 0.17/0.45 ! [A: complex,B2: complex,C: complex] :
% 0.17/0.45 ( ( A
% 0.17/0.45 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.17/0.45 = ( ( ( C != zero_zero_complex )
% 0.17/0.45 => ( ( times_times_complex @ A @ C )
% 0.17/0.45 = B2 ) )
% 0.17/0.45 & ( ( C = zero_zero_complex )
% 0.17/0.45 => ( A = zero_zero_complex ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_divide_eq
% 0.17/0.45 thf(fact_2557_eq__divide__eq,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real] :
% 0.17/0.45 ( ( A
% 0.17/0.45 = ( divide_divide_real @ B2 @ C ) )
% 0.17/0.45 = ( ( ( C != zero_zero_real )
% 0.17/0.45 => ( ( times_times_real @ A @ C )
% 0.17/0.45 = B2 ) )
% 0.17/0.45 & ( ( C = zero_zero_real )
% 0.17/0.45 => ( A = zero_zero_real ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_divide_eq
% 0.17/0.45 thf(fact_2558_eq__divide__eq,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.45 ( ( A
% 0.17/0.45 = ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.45 = ( ( ( C != zero_zero_rat )
% 0.17/0.45 => ( ( times_times_rat @ A @ C )
% 0.17/0.45 = B2 ) )
% 0.17/0.45 & ( ( C = zero_zero_rat )
% 0.17/0.45 => ( A = zero_zero_rat ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_divide_eq
% 0.17/0.45 thf(fact_2559_divide__eq__imp,axiom,
% 0.17/0.45 ! [C: complex,B2: complex,A: complex] :
% 0.17/0.45 ( ( C != zero_zero_complex )
% 0.17/0.45 => ( ( B2
% 0.17/0.45 = ( times_times_complex @ A @ C ) )
% 0.17/0.45 => ( ( divide1717551699836669952omplex @ B2 @ C )
% 0.17/0.45 = A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_eq_imp
% 0.17/0.45 thf(fact_2560_divide__eq__imp,axiom,
% 0.17/0.45 ! [C: real,B2: real,A: real] :
% 0.17/0.45 ( ( C != zero_zero_real )
% 0.17/0.45 => ( ( B2
% 0.17/0.45 = ( times_times_real @ A @ C ) )
% 0.17/0.45 => ( ( divide_divide_real @ B2 @ C )
% 0.17/0.45 = A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_eq_imp
% 0.17/0.45 thf(fact_2561_divide__eq__imp,axiom,
% 0.17/0.45 ! [C: rat,B2: rat,A: rat] :
% 0.17/0.45 ( ( C != zero_zero_rat )
% 0.17/0.45 => ( ( B2
% 0.17/0.45 = ( times_times_rat @ A @ C ) )
% 0.17/0.45 => ( ( divide_divide_rat @ B2 @ C )
% 0.17/0.45 = A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_eq_imp
% 0.17/0.45 thf(fact_2562_eq__divide__imp,axiom,
% 0.17/0.45 ! [C: complex,A: complex,B2: complex] :
% 0.17/0.45 ( ( C != zero_zero_complex )
% 0.17/0.45 => ( ( ( times_times_complex @ A @ C )
% 0.17/0.45 = B2 )
% 0.17/0.45 => ( A
% 0.17/0.45 = ( divide1717551699836669952omplex @ B2 @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_divide_imp
% 0.17/0.45 thf(fact_2563_eq__divide__imp,axiom,
% 0.17/0.45 ! [C: real,A: real,B2: real] :
% 0.17/0.45 ( ( C != zero_zero_real )
% 0.17/0.45 => ( ( ( times_times_real @ A @ C )
% 0.17/0.45 = B2 )
% 0.17/0.45 => ( A
% 0.17/0.45 = ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_divide_imp
% 0.17/0.45 thf(fact_2564_eq__divide__imp,axiom,
% 0.17/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( C != zero_zero_rat )
% 0.17/0.45 => ( ( ( times_times_rat @ A @ C )
% 0.17/0.45 = B2 )
% 0.17/0.45 => ( A
% 0.17/0.45 = ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_divide_imp
% 0.17/0.45 thf(fact_2565_nonzero__divide__eq__eq,axiom,
% 0.17/0.45 ! [C: complex,B2: complex,A: complex] :
% 0.17/0.45 ( ( C != zero_zero_complex )
% 0.17/0.45 => ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 0.17/0.45 = A )
% 0.17/0.45 = ( B2
% 0.17/0.45 = ( times_times_complex @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nonzero_divide_eq_eq
% 0.17/0.45 thf(fact_2566_nonzero__divide__eq__eq,axiom,
% 0.17/0.45 ! [C: real,B2: real,A: real] :
% 0.17/0.45 ( ( C != zero_zero_real )
% 0.17/0.45 => ( ( ( divide_divide_real @ B2 @ C )
% 0.17/0.45 = A )
% 0.17/0.45 = ( B2
% 0.17/0.45 = ( times_times_real @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nonzero_divide_eq_eq
% 0.17/0.45 thf(fact_2567_nonzero__divide__eq__eq,axiom,
% 0.17/0.45 ! [C: rat,B2: rat,A: rat] :
% 0.17/0.45 ( ( C != zero_zero_rat )
% 0.17/0.45 => ( ( ( divide_divide_rat @ B2 @ C )
% 0.17/0.45 = A )
% 0.17/0.45 = ( B2
% 0.17/0.45 = ( times_times_rat @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nonzero_divide_eq_eq
% 0.17/0.45 thf(fact_2568_nonzero__eq__divide__eq,axiom,
% 0.17/0.45 ! [C: complex,A: complex,B2: complex] :
% 0.17/0.45 ( ( C != zero_zero_complex )
% 0.17/0.45 => ( ( A
% 0.17/0.45 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.17/0.45 = ( ( times_times_complex @ A @ C )
% 0.17/0.45 = B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nonzero_eq_divide_eq
% 0.17/0.45 thf(fact_2569_nonzero__eq__divide__eq,axiom,
% 0.17/0.45 ! [C: real,A: real,B2: real] :
% 0.17/0.45 ( ( C != zero_zero_real )
% 0.17/0.45 => ( ( A
% 0.17/0.45 = ( divide_divide_real @ B2 @ C ) )
% 0.17/0.45 = ( ( times_times_real @ A @ C )
% 0.17/0.45 = B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nonzero_eq_divide_eq
% 0.17/0.45 thf(fact_2570_nonzero__eq__divide__eq,axiom,
% 0.17/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( C != zero_zero_rat )
% 0.17/0.45 => ( ( A
% 0.17/0.45 = ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.45 = ( ( times_times_rat @ A @ C )
% 0.17/0.45 = B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nonzero_eq_divide_eq
% 0.17/0.45 thf(fact_2571_right__inverse__eq,axiom,
% 0.17/0.45 ! [B2: complex,A: complex] :
% 0.17/0.45 ( ( B2 != zero_zero_complex )
% 0.17/0.45 => ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 0.17/0.45 = one_one_complex )
% 0.17/0.45 = ( A = B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % right_inverse_eq
% 0.17/0.45 thf(fact_2572_right__inverse__eq,axiom,
% 0.17/0.45 ! [B2: real,A: real] :
% 0.17/0.45 ( ( B2 != zero_zero_real )
% 0.17/0.45 => ( ( ( divide_divide_real @ A @ B2 )
% 0.17/0.45 = one_one_real )
% 0.17/0.45 = ( A = B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % right_inverse_eq
% 0.17/0.45 thf(fact_2573_right__inverse__eq,axiom,
% 0.17/0.45 ! [B2: rat,A: rat] :
% 0.17/0.45 ( ( B2 != zero_zero_rat )
% 0.17/0.45 => ( ( ( divide_divide_rat @ A @ B2 )
% 0.17/0.45 = one_one_rat )
% 0.17/0.45 = ( A = B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % right_inverse_eq
% 0.17/0.45 thf(fact_2574_not__exp__less__eq__0__int,axiom,
% 0.17/0.45 ! [N: nat] :
% 0.17/0.45 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 0.17/0.45
% 0.17/0.45 % not_exp_less_eq_0_int
% 0.17/0.45 thf(fact_2575_pos__zdiv__mult__2,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.17/0.45 = ( divide_divide_int @ B2 @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % pos_zdiv_mult_2
% 0.17/0.45 thf(fact_2576_neg__zdiv__mult__2,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.45 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.17/0.45 = ( divide_divide_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % neg_zdiv_mult_2
% 0.17/0.45 thf(fact_2577_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.17/0.45 ! [A: code_integer,B2: code_integer] :
% 0.17/0.45 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.17/0.45 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ A ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.17/0.45 thf(fact_2578_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.17/0.45 ! [A: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B2 ) @ A ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.17/0.45 thf(fact_2579_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ A ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.17/0.45 thf(fact_2580_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.17/0.45 ! [B2: nat,A: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.17/0.45 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.17/0.45 thf(fact_2581_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.17/0.45 ! [B2: int,A: int] :
% 0.17/0.45 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.45 => ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.17/0.45 thf(fact_2582_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.17/0.45 ! [B2: code_integer,A: code_integer] :
% 0.17/0.45 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.17/0.45 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.17/0.45 thf(fact_2583_power__0,axiom,
% 0.17/0.45 ! [A: rat] :
% 0.17/0.45 ( ( power_power_rat @ A @ zero_zero_nat )
% 0.17/0.45 = one_one_rat ) ).
% 0.17/0.45
% 0.17/0.45 % power_0
% 0.17/0.45 thf(fact_2584_power__0,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( power_power_nat @ A @ zero_zero_nat )
% 0.17/0.45 = one_one_nat ) ).
% 0.17/0.45
% 0.17/0.45 % power_0
% 0.17/0.45 thf(fact_2585_power__0,axiom,
% 0.17/0.45 ! [A: real] :
% 0.17/0.45 ( ( power_power_real @ A @ zero_zero_nat )
% 0.17/0.45 = one_one_real ) ).
% 0.17/0.45
% 0.17/0.45 % power_0
% 0.17/0.45 thf(fact_2586_power__0,axiom,
% 0.17/0.45 ! [A: int] :
% 0.17/0.45 ( ( power_power_int @ A @ zero_zero_nat )
% 0.17/0.45 = one_one_int ) ).
% 0.17/0.45
% 0.17/0.45 % power_0
% 0.17/0.45 thf(fact_2587_power__0,axiom,
% 0.17/0.45 ! [A: complex] :
% 0.17/0.45 ( ( power_power_complex @ A @ zero_zero_nat )
% 0.17/0.45 = one_one_complex ) ).
% 0.17/0.45
% 0.17/0.45 % power_0
% 0.17/0.45 thf(fact_2588_mod__eq__self__iff__div__eq__0,axiom,
% 0.17/0.45 ! [A: nat,B2: nat] :
% 0.17/0.45 ( ( ( modulo_modulo_nat @ A @ B2 )
% 0.17/0.45 = A )
% 0.17/0.45 = ( ( divide_divide_nat @ A @ B2 )
% 0.17/0.45 = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod_eq_self_iff_div_eq_0
% 0.17/0.45 thf(fact_2589_mod__eq__self__iff__div__eq__0,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ( ( modulo_modulo_int @ A @ B2 )
% 0.17/0.45 = A )
% 0.17/0.45 = ( ( divide_divide_int @ A @ B2 )
% 0.17/0.45 = zero_zero_int ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod_eq_self_iff_div_eq_0
% 0.17/0.45 thf(fact_2590_mod__eq__self__iff__div__eq__0,axiom,
% 0.17/0.45 ! [A: code_integer,B2: code_integer] :
% 0.17/0.45 ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.17/0.45 = A )
% 0.17/0.45 = ( ( divide6298287555418463151nteger @ A @ B2 )
% 0.17/0.45 = zero_z3403309356797280102nteger ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod_eq_self_iff_div_eq_0
% 0.17/0.45 thf(fact_2591_less__Suc__eq__0__disj,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.17/0.45 = ( ( M = zero_zero_nat )
% 0.17/0.45 | ? [J3: nat] :
% 0.17/0.45 ( ( M
% 0.17/0.45 = ( suc @ J3 ) )
% 0.17/0.45 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % less_Suc_eq_0_disj
% 0.17/0.45 thf(fact_2592_gr0__implies__Suc,axiom,
% 0.17/0.45 ! [N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.45 => ? [M4: nat] :
% 0.17/0.45 ( N
% 0.17/0.45 = ( suc @ M4 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % gr0_implies_Suc
% 0.17/0.45 thf(fact_2593_All__less__Suc2,axiom,
% 0.17/0.45 ! [N: nat,P: nat > $o] :
% 0.17/0.45 ( ( ! [I2: nat] :
% 0.17/0.45 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.17/0.45 => ( P @ I2 ) ) )
% 0.17/0.45 = ( ( P @ zero_zero_nat )
% 0.17/0.45 & ! [I2: nat] :
% 0.17/0.45 ( ( ord_less_nat @ I2 @ N )
% 0.17/0.45 => ( P @ ( suc @ I2 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % All_less_Suc2
% 0.17/0.45 thf(fact_2594_gr0__conv__Suc,axiom,
% 0.17/0.45 ! [N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.45 = ( ? [M6: nat] :
% 0.17/0.45 ( N
% 0.17/0.45 = ( suc @ M6 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % gr0_conv_Suc
% 0.17/0.45 thf(fact_2595_Ex__less__Suc2,axiom,
% 0.17/0.45 ! [N: nat,P: nat > $o] :
% 0.17/0.45 ( ( ? [I2: nat] :
% 0.17/0.45 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.17/0.45 & ( P @ I2 ) ) )
% 0.17/0.45 = ( ( P @ zero_zero_nat )
% 0.17/0.45 | ? [I2: nat] :
% 0.17/0.45 ( ( ord_less_nat @ I2 @ N )
% 0.17/0.45 & ( P @ ( suc @ I2 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % Ex_less_Suc2
% 0.17/0.45 thf(fact_2596_add__is__1,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ( plus_plus_nat @ M @ N )
% 0.17/0.45 = ( suc @ zero_zero_nat ) )
% 0.17/0.45 = ( ( ( M
% 0.17/0.45 = ( suc @ zero_zero_nat ) )
% 0.17/0.45 & ( N = zero_zero_nat ) )
% 0.17/0.45 | ( ( M = zero_zero_nat )
% 0.17/0.45 & ( N
% 0.17/0.45 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_is_1
% 0.17/0.45 thf(fact_2597_one__is__add,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ( suc @ zero_zero_nat )
% 0.17/0.45 = ( plus_plus_nat @ M @ N ) )
% 0.17/0.45 = ( ( ( M
% 0.17/0.45 = ( suc @ zero_zero_nat ) )
% 0.17/0.45 & ( N = zero_zero_nat ) )
% 0.17/0.45 | ( ( M = zero_zero_nat )
% 0.17/0.45 & ( N
% 0.17/0.45 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % one_is_add
% 0.17/0.45 thf(fact_2598_ex__least__nat__le,axiom,
% 0.17/0.45 ! [P: nat > $o,N: nat] :
% 0.17/0.45 ( ( P @ N )
% 0.17/0.45 => ( ~ ( P @ zero_zero_nat )
% 0.17/0.45 => ? [K2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ K2 @ N )
% 0.17/0.45 & ! [I4: nat] :
% 0.17/0.45 ( ( ord_less_nat @ I4 @ K2 )
% 0.17/0.45 => ~ ( P @ I4 ) )
% 0.17/0.45 & ( P @ K2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ex_least_nat_le
% 0.17/0.45 thf(fact_2599_less__imp__add__positive,axiom,
% 0.17/0.45 ! [I: nat,J: nat] :
% 0.17/0.45 ( ( ord_less_nat @ I @ J )
% 0.17/0.45 => ? [K2: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 0.17/0.45 & ( ( plus_plus_nat @ I @ K2 )
% 0.17/0.45 = J ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % less_imp_add_positive
% 0.17/0.45 thf(fact_2600_diff__less,axiom,
% 0.17/0.45 ! [N: nat,M: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.45 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.17/0.45 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_less
% 0.17/0.45 thf(fact_2601_nat__mult__less__cancel1,axiom,
% 0.17/0.45 ! [K: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.45 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.17/0.45 = ( ord_less_nat @ M @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_mult_less_cancel1
% 0.17/0.45 thf(fact_2602_nat__mult__eq__cancel1,axiom,
% 0.17/0.45 ! [K: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.45 => ( ( ( times_times_nat @ K @ M )
% 0.17/0.45 = ( times_times_nat @ K @ N ) )
% 0.17/0.45 = ( M = N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_mult_eq_cancel1
% 0.17/0.45 thf(fact_2603_mult__less__mono2,axiom,
% 0.17/0.45 ! [I: nat,J: nat,K: nat] :
% 0.17/0.45 ( ( ord_less_nat @ I @ J )
% 0.17/0.45 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.45 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_mono2
% 0.17/0.45 thf(fact_2604_mult__less__mono1,axiom,
% 0.17/0.45 ! [I: nat,J: nat,K: nat] :
% 0.17/0.45 ( ( ord_less_nat @ I @ J )
% 0.17/0.45 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.45 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_mono1
% 0.17/0.45 thf(fact_2605_One__nat__def,axiom,
% 0.17/0.45 ( one_one_nat
% 0.17/0.45 = ( suc @ zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % One_nat_def
% 0.17/0.45 thf(fact_2606_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ( divide_divide_nat @ M @ N )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 = ( ( ord_less_nat @ M @ N )
% 0.17/0.45 | ( N = zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % Euclidean_Division.div_eq_0_iff
% 0.17/0.45 thf(fact_2607_diff__add__0,axiom,
% 0.17/0.45 ! [N: nat,M: nat] :
% 0.17/0.45 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.17/0.45 = zero_zero_nat ) ).
% 0.17/0.45
% 0.17/0.45 % diff_add_0
% 0.17/0.45 thf(fact_2608_nat__power__less__imp__less,axiom,
% 0.17/0.45 ! [I: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ I )
% 0.17/0.45 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.17/0.45 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_power_less_imp_less
% 0.17/0.45 thf(fact_2609_mult__eq__self__implies__10,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( M
% 0.17/0.45 = ( times_times_nat @ M @ N ) )
% 0.17/0.45 => ( ( N = one_one_nat )
% 0.17/0.45 | ( M = zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_eq_self_implies_10
% 0.17/0.45 thf(fact_2610_mod__Suc,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.17/0.45 = N )
% 0.17/0.45 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.17/0.45 = zero_zero_nat ) )
% 0.17/0.45 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.17/0.45 != N )
% 0.17/0.45 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.17/0.45 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod_Suc
% 0.17/0.45 thf(fact_2611_mod__less__divisor,axiom,
% 0.17/0.45 ! [N: nat,M: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.45 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod_less_divisor
% 0.17/0.45 thf(fact_2612_mod2__eq__if,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.45 = zero_zero_nat ) )
% 0.17/0.45 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.45 = one_one_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod2_eq_if
% 0.17/0.45 thf(fact_2613_mod2__eq__if,axiom,
% 0.17/0.45 ! [A: int] :
% 0.17/0.45 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.45 = zero_zero_int ) )
% 0.17/0.45 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.45 = one_one_int ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod2_eq_if
% 0.17/0.45 thf(fact_2614_mod2__eq__if,axiom,
% 0.17/0.45 ! [A: code_integer] :
% 0.17/0.45 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.45 = zero_z3403309356797280102nteger ) )
% 0.17/0.45 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.45 = one_one_Code_integer ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod2_eq_if
% 0.17/0.45 thf(fact_2615_parity__cases,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.45 != zero_zero_nat ) )
% 0.17/0.45 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.45 != one_one_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % parity_cases
% 0.17/0.45 thf(fact_2616_parity__cases,axiom,
% 0.17/0.45 ! [A: int] :
% 0.17/0.45 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.45 != zero_zero_int ) )
% 0.17/0.45 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.45 != one_one_int ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % parity_cases
% 0.17/0.45 thf(fact_2617_parity__cases,axiom,
% 0.17/0.45 ! [A: code_integer] :
% 0.17/0.45 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.45 != zero_z3403309356797280102nteger ) )
% 0.17/0.45 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.45 != one_one_Code_integer ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % parity_cases
% 0.17/0.45 thf(fact_2618_pow_Osimps_I1_J,axiom,
% 0.17/0.45 ! [X4: num] :
% 0.17/0.45 ( ( pow @ X4 @ one )
% 0.17/0.45 = X4 ) ).
% 0.17/0.45
% 0.17/0.45 % pow.simps(1)
% 0.17/0.45 thf(fact_2619_zero__le__power__eq,axiom,
% 0.17/0.45 ! [A: real,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.17/0.45 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_power_eq
% 0.17/0.45 thf(fact_2620_zero__le__power__eq,axiom,
% 0.17/0.45 ! [A: rat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.17/0.45 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_power_eq
% 0.17/0.45 thf(fact_2621_zero__le__power__eq,axiom,
% 0.17/0.45 ! [A: int,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.17/0.45 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_power_eq
% 0.17/0.45 thf(fact_2622_zero__le__odd__power,axiom,
% 0.17/0.45 ! [N: nat,A: real] :
% 0.17/0.45 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.17/0.45 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_odd_power
% 0.17/0.45 thf(fact_2623_zero__le__odd__power,axiom,
% 0.17/0.45 ! [N: nat,A: rat] :
% 0.17/0.45 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.17/0.45 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_odd_power
% 0.17/0.45 thf(fact_2624_zero__le__odd__power,axiom,
% 0.17/0.45 ! [N: nat,A: int] :
% 0.17/0.45 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.17/0.45 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_odd_power
% 0.17/0.45 thf(fact_2625_zero__le__even__power,axiom,
% 0.17/0.45 ! [N: nat,A: real] :
% 0.17/0.45 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_even_power
% 0.17/0.45 thf(fact_2626_zero__le__even__power,axiom,
% 0.17/0.45 ! [N: nat,A: rat] :
% 0.17/0.45 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_even_power
% 0.17/0.45 thf(fact_2627_zero__le__even__power,axiom,
% 0.17/0.45 ! [N: nat,A: int] :
% 0.17/0.45 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % zero_le_even_power
% 0.17/0.45 thf(fact_2628_mod__eq__0D,axiom,
% 0.17/0.45 ! [M: nat,D: nat] :
% 0.17/0.45 ( ( ( modulo_modulo_nat @ M @ D )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 => ? [Q3: nat] :
% 0.17/0.45 ( M
% 0.17/0.45 = ( times_times_nat @ D @ Q3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod_eq_0D
% 0.17/0.45 thf(fact_2629_div2__even__ext__nat,axiom,
% 0.17/0.45 ! [X4: nat,Y3: nat] :
% 0.17/0.45 ( ( ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.45 = ( divide_divide_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.17/0.45 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 )
% 0.17/0.45 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y3 ) )
% 0.17/0.45 => ( X4 = Y3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div2_even_ext_nat
% 0.17/0.45 thf(fact_2630_even__numeral,axiom,
% 0.17/0.45 ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % even_numeral
% 0.17/0.45 thf(fact_2631_even__numeral,axiom,
% 0.17/0.45 ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % even_numeral
% 0.17/0.45 thf(fact_2632_even__numeral,axiom,
% 0.17/0.45 ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % even_numeral
% 0.17/0.45 thf(fact_2633_signed__take__bit__int__less__eq,axiom,
% 0.17/0.45 ! [N: nat,K: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.17/0.45 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % signed_take_bit_int_less_eq
% 0.17/0.45 thf(fact_2634_unit__eq__div1,axiom,
% 0.17/0.45 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.17/0.45 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.45 => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 0.17/0.45 = C )
% 0.17/0.45 = ( A
% 0.17/0.45 = ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_eq_div1
% 0.17/0.45 thf(fact_2635_unit__eq__div1,axiom,
% 0.17/0.45 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.45 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.45 => ( ( ( divide_divide_nat @ A @ B2 )
% 0.17/0.45 = C )
% 0.17/0.45 = ( A
% 0.17/0.45 = ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_eq_div1
% 0.17/0.45 thf(fact_2636_unit__eq__div1,axiom,
% 0.17/0.45 ! [B2: int,A: int,C: int] :
% 0.17/0.45 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.45 => ( ( ( divide_divide_int @ A @ B2 )
% 0.17/0.45 = C )
% 0.17/0.45 = ( A
% 0.17/0.45 = ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_eq_div1
% 0.17/0.45 thf(fact_2637_unit__eq__div2,axiom,
% 0.17/0.45 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.17/0.45 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.45 => ( ( A
% 0.17/0.45 = ( divide6298287555418463151nteger @ C @ B2 ) )
% 0.17/0.45 = ( ( times_3573771949741848930nteger @ A @ B2 )
% 0.17/0.45 = C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_eq_div2
% 0.17/0.45 thf(fact_2638_unit__eq__div2,axiom,
% 0.17/0.45 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.45 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.45 => ( ( A
% 0.17/0.45 = ( divide_divide_nat @ C @ B2 ) )
% 0.17/0.45 = ( ( times_times_nat @ A @ B2 )
% 0.17/0.45 = C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_eq_div2
% 0.17/0.45 thf(fact_2639_unit__eq__div2,axiom,
% 0.17/0.45 ! [B2: int,A: int,C: int] :
% 0.17/0.45 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.45 => ( ( A
% 0.17/0.45 = ( divide_divide_int @ C @ B2 ) )
% 0.17/0.45 = ( ( times_times_int @ A @ B2 )
% 0.17/0.45 = C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_eq_div2
% 0.17/0.45 thf(fact_2640_div__mult__unit2,axiom,
% 0.17/0.45 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.17/0.45 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.17/0.45 => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.17/0.45 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.17/0.45 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_mult_unit2
% 0.17/0.45 thf(fact_2641_div__mult__unit2,axiom,
% 0.17/0.45 ! [C: nat,B2: nat,A: nat] :
% 0.17/0.45 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.17/0.45 => ( ( dvd_dvd_nat @ B2 @ A )
% 0.17/0.45 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.17/0.45 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_mult_unit2
% 0.17/0.45 thf(fact_2642_div__mult__unit2,axiom,
% 0.17/0.45 ! [C: int,B2: int,A: int] :
% 0.17/0.45 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.17/0.45 => ( ( dvd_dvd_int @ B2 @ A )
% 0.17/0.45 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.17/0.45 = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_mult_unit2
% 0.17/0.45 thf(fact_2643_unit__div__commute,axiom,
% 0.17/0.45 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.17/0.45 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.45 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 0.17/0.45 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_div_commute
% 0.17/0.45 thf(fact_2644_unit__div__commute,axiom,
% 0.17/0.45 ! [B2: nat,A: nat,C: nat] :
% 0.17/0.45 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.45 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 0.17/0.45 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_div_commute
% 0.17/0.45 thf(fact_2645_unit__div__commute,axiom,
% 0.17/0.45 ! [B2: int,A: int,C: int] :
% 0.17/0.45 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.45 => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 0.17/0.45 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_div_commute
% 0.17/0.45 thf(fact_2646_unit__div__mult__swap,axiom,
% 0.17/0.45 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.17/0.45 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.17/0.45 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.17/0.45 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_div_mult_swap
% 0.17/0.45 thf(fact_2647_unit__div__mult__swap,axiom,
% 0.17/0.45 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.45 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.17/0.45 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 0.17/0.45 = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_div_mult_swap
% 0.17/0.45 thf(fact_2648_unit__div__mult__swap,axiom,
% 0.17/0.45 ! [C: int,A: int,B2: int] :
% 0.17/0.45 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.17/0.45 => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 0.17/0.45 = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unit_div_mult_swap
% 0.17/0.45 thf(fact_2649_is__unit__div__mult2__eq,axiom,
% 0.17/0.45 ! [B2: code_integer,C: code_integer,A: code_integer] :
% 0.17/0.45 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.17/0.45 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.17/0.45 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.17/0.45 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % is_unit_div_mult2_eq
% 0.17/0.45 thf(fact_2650_is__unit__div__mult2__eq,axiom,
% 0.17/0.45 ! [B2: nat,C: nat,A: nat] :
% 0.17/0.45 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.17/0.45 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.17/0.45 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.17/0.45 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % is_unit_div_mult2_eq
% 0.17/0.45 thf(fact_2651_is__unit__div__mult2__eq,axiom,
% 0.17/0.45 ! [B2: int,C: int,A: int] :
% 0.17/0.45 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.17/0.45 => ( ( dvd_dvd_int @ C @ one_one_int )
% 0.17/0.45 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.17/0.45 = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % is_unit_div_mult2_eq
% 0.17/0.45 thf(fact_2652_mod__eq__dvd__iff__nat,axiom,
% 0.17/0.45 ! [N: nat,M: nat,Q2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.45 => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.17/0.45 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.17/0.45 = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod_eq_dvd_iff_nat
% 0.17/0.45 thf(fact_2653_mult__less__le__imp__less,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.45 ( ( ord_less_real @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_real @ C @ D )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_le_imp_less
% 0.17/0.45 thf(fact_2654_mult__less__le__imp__less,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.45 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ C @ D )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_le_imp_less
% 0.17/0.45 thf(fact_2655_mult__less__le__imp__less,axiom,
% 0.17/0.45 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.17/0.45 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_nat @ C @ D )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.17/0.45 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_le_imp_less
% 0.17/0.45 thf(fact_2656_mult__less__le__imp__less,axiom,
% 0.17/0.45 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.45 ( ( ord_less_int @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_int @ C @ D )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_le_imp_less
% 0.17/0.45 thf(fact_2657_mult__le__less__imp__less,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_real @ C @ D )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_less_imp_less
% 0.17/0.45 thf(fact_2658_mult__le__less__imp__less,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_rat @ C @ D )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_less_imp_less
% 0.17/0.45 thf(fact_2659_mult__le__less__imp__less,axiom,
% 0.17/0.45 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_nat @ C @ D )
% 0.17/0.45 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.45 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_less_imp_less
% 0.17/0.45 thf(fact_2660_mult__le__less__imp__less,axiom,
% 0.17/0.45 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_int @ C @ D )
% 0.17/0.45 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_less_imp_less
% 0.17/0.45 thf(fact_2661_mult__right__le__imp__le,axiom,
% 0.17/0.45 ! [A: real,C: real,B2: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_le_imp_le
% 0.17/0.45 thf(fact_2662_mult__right__le__imp__le,axiom,
% 0.17/0.45 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_le_imp_le
% 0.17/0.45 thf(fact_2663_mult__right__le__imp__le,axiom,
% 0.17/0.45 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 0.17/0.45 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.17/0.45 => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_le_imp_le
% 0.17/0.45 thf(fact_2664_mult__right__le__imp__le,axiom,
% 0.17/0.45 ! [A: int,C: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.17/0.45 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_le_imp_le
% 0.17/0.45 thf(fact_2665_mult__left__le__imp__le,axiom,
% 0.17/0.45 ! [C: real,A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_le_imp_le
% 0.17/0.45 thf(fact_2666_mult__left__le__imp__le,axiom,
% 0.17/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_le_imp_le
% 0.17/0.45 thf(fact_2667_mult__left__le__imp__le,axiom,
% 0.17/0.45 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 0.17/0.45 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.17/0.45 => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_le_imp_le
% 0.17/0.45 thf(fact_2668_mult__left__le__imp__le,axiom,
% 0.17/0.45 ! [C: int,A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.45 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_le_imp_le
% 0.17/0.45 thf(fact_2669_mult__le__cancel__left__pos,axiom,
% 0.17/0.45 ! [C: real,A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.45 = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_left_pos
% 0.17/0.45 thf(fact_2670_mult__le__cancel__left__pos,axiom,
% 0.17/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.45 = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_left_pos
% 0.17/0.45 thf(fact_2671_mult__le__cancel__left__pos,axiom,
% 0.17/0.45 ! [C: int,A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.45 = ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_left_pos
% 0.17/0.45 thf(fact_2672_mult__le__cancel__left__neg,axiom,
% 0.17/0.45 ! [C: real,A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.45 = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_left_neg
% 0.17/0.45 thf(fact_2673_mult__le__cancel__left__neg,axiom,
% 0.17/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.45 = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_left_neg
% 0.17/0.45 thf(fact_2674_mult__le__cancel__left__neg,axiom,
% 0.17/0.45 ! [C: int,A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_int @ C @ zero_zero_int )
% 0.17/0.45 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.45 = ( ord_less_eq_int @ B2 @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_left_neg
% 0.17/0.45 thf(fact_2675_mult__less__cancel__right,axiom,
% 0.17/0.45 ! [A: real,C: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_right
% 0.17/0.45 thf(fact_2676_mult__less__cancel__right,axiom,
% 0.17/0.45 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_right
% 0.17/0.45 thf(fact_2677_mult__less__cancel__right,axiom,
% 0.17/0.45 ! [A: int,C: int,B2: int] :
% 0.17/0.45 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_int @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.17/0.45 => ( ord_less_int @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_right
% 0.17/0.45 thf(fact_2678_mult__strict__mono_H,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.45 ( ( ord_less_real @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_real @ C @ D )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_strict_mono'
% 0.17/0.45 thf(fact_2679_mult__strict__mono_H,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.45 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_rat @ C @ D )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_strict_mono'
% 0.17/0.45 thf(fact_2680_mult__strict__mono_H,axiom,
% 0.17/0.45 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.17/0.45 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_nat @ C @ D )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.45 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_strict_mono'
% 0.17/0.45 thf(fact_2681_mult__strict__mono_H,axiom,
% 0.17/0.45 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.45 ( ( ord_less_int @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_int @ C @ D )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_strict_mono'
% 0.17/0.45 thf(fact_2682_mult__right__less__imp__less,axiom,
% 0.17/0.45 ! [A: real,C: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_less_imp_less
% 0.17/0.45 thf(fact_2683_mult__right__less__imp__less,axiom,
% 0.17/0.45 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_less_imp_less
% 0.17/0.45 thf(fact_2684_mult__right__less__imp__less,axiom,
% 0.17/0.45 ! [A: nat,C: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.45 => ( ord_less_nat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_less_imp_less
% 0.17/0.45 thf(fact_2685_mult__right__less__imp__less,axiom,
% 0.17/0.45 ! [A: int,C: int,B2: int] :
% 0.17/0.45 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_int @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_less_imp_less
% 0.17/0.45 thf(fact_2686_mult__less__cancel__left,axiom,
% 0.17/0.45 ! [C: real,A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.45 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_left
% 0.17/0.45 thf(fact_2687_mult__less__cancel__left,axiom,
% 0.17/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.45 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_left
% 0.17/0.45 thf(fact_2688_mult__less__cancel__left,axiom,
% 0.17/0.45 ! [C: int,A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.45 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_int @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.17/0.45 => ( ord_less_int @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_left
% 0.17/0.45 thf(fact_2689_mult__strict__mono,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real,D: real] :
% 0.17/0.45 ( ( ord_less_real @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_real @ C @ D )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_strict_mono
% 0.17/0.45 thf(fact_2690_mult__strict__mono,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.17/0.45 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_rat @ C @ D )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_strict_mono
% 0.17/0.45 thf(fact_2691_mult__strict__mono,axiom,
% 0.17/0.45 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.17/0.45 ( ( ord_less_nat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_nat @ C @ D )
% 0.17/0.45 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.45 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_strict_mono
% 0.17/0.45 thf(fact_2692_mult__strict__mono,axiom,
% 0.17/0.45 ! [A: int,B2: int,C: int,D: int] :
% 0.17/0.45 ( ( ord_less_int @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_int @ C @ D )
% 0.17/0.45 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_strict_mono
% 0.17/0.45 thf(fact_2693_mult__left__less__imp__less,axiom,
% 0.17/0.45 ! [C: real,A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_less_imp_less
% 0.17/0.45 thf(fact_2694_mult__left__less__imp__less,axiom,
% 0.17/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_less_imp_less
% 0.17/0.45 thf(fact_2695_mult__left__less__imp__less,axiom,
% 0.17/0.45 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.45 => ( ord_less_nat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_less_imp_less
% 0.17/0.45 thf(fact_2696_mult__left__less__imp__less,axiom,
% 0.17/0.45 ! [C: int,A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_int @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_less_imp_less
% 0.17/0.45 thf(fact_2697_mult__le__cancel__right,axiom,
% 0.17/0.45 ! [A: real,C: real,B2: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_eq_real @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_right
% 0.17/0.45 thf(fact_2698_mult__le__cancel__right,axiom,
% 0.17/0.45 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_eq_rat @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_right
% 0.17/0.45 thf(fact_2699_mult__le__cancel__right,axiom,
% 0.17/0.45 ! [A: int,C: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_eq_int @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.17/0.45 => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_right
% 0.17/0.45 thf(fact_2700_mult__le__cancel__left,axiom,
% 0.17/0.45 ! [C: real,A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.17/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_eq_real @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_left
% 0.17/0.45 thf(fact_2701_mult__le__cancel__left,axiom,
% 0.17/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.17/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_eq_rat @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_left
% 0.17/0.45 thf(fact_2702_mult__le__cancel__left,axiom,
% 0.17/0.45 ! [C: int,A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.17/0.45 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_eq_int @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.17/0.45 => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_cancel_left
% 0.17/0.45 thf(fact_2703_add__strict__increasing2,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( ord_less_real @ B2 @ C )
% 0.17/0.45 => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_strict_increasing2
% 0.17/0.45 thf(fact_2704_add__strict__increasing2,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ( ord_less_rat @ B2 @ C )
% 0.17/0.45 => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_strict_increasing2
% 0.17/0.45 thf(fact_2705_add__strict__increasing2,axiom,
% 0.17/0.45 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_nat @ B2 @ C )
% 0.17/0.45 => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_strict_increasing2
% 0.17/0.45 thf(fact_2706_add__strict__increasing2,axiom,
% 0.17/0.45 ! [A: int,B2: int,C: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( ord_less_int @ B2 @ C )
% 0.17/0.45 => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_strict_increasing2
% 0.17/0.45 thf(fact_2707_add__strict__increasing,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( ord_less_eq_real @ B2 @ C )
% 0.17/0.45 => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_strict_increasing
% 0.17/0.45 thf(fact_2708_add__strict__increasing,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.17/0.45 => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_strict_increasing
% 0.17/0.45 thf(fact_2709_add__strict__increasing,axiom,
% 0.17/0.45 ! [A: nat,B2: nat,C: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.17/0.45 => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_strict_increasing
% 0.17/0.45 thf(fact_2710_add__strict__increasing,axiom,
% 0.17/0.45 ! [A: int,B2: int,C: int] :
% 0.17/0.45 ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( ord_less_eq_int @ B2 @ C )
% 0.17/0.45 => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_strict_increasing
% 0.17/0.45 thf(fact_2711_add__pos__nonneg,axiom,
% 0.17/0.45 ! [A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.45 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_pos_nonneg
% 0.17/0.45 thf(fact_2712_add__pos__nonneg,axiom,
% 0.17/0.45 ! [A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.45 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_pos_nonneg
% 0.17/0.45 thf(fact_2713_add__pos__nonneg,axiom,
% 0.17/0.45 ! [A: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.45 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_pos_nonneg
% 0.17/0.45 thf(fact_2714_add__pos__nonneg,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.45 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_pos_nonneg
% 0.17/0.45 thf(fact_2715_add__nonpos__neg,axiom,
% 0.17/0.45 ! [A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonpos_neg
% 0.17/0.45 thf(fact_2716_add__nonpos__neg,axiom,
% 0.17/0.45 ! [A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonpos_neg
% 0.17/0.45 thf(fact_2717_add__nonpos__neg,axiom,
% 0.17/0.45 ! [A: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.17/0.45 => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 0.17/0.45 => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonpos_neg
% 0.17/0.45 thf(fact_2718_add__nonpos__neg,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.17/0.45 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.17/0.45 => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonpos_neg
% 0.17/0.45 thf(fact_2719_add__nonneg__pos,axiom,
% 0.17/0.45 ! [A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.17/0.45 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonneg_pos
% 0.17/0.45 thf(fact_2720_add__nonneg__pos,axiom,
% 0.17/0.45 ! [A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.17/0.45 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonneg_pos
% 0.17/0.45 thf(fact_2721_add__nonneg__pos,axiom,
% 0.17/0.45 ! [A: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.17/0.45 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonneg_pos
% 0.17/0.45 thf(fact_2722_add__nonneg__pos,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.45 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_nonneg_pos
% 0.17/0.45 thf(fact_2723_add__neg__nonpos,axiom,
% 0.17/0.45 ! [A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_neg_nonpos
% 0.17/0.45 thf(fact_2724_add__neg__nonpos,axiom,
% 0.17/0.45 ! [A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_neg_nonpos
% 0.17/0.45 thf(fact_2725_add__neg__nonpos,axiom,
% 0.17/0.45 ! [A: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.17/0.45 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 0.17/0.45 => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_neg_nonpos
% 0.17/0.45 thf(fact_2726_add__neg__nonpos,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_int @ A @ zero_zero_int )
% 0.17/0.45 => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.17/0.45 => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_neg_nonpos
% 0.17/0.45 thf(fact_2727_field__le__epsilon,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ! [E2: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.17/0.45 => ( ord_less_eq_real @ X4 @ ( plus_plus_real @ Y3 @ E2 ) ) )
% 0.17/0.45 => ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 0.17/0.45
% 0.17/0.45 % field_le_epsilon
% 0.17/0.45 thf(fact_2728_field__le__epsilon,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ! [E2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.17/0.45 => ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ Y3 @ E2 ) ) )
% 0.17/0.45 => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).
% 0.17/0.45
% 0.17/0.45 % field_le_epsilon
% 0.17/0.45 thf(fact_2729_div__positive,axiom,
% 0.17/0.45 ! [B2: nat,A: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_nat @ B2 @ A )
% 0.17/0.45 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_positive
% 0.17/0.45 thf(fact_2730_div__positive,axiom,
% 0.17/0.45 ! [B2: int,A: int] :
% 0.17/0.45 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_int @ B2 @ A )
% 0.17/0.45 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_positive
% 0.17/0.45 thf(fact_2731_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.17/0.45 ! [A: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_nat @ A @ B2 )
% 0.17/0.45 => ( ( divide_divide_nat @ A @ B2 )
% 0.17/0.45 = zero_zero_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.div_less
% 0.17/0.45 thf(fact_2732_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( ord_less_int @ A @ B2 )
% 0.17/0.45 => ( ( divide_divide_int @ A @ B2 )
% 0.17/0.45 = zero_zero_int ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.div_less
% 0.17/0.45 thf(fact_2733_divide__nonpos__pos,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonpos_pos
% 0.17/0.45 thf(fact_2734_divide__nonpos__pos,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.17/0.45 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonpos_pos
% 0.17/0.45 thf(fact_2735_divide__nonpos__neg,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.17/0.45 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonpos_neg
% 0.17/0.45 thf(fact_2736_divide__nonpos__neg,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonpos_neg
% 0.17/0.45 thf(fact_2737_divide__nonneg__pos,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonneg_pos
% 0.17/0.45 thf(fact_2738_divide__nonneg__pos,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.17/0.45 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonneg_pos
% 0.17/0.45 thf(fact_2739_divide__nonneg__neg,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.17/0.45 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.17/0.45 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonneg_neg
% 0.17/0.45 thf(fact_2740_divide__nonneg__neg,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.17/0.45 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_nonneg_neg
% 0.17/0.45 thf(fact_2741_divide__le__cancel,axiom,
% 0.17/0.45 ! [A: real,C: real,B2: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_eq_real @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_le_cancel
% 0.17/0.45 thf(fact_2742_divide__le__cancel,axiom,
% 0.17/0.45 ! [A: rat,C: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_eq_rat @ A @ B2 ) )
% 0.17/0.45 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_le_cancel
% 0.17/0.45 thf(fact_2743_frac__less2,axiom,
% 0.17/0.45 ! [X4: real,Y3: real,W: real,Z: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.17/0.45 => ( ( ord_less_real @ W @ Z )
% 0.17/0.45 => ( ord_less_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % frac_less2
% 0.17/0.45 thf(fact_2744_frac__less2,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat,W: rat,Z: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.17/0.45 => ( ( ord_less_rat @ W @ Z )
% 0.17/0.45 => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % frac_less2
% 0.17/0.45 thf(fact_2745_frac__less,axiom,
% 0.17/0.45 ! [X4: real,Y3: real,W: real,Z: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.17/0.45 => ( ( ord_less_real @ X4 @ Y3 )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.17/0.45 => ( ( ord_less_eq_real @ W @ Z )
% 0.17/0.45 => ( ord_less_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % frac_less
% 0.17/0.45 thf(fact_2746_frac__less,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat,W: rat,Z: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.17/0.45 => ( ( ord_less_rat @ X4 @ Y3 )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.17/0.45 => ( ( ord_less_eq_rat @ W @ Z )
% 0.17/0.45 => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % frac_less
% 0.17/0.45 thf(fact_2747_frac__le,axiom,
% 0.17/0.45 ! [Y3: real,X4: real,W: real,Z: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.17/0.45 => ( ( ord_less_eq_real @ W @ Z )
% 0.17/0.45 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % frac_le
% 0.17/0.45 thf(fact_2748_frac__le,axiom,
% 0.17/0.45 ! [Y3: rat,X4: rat,W: rat,Z: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.17/0.45 => ( ( ord_less_eq_rat @ W @ Z )
% 0.17/0.45 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % frac_le
% 0.17/0.45 thf(fact_2749_mult__left__le__one__le,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.17/0.45 => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X4 ) @ X4 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_le_one_le
% 0.17/0.45 thf(fact_2750_mult__left__le__one__le,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X4 ) @ X4 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_le_one_le
% 0.17/0.45 thf(fact_2751_mult__left__le__one__le,axiom,
% 0.17/0.45 ! [X4: int,Y3: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.17/0.45 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.17/0.45 => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X4 ) @ X4 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_le_one_le
% 0.17/0.45 thf(fact_2752_mult__right__le__one__le,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.17/0.45 => ( ord_less_eq_real @ ( times_times_real @ X4 @ Y3 ) @ X4 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_le_one_le
% 0.17/0.45 thf(fact_2753_mult__right__le__one__le,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ ( times_times_rat @ X4 @ Y3 ) @ X4 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_le_one_le
% 0.17/0.45 thf(fact_2754_mult__right__le__one__le,axiom,
% 0.17/0.45 ! [X4: int,Y3: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.17/0.45 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.17/0.45 => ( ord_less_eq_int @ ( times_times_int @ X4 @ Y3 ) @ X4 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_right_le_one_le
% 0.17/0.45 thf(fact_2755_mult__le__one,axiom,
% 0.17/0.45 ! [A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ A @ one_one_real )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_real @ B2 @ one_one_real )
% 0.17/0.45 => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ one_one_real ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_one
% 0.17/0.45 thf(fact_2756_mult__le__one,axiom,
% 0.17/0.45 ! [A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_rat @ B2 @ one_one_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ one_one_rat ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_one
% 0.17/0.45 thf(fact_2757_mult__le__one,axiom,
% 0.17/0.45 ! [A: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_nat @ B2 @ one_one_nat )
% 0.17/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_one
% 0.17/0.45 thf(fact_2758_mult__le__one,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ A @ one_one_int )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.45 => ( ( ord_less_eq_int @ B2 @ one_one_int )
% 0.17/0.45 => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_le_one
% 0.17/0.45 thf(fact_2759_mult__left__le,axiom,
% 0.17/0.45 ! [C: real,A: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ C @ one_one_real )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_le
% 0.17/0.45 thf(fact_2760_mult__left__le,axiom,
% 0.17/0.45 ! [C: rat,A: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_le
% 0.17/0.45 thf(fact_2761_mult__left__le,axiom,
% 0.17/0.45 ! [C: nat,A: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_le
% 0.17/0.45 thf(fact_2762_mult__left__le,axiom,
% 0.17/0.45 ! [C: int,A: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ C @ one_one_int )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_left_le
% 0.17/0.45 thf(fact_2763_sum__squares__le__zero__iff,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 0.17/0.45 = ( ( X4 = zero_zero_real )
% 0.17/0.45 & ( Y3 = zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % sum_squares_le_zero_iff
% 0.17/0.45 thf(fact_2764_sum__squares__le__zero__iff,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 0.17/0.45 = ( ( X4 = zero_zero_rat )
% 0.17/0.45 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % sum_squares_le_zero_iff
% 0.17/0.45 thf(fact_2765_sum__squares__le__zero__iff,axiom,
% 0.17/0.45 ! [X4: int,Y3: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 0.17/0.45 = ( ( X4 = zero_zero_int )
% 0.17/0.45 & ( Y3 = zero_zero_int ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % sum_squares_le_zero_iff
% 0.17/0.45 thf(fact_2766_sum__squares__ge__zero,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % sum_squares_ge_zero
% 0.17/0.45 thf(fact_2767_sum__squares__ge__zero,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % sum_squares_ge_zero
% 0.17/0.45 thf(fact_2768_sum__squares__ge__zero,axiom,
% 0.17/0.45 ! [X4: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % sum_squares_ge_zero
% 0.17/0.45 thf(fact_2769_power__less__imp__less__base,axiom,
% 0.17/0.45 ! [A: real,N: nat,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.45 => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_less_imp_less_base
% 0.17/0.45 thf(fact_2770_power__less__imp__less__base,axiom,
% 0.17/0.45 ! [A: rat,N: nat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.45 => ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_less_imp_less_base
% 0.17/0.45 thf(fact_2771_power__less__imp__less__base,axiom,
% 0.17/0.45 ! [A: nat,N: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.45 => ( ord_less_nat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_less_imp_less_base
% 0.17/0.45 thf(fact_2772_power__less__imp__less__base,axiom,
% 0.17/0.45 ! [A: int,N: nat,B2: int] :
% 0.17/0.45 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.45 => ( ord_less_int @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_less_imp_less_base
% 0.17/0.45 thf(fact_2773_sum__squares__gt__zero__iff,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 0.17/0.45 = ( ( X4 != zero_zero_real )
% 0.17/0.45 | ( Y3 != zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % sum_squares_gt_zero_iff
% 0.17/0.45 thf(fact_2774_sum__squares__gt__zero__iff,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 0.17/0.45 = ( ( X4 != zero_zero_rat )
% 0.17/0.45 | ( Y3 != zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % sum_squares_gt_zero_iff
% 0.17/0.45 thf(fact_2775_sum__squares__gt__zero__iff,axiom,
% 0.17/0.45 ! [X4: int,Y3: int] :
% 0.17/0.45 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 0.17/0.45 = ( ( X4 != zero_zero_int )
% 0.17/0.45 | ( Y3 != zero_zero_int ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % sum_squares_gt_zero_iff
% 0.17/0.45 thf(fact_2776_not__sum__squares__lt__zero,axiom,
% 0.17/0.45 ! [X4: real,Y3: real] :
% 0.17/0.45 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 0.17/0.45
% 0.17/0.45 % not_sum_squares_lt_zero
% 0.17/0.45 thf(fact_2777_not__sum__squares__lt__zero,axiom,
% 0.17/0.45 ! [X4: rat,Y3: rat] :
% 0.17/0.45 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 0.17/0.45
% 0.17/0.45 % not_sum_squares_lt_zero
% 0.17/0.45 thf(fact_2778_not__sum__squares__lt__zero,axiom,
% 0.17/0.45 ! [X4: int,Y3: int] :
% 0.17/0.45 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 0.17/0.45
% 0.17/0.45 % not_sum_squares_lt_zero
% 0.17/0.45 thf(fact_2779_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.17/0.45 ! [C: nat,A: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.17/0.45 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.17/0.45 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.17/0.45 thf(fact_2780_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.17/0.45 ! [C: int,A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.17/0.45 = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.17/0.45 thf(fact_2781_zero__less__two,axiom,
% 0.17/0.45 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.17/0.45
% 0.17/0.45 % zero_less_two
% 0.17/0.45 thf(fact_2782_zero__less__two,axiom,
% 0.17/0.45 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.17/0.45
% 0.17/0.45 % zero_less_two
% 0.17/0.45 thf(fact_2783_zero__less__two,axiom,
% 0.17/0.45 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.17/0.45
% 0.17/0.45 % zero_less_two
% 0.17/0.45 thf(fact_2784_zero__less__two,axiom,
% 0.17/0.45 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.17/0.45
% 0.17/0.45 % zero_less_two
% 0.17/0.45 thf(fact_2785_divide__strict__left__mono__neg,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real] :
% 0.17/0.45 ( ( ord_less_real @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.17/0.45 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_strict_left_mono_neg
% 0.17/0.45 thf(fact_2786_divide__strict__left__mono__neg,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.45 ( ( ord_less_rat @ A @ B2 )
% 0.17/0.45 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.17/0.45 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_strict_left_mono_neg
% 0.17/0.45 thf(fact_2787_divide__strict__left__mono,axiom,
% 0.17/0.45 ! [B2: real,A: real,C: real] :
% 0.17/0.45 ( ( ord_less_real @ B2 @ A )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.17/0.45 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_strict_left_mono
% 0.17/0.45 thf(fact_2788_divide__strict__left__mono,axiom,
% 0.17/0.45 ! [B2: rat,A: rat,C: rat] :
% 0.17/0.45 ( ( ord_less_rat @ B2 @ A )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.17/0.45 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_strict_left_mono
% 0.17/0.45 thf(fact_2789_mult__imp__less__div__pos,axiom,
% 0.17/0.45 ! [Y3: real,Z: real,X4: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X4 )
% 0.17/0.45 => ( ord_less_real @ Z @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_imp_less_div_pos
% 0.17/0.45 thf(fact_2790_mult__imp__less__div__pos,axiom,
% 0.17/0.45 ! [Y3: rat,Z: rat,X4: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.17/0.45 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X4 )
% 0.17/0.45 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_imp_less_div_pos
% 0.17/0.45 thf(fact_2791_mult__imp__div__pos__less,axiom,
% 0.17/0.45 ! [Y3: real,X4: real,Z: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ( ord_less_real @ X4 @ ( times_times_real @ Z @ Y3 ) )
% 0.17/0.45 => ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_imp_div_pos_less
% 0.17/0.45 thf(fact_2792_mult__imp__div__pos__less,axiom,
% 0.17/0.45 ! [Y3: rat,X4: rat,Z: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.17/0.45 => ( ( ord_less_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) )
% 0.17/0.45 => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_imp_div_pos_less
% 0.17/0.45 thf(fact_2793_pos__less__divide__eq,axiom,
% 0.17/0.45 ! [C: real,A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.17/0.45 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % pos_less_divide_eq
% 0.17/0.45 thf(fact_2794_pos__less__divide__eq,axiom,
% 0.17/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.45 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % pos_less_divide_eq
% 0.17/0.45 thf(fact_2795_pos__divide__less__eq,axiom,
% 0.17/0.45 ! [C: real,B2: real,A: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.17/0.45 = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % pos_divide_less_eq
% 0.17/0.45 thf(fact_2796_pos__divide__less__eq,axiom,
% 0.17/0.45 ! [C: rat,B2: rat,A: rat] :
% 0.17/0.45 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.17/0.45 = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % pos_divide_less_eq
% 0.17/0.45 thf(fact_2797_neg__less__divide__eq,axiom,
% 0.17/0.45 ! [C: real,A: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.17/0.45 = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % neg_less_divide_eq
% 0.17/0.45 thf(fact_2798_neg__less__divide__eq,axiom,
% 0.17/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.45 = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % neg_less_divide_eq
% 0.17/0.45 thf(fact_2799_neg__divide__less__eq,axiom,
% 0.17/0.45 ! [C: real,B2: real,A: real] :
% 0.17/0.45 ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.17/0.45 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % neg_divide_less_eq
% 0.17/0.45 thf(fact_2800_neg__divide__less__eq,axiom,
% 0.17/0.45 ! [C: rat,B2: rat,A: rat] :
% 0.17/0.45 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.17/0.45 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % neg_divide_less_eq
% 0.17/0.45 thf(fact_2801_less__divide__eq,axiom,
% 0.17/0.45 ! [A: real,B2: real,C: real] :
% 0.17/0.45 ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 0.17/0.45 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 0.17/0.45 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % less_divide_eq
% 0.17/0.45 thf(fact_2802_less__divide__eq,axiom,
% 0.17/0.45 ! [A: rat,B2: rat,C: rat] :
% 0.17/0.45 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 0.17/0.45 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 0.17/0.45 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % less_divide_eq
% 0.17/0.45 thf(fact_2803_divide__less__eq,axiom,
% 0.17/0.45 ! [B2: real,C: real,A: real] :
% 0.17/0.45 ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.17/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 0.17/0.45 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 0.17/0.45 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_less_eq
% 0.17/0.45 thf(fact_2804_divide__less__eq,axiom,
% 0.17/0.45 ! [B2: rat,C: rat,A: rat] :
% 0.17/0.45 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.17/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 0.17/0.45 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 0.17/0.45 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_less_eq
% 0.17/0.45 thf(fact_2805_less__divide__eq__1,axiom,
% 0.17/0.45 ! [B2: real,A: real] :
% 0.17/0.45 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.17/0.45 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.45 & ( ord_less_real @ A @ B2 ) )
% 0.17/0.45 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.45 & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % less_divide_eq_1
% 0.17/0.45 thf(fact_2806_less__divide__eq__1,axiom,
% 0.17/0.45 ! [B2: rat,A: rat] :
% 0.17/0.45 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.17/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.45 & ( ord_less_rat @ A @ B2 ) )
% 0.17/0.45 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.45 & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % less_divide_eq_1
% 0.17/0.45 thf(fact_2807_divide__less__eq__1,axiom,
% 0.17/0.45 ! [B2: real,A: real] :
% 0.17/0.45 ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.17/0.45 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.45 & ( ord_less_real @ B2 @ A ) )
% 0.17/0.45 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.17/0.45 & ( ord_less_real @ A @ B2 ) )
% 0.17/0.45 | ( A = zero_zero_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_less_eq_1
% 0.17/0.45 thf(fact_2808_divide__less__eq__1,axiom,
% 0.17/0.45 ! [B2: rat,A: rat] :
% 0.17/0.45 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.17/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.17/0.45 & ( ord_less_rat @ B2 @ A ) )
% 0.17/0.45 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.17/0.45 & ( ord_less_rat @ A @ B2 ) )
% 0.17/0.45 | ( A = zero_zero_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_less_eq_1
% 0.17/0.45 thf(fact_2809_power__le__one,axiom,
% 0.17/0.45 ! [A: real,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.17/0.45 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_le_one
% 0.17/0.45 thf(fact_2810_power__le__one,axiom,
% 0.17/0.45 ! [A: rat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.17/0.45 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_le_one
% 0.17/0.45 thf(fact_2811_power__le__one,axiom,
% 0.17/0.45 ! [A: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.17/0.45 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_le_one
% 0.17/0.45 thf(fact_2812_power__le__one,axiom,
% 0.17/0.45 ! [A: int,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.17/0.45 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_le_one
% 0.17/0.45 thf(fact_2813_divide__eq__eq__numeral_I1_J,axiom,
% 0.17/0.45 ! [B2: complex,C: complex,W: num] :
% 0.17/0.45 ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 0.17/0.45 = ( numera6690914467698888265omplex @ W ) )
% 0.17/0.45 = ( ( ( C != zero_zero_complex )
% 0.17/0.45 => ( B2
% 0.17/0.45 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 0.17/0.45 & ( ( C = zero_zero_complex )
% 0.17/0.45 => ( ( numera6690914467698888265omplex @ W )
% 0.17/0.45 = zero_zero_complex ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_eq_eq_numeral(1)
% 0.17/0.45 thf(fact_2814_divide__eq__eq__numeral_I1_J,axiom,
% 0.17/0.45 ! [B2: real,C: real,W: num] :
% 0.17/0.45 ( ( ( divide_divide_real @ B2 @ C )
% 0.17/0.45 = ( numeral_numeral_real @ W ) )
% 0.17/0.45 = ( ( ( C != zero_zero_real )
% 0.17/0.45 => ( B2
% 0.17/0.45 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.17/0.45 & ( ( C = zero_zero_real )
% 0.17/0.45 => ( ( numeral_numeral_real @ W )
% 0.17/0.45 = zero_zero_real ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_eq_eq_numeral(1)
% 0.17/0.45 thf(fact_2815_divide__eq__eq__numeral_I1_J,axiom,
% 0.17/0.45 ! [B2: rat,C: rat,W: num] :
% 0.17/0.45 ( ( ( divide_divide_rat @ B2 @ C )
% 0.17/0.45 = ( numeral_numeral_rat @ W ) )
% 0.17/0.45 = ( ( ( C != zero_zero_rat )
% 0.17/0.45 => ( B2
% 0.17/0.45 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.17/0.45 & ( ( C = zero_zero_rat )
% 0.17/0.45 => ( ( numeral_numeral_rat @ W )
% 0.17/0.45 = zero_zero_rat ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_eq_eq_numeral(1)
% 0.17/0.45 thf(fact_2816_eq__divide__eq__numeral_I1_J,axiom,
% 0.17/0.45 ! [W: num,B2: complex,C: complex] :
% 0.17/0.45 ( ( ( numera6690914467698888265omplex @ W )
% 0.17/0.45 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.17/0.45 = ( ( ( C != zero_zero_complex )
% 0.17/0.45 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 0.17/0.45 = B2 ) )
% 0.17/0.45 & ( ( C = zero_zero_complex )
% 0.17/0.45 => ( ( numera6690914467698888265omplex @ W )
% 0.17/0.45 = zero_zero_complex ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_divide_eq_numeral(1)
% 0.17/0.45 thf(fact_2817_eq__divide__eq__numeral_I1_J,axiom,
% 0.17/0.45 ! [W: num,B2: real,C: real] :
% 0.17/0.45 ( ( ( numeral_numeral_real @ W )
% 0.17/0.45 = ( divide_divide_real @ B2 @ C ) )
% 0.17/0.45 = ( ( ( C != zero_zero_real )
% 0.17/0.45 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 0.17/0.45 = B2 ) )
% 0.17/0.45 & ( ( C = zero_zero_real )
% 0.17/0.45 => ( ( numeral_numeral_real @ W )
% 0.17/0.45 = zero_zero_real ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_divide_eq_numeral(1)
% 0.17/0.45 thf(fact_2818_eq__divide__eq__numeral_I1_J,axiom,
% 0.17/0.45 ! [W: num,B2: rat,C: rat] :
% 0.17/0.45 ( ( ( numeral_numeral_rat @ W )
% 0.17/0.45 = ( divide_divide_rat @ B2 @ C ) )
% 0.17/0.45 = ( ( ( C != zero_zero_rat )
% 0.17/0.45 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 0.17/0.45 = B2 ) )
% 0.17/0.45 & ( ( C = zero_zero_rat )
% 0.17/0.45 => ( ( numeral_numeral_rat @ W )
% 0.17/0.45 = zero_zero_rat ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % eq_divide_eq_numeral(1)
% 0.17/0.45 thf(fact_2819_add__divide__eq__if__simps_I2_J,axiom,
% 0.17/0.45 ! [Z: complex,A: complex,B2: complex] :
% 0.17/0.45 ( ( ( Z = zero_zero_complex )
% 0.17/0.45 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 0.17/0.45 = B2 ) )
% 0.17/0.45 & ( ( Z != zero_zero_complex )
% 0.17/0.45 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 0.17/0.45 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_if_simps(2)
% 0.17/0.45 thf(fact_2820_add__divide__eq__if__simps_I2_J,axiom,
% 0.17/0.45 ! [Z: real,A: real,B2: real] :
% 0.17/0.45 ( ( ( Z = zero_zero_real )
% 0.17/0.45 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 0.17/0.45 = B2 ) )
% 0.17/0.45 & ( ( Z != zero_zero_real )
% 0.17/0.45 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 0.17/0.45 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_if_simps(2)
% 0.17/0.45 thf(fact_2821_add__divide__eq__if__simps_I2_J,axiom,
% 0.17/0.45 ! [Z: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( ( Z = zero_zero_rat )
% 0.17/0.45 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 0.17/0.45 = B2 ) )
% 0.17/0.45 & ( ( Z != zero_zero_rat )
% 0.17/0.45 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 0.17/0.45 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_if_simps(2)
% 0.17/0.45 thf(fact_2822_add__divide__eq__if__simps_I1_J,axiom,
% 0.17/0.45 ! [Z: complex,A: complex,B2: complex] :
% 0.17/0.45 ( ( ( Z = zero_zero_complex )
% 0.17/0.45 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 0.17/0.45 = A ) )
% 0.17/0.45 & ( ( Z != zero_zero_complex )
% 0.17/0.45 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 0.17/0.45 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_if_simps(1)
% 0.17/0.45 thf(fact_2823_add__divide__eq__if__simps_I1_J,axiom,
% 0.17/0.45 ! [Z: real,A: real,B2: real] :
% 0.17/0.45 ( ( ( Z = zero_zero_real )
% 0.17/0.45 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 0.17/0.45 = A ) )
% 0.17/0.45 & ( ( Z != zero_zero_real )
% 0.17/0.45 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 0.17/0.45 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_if_simps(1)
% 0.17/0.45 thf(fact_2824_add__divide__eq__if__simps_I1_J,axiom,
% 0.17/0.45 ! [Z: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( ( Z = zero_zero_rat )
% 0.17/0.45 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 0.17/0.45 = A ) )
% 0.17/0.45 & ( ( Z != zero_zero_rat )
% 0.17/0.45 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 0.17/0.45 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_if_simps(1)
% 0.17/0.45 thf(fact_2825_add__frac__eq,axiom,
% 0.17/0.45 ! [Y3: complex,Z: complex,X4: complex,W: complex] :
% 0.17/0.45 ( ( Y3 != zero_zero_complex )
% 0.17/0.45 => ( ( Z != zero_zero_complex )
% 0.17/0.45 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.17/0.45 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_frac_eq
% 0.17/0.45 thf(fact_2826_add__frac__eq,axiom,
% 0.17/0.45 ! [Y3: real,Z: real,X4: real,W: real] :
% 0.17/0.45 ( ( Y3 != zero_zero_real )
% 0.17/0.45 => ( ( Z != zero_zero_real )
% 0.17/0.45 => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.17/0.45 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_frac_eq
% 0.17/0.45 thf(fact_2827_add__frac__eq,axiom,
% 0.17/0.45 ! [Y3: rat,Z: rat,X4: rat,W: rat] :
% 0.17/0.45 ( ( Y3 != zero_zero_rat )
% 0.17/0.45 => ( ( Z != zero_zero_rat )
% 0.17/0.45 => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.17/0.45 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_frac_eq
% 0.17/0.45 thf(fact_2828_add__frac__num,axiom,
% 0.17/0.45 ! [Y3: complex,X4: complex,Z: complex] :
% 0.17/0.45 ( ( Y3 != zero_zero_complex )
% 0.17/0.45 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ Z )
% 0.17/0.45 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_frac_num
% 0.17/0.45 thf(fact_2829_add__frac__num,axiom,
% 0.17/0.45 ! [Y3: real,X4: real,Z: real] :
% 0.17/0.45 ( ( Y3 != zero_zero_real )
% 0.17/0.45 => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Y3 ) @ Z )
% 0.17/0.45 = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_frac_num
% 0.17/0.45 thf(fact_2830_add__frac__num,axiom,
% 0.17/0.45 ! [Y3: rat,X4: rat,Z: rat] :
% 0.17/0.45 ( ( Y3 != zero_zero_rat )
% 0.17/0.45 => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ Z )
% 0.17/0.45 = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_frac_num
% 0.17/0.45 thf(fact_2831_add__num__frac,axiom,
% 0.17/0.45 ! [Y3: complex,Z: complex,X4: complex] :
% 0.17/0.45 ( ( Y3 != zero_zero_complex )
% 0.17/0.45 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
% 0.17/0.45 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_num_frac
% 0.17/0.45 thf(fact_2832_add__num__frac,axiom,
% 0.17/0.45 ! [Y3: real,Z: real,X4: real] :
% 0.17/0.45 ( ( Y3 != zero_zero_real )
% 0.17/0.45 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X4 @ Y3 ) )
% 0.17/0.45 = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_num_frac
% 0.17/0.45 thf(fact_2833_add__num__frac,axiom,
% 0.17/0.45 ! [Y3: rat,Z: rat,X4: rat] :
% 0.17/0.45 ( ( Y3 != zero_zero_rat )
% 0.17/0.45 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X4 @ Y3 ) )
% 0.17/0.45 = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_num_frac
% 0.17/0.45 thf(fact_2834_add__divide__eq__iff,axiom,
% 0.17/0.45 ! [Z: complex,X4: complex,Y3: complex] :
% 0.17/0.45 ( ( Z != zero_zero_complex )
% 0.17/0.45 => ( ( plus_plus_complex @ X4 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 0.17/0.45 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_iff
% 0.17/0.45 thf(fact_2835_add__divide__eq__iff,axiom,
% 0.17/0.45 ! [Z: real,X4: real,Y3: real] :
% 0.17/0.45 ( ( Z != zero_zero_real )
% 0.17/0.45 => ( ( plus_plus_real @ X4 @ ( divide_divide_real @ Y3 @ Z ) )
% 0.17/0.45 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_iff
% 0.17/0.45 thf(fact_2836_add__divide__eq__iff,axiom,
% 0.17/0.45 ! [Z: rat,X4: rat,Y3: rat] :
% 0.17/0.45 ( ( Z != zero_zero_rat )
% 0.17/0.45 => ( ( plus_plus_rat @ X4 @ ( divide_divide_rat @ Y3 @ Z ) )
% 0.17/0.45 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_iff
% 0.17/0.45 thf(fact_2837_divide__add__eq__iff,axiom,
% 0.17/0.45 ! [Z: complex,X4: complex,Y3: complex] :
% 0.17/0.45 ( ( Z != zero_zero_complex )
% 0.17/0.45 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Z ) @ Y3 )
% 0.17/0.45 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_add_eq_iff
% 0.17/0.45 thf(fact_2838_divide__add__eq__iff,axiom,
% 0.17/0.45 ! [Z: real,X4: real,Y3: real] :
% 0.17/0.45 ( ( Z != zero_zero_real )
% 0.17/0.45 => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Z ) @ Y3 )
% 0.17/0.45 = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_add_eq_iff
% 0.17/0.45 thf(fact_2839_divide__add__eq__iff,axiom,
% 0.17/0.45 ! [Z: rat,X4: rat,Y3: rat] :
% 0.17/0.45 ( ( Z != zero_zero_rat )
% 0.17/0.45 => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Z ) @ Y3 )
% 0.17/0.45 = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_add_eq_iff
% 0.17/0.45 thf(fact_2840_power__le__imp__le__base,axiom,
% 0.17/0.45 ! [A: real,N: nat,B2: real] :
% 0.17/0.45 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B2 @ ( suc @ N ) ) )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.45 => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_le_imp_le_base
% 0.17/0.45 thf(fact_2841_power__le__imp__le__base,axiom,
% 0.17/0.45 ! [A: rat,N: nat,B2: rat] :
% 0.17/0.45 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B2 @ ( suc @ N ) ) )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.45 => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_le_imp_le_base
% 0.17/0.45 thf(fact_2842_power__le__imp__le__base,axiom,
% 0.17/0.45 ! [A: nat,N: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B2 @ ( suc @ N ) ) )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.45 => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_le_imp_le_base
% 0.17/0.45 thf(fact_2843_power__le__imp__le__base,axiom,
% 0.17/0.45 ! [A: int,N: nat,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B2 @ ( suc @ N ) ) )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.45 => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_le_imp_le_base
% 0.17/0.45 thf(fact_2844_power__inject__base,axiom,
% 0.17/0.45 ! [A: real,N: nat,B2: real] :
% 0.17/0.45 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 0.17/0.45 = ( power_power_real @ B2 @ ( suc @ N ) ) )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.17/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.17/0.45 => ( A = B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_inject_base
% 0.17/0.45 thf(fact_2845_power__inject__base,axiom,
% 0.17/0.45 ! [A: rat,N: nat,B2: rat] :
% 0.17/0.45 ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.17/0.45 = ( power_power_rat @ B2 @ ( suc @ N ) ) )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.17/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.17/0.45 => ( A = B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_inject_base
% 0.17/0.45 thf(fact_2846_power__inject__base,axiom,
% 0.17/0.45 ! [A: nat,N: nat,B2: nat] :
% 0.17/0.45 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.17/0.45 = ( power_power_nat @ B2 @ ( suc @ N ) ) )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.17/0.45 => ( A = B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_inject_base
% 0.17/0.45 thf(fact_2847_power__inject__base,axiom,
% 0.17/0.45 ! [A: int,N: nat,B2: int] :
% 0.17/0.45 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 0.17/0.45 = ( power_power_int @ B2 @ ( suc @ N ) ) )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.17/0.45 => ( A = B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_inject_base
% 0.17/0.45 thf(fact_2848_div__add__self1,axiom,
% 0.17/0.45 ! [B2: nat,A: nat] :
% 0.17/0.45 ( ( B2 != zero_zero_nat )
% 0.17/0.45 => ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 0.17/0.45 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_add_self1
% 0.17/0.45 thf(fact_2849_div__add__self1,axiom,
% 0.17/0.45 ! [B2: int,A: int] :
% 0.17/0.45 ( ( B2 != zero_zero_int )
% 0.17/0.45 => ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 0.17/0.45 = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_add_self1
% 0.17/0.45 thf(fact_2850_div__add__self2,axiom,
% 0.17/0.45 ! [B2: nat,A: nat] :
% 0.17/0.45 ( ( B2 != zero_zero_nat )
% 0.17/0.45 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 0.17/0.45 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_add_self2
% 0.17/0.45 thf(fact_2851_div__add__self2,axiom,
% 0.17/0.45 ! [B2: int,A: int] :
% 0.17/0.45 ( ( B2 != zero_zero_int )
% 0.17/0.45 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.17/0.45 = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_add_self2
% 0.17/0.45 thf(fact_2852_add__divide__eq__if__simps_I4_J,axiom,
% 0.17/0.45 ! [Z: complex,A: complex,B2: complex] :
% 0.17/0.45 ( ( ( Z = zero_zero_complex )
% 0.17/0.45 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 0.17/0.45 = A ) )
% 0.17/0.45 & ( ( Z != zero_zero_complex )
% 0.17/0.45 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 0.17/0.45 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_if_simps(4)
% 0.17/0.45 thf(fact_2853_add__divide__eq__if__simps_I4_J,axiom,
% 0.17/0.45 ! [Z: real,A: real,B2: real] :
% 0.17/0.45 ( ( ( Z = zero_zero_real )
% 0.17/0.45 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 0.17/0.45 = A ) )
% 0.17/0.45 & ( ( Z != zero_zero_real )
% 0.17/0.45 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 0.17/0.45 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_if_simps(4)
% 0.17/0.45 thf(fact_2854_add__divide__eq__if__simps_I4_J,axiom,
% 0.17/0.45 ! [Z: rat,A: rat,B2: rat] :
% 0.17/0.45 ( ( ( Z = zero_zero_rat )
% 0.17/0.45 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 0.17/0.45 = A ) )
% 0.17/0.45 & ( ( Z != zero_zero_rat )
% 0.17/0.45 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 0.17/0.45 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % add_divide_eq_if_simps(4)
% 0.17/0.45 thf(fact_2855_diff__frac__eq,axiom,
% 0.17/0.45 ! [Y3: complex,Z: complex,X4: complex,W: complex] :
% 0.17/0.45 ( ( Y3 != zero_zero_complex )
% 0.17/0.45 => ( ( Z != zero_zero_complex )
% 0.17/0.45 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.17/0.45 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_frac_eq
% 0.17/0.45 thf(fact_2856_diff__frac__eq,axiom,
% 0.17/0.45 ! [Y3: real,Z: real,X4: real,W: real] :
% 0.17/0.45 ( ( Y3 != zero_zero_real )
% 0.17/0.45 => ( ( Z != zero_zero_real )
% 0.17/0.45 => ( ( minus_minus_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.17/0.45 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_frac_eq
% 0.17/0.45 thf(fact_2857_diff__frac__eq,axiom,
% 0.17/0.45 ! [Y3: rat,Z: rat,X4: rat,W: rat] :
% 0.17/0.45 ( ( Y3 != zero_zero_rat )
% 0.17/0.45 => ( ( Z != zero_zero_rat )
% 0.17/0.45 => ( ( minus_minus_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.17/0.45 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_frac_eq
% 0.17/0.45 thf(fact_2858_diff__divide__eq__iff,axiom,
% 0.17/0.45 ! [Z: complex,X4: complex,Y3: complex] :
% 0.17/0.45 ( ( Z != zero_zero_complex )
% 0.17/0.45 => ( ( minus_minus_complex @ X4 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 0.17/0.45 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_divide_eq_iff
% 0.17/0.45 thf(fact_2859_diff__divide__eq__iff,axiom,
% 0.17/0.45 ! [Z: real,X4: real,Y3: real] :
% 0.17/0.45 ( ( Z != zero_zero_real )
% 0.17/0.45 => ( ( minus_minus_real @ X4 @ ( divide_divide_real @ Y3 @ Z ) )
% 0.17/0.45 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_divide_eq_iff
% 0.17/0.45 thf(fact_2860_diff__divide__eq__iff,axiom,
% 0.17/0.45 ! [Z: rat,X4: rat,Y3: rat] :
% 0.17/0.45 ( ( Z != zero_zero_rat )
% 0.17/0.45 => ( ( minus_minus_rat @ X4 @ ( divide_divide_rat @ Y3 @ Z ) )
% 0.17/0.45 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_divide_eq_iff
% 0.17/0.45 thf(fact_2861_divide__diff__eq__iff,axiom,
% 0.17/0.45 ! [Z: complex,X4: complex,Y3: complex] :
% 0.17/0.45 ( ( Z != zero_zero_complex )
% 0.17/0.45 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X4 @ Z ) @ Y3 )
% 0.17/0.45 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X4 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_diff_eq_iff
% 0.17/0.45 thf(fact_2862_divide__diff__eq__iff,axiom,
% 0.17/0.45 ! [Z: real,X4: real,Y3: real] :
% 0.17/0.45 ( ( Z != zero_zero_real )
% 0.17/0.45 => ( ( minus_minus_real @ ( divide_divide_real @ X4 @ Z ) @ Y3 )
% 0.17/0.45 = ( divide_divide_real @ ( minus_minus_real @ X4 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_diff_eq_iff
% 0.17/0.45 thf(fact_2863_divide__diff__eq__iff,axiom,
% 0.17/0.45 ! [Z: rat,X4: rat,Y3: rat] :
% 0.17/0.45 ( ( Z != zero_zero_rat )
% 0.17/0.45 => ( ( minus_minus_rat @ ( divide_divide_rat @ X4 @ Z ) @ Y3 )
% 0.17/0.45 = ( divide_divide_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % divide_diff_eq_iff
% 0.17/0.45 thf(fact_2864_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.17/0.45 ! [B2: code_integer,A: code_integer] :
% 0.17/0.45 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.17/0.45 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.17/0.45 thf(fact_2865_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.17/0.45 ! [B2: nat,A: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.17/0.45 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.17/0.45 thf(fact_2866_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.17/0.45 ! [B2: int,A: int] :
% 0.17/0.45 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.17/0.45 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.17/0.45 thf(fact_2867_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.17/0.45 ! [A: code_integer,B2: code_integer] :
% 0.17/0.45 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.17/0.45 => ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 0.17/0.45 => ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.17/0.45 = A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.mod_less
% 0.17/0.45 thf(fact_2868_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.17/0.45 ! [A: nat,B2: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.17/0.45 => ( ( ord_less_nat @ A @ B2 )
% 0.17/0.45 => ( ( modulo_modulo_nat @ A @ B2 )
% 0.17/0.45 = A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.mod_less
% 0.17/0.45 thf(fact_2869_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.17/0.45 => ( ( ord_less_int @ A @ B2 )
% 0.17/0.45 => ( ( modulo_modulo_int @ A @ B2 )
% 0.17/0.45 = A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % unique_euclidean_semiring_numeral_class.mod_less
% 0.17/0.45 thf(fact_2870_cong__exp__iff__simps_I2_J,axiom,
% 0.17/0.45 ! [N: num,Q2: num] :
% 0.17/0.45 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.17/0.45 = zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % cong_exp_iff_simps(2)
% 0.17/0.45 thf(fact_2871_cong__exp__iff__simps_I2_J,axiom,
% 0.17/0.45 ! [N: num,Q2: num] :
% 0.17/0.45 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.17/0.45 = zero_zero_int )
% 0.17/0.45 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 0.17/0.45 = zero_zero_int ) ) ).
% 0.17/0.45
% 0.17/0.45 % cong_exp_iff_simps(2)
% 0.17/0.45 thf(fact_2872_cong__exp__iff__simps_I2_J,axiom,
% 0.17/0.45 ! [N: num,Q2: num] :
% 0.17/0.45 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.17/0.45 = zero_z3403309356797280102nteger )
% 0.17/0.45 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.17/0.45 = zero_z3403309356797280102nteger ) ) ).
% 0.17/0.45
% 0.17/0.45 % cong_exp_iff_simps(2)
% 0.17/0.45 thf(fact_2873_cong__exp__iff__simps_I1_J,axiom,
% 0.17/0.45 ! [N: num] :
% 0.17/0.45 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 0.17/0.45 = zero_zero_nat ) ).
% 0.17/0.45
% 0.17/0.45 % cong_exp_iff_simps(1)
% 0.17/0.45 thf(fact_2874_cong__exp__iff__simps_I1_J,axiom,
% 0.17/0.45 ! [N: num] :
% 0.17/0.45 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 0.17/0.45 = zero_zero_int ) ).
% 0.17/0.45
% 0.17/0.45 % cong_exp_iff_simps(1)
% 0.17/0.45 thf(fact_2875_cong__exp__iff__simps_I1_J,axiom,
% 0.17/0.45 ! [N: num] :
% 0.17/0.45 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 0.17/0.45 = zero_z3403309356797280102nteger ) ).
% 0.17/0.45
% 0.17/0.45 % cong_exp_iff_simps(1)
% 0.17/0.45 thf(fact_2876_numeral__1__eq__Suc__0,axiom,
% 0.17/0.45 ( ( numeral_numeral_nat @ one )
% 0.17/0.45 = ( suc @ zero_zero_nat ) ) ).
% 0.17/0.45
% 0.17/0.45 % numeral_1_eq_Suc_0
% 0.17/0.45 thf(fact_2877_ex__least__nat__less,axiom,
% 0.17/0.45 ! [P: nat > $o,N: nat] :
% 0.17/0.45 ( ( P @ N )
% 0.17/0.45 => ( ~ ( P @ zero_zero_nat )
% 0.17/0.45 => ? [K2: nat] :
% 0.17/0.45 ( ( ord_less_nat @ K2 @ N )
% 0.17/0.45 & ! [I4: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ I4 @ K2 )
% 0.17/0.45 => ~ ( P @ I4 ) )
% 0.17/0.45 & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % ex_least_nat_less
% 0.17/0.45 thf(fact_2878_diff__Suc__less,axiom,
% 0.17/0.45 ! [N: nat,I: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.45 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 0.17/0.45
% 0.17/0.45 % diff_Suc_less
% 0.17/0.45 thf(fact_2879_n__less__n__mult__m,axiom,
% 0.17/0.45 ! [N: nat,M: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.45 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.17/0.45 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % n_less_n_mult_m
% 0.17/0.45 thf(fact_2880_n__less__m__mult__n,axiom,
% 0.17/0.45 ! [N: nat,M: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.45 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.17/0.45 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % n_less_m_mult_n
% 0.17/0.45 thf(fact_2881_one__less__mult,axiom,
% 0.17/0.45 ! [N: nat,M: nat] :
% 0.17/0.45 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.17/0.45 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.17/0.45 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % one_less_mult
% 0.17/0.45 thf(fact_2882_realpow__pos__nth2,axiom,
% 0.17/0.45 ! [A: real,N: nat] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ A )
% 0.17/0.45 => ? [R3: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.17/0.45 & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 0.17/0.45 = A ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % realpow_pos_nth2
% 0.17/0.45 thf(fact_2883_nat__induct__non__zero,axiom,
% 0.17/0.45 ! [N: nat,P: nat > $o] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.45 => ( ( P @ one_one_nat )
% 0.17/0.45 => ( ! [N4: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.17/0.45 => ( ( P @ N4 )
% 0.17/0.45 => ( P @ ( suc @ N4 ) ) ) )
% 0.17/0.45 => ( P @ N ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_induct_non_zero
% 0.17/0.45 thf(fact_2884_nat__mult__le__cancel1,axiom,
% 0.17/0.45 ! [K: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.45 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.17/0.45 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_mult_le_cancel1
% 0.17/0.45 thf(fact_2885_div__greater__zero__iff,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 0.17/0.45 = ( ( ord_less_eq_nat @ N @ M )
% 0.17/0.45 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_greater_zero_iff
% 0.17/0.45 thf(fact_2886_div__le__mono2,axiom,
% 0.17/0.45 ! [M: nat,N: nat,K: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.17/0.45 => ( ( ord_less_eq_nat @ M @ N )
% 0.17/0.45 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_le_mono2
% 0.17/0.45 thf(fact_2887_power__gt__expt,axiom,
% 0.17/0.45 ! [N: nat,K: nat] :
% 0.17/0.45 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.17/0.45 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_gt_expt
% 0.17/0.45 thf(fact_2888_nat__diff__split,axiom,
% 0.17/0.45 ! [P: nat > $o,A: nat,B2: nat] :
% 0.17/0.45 ( ( P @ ( minus_minus_nat @ A @ B2 ) )
% 0.17/0.45 = ( ( ( ord_less_nat @ A @ B2 )
% 0.17/0.45 => ( P @ zero_zero_nat ) )
% 0.17/0.45 & ! [D2: nat] :
% 0.17/0.45 ( ( A
% 0.17/0.45 = ( plus_plus_nat @ B2 @ D2 ) )
% 0.17/0.45 => ( P @ D2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_diff_split
% 0.17/0.45 thf(fact_2889_nat__diff__split__asm,axiom,
% 0.17/0.45 ! [P: nat > $o,A: nat,B2: nat] :
% 0.17/0.45 ( ( P @ ( minus_minus_nat @ A @ B2 ) )
% 0.17/0.45 = ( ~ ( ( ( ord_less_nat @ A @ B2 )
% 0.17/0.45 & ~ ( P @ zero_zero_nat ) )
% 0.17/0.45 | ? [D2: nat] :
% 0.17/0.45 ( ( A
% 0.17/0.45 = ( plus_plus_nat @ B2 @ D2 ) )
% 0.17/0.45 & ~ ( P @ D2 ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_diff_split_asm
% 0.17/0.45 thf(fact_2890_nat__one__le__power,axiom,
% 0.17/0.45 ! [I: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 0.17/0.45 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_one_le_power
% 0.17/0.45 thf(fact_2891_real__arch__pow__inv,axiom,
% 0.17/0.45 ! [Y3: real,X4: real] :
% 0.17/0.45 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.17/0.45 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.17/0.45 => ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X4 @ N4 ) @ Y3 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % real_arch_pow_inv
% 0.17/0.45 thf(fact_2892_div__less__iff__less__mult,axiom,
% 0.17/0.45 ! [Q2: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.17/0.45 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
% 0.17/0.45 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_less_iff_less_mult
% 0.17/0.45 thf(fact_2893_nat__mult__div__cancel1,axiom,
% 0.17/0.45 ! [K: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.17/0.45 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.17/0.45 = ( divide_divide_nat @ M @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % nat_mult_div_cancel1
% 0.17/0.45 thf(fact_2894_div__eq__dividend__iff,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.17/0.45 => ( ( ( divide_divide_nat @ M @ N )
% 0.17/0.45 = M )
% 0.17/0.45 = ( N = one_one_nat ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_eq_dividend_iff
% 0.17/0.45 thf(fact_2895_div__less__dividend,axiom,
% 0.17/0.45 ! [N: nat,M: nat] :
% 0.17/0.45 ( ( ord_less_nat @ one_one_nat @ N )
% 0.17/0.45 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.17/0.45 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_less_dividend
% 0.17/0.45 thf(fact_2896_mod__le__divisor,axiom,
% 0.17/0.45 ! [N: nat,M: nat] :
% 0.17/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.45 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod_le_divisor
% 0.17/0.45 thf(fact_2897_div__less__mono,axiom,
% 0.17/0.45 ! [A3: nat,B4: nat,N: nat] :
% 0.17/0.45 ( ( ord_less_nat @ A3 @ B4 )
% 0.17/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.17/0.45 => ( ( ( modulo_modulo_nat @ A3 @ N )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 => ( ( ( modulo_modulo_nat @ B4 @ N )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N ) @ ( divide_divide_nat @ B4 @ N ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % div_less_mono
% 0.17/0.45 thf(fact_2898_evenE,axiom,
% 0.17/0.45 ! [A: code_integer] :
% 0.17/0.45 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ~ ! [B5: code_integer] :
% 0.17/0.45 ( A
% 0.17/0.45 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % evenE
% 0.17/0.45 thf(fact_2899_evenE,axiom,
% 0.17/0.45 ! [A: nat] :
% 0.17/0.45 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ~ ! [B5: nat] :
% 0.17/0.45 ( A
% 0.17/0.45 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % evenE
% 0.17/0.45 thf(fact_2900_evenE,axiom,
% 0.17/0.45 ! [A: int] :
% 0.17/0.45 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ~ ! [B5: int] :
% 0.17/0.45 ( A
% 0.17/0.45 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % evenE
% 0.17/0.45 thf(fact_2901_odd__one,axiom,
% 0.17/0.45 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 0.17/0.45
% 0.17/0.45 % odd_one
% 0.17/0.45 thf(fact_2902_odd__one,axiom,
% 0.17/0.45 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 0.17/0.45
% 0.17/0.45 % odd_one
% 0.17/0.45 thf(fact_2903_odd__one,axiom,
% 0.17/0.45 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 0.17/0.45
% 0.17/0.45 % odd_one
% 0.17/0.45 thf(fact_2904_odd__even__add,axiom,
% 0.17/0.45 ! [A: code_integer,B2: code_integer] :
% 0.17/0.45 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 )
% 0.17/0.45 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % odd_even_add
% 0.17/0.45 thf(fact_2905_odd__even__add,axiom,
% 0.17/0.45 ! [A: nat,B2: nat] :
% 0.17/0.45 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.17/0.45 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % odd_even_add
% 0.17/0.45 thf(fact_2906_odd__even__add,axiom,
% 0.17/0.45 ! [A: int,B2: int] :
% 0.17/0.45 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.17/0.45 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 )
% 0.17/0.45 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % odd_even_add
% 0.17/0.45 thf(fact_2907_bit__eq__rec,axiom,
% 0.17/0.45 ( ( ^ [Y6: code_integer,Z4: code_integer] : ( Y6 = Z4 ) )
% 0.17/0.45 = ( ^ [A4: code_integer,B3: code_integer] :
% 0.17/0.45 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 )
% 0.17/0.45 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) )
% 0.17/0.45 & ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.17/0.45 = ( divide6298287555418463151nteger @ B3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % bit_eq_rec
% 0.17/0.45 thf(fact_2908_bit__eq__rec,axiom,
% 0.17/0.45 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.17/0.45 = ( ^ [A4: nat,B3: nat] :
% 0.17/0.45 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 )
% 0.17/0.45 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) )
% 0.17/0.45 & ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.17/0.45 = ( divide_divide_nat @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % bit_eq_rec
% 0.17/0.45 thf(fact_2909_bit__eq__rec,axiom,
% 0.17/0.45 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.17/0.45 = ( ^ [A4: int,B3: int] :
% 0.17/0.45 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 )
% 0.17/0.45 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) )
% 0.17/0.45 & ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.17/0.45 = ( divide_divide_int @ B3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % bit_eq_rec
% 0.17/0.45 thf(fact_2910_even__mask__div__iff,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( 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 ) ) )
% 0.17/0.45 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 = zero_z3403309356797280102nteger )
% 0.17/0.45 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % even_mask_div_iff
% 0.17/0.45 thf(fact_2911_even__mask__div__iff,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( 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 ) ) )
% 0.17/0.45 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 = zero_zero_nat )
% 0.17/0.45 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % even_mask_div_iff
% 0.17/0.45 thf(fact_2912_even__mask__div__iff,axiom,
% 0.17/0.45 ! [M: nat,N: nat] :
% 0.17/0.45 ( ( 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 ) ) )
% 0.17/0.45 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.17/0.45 = zero_zero_int )
% 0.17/0.45 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % even_mask_div_iff
% 0.17/0.45 thf(fact_2913_dvd__minus__add,axiom,
% 0.17/0.45 ! [Q2: nat,N: nat,R2: nat,M: nat] :
% 0.17/0.45 ( ( ord_less_eq_nat @ Q2 @ N )
% 0.17/0.45 => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M ) )
% 0.17/0.45 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
% 0.17/0.45 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % dvd_minus_add
% 0.17/0.45 thf(fact_2914_power__dvd__imp__le,axiom,
% 0.17/0.45 ! [I: nat,M: nat,N: nat] :
% 0.17/0.45 ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.17/0.45 => ( ( ord_less_nat @ one_one_nat @ I )
% 0.17/0.45 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % power_dvd_imp_le
% 0.17/0.45 thf(fact_2915_mod__nat__eqI,axiom,
% 0.17/0.45 ! [R2: nat,N: nat,M: nat] :
% 0.17/0.45 ( ( ord_less_nat @ R2 @ N )
% 0.17/0.45 => ( ( ord_less_eq_nat @ R2 @ M )
% 0.17/0.45 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 0.17/0.45 => ( ( modulo_modulo_nat @ M @ N )
% 0.17/0.45 = R2 ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mod_nat_eqI
% 0.17/0.45 thf(fact_2916_mult__less__cancel__right2,axiom,
% 0.17/0.45 ! [A: real,C: real] :
% 0.17/0.45 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 0.17/0.45 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ A @ one_one_real ) )
% 0.17/0.45 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_right2
% 0.17/0.45 thf(fact_2917_mult__less__cancel__right2,axiom,
% 0.17/0.45 ! [A: rat,C: rat] :
% 0.17/0.45 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.17/0.45 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.17/0.45 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_right2
% 0.17/0.45 thf(fact_2918_mult__less__cancel__right2,axiom,
% 0.17/0.45 ! [A: int,C: int] :
% 0.17/0.45 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 0.17/0.45 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_int @ A @ one_one_int ) )
% 0.17/0.45 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.17/0.45 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_right2
% 0.17/0.45 thf(fact_2919_mult__less__cancel__right1,axiom,
% 0.17/0.45 ! [C: real,B2: real] :
% 0.17/0.45 ( ( ord_less_real @ C @ ( times_times_real @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ one_one_real @ B2 ) )
% 0.17/0.45 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_right1
% 0.17/0.45 thf(fact_2920_mult__less__cancel__right1,axiom,
% 0.17/0.45 ! [C: rat,B2: rat] :
% 0.17/0.45 ( ( ord_less_rat @ C @ ( times_times_rat @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.17/0.45 => ( ord_less_rat @ one_one_rat @ B2 ) )
% 0.17/0.45 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.17/0.45 => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_right1
% 0.17/0.45 thf(fact_2921_mult__less__cancel__right1,axiom,
% 0.17/0.45 ! [C: int,B2: int] :
% 0.17/0.45 ( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
% 0.17/0.45 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.17/0.45 => ( ord_less_int @ one_one_int @ B2 ) )
% 0.17/0.45 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.17/0.45 => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_right1
% 0.17/0.45 thf(fact_2922_mult__less__cancel__left2,axiom,
% 0.17/0.45 ! [C: real,A: real] :
% 0.17/0.45 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 0.17/0.45 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.17/0.45 => ( ord_less_real @ A @ one_one_real ) )
% 0.17/0.45 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.17/0.45 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 % mult_less_cancel_left2
% 0.17/0.45 thf(fact_2923_mult__less__cancel__left2,axiom,
% 0.17/0.45 ! [C: rat,A: rat] :
% 0.17/0.45 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.17/0.45 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.23/0.45 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_less_cancel_left2
% 0.23/0.45 thf(fact_2924_mult__less__cancel__left2,axiom,
% 0.23/0.45 ! [C: int,A: int] :
% 0.23/0.45 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 0.23/0.45 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.23/0.45 => ( ord_less_int @ A @ one_one_int ) )
% 0.23/0.45 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.23/0.45 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_less_cancel_left2
% 0.23/0.45 thf(fact_2925_mult__less__cancel__left1,axiom,
% 0.23/0.45 ! [C: real,B2: real] :
% 0.23/0.45 ( ( ord_less_real @ C @ ( times_times_real @ C @ B2 ) )
% 0.23/0.45 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ord_less_real @ one_one_real @ B2 ) )
% 0.23/0.45 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_less_cancel_left1
% 0.23/0.45 thf(fact_2926_mult__less__cancel__left1,axiom,
% 0.23/0.45 ! [C: rat,B2: rat] :
% 0.23/0.45 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B2 ) )
% 0.23/0.45 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ord_less_rat @ one_one_rat @ B2 ) )
% 0.23/0.45 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_less_cancel_left1
% 0.23/0.45 thf(fact_2927_mult__less__cancel__left1,axiom,
% 0.23/0.45 ! [C: int,B2: int] :
% 0.23/0.45 ( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
% 0.23/0.45 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.23/0.45 => ( ord_less_int @ one_one_int @ B2 ) )
% 0.23/0.45 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.23/0.45 => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_less_cancel_left1
% 0.23/0.45 thf(fact_2928_mult__le__cancel__right2,axiom,
% 0.23/0.45 ! [A: real,C: real] :
% 0.23/0.45 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 0.23/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.23/0.45 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_right2
% 0.23/0.45 thf(fact_2929_mult__le__cancel__right2,axiom,
% 0.23/0.45 ! [A: rat,C: rat] :
% 0.23/0.45 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.23/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.23/0.45 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_right2
% 0.23/0.45 thf(fact_2930_mult__le__cancel__right2,axiom,
% 0.23/0.45 ! [A: int,C: int] :
% 0.23/0.45 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 0.23/0.45 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.23/0.45 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.23/0.45 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.23/0.45 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_right2
% 0.23/0.45 thf(fact_2931_mult__le__cancel__right1,axiom,
% 0.23/0.45 ! [C: real,B2: real] :
% 0.23/0.45 ( ( ord_less_eq_real @ C @ ( times_times_real @ B2 @ C ) )
% 0.23/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ord_less_eq_real @ one_one_real @ B2 ) )
% 0.23/0.45 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_right1
% 0.23/0.45 thf(fact_2932_mult__le__cancel__right1,axiom,
% 0.23/0.45 ! [C: rat,B2: rat] :
% 0.23/0.45 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B2 @ C ) )
% 0.23/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
% 0.23/0.45 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_right1
% 0.23/0.45 thf(fact_2933_mult__le__cancel__right1,axiom,
% 0.23/0.45 ! [C: int,B2: int] :
% 0.23/0.45 ( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
% 0.23/0.45 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.23/0.45 => ( ord_less_eq_int @ one_one_int @ B2 ) )
% 0.23/0.45 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.23/0.45 => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_right1
% 0.23/0.45 thf(fact_2934_mult__le__cancel__left2,axiom,
% 0.23/0.45 ! [C: real,A: real] :
% 0.23/0.45 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 0.23/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.23/0.45 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_left2
% 0.23/0.45 thf(fact_2935_mult__le__cancel__left2,axiom,
% 0.23/0.45 ! [C: rat,A: rat] :
% 0.23/0.45 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.23/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.23/0.45 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_left2
% 0.23/0.45 thf(fact_2936_mult__le__cancel__left2,axiom,
% 0.23/0.45 ! [C: int,A: int] :
% 0.23/0.45 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 0.23/0.45 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.23/0.45 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.23/0.45 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.23/0.45 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_left2
% 0.23/0.45 thf(fact_2937_mult__le__cancel__left1,axiom,
% 0.23/0.45 ! [C: real,B2: real] :
% 0.23/0.45 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B2 ) )
% 0.23/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ord_less_eq_real @ one_one_real @ B2 ) )
% 0.23/0.45 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_left1
% 0.23/0.45 thf(fact_2938_mult__le__cancel__left1,axiom,
% 0.23/0.45 ! [C: rat,B2: rat] :
% 0.23/0.45 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B2 ) )
% 0.23/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
% 0.23/0.45 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_left1
% 0.23/0.45 thf(fact_2939_mult__le__cancel__left1,axiom,
% 0.23/0.45 ! [C: int,B2: int] :
% 0.23/0.45 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
% 0.23/0.45 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.23/0.45 => ( ord_less_eq_int @ one_one_int @ B2 ) )
% 0.23/0.45 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.23/0.45 => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_le_cancel_left1
% 0.23/0.45 thf(fact_2940_field__le__mult__one__interval,axiom,
% 0.23/0.45 ! [X4: real,Y3: real] :
% 0.23/0.45 ( ! [Z2: real] :
% 0.23/0.45 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.23/0.45 => ( ( ord_less_real @ Z2 @ one_one_real )
% 0.23/0.45 => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X4 ) @ Y3 ) ) )
% 0.23/0.45 => ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 0.23/0.45
% 0.23/0.45 % field_le_mult_one_interval
% 0.23/0.45 thf(fact_2941_field__le__mult__one__interval,axiom,
% 0.23/0.45 ! [X4: rat,Y3: rat] :
% 0.23/0.45 ( ! [Z2: rat] :
% 0.23/0.45 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.23/0.45 => ( ( ord_less_rat @ Z2 @ one_one_rat )
% 0.23/0.45 => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X4 ) @ Y3 ) ) )
% 0.23/0.45 => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).
% 0.23/0.45
% 0.23/0.45 % field_le_mult_one_interval
% 0.23/0.45 thf(fact_2942_divide__left__mono__neg,axiom,
% 0.23/0.45 ! [A: real,B2: real,C: real] :
% 0.23/0.45 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.45 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.23/0.45 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % divide_left_mono_neg
% 0.23/0.45 thf(fact_2943_divide__left__mono__neg,axiom,
% 0.23/0.45 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.45 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.45 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.23/0.45 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % divide_left_mono_neg
% 0.23/0.45 thf(fact_2944_mult__imp__le__div__pos,axiom,
% 0.23/0.45 ! [Y3: real,Z: real,X4: real] :
% 0.23/0.45 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.45 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X4 )
% 0.23/0.45 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_imp_le_div_pos
% 0.23/0.45 thf(fact_2945_mult__imp__le__div__pos,axiom,
% 0.23/0.45 ! [Y3: rat,Z: rat,X4: rat] :
% 0.23/0.45 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.23/0.45 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X4 )
% 0.23/0.45 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_imp_le_div_pos
% 0.23/0.45 thf(fact_2946_mult__imp__div__pos__le,axiom,
% 0.23/0.45 ! [Y3: real,X4: real,Z: real] :
% 0.23/0.45 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.45 => ( ( ord_less_eq_real @ X4 @ ( times_times_real @ Z @ Y3 ) )
% 0.23/0.45 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ Z ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_imp_div_pos_le
% 0.23/0.45 thf(fact_2947_mult__imp__div__pos__le,axiom,
% 0.23/0.45 ! [Y3: rat,X4: rat,Z: rat] :
% 0.23/0.45 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.23/0.45 => ( ( ord_less_eq_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) )
% 0.23/0.45 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ Z ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % mult_imp_div_pos_le
% 0.23/0.45 thf(fact_2948_pos__le__divide__eq,axiom,
% 0.23/0.45 ! [C: real,A: real,B2: real] :
% 0.23/0.45 ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.23/0.45 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % pos_le_divide_eq
% 0.23/0.45 thf(fact_2949_pos__le__divide__eq,axiom,
% 0.23/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.23/0.45 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.23/0.45 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % pos_le_divide_eq
% 0.23/0.45 thf(fact_2950_pos__divide__le__eq,axiom,
% 0.23/0.45 ! [C: real,B2: real,A: real] :
% 0.23/0.45 ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.23/0.45 = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % pos_divide_le_eq
% 0.23/0.45 thf(fact_2951_pos__divide__le__eq,axiom,
% 0.23/0.45 ! [C: rat,B2: rat,A: rat] :
% 0.23/0.45 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.23/0.45 = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % pos_divide_le_eq
% 0.23/0.45 thf(fact_2952_neg__le__divide__eq,axiom,
% 0.23/0.45 ! [C: real,A: real,B2: real] :
% 0.23/0.45 ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.23/0.45 = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % neg_le_divide_eq
% 0.23/0.45 thf(fact_2953_neg__le__divide__eq,axiom,
% 0.23/0.45 ! [C: rat,A: rat,B2: rat] :
% 0.23/0.45 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.23/0.45 = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % neg_le_divide_eq
% 0.23/0.45 thf(fact_2954_neg__divide__le__eq,axiom,
% 0.23/0.45 ! [C: real,B2: real,A: real] :
% 0.23/0.45 ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.23/0.45 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % neg_divide_le_eq
% 0.23/0.45 thf(fact_2955_neg__divide__le__eq,axiom,
% 0.23/0.45 ! [C: rat,B2: rat,A: rat] :
% 0.23/0.45 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.23/0.45 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % neg_divide_le_eq
% 0.23/0.45 thf(fact_2956_divide__left__mono,axiom,
% 0.23/0.45 ! [B2: real,A: real,C: real] :
% 0.23/0.45 ( ( ord_less_eq_real @ B2 @ A )
% 0.23/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.23/0.45 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % divide_left_mono
% 0.23/0.45 thf(fact_2957_divide__left__mono,axiom,
% 0.23/0.45 ! [B2: rat,A: rat,C: rat] :
% 0.23/0.45 ( ( ord_less_eq_rat @ B2 @ A )
% 0.23/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.23/0.45 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % divide_left_mono
% 0.23/0.45 thf(fact_2958_le__divide__eq,axiom,
% 0.23/0.45 ! [A: real,B2: real,C: real] :
% 0.23/0.45 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.23/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 0.23/0.45 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 0.23/0.45 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % le_divide_eq
% 0.23/0.45 thf(fact_2959_le__divide__eq,axiom,
% 0.23/0.45 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.45 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.23/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 0.23/0.45 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 0.23/0.45 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % le_divide_eq
% 0.23/0.45 thf(fact_2960_divide__le__eq,axiom,
% 0.23/0.45 ! [B2: real,C: real,A: real] :
% 0.23/0.45 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.23/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 0.23/0.45 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 0.23/0.45 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % divide_le_eq
% 0.23/0.45 thf(fact_2961_divide__le__eq,axiom,
% 0.23/0.45 ! [B2: rat,C: rat,A: rat] :
% 0.23/0.45 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.23/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 0.23/0.45 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 0.23/0.45 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % divide_le_eq
% 0.23/0.45 thf(fact_2962_le__divide__eq__1,axiom,
% 0.23/0.45 ! [B2: real,A: real] :
% 0.23/0.45 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.23/0.45 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.45 & ( ord_less_eq_real @ A @ B2 ) )
% 0.23/0.45 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.23/0.45 & ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % le_divide_eq_1
% 0.23/0.45 thf(fact_2963_le__divide__eq__1,axiom,
% 0.23/0.45 ! [B2: rat,A: rat] :
% 0.23/0.45 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.23/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.45 & ( ord_less_eq_rat @ A @ B2 ) )
% 0.23/0.45 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.23/0.45 & ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % le_divide_eq_1
% 0.23/0.45 thf(fact_2964_divide__le__eq__1,axiom,
% 0.23/0.45 ! [B2: real,A: real] :
% 0.23/0.45 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.23/0.45 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.45 & ( ord_less_eq_real @ B2 @ A ) )
% 0.23/0.45 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.23/0.45 & ( ord_less_eq_real @ A @ B2 ) )
% 0.23/0.45 | ( A = zero_zero_real ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % divide_le_eq_1
% 0.23/0.45 thf(fact_2965_divide__le__eq__1,axiom,
% 0.23/0.45 ! [B2: rat,A: rat] :
% 0.23/0.45 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.23/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.45 & ( ord_less_eq_rat @ B2 @ A ) )
% 0.23/0.45 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.23/0.45 & ( ord_less_eq_rat @ A @ B2 ) )
% 0.23/0.45 | ( A = zero_zero_rat ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % divide_le_eq_1
% 0.23/0.45 thf(fact_2966_convex__bound__le,axiom,
% 0.23/0.45 ! [X4: real,A: real,Y3: real,U: real,V: real] :
% 0.23/0.45 ( ( ord_less_eq_real @ X4 @ A )
% 0.23/0.45 => ( ( ord_less_eq_real @ Y3 @ A )
% 0.23/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.23/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.23/0.45 => ( ( ( plus_plus_real @ U @ V )
% 0.23/0.45 = one_one_real )
% 0.23/0.45 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % convex_bound_le
% 0.23/0.45 thf(fact_2967_convex__bound__le,axiom,
% 0.23/0.45 ! [X4: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 0.23/0.45 ( ( ord_less_eq_rat @ X4 @ A )
% 0.23/0.45 => ( ( ord_less_eq_rat @ Y3 @ A )
% 0.23/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.23/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.23/0.45 => ( ( ( plus_plus_rat @ U @ V )
% 0.23/0.45 = one_one_rat )
% 0.23/0.45 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X4 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % convex_bound_le
% 0.23/0.45 thf(fact_2968_convex__bound__le,axiom,
% 0.23/0.45 ! [X4: int,A: int,Y3: int,U: int,V: int] :
% 0.23/0.45 ( ( ord_less_eq_int @ X4 @ A )
% 0.23/0.45 => ( ( ord_less_eq_int @ Y3 @ A )
% 0.23/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.23/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.23/0.45 => ( ( ( plus_plus_int @ U @ V )
% 0.23/0.45 = one_one_int )
% 0.23/0.45 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % convex_bound_le
% 0.23/0.45 thf(fact_2969_divide__less__eq__numeral_I1_J,axiom,
% 0.23/0.45 ! [B2: real,C: real,W: num] :
% 0.23/0.45 ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
% 0.23/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.23/0.45 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 0.23/0.45 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % divide_less_eq_numeral(1)
% 0.23/0.45 thf(fact_2970_divide__less__eq__numeral_I1_J,axiom,
% 0.23/0.45 ! [B2: rat,C: rat,W: num] :
% 0.23/0.45 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.23/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.23/0.45 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 0.23/0.45 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % divide_less_eq_numeral(1)
% 0.23/0.45 thf(fact_2971_less__divide__eq__numeral_I1_J,axiom,
% 0.23/0.45 ! [W: num,B2: real,C: real] :
% 0.23/0.45 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
% 0.23/0.45 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 0.23/0.45 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.45 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.23/0.45 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.45 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % less_divide_eq_numeral(1)
% 0.23/0.45 thf(fact_2972_less__divide__eq__numeral_I1_J,axiom,
% 0.23/0.45 ! [W: num,B2: rat,C: rat] :
% 0.23/0.45 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.23/0.45 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 0.23/0.45 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.45 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.23/0.45 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.45 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % less_divide_eq_numeral(1)
% 0.23/0.45 thf(fact_2973_frac__le__eq,axiom,
% 0.23/0.45 ! [Y3: real,Z: real,X4: real,W: real] :
% 0.23/0.45 ( ( Y3 != zero_zero_real )
% 0.23/0.45 => ( ( Z != zero_zero_real )
% 0.23/0.45 => ( ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.23/0.45 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.23/0.45
% 0.23/0.45 % frac_le_eq
% 0.23/0.45 thf(fact_2974_frac__le__eq,axiom,
% 0.23/0.45 ! [Y3: rat,Z: rat,X4: rat,W: rat] :
% 0.23/0.45 ( ( Y3 != zero_zero_rat )
% 0.23/0.45 => ( ( Z != zero_zero_rat )
% 0.23/0.46 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.23/0.46 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % frac_le_eq
% 0.23/0.46 thf(fact_2975_frac__less__eq,axiom,
% 0.23/0.46 ! [Y3: real,Z: real,X4: real,W: real] :
% 0.23/0.46 ( ( Y3 != zero_zero_real )
% 0.23/0.46 => ( ( Z != zero_zero_real )
% 0.23/0.46 => ( ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.23/0.46 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % frac_less_eq
% 0.23/0.46 thf(fact_2976_frac__less__eq,axiom,
% 0.23/0.46 ! [Y3: rat,Z: rat,X4: rat,W: rat] :
% 0.23/0.46 ( ( Y3 != zero_zero_rat )
% 0.23/0.46 => ( ( Z != zero_zero_rat )
% 0.23/0.46 => ( ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.23/0.46 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % frac_less_eq
% 0.23/0.46 thf(fact_2977_power__Suc__less,axiom,
% 0.23/0.46 ! [A: real,N: nat] :
% 0.23/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.46 => ( ( ord_less_real @ A @ one_one_real )
% 0.23/0.46 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_less
% 0.23/0.46 thf(fact_2978_power__Suc__less,axiom,
% 0.23/0.46 ! [A: rat,N: nat] :
% 0.23/0.46 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.46 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.23/0.46 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_less
% 0.23/0.46 thf(fact_2979_power__Suc__less,axiom,
% 0.23/0.46 ! [A: nat,N: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.23/0.46 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.23/0.46 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_less
% 0.23/0.46 thf(fact_2980_power__Suc__less,axiom,
% 0.23/0.46 ! [A: int,N: nat] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ A )
% 0.23/0.46 => ( ( ord_less_int @ A @ one_one_int )
% 0.23/0.46 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_less
% 0.23/0.46 thf(fact_2981_power__Suc__le__self,axiom,
% 0.23/0.46 ! [A: real,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.46 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.23/0.46 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_le_self
% 0.23/0.46 thf(fact_2982_power__Suc__le__self,axiom,
% 0.23/0.46 ! [A: rat,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.46 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.23/0.46 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_le_self
% 0.23/0.46 thf(fact_2983_power__Suc__le__self,axiom,
% 0.23/0.46 ! [A: nat,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.23/0.46 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.23/0.46 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_le_self
% 0.23/0.46 thf(fact_2984_power__Suc__le__self,axiom,
% 0.23/0.46 ! [A: int,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.23/0.46 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.23/0.46 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_le_self
% 0.23/0.46 thf(fact_2985_power__Suc__less__one,axiom,
% 0.23/0.46 ! [A: real,N: nat] :
% 0.23/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.46 => ( ( ord_less_real @ A @ one_one_real )
% 0.23/0.46 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_less_one
% 0.23/0.46 thf(fact_2986_power__Suc__less__one,axiom,
% 0.23/0.46 ! [A: rat,N: nat] :
% 0.23/0.46 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.46 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.23/0.46 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_less_one
% 0.23/0.46 thf(fact_2987_power__Suc__less__one,axiom,
% 0.23/0.46 ! [A: nat,N: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.23/0.46 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.23/0.46 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_less_one
% 0.23/0.46 thf(fact_2988_power__Suc__less__one,axiom,
% 0.23/0.46 ! [A: int,N: nat] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ A )
% 0.23/0.46 => ( ( ord_less_int @ A @ one_one_int )
% 0.23/0.46 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_Suc_less_one
% 0.23/0.46 thf(fact_2989_power__strict__decreasing,axiom,
% 0.23/0.46 ! [N: nat,N3: nat,A: real] :
% 0.23/0.46 ( ( ord_less_nat @ N @ N3 )
% 0.23/0.46 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.46 => ( ( ord_less_real @ A @ one_one_real )
% 0.23/0.46 => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_strict_decreasing
% 0.23/0.46 thf(fact_2990_power__strict__decreasing,axiom,
% 0.23/0.46 ! [N: nat,N3: nat,A: rat] :
% 0.23/0.46 ( ( ord_less_nat @ N @ N3 )
% 0.23/0.46 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.46 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.23/0.46 => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_strict_decreasing
% 0.23/0.46 thf(fact_2991_power__strict__decreasing,axiom,
% 0.23/0.46 ! [N: nat,N3: nat,A: nat] :
% 0.23/0.46 ( ( ord_less_nat @ N @ N3 )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.23/0.46 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.23/0.46 => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_strict_decreasing
% 0.23/0.46 thf(fact_2992_power__strict__decreasing,axiom,
% 0.23/0.46 ! [N: nat,N3: nat,A: int] :
% 0.23/0.46 ( ( ord_less_nat @ N @ N3 )
% 0.23/0.46 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.23/0.46 => ( ( ord_less_int @ A @ one_one_int )
% 0.23/0.46 => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_strict_decreasing
% 0.23/0.46 thf(fact_2993_power__decreasing,axiom,
% 0.23/0.46 ! [N: nat,N3: nat,A: real] :
% 0.23/0.46 ( ( ord_less_eq_nat @ N @ N3 )
% 0.23/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.46 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.23/0.46 => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_decreasing
% 0.23/0.46 thf(fact_2994_power__decreasing,axiom,
% 0.23/0.46 ! [N: nat,N3: nat,A: rat] :
% 0.23/0.46 ( ( ord_less_eq_nat @ N @ N3 )
% 0.23/0.46 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.46 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.23/0.46 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_decreasing
% 0.23/0.46 thf(fact_2995_power__decreasing,axiom,
% 0.23/0.46 ! [N: nat,N3: nat,A: nat] :
% 0.23/0.46 ( ( ord_less_eq_nat @ N @ N3 )
% 0.23/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.23/0.46 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.23/0.46 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_decreasing
% 0.23/0.46 thf(fact_2996_power__decreasing,axiom,
% 0.23/0.46 ! [N: nat,N3: nat,A: int] :
% 0.23/0.46 ( ( ord_less_eq_nat @ N @ N3 )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.23/0.46 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.23/0.46 => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_decreasing
% 0.23/0.46 thf(fact_2997_zero__power2,axiom,
% 0.23/0.46 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = zero_zero_rat ) ).
% 0.23/0.46
% 0.23/0.46 % zero_power2
% 0.23/0.46 thf(fact_2998_zero__power2,axiom,
% 0.23/0.46 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = zero_zero_nat ) ).
% 0.23/0.46
% 0.23/0.46 % zero_power2
% 0.23/0.46 thf(fact_2999_zero__power2,axiom,
% 0.23/0.46 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = zero_zero_real ) ).
% 0.23/0.46
% 0.23/0.46 % zero_power2
% 0.23/0.46 thf(fact_3000_zero__power2,axiom,
% 0.23/0.46 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = zero_zero_int ) ).
% 0.23/0.46
% 0.23/0.46 % zero_power2
% 0.23/0.46 thf(fact_3001_zero__power2,axiom,
% 0.23/0.46 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = zero_zero_complex ) ).
% 0.23/0.46
% 0.23/0.46 % zero_power2
% 0.23/0.46 thf(fact_3002_self__le__power,axiom,
% 0.23/0.46 ! [A: real,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % self_le_power
% 0.23/0.46 thf(fact_3003_self__le__power,axiom,
% 0.23/0.46 ! [A: rat,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % self_le_power
% 0.23/0.46 thf(fact_3004_self__le__power,axiom,
% 0.23/0.46 ! [A: nat,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % self_le_power
% 0.23/0.46 thf(fact_3005_self__le__power,axiom,
% 0.23/0.46 ! [A: int,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % self_le_power
% 0.23/0.46 thf(fact_3006_one__less__power,axiom,
% 0.23/0.46 ! [A: real,N: nat] :
% 0.23/0.46 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % one_less_power
% 0.23/0.46 thf(fact_3007_one__less__power,axiom,
% 0.23/0.46 ! [A: rat,N: nat] :
% 0.23/0.46 ( ( ord_less_rat @ one_one_rat @ A )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % one_less_power
% 0.23/0.46 thf(fact_3008_one__less__power,axiom,
% 0.23/0.46 ! [A: nat,N: nat] :
% 0.23/0.46 ( ( ord_less_nat @ one_one_nat @ A )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % one_less_power
% 0.23/0.46 thf(fact_3009_one__less__power,axiom,
% 0.23/0.46 ! [A: int,N: nat] :
% 0.23/0.46 ( ( ord_less_int @ one_one_int @ A )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % one_less_power
% 0.23/0.46 thf(fact_3010_numeral__2__eq__2,axiom,
% 0.23/0.46 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.23/0.46 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % numeral_2_eq_2
% 0.23/0.46 thf(fact_3011_pos2,axiom,
% 0.23/0.46 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 0.23/0.46
% 0.23/0.46 % pos2
% 0.23/0.46 thf(fact_3012_power__diff,axiom,
% 0.23/0.46 ! [A: complex,N: nat,M: nat] :
% 0.23/0.46 ( ( A != zero_zero_complex )
% 0.23/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.46 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 0.23/0.46 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_diff
% 0.23/0.46 thf(fact_3013_power__diff,axiom,
% 0.23/0.46 ! [A: real,N: nat,M: nat] :
% 0.23/0.46 ( ( A != zero_zero_real )
% 0.23/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.46 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 0.23/0.46 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_diff
% 0.23/0.46 thf(fact_3014_power__diff,axiom,
% 0.23/0.46 ! [A: rat,N: nat,M: nat] :
% 0.23/0.46 ( ( A != zero_zero_rat )
% 0.23/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.46 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.23/0.46 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_diff
% 0.23/0.46 thf(fact_3015_power__diff,axiom,
% 0.23/0.46 ! [A: nat,N: nat,M: nat] :
% 0.23/0.46 ( ( A != zero_zero_nat )
% 0.23/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.46 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.23/0.46 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_diff
% 0.23/0.46 thf(fact_3016_power__diff,axiom,
% 0.23/0.46 ! [A: int,N: nat,M: nat] :
% 0.23/0.46 ( ( A != zero_zero_int )
% 0.23/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.46 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 0.23/0.46 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_diff
% 0.23/0.46 thf(fact_3017_even__mult__exp__div__exp__iff,axiom,
% 0.23/0.46 ! [A: code_integer,M: nat,N: nat] :
% 0.23/0.46 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.46 = ( ( ord_less_nat @ N @ M )
% 0.23/0.46 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.23/0.46 = zero_z3403309356797280102nteger )
% 0.23/0.46 | ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.46 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_mult_exp_div_exp_iff
% 0.23/0.46 thf(fact_3018_even__mult__exp__div__exp__iff,axiom,
% 0.23/0.46 ! [A: nat,M: nat,N: nat] :
% 0.23/0.46 ( ( 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 ) ) )
% 0.23/0.46 = ( ( ord_less_nat @ N @ M )
% 0.23/0.46 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.46 = zero_zero_nat )
% 0.23/0.46 | ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.46 & ( 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 ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_mult_exp_div_exp_iff
% 0.23/0.46 thf(fact_3019_even__mult__exp__div__exp__iff,axiom,
% 0.23/0.46 ! [A: int,M: nat,N: nat] :
% 0.23/0.46 ( ( 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 ) ) )
% 0.23/0.46 = ( ( ord_less_nat @ N @ M )
% 0.23/0.46 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.23/0.46 = zero_zero_int )
% 0.23/0.46 | ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.46 & ( 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 ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_mult_exp_div_exp_iff
% 0.23/0.46 thf(fact_3020_div__if,axiom,
% 0.23/0.46 ( divide_divide_nat
% 0.23/0.46 = ( ^ [M6: nat,N2: nat] :
% 0.23/0.46 ( if_nat
% 0.23/0.46 @ ( ( ord_less_nat @ M6 @ N2 )
% 0.23/0.46 | ( N2 = zero_zero_nat ) )
% 0.23/0.46 @ zero_zero_nat
% 0.23/0.46 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % div_if
% 0.23/0.46 thf(fact_3021_div__geq,axiom,
% 0.23/0.46 ! [N: nat,M: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ~ ( ord_less_nat @ M @ N )
% 0.23/0.46 => ( ( divide_divide_nat @ M @ N )
% 0.23/0.46 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % div_geq
% 0.23/0.46 thf(fact_3022_Suc__pred_H,axiom,
% 0.23/0.46 ! [N: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( N
% 0.23/0.46 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % Suc_pred'
% 0.23/0.46 thf(fact_3023_Suc__diff__eq__diff__pred,axiom,
% 0.23/0.46 ! [N: nat,M: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.23/0.46 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % Suc_diff_eq_diff_pred
% 0.23/0.46 thf(fact_3024_less__eq__div__iff__mult__less__eq,axiom,
% 0.23/0.46 ! [Q2: nat,M: nat,N: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.23/0.46 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
% 0.23/0.46 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % less_eq_div_iff_mult_less_eq
% 0.23/0.46 thf(fact_3025_add__eq__if,axiom,
% 0.23/0.46 ( plus_plus_nat
% 0.23/0.46 = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % add_eq_if
% 0.23/0.46 thf(fact_3026_dividend__less__times__div,axiom,
% 0.23/0.46 ! [N: nat,M: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % dividend_less_times_div
% 0.23/0.46 thf(fact_3027_dividend__less__div__times,axiom,
% 0.23/0.46 ! [N: nat,M: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % dividend_less_div_times
% 0.23/0.46 thf(fact_3028_split__div,axiom,
% 0.23/0.46 ! [P: nat > $o,M: nat,N: nat] :
% 0.23/0.46 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.23/0.46 = ( ( ( N = zero_zero_nat )
% 0.23/0.46 => ( P @ zero_zero_nat ) )
% 0.23/0.46 & ( ( N != zero_zero_nat )
% 0.23/0.46 => ! [I2: nat,J3: nat] :
% 0.23/0.46 ( ( ord_less_nat @ J3 @ N )
% 0.23/0.46 => ( ( M
% 0.23/0.46 = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
% 0.23/0.46 => ( P @ I2 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % split_div
% 0.23/0.46 thf(fact_3029_mult__eq__if,axiom,
% 0.23/0.46 ( times_times_nat
% 0.23/0.46 = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_eq_if
% 0.23/0.46 thf(fact_3030_split__mod,axiom,
% 0.23/0.46 ! [P: nat > $o,M: nat,N: nat] :
% 0.23/0.46 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 0.23/0.46 = ( ( ( N = zero_zero_nat )
% 0.23/0.46 => ( P @ M ) )
% 0.23/0.46 & ( ( N != zero_zero_nat )
% 0.23/0.46 => ! [I2: nat,J3: nat] :
% 0.23/0.46 ( ( ord_less_nat @ J3 @ N )
% 0.23/0.46 => ( ( M
% 0.23/0.46 = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
% 0.23/0.46 => ( P @ J3 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % split_mod
% 0.23/0.46 thf(fact_3031_even__two__times__div__two,axiom,
% 0.23/0.46 ! [A: code_integer] :
% 0.23/0.46 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.23/0.46 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.23/0.46 = A ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_two_times_div_two
% 0.23/0.46 thf(fact_3032_even__two__times__div__two,axiom,
% 0.23/0.46 ! [A: nat] :
% 0.23/0.46 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.23/0.46 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 = A ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_two_times_div_two
% 0.23/0.46 thf(fact_3033_even__two__times__div__two,axiom,
% 0.23/0.46 ! [A: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.23/0.46 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.23/0.46 = A ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_two_times_div_two
% 0.23/0.46 thf(fact_3034_odd__iff__mod__2__eq__one,axiom,
% 0.23/0.46 ! [A: nat] :
% 0.23/0.46 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.23/0.46 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = one_one_nat ) ) ).
% 0.23/0.46
% 0.23/0.46 % odd_iff_mod_2_eq_one
% 0.23/0.46 thf(fact_3035_odd__iff__mod__2__eq__one,axiom,
% 0.23/0.46 ! [A: int] :
% 0.23/0.46 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.23/0.46 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.46 = one_one_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % odd_iff_mod_2_eq_one
% 0.23/0.46 thf(fact_3036_odd__iff__mod__2__eq__one,axiom,
% 0.23/0.46 ! [A: code_integer] :
% 0.23/0.46 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.23/0.46 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.46 = one_one_Code_integer ) ) ).
% 0.23/0.46
% 0.23/0.46 % odd_iff_mod_2_eq_one
% 0.23/0.46 thf(fact_3037_power__mono__odd,axiom,
% 0.23/0.46 ! [N: nat,A: real,B2: real] :
% 0.23/0.46 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.46 => ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.46 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_mono_odd
% 0.23/0.46 thf(fact_3038_power__mono__odd,axiom,
% 0.23/0.46 ! [N: nat,A: rat,B2: rat] :
% 0.23/0.46 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.46 => ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.46 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_mono_odd
% 0.23/0.46 thf(fact_3039_power__mono__odd,axiom,
% 0.23/0.46 ! [N: nat,A: int,B2: int] :
% 0.23/0.46 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.46 => ( ( ord_less_eq_int @ A @ B2 )
% 0.23/0.46 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_mono_odd
% 0.23/0.46 thf(fact_3040_dvd__power__iff__le,axiom,
% 0.23/0.46 ! [K: nat,M: nat,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.23/0.46 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 0.23/0.46 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % dvd_power_iff_le
% 0.23/0.46 thf(fact_3041_convex__bound__lt,axiom,
% 0.23/0.46 ! [X4: real,A: real,Y3: real,U: real,V: real] :
% 0.23/0.46 ( ( ord_less_real @ X4 @ A )
% 0.23/0.46 => ( ( ord_less_real @ Y3 @ A )
% 0.23/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.23/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.23/0.46 => ( ( ( plus_plus_real @ U @ V )
% 0.23/0.46 = one_one_real )
% 0.23/0.46 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % convex_bound_lt
% 0.23/0.46 thf(fact_3042_convex__bound__lt,axiom,
% 0.23/0.46 ! [X4: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X4 @ A )
% 0.23/0.46 => ( ( ord_less_rat @ Y3 @ A )
% 0.23/0.46 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.23/0.46 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.23/0.46 => ( ( ( plus_plus_rat @ U @ V )
% 0.23/0.46 = one_one_rat )
% 0.23/0.46 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X4 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % convex_bound_lt
% 0.23/0.46 thf(fact_3043_convex__bound__lt,axiom,
% 0.23/0.46 ! [X4: int,A: int,Y3: int,U: int,V: int] :
% 0.23/0.46 ( ( ord_less_int @ X4 @ A )
% 0.23/0.46 => ( ( ord_less_int @ Y3 @ A )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.23/0.46 => ( ( ( plus_plus_int @ U @ V )
% 0.23/0.46 = one_one_int )
% 0.23/0.46 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % convex_bound_lt
% 0.23/0.46 thf(fact_3044_le__divide__eq__numeral_I1_J,axiom,
% 0.23/0.46 ! [W: num,B2: real,C: real] :
% 0.23/0.46 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
% 0.23/0.46 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.46 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 0.23/0.46 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.46 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.46 => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.23/0.46 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.46 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % le_divide_eq_numeral(1)
% 0.23/0.46 thf(fact_3045_le__divide__eq__numeral_I1_J,axiom,
% 0.23/0.46 ! [W: num,B2: rat,C: rat] :
% 0.23/0.46 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.23/0.46 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.46 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 0.23/0.46 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.46 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.46 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.23/0.46 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.46 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % le_divide_eq_numeral(1)
% 0.23/0.46 thf(fact_3046_divide__le__eq__numeral_I1_J,axiom,
% 0.23/0.46 ! [B2: real,C: real,W: num] :
% 0.23/0.46 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
% 0.23/0.46 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.46 => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.23/0.46 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.46 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.46 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 0.23/0.46 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % divide_le_eq_numeral(1)
% 0.23/0.46 thf(fact_3047_divide__le__eq__numeral_I1_J,axiom,
% 0.23/0.46 ! [B2: rat,C: rat,W: num] :
% 0.23/0.46 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.23/0.46 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.46 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.23/0.46 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.46 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.46 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 0.23/0.46 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.46 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % divide_le_eq_numeral(1)
% 0.23/0.46 thf(fact_3048_half__gt__zero,axiom,
% 0.23/0.46 ! [A: real] :
% 0.23/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.46 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % half_gt_zero
% 0.23/0.46 thf(fact_3049_half__gt__zero,axiom,
% 0.23/0.46 ! [A: rat] :
% 0.23/0.46 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.46 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % half_gt_zero
% 0.23/0.46 thf(fact_3050_half__gt__zero__iff,axiom,
% 0.23/0.46 ! [A: real] :
% 0.23/0.46 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.46 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.23/0.46
% 0.23/0.46 % half_gt_zero_iff
% 0.23/0.46 thf(fact_3051_half__gt__zero__iff,axiom,
% 0.23/0.46 ! [A: rat] :
% 0.23/0.46 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.23/0.46
% 0.23/0.46 % half_gt_zero_iff
% 0.23/0.46 thf(fact_3052_scaling__mono,axiom,
% 0.23/0.46 ! [U: real,V: real,R2: real,S: real] :
% 0.23/0.46 ( ( ord_less_eq_real @ U @ V )
% 0.23/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.23/0.46 => ( ( ord_less_eq_real @ R2 @ S )
% 0.23/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % scaling_mono
% 0.23/0.46 thf(fact_3053_scaling__mono,axiom,
% 0.23/0.46 ! [U: rat,V: rat,R2: rat,S: rat] :
% 0.23/0.46 ( ( ord_less_eq_rat @ U @ V )
% 0.23/0.46 => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 0.23/0.46 => ( ( ord_less_eq_rat @ R2 @ S )
% 0.23/0.46 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % scaling_mono
% 0.23/0.46 thf(fact_3054_zero__le__power2,axiom,
% 0.23/0.46 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zero_le_power2
% 0.23/0.46 thf(fact_3055_zero__le__power2,axiom,
% 0.23/0.46 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zero_le_power2
% 0.23/0.46 thf(fact_3056_zero__le__power2,axiom,
% 0.23/0.46 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zero_le_power2
% 0.23/0.46 thf(fact_3057_power2__eq__imp__eq,axiom,
% 0.23/0.46 ! [X4: real,Y3: real] :
% 0.23/0.46 ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.46 => ( X4 = Y3 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_eq_imp_eq
% 0.23/0.46 thf(fact_3058_power2__eq__imp__eq,axiom,
% 0.23/0.46 ! [X4: rat,Y3: rat] :
% 0.23/0.46 ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.23/0.46 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.23/0.46 => ( X4 = Y3 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_eq_imp_eq
% 0.23/0.46 thf(fact_3059_power2__eq__imp__eq,axiom,
% 0.23/0.46 ! [X4: nat,Y3: nat] :
% 0.23/0.46 ( ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
% 0.23/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.23/0.46 => ( X4 = Y3 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_eq_imp_eq
% 0.23/0.46 thf(fact_3060_power2__eq__imp__eq,axiom,
% 0.23/0.46 ! [X4: int,Y3: int] :
% 0.23/0.46 ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.46 => ( X4 = Y3 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_eq_imp_eq
% 0.23/0.46 thf(fact_3061_power2__le__imp__le,axiom,
% 0.23/0.46 ! [X4: real,Y3: real] :
% 0.23/0.46 ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.46 => ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_le_imp_le
% 0.23/0.46 thf(fact_3062_power2__le__imp__le,axiom,
% 0.23/0.46 ! [X4: rat,Y3: rat] :
% 0.23/0.46 ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.23/0.46 => ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_le_imp_le
% 0.23/0.46 thf(fact_3063_power2__le__imp__le,axiom,
% 0.23/0.46 ! [X4: nat,Y3: nat] :
% 0.23/0.46 ( ( ord_less_eq_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.23/0.46 => ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_le_imp_le
% 0.23/0.46 thf(fact_3064_power2__le__imp__le,axiom,
% 0.23/0.46 ! [X4: int,Y3: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.46 => ( ord_less_eq_int @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_le_imp_le
% 0.23/0.46 thf(fact_3065_power2__less__0,axiom,
% 0.23/0.46 ! [A: real] :
% 0.23/0.46 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 0.23/0.46
% 0.23/0.46 % power2_less_0
% 0.23/0.46 thf(fact_3066_power2__less__0,axiom,
% 0.23/0.46 ! [A: rat] :
% 0.23/0.46 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 0.23/0.46
% 0.23/0.46 % power2_less_0
% 0.23/0.46 thf(fact_3067_power2__less__0,axiom,
% 0.23/0.46 ! [A: int] :
% 0.23/0.46 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 0.23/0.46
% 0.23/0.46 % power2_less_0
% 0.23/0.46 thf(fact_3068_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.23/0.46 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.23/0.46 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.23/0.46 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.23/0.46 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.23/0.46 thf(fact_3069_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.23/0.46 ! [C: nat,A: nat,B2: nat] :
% 0.23/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.23/0.46 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.23/0.46 = ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.23/0.46 thf(fact_3070_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.23/0.46 ! [C: int,A: int,B2: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.23/0.46 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.23/0.46 = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.23/0.46 thf(fact_3071_exp__add__not__zero__imp__left,axiom,
% 0.23/0.46 ! [M: nat,N: nat] :
% 0.23/0.46 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.23/0.46 != zero_zero_nat )
% 0.23/0.46 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.23/0.46 != zero_zero_nat ) ) ).
% 0.23/0.46
% 0.23/0.46 % exp_add_not_zero_imp_left
% 0.23/0.46 thf(fact_3072_exp__add__not__zero__imp__left,axiom,
% 0.23/0.46 ! [M: nat,N: nat] :
% 0.23/0.46 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.23/0.46 != zero_zero_int )
% 0.23/0.46 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.23/0.46 != zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % exp_add_not_zero_imp_left
% 0.23/0.46 thf(fact_3073_exp__add__not__zero__imp__right,axiom,
% 0.23/0.46 ! [M: nat,N: nat] :
% 0.23/0.46 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.23/0.46 != zero_zero_nat )
% 0.23/0.46 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.46 != zero_zero_nat ) ) ).
% 0.23/0.46
% 0.23/0.46 % exp_add_not_zero_imp_right
% 0.23/0.46 thf(fact_3074_exp__add__not__zero__imp__right,axiom,
% 0.23/0.46 ! [M: nat,N: nat] :
% 0.23/0.46 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.23/0.46 != zero_zero_int )
% 0.23/0.46 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.23/0.46 != zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % exp_add_not_zero_imp_right
% 0.23/0.46 thf(fact_3075_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.23/0.46 ! [N: nat,M: nat] :
% 0.23/0.46 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.46 != zero_zero_nat )
% 0.23/0.46 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.23/0.46 != zero_zero_nat ) ) ).
% 0.23/0.46
% 0.23/0.46 % exp_not_zero_imp_exp_diff_not_zero
% 0.23/0.46 thf(fact_3076_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.23/0.46 ! [N: nat,M: nat] :
% 0.23/0.46 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.23/0.46 != zero_zero_int )
% 0.23/0.46 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.23/0.46 != zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % exp_not_zero_imp_exp_diff_not_zero
% 0.23/0.46 thf(fact_3077_power__diff__power__eq,axiom,
% 0.23/0.46 ! [A: nat,N: nat,M: nat] :
% 0.23/0.46 ( ( A != zero_zero_nat )
% 0.23/0.46 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.46 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.23/0.46 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.23/0.46 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.23/0.46 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.23/0.46 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_diff_power_eq
% 0.23/0.46 thf(fact_3078_power__diff__power__eq,axiom,
% 0.23/0.46 ! [A: int,N: nat,M: nat] :
% 0.23/0.46 ( ( A != zero_zero_int )
% 0.23/0.46 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.46 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.23/0.46 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.23/0.46 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.23/0.46 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.23/0.46 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_diff_power_eq
% 0.23/0.46 thf(fact_3079_less__2__cases,axiom,
% 0.23/0.46 ! [N: nat] :
% 0.23/0.46 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 => ( ( N = zero_zero_nat )
% 0.23/0.46 | ( N
% 0.23/0.46 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % less_2_cases
% 0.23/0.46 thf(fact_3080_less__2__cases__iff,axiom,
% 0.23/0.46 ! [N: nat] :
% 0.23/0.46 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = ( ( N = zero_zero_nat )
% 0.23/0.46 | ( N
% 0.23/0.46 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % less_2_cases_iff
% 0.23/0.46 thf(fact_3081_nat__induct2,axiom,
% 0.23/0.46 ! [P: nat > $o,N: nat] :
% 0.23/0.46 ( ( P @ zero_zero_nat )
% 0.23/0.46 => ( ( P @ one_one_nat )
% 0.23/0.46 => ( ! [N4: nat] :
% 0.23/0.46 ( ( P @ N4 )
% 0.23/0.46 => ( P @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.46 => ( P @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % nat_induct2
% 0.23/0.46 thf(fact_3082_power__eq__if,axiom,
% 0.23/0.46 ( power_power_complex
% 0.23/0.46 = ( ^ [P3: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P3 @ ( power_power_complex @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_eq_if
% 0.23/0.46 thf(fact_3083_power__eq__if,axiom,
% 0.23/0.46 ( power_power_real
% 0.23/0.46 = ( ^ [P3: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P3 @ ( power_power_real @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_eq_if
% 0.23/0.46 thf(fact_3084_power__eq__if,axiom,
% 0.23/0.46 ( power_power_rat
% 0.23/0.46 = ( ^ [P3: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P3 @ ( power_power_rat @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_eq_if
% 0.23/0.46 thf(fact_3085_power__eq__if,axiom,
% 0.23/0.46 ( power_power_nat
% 0.23/0.46 = ( ^ [P3: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_eq_if
% 0.23/0.46 thf(fact_3086_power__eq__if,axiom,
% 0.23/0.46 ( power_power_int
% 0.23/0.46 = ( ^ [P3: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_eq_if
% 0.23/0.46 thf(fact_3087_power__minus__mult,axiom,
% 0.23/0.46 ! [N: nat,A: complex] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.23/0.46 = ( power_power_complex @ A @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_minus_mult
% 0.23/0.46 thf(fact_3088_power__minus__mult,axiom,
% 0.23/0.46 ! [N: nat,A: real] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.23/0.46 = ( power_power_real @ A @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_minus_mult
% 0.23/0.46 thf(fact_3089_power__minus__mult,axiom,
% 0.23/0.46 ! [N: nat,A: rat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.23/0.46 = ( power_power_rat @ A @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_minus_mult
% 0.23/0.46 thf(fact_3090_power__minus__mult,axiom,
% 0.23/0.46 ! [N: nat,A: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.23/0.46 = ( power_power_nat @ A @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_minus_mult
% 0.23/0.46 thf(fact_3091_power__minus__mult,axiom,
% 0.23/0.46 ! [N: nat,A: int] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.23/0.46 = ( power_power_int @ A @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power_minus_mult
% 0.23/0.46 thf(fact_3092_le__div__geq,axiom,
% 0.23/0.46 ! [N: nat,M: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.46 => ( ( divide_divide_nat @ M @ N )
% 0.23/0.46 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % le_div_geq
% 0.23/0.46 thf(fact_3093_split__div_H,axiom,
% 0.23/0.46 ! [P: nat > $o,M: nat,N: nat] :
% 0.23/0.46 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.23/0.46 = ( ( ( N = zero_zero_nat )
% 0.23/0.46 & ( P @ zero_zero_nat ) )
% 0.23/0.46 | ? [Q4: nat] :
% 0.23/0.46 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
% 0.23/0.46 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
% 0.23/0.46 & ( P @ Q4 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % split_div'
% 0.23/0.46 thf(fact_3094_Suc__times__mod__eq,axiom,
% 0.23/0.46 ! [M: nat,N: nat] :
% 0.23/0.46 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.23/0.46 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 0.23/0.46 = one_one_nat ) ) ).
% 0.23/0.46
% 0.23/0.46 % Suc_times_mod_eq
% 0.23/0.46 thf(fact_3095_oddE,axiom,
% 0.23/0.46 ! [A: code_integer] :
% 0.23/0.46 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.23/0.46 => ~ ! [B5: code_integer] :
% 0.23/0.46 ( A
% 0.23/0.46 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) @ one_one_Code_integer ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % oddE
% 0.23/0.46 thf(fact_3096_oddE,axiom,
% 0.23/0.46 ! [A: nat] :
% 0.23/0.46 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.23/0.46 => ~ ! [B5: nat] :
% 0.23/0.46 ( A
% 0.23/0.46 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) @ one_one_nat ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % oddE
% 0.23/0.46 thf(fact_3097_oddE,axiom,
% 0.23/0.46 ! [A: int] :
% 0.23/0.46 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.23/0.46 => ~ ! [B5: int] :
% 0.23/0.46 ( A
% 0.23/0.46 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) @ one_one_int ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % oddE
% 0.23/0.46 thf(fact_3098_power2__less__imp__less,axiom,
% 0.23/0.46 ! [X4: real,Y3: real] :
% 0.23/0.46 ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.46 => ( ord_less_real @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_less_imp_less
% 0.23/0.46 thf(fact_3099_power2__less__imp__less,axiom,
% 0.23/0.46 ! [X4: rat,Y3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.23/0.46 => ( ord_less_rat @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_less_imp_less
% 0.23/0.46 thf(fact_3100_power2__less__imp__less,axiom,
% 0.23/0.46 ! [X4: nat,Y3: nat] :
% 0.23/0.46 ( ( ord_less_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.23/0.46 => ( ord_less_nat @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_less_imp_less
% 0.23/0.46 thf(fact_3101_power2__less__imp__less,axiom,
% 0.23/0.46 ! [X4: int,Y3: int] :
% 0.23/0.46 ( ( ord_less_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.46 => ( ord_less_int @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % power2_less_imp_less
% 0.23/0.46 thf(fact_3102_sum__power2__ge__zero,axiom,
% 0.23/0.46 ! [X4: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % sum_power2_ge_zero
% 0.23/0.46 thf(fact_3103_sum__power2__ge__zero,axiom,
% 0.23/0.46 ! [X4: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % sum_power2_ge_zero
% 0.23/0.46 thf(fact_3104_sum__power2__ge__zero,axiom,
% 0.23/0.46 ! [X4: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % sum_power2_ge_zero
% 0.23/0.46 thf(fact_3105_sum__power2__le__zero__iff,axiom,
% 0.23/0.46 ! [X4: real,Y3: real] :
% 0.23/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 0.23/0.46 = ( ( X4 = zero_zero_real )
% 0.23/0.46 & ( Y3 = zero_zero_real ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % sum_power2_le_zero_iff
% 0.23/0.46 thf(fact_3106_sum__power2__le__zero__iff,axiom,
% 0.23/0.46 ! [X4: rat,Y3: rat] :
% 0.23/0.46 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 0.23/0.46 = ( ( X4 = zero_zero_rat )
% 0.23/0.46 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % sum_power2_le_zero_iff
% 0.23/0.46 thf(fact_3107_sum__power2__le__zero__iff,axiom,
% 0.23/0.46 ! [X4: int,Y3: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 0.23/0.46 = ( ( X4 = zero_zero_int )
% 0.23/0.46 & ( Y3 = zero_zero_int ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % sum_power2_le_zero_iff
% 0.23/0.46 thf(fact_3108_not__sum__power2__lt__zero,axiom,
% 0.23/0.46 ! [X4: real,Y3: real] :
% 0.23/0.46 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 0.23/0.46
% 0.23/0.46 % not_sum_power2_lt_zero
% 0.23/0.46 thf(fact_3109_not__sum__power2__lt__zero,axiom,
% 0.23/0.46 ! [X4: rat,Y3: rat] :
% 0.23/0.46 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 0.23/0.46
% 0.23/0.46 % not_sum_power2_lt_zero
% 0.23/0.46 thf(fact_3110_not__sum__power2__lt__zero,axiom,
% 0.23/0.46 ! [X4: int,Y3: int] :
% 0.23/0.46 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 0.23/0.46
% 0.23/0.46 % not_sum_power2_lt_zero
% 0.23/0.46 thf(fact_3111_sum__power2__gt__zero__iff,axiom,
% 0.23/0.46 ! [X4: real,Y3: real] :
% 0.23/0.46 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.46 = ( ( X4 != zero_zero_real )
% 0.23/0.46 | ( Y3 != zero_zero_real ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % sum_power2_gt_zero_iff
% 0.23/0.46 thf(fact_3112_sum__power2__gt__zero__iff,axiom,
% 0.23/0.46 ! [X4: rat,Y3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.46 = ( ( X4 != zero_zero_rat )
% 0.23/0.46 | ( Y3 != zero_zero_rat ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % sum_power2_gt_zero_iff
% 0.23/0.46 thf(fact_3113_sum__power2__gt__zero__iff,axiom,
% 0.23/0.46 ! [X4: int,Y3: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.46 = ( ( X4 != zero_zero_int )
% 0.23/0.46 | ( Y3 != zero_zero_int ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % sum_power2_gt_zero_iff
% 0.23/0.46 thf(fact_3114_divmod__digit__0_I2_J,axiom,
% 0.23/0.46 ! [B2: nat,A: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.23/0.46 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.23/0.46 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
% 0.23/0.46 = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % divmod_digit_0(2)
% 0.23/0.46 thf(fact_3115_divmod__digit__0_I2_J,axiom,
% 0.23/0.46 ! [B2: int,A: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.46 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.23/0.46 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
% 0.23/0.46 = ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % divmod_digit_0(2)
% 0.23/0.46 thf(fact_3116_divmod__digit__0_I2_J,axiom,
% 0.23/0.46 ! [B2: code_integer,A: code_integer] :
% 0.23/0.46 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.23/0.46 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.23/0.46 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
% 0.23/0.46 = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % divmod_digit_0(2)
% 0.23/0.46 thf(fact_3117_bits__stable__imp__add__self,axiom,
% 0.23/0.46 ! [A: nat] :
% 0.23/0.46 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.46 = A )
% 0.23/0.46 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.46 = zero_zero_nat ) ) ).
% 0.23/0.46
% 0.23/0.46 % bits_stable_imp_add_self
% 0.23/0.46 thf(fact_3118_bits__stable__imp__add__self,axiom,
% 0.23/0.46 ! [A: int] :
% 0.23/0.46 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.46 = A )
% 0.23/0.46 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.23/0.46 = zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % bits_stable_imp_add_self
% 0.23/0.46 thf(fact_3119_bits__stable__imp__add__self,axiom,
% 0.23/0.46 ! [A: code_integer] :
% 0.23/0.46 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.46 = A )
% 0.23/0.46 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.23/0.46 = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.46
% 0.23/0.46 % bits_stable_imp_add_self
% 0.23/0.46 thf(fact_3120_zero__le__even__power_H,axiom,
% 0.23/0.46 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zero_le_even_power'
% 0.23/0.46 thf(fact_3121_zero__le__even__power_H,axiom,
% 0.23/0.46 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zero_le_even_power'
% 0.23/0.46 thf(fact_3122_zero__le__even__power_H,axiom,
% 0.23/0.46 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zero_le_even_power'
% 0.23/0.46 thf(fact_3123_nat__bit__induct,axiom,
% 0.23/0.46 ! [P: nat > $o,N: nat] :
% 0.23/0.46 ( ( P @ zero_zero_nat )
% 0.23/0.46 => ( ! [N4: nat] :
% 0.23/0.46 ( ( P @ N4 )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.23/0.46 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 0.23/0.46 => ( ! [N4: nat] :
% 0.23/0.46 ( ( P @ N4 )
% 0.23/0.46 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 0.23/0.46 => ( P @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % nat_bit_induct
% 0.23/0.46 thf(fact_3124_div__2__gt__zero,axiom,
% 0.23/0.46 ! [N: nat] :
% 0.23/0.46 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.23/0.46 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % div_2_gt_zero
% 0.23/0.46 thf(fact_3125_Suc__n__div__2__gt__zero,axiom,
% 0.23/0.46 ! [N: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % Suc_n_div_2_gt_zero
% 0.23/0.46 thf(fact_3126_verit__le__mono__div,axiom,
% 0.23/0.46 ! [A3: nat,B4: nat,N: nat] :
% 0.23/0.46 ( ( ord_less_nat @ A3 @ B4 )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ord_less_eq_nat
% 0.23/0.46 @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N )
% 0.23/0.46 @ ( if_nat
% 0.23/0.46 @ ( ( modulo_modulo_nat @ B4 @ N )
% 0.23/0.46 = zero_zero_nat )
% 0.23/0.46 @ one_one_nat
% 0.23/0.46 @ zero_zero_nat ) )
% 0.23/0.46 @ ( divide_divide_nat @ B4 @ N ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % verit_le_mono_div
% 0.23/0.46 thf(fact_3127_divmod__digit__0_I1_J,axiom,
% 0.23/0.46 ! [B2: nat,A: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.23/0.46 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.23/0.46 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.23/0.46 = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % divmod_digit_0(1)
% 0.23/0.46 thf(fact_3128_divmod__digit__0_I1_J,axiom,
% 0.23/0.46 ! [B2: int,A: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.46 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.23/0.46 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.23/0.46 = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % divmod_digit_0(1)
% 0.23/0.46 thf(fact_3129_divmod__digit__0_I1_J,axiom,
% 0.23/0.46 ! [B2: code_integer,A: code_integer] :
% 0.23/0.46 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.23/0.46 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.23/0.46 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.23/0.46 = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % divmod_digit_0(1)
% 0.23/0.46 thf(fact_3130_odd__0__le__power__imp__0__le,axiom,
% 0.23/0.46 ! [A: real,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.23/0.46 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.23/0.46
% 0.23/0.46 % odd_0_le_power_imp_0_le
% 0.23/0.46 thf(fact_3131_odd__0__le__power__imp__0__le,axiom,
% 0.23/0.46 ! [A: rat,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.23/0.46 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.23/0.46
% 0.23/0.46 % odd_0_le_power_imp_0_le
% 0.23/0.46 thf(fact_3132_odd__0__le__power__imp__0__le,axiom,
% 0.23/0.46 ! [A: int,N: nat] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.23/0.46 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.23/0.46
% 0.23/0.46 % odd_0_le_power_imp_0_le
% 0.23/0.46 thf(fact_3133_odd__power__less__zero,axiom,
% 0.23/0.46 ! [A: real,N: nat] :
% 0.23/0.46 ( ( ord_less_real @ A @ zero_zero_real )
% 0.23/0.46 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 0.23/0.46
% 0.23/0.46 % odd_power_less_zero
% 0.23/0.46 thf(fact_3134_odd__power__less__zero,axiom,
% 0.23/0.46 ! [A: rat,N: nat] :
% 0.23/0.46 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.23/0.46 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 0.23/0.46
% 0.23/0.46 % odd_power_less_zero
% 0.23/0.46 thf(fact_3135_odd__power__less__zero,axiom,
% 0.23/0.46 ! [A: int,N: nat] :
% 0.23/0.46 ( ( ord_less_int @ A @ zero_zero_int )
% 0.23/0.46 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % odd_power_less_zero
% 0.23/0.46 thf(fact_3136_even__mask__div__iff_H,axiom,
% 0.23/0.46 ! [M: nat,N: nat] :
% 0.23/0.46 ( ( 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 ) ) )
% 0.23/0.46 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_mask_div_iff'
% 0.23/0.46 thf(fact_3137_even__mask__div__iff_H,axiom,
% 0.23/0.46 ! [M: nat,N: nat] :
% 0.23/0.46 ( ( 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 ) ) )
% 0.23/0.46 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_mask_div_iff'
% 0.23/0.46 thf(fact_3138_even__mask__div__iff_H,axiom,
% 0.23/0.46 ! [M: nat,N: nat] :
% 0.23/0.46 ( ( 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 ) ) )
% 0.23/0.46 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_mask_div_iff'
% 0.23/0.46 thf(fact_3139_mod__double__modulus,axiom,
% 0.23/0.46 ! [M: code_integer,X4: code_integer] :
% 0.23/0.46 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 0.23/0.46 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 0.23/0.46 => ( ( ( modulo364778990260209775nteger @ X4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.46 = ( modulo364778990260209775nteger @ X4 @ M ) )
% 0.23/0.46 | ( ( modulo364778990260209775nteger @ X4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.46 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X4 @ M ) @ M ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mod_double_modulus
% 0.23/0.46 thf(fact_3140_mod__double__modulus,axiom,
% 0.23/0.46 ! [M: nat,X4: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
% 0.23/0.46 => ( ( ( modulo_modulo_nat @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.46 = ( modulo_modulo_nat @ X4 @ M ) )
% 0.23/0.46 | ( ( modulo_modulo_nat @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.46 = ( plus_plus_nat @ ( modulo_modulo_nat @ X4 @ M ) @ M ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mod_double_modulus
% 0.23/0.46 thf(fact_3141_mod__double__modulus,axiom,
% 0.23/0.46 ! [M: int,X4: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ M )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.46 => ( ( ( modulo_modulo_int @ X4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.46 = ( modulo_modulo_int @ X4 @ M ) )
% 0.23/0.46 | ( ( modulo_modulo_int @ X4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.46 = ( plus_plus_int @ ( modulo_modulo_int @ X4 @ M ) @ M ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mod_double_modulus
% 0.23/0.46 thf(fact_3142_divmod__digit__1_I2_J,axiom,
% 0.23/0.46 ! [A: code_integer,B2: code_integer] :
% 0.23/0.46 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.23/0.46 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.23/0.46 => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.23/0.46 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.23/0.46 = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % divmod_digit_1(2)
% 0.23/0.46 thf(fact_3143_divmod__digit__1_I2_J,axiom,
% 0.23/0.46 ! [A: nat,B2: nat] :
% 0.23/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.23/0.46 => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.23/0.46 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.23/0.46 = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % divmod_digit_1(2)
% 0.23/0.46 thf(fact_3144_divmod__digit__1_I2_J,axiom,
% 0.23/0.46 ! [A: int,B2: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.23/0.46 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.46 => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.23/0.46 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.23/0.46 = ( modulo_modulo_int @ A @ B2 ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % divmod_digit_1(2)
% 0.23/0.46 thf(fact_3145_even__mod__4__div__2,axiom,
% 0.23/0.46 ! [N: nat] :
% 0.23/0.46 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.23/0.46 = ( suc @ zero_zero_nat ) )
% 0.23/0.46 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_mod_4_div_2
% 0.23/0.46 thf(fact_3146_pow__divides__pow__iff,axiom,
% 0.23/0.46 ! [N: nat,A: nat,B2: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 0.23/0.46 = ( dvd_dvd_nat @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pow_divides_pow_iff
% 0.23/0.46 thf(fact_3147_pow__divides__pow__iff,axiom,
% 0.23/0.46 ! [N: nat,A: int,B2: int] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 0.23/0.46 = ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pow_divides_pow_iff
% 0.23/0.46 thf(fact_3148_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 0.23/0.46 ! [X4: nat,N: nat,M: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.46 => ( ord_less_nat @ ( vEBT_VEBT_low @ X4 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % VEBT_internal.exp_split_high_low(2)
% 0.23/0.46 thf(fact_3149_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 0.23/0.46 ! [X4: nat,N: nat,M: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.46 => ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % VEBT_internal.exp_split_high_low(1)
% 0.23/0.46 thf(fact_3150_even__even__mod__4__iff,axiom,
% 0.23/0.46 ! [N: nat] :
% 0.23/0.46 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.46 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_even_mod_4_iff
% 0.23/0.46 thf(fact_3151_double__eq__0__iff,axiom,
% 0.23/0.46 ! [A: real] :
% 0.23/0.46 ( ( ( plus_plus_real @ A @ A )
% 0.23/0.46 = zero_zero_real )
% 0.23/0.46 = ( A = zero_zero_real ) ) ).
% 0.23/0.46
% 0.23/0.46 % double_eq_0_iff
% 0.23/0.46 thf(fact_3152_double__eq__0__iff,axiom,
% 0.23/0.46 ! [A: rat] :
% 0.23/0.46 ( ( ( plus_plus_rat @ A @ A )
% 0.23/0.46 = zero_zero_rat )
% 0.23/0.46 = ( A = zero_zero_rat ) ) ).
% 0.23/0.46
% 0.23/0.46 % double_eq_0_iff
% 0.23/0.46 thf(fact_3153_double__eq__0__iff,axiom,
% 0.23/0.46 ! [A: int] :
% 0.23/0.46 ( ( ( plus_plus_int @ A @ A )
% 0.23/0.46 = zero_zero_int )
% 0.23/0.46 = ( A = zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % double_eq_0_iff
% 0.23/0.46 thf(fact_3154_inf__period_I3_J,axiom,
% 0.23/0.46 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 0.23/0.46 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 0.23/0.46 => ! [X5: code_integer,K4: code_integer] :
% 0.23/0.46 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
% 0.23/0.46 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(3)
% 0.23/0.46 thf(fact_3155_inf__period_I3_J,axiom,
% 0.23/0.46 ! [D: real,D4: real,T: real] :
% 0.23/0.46 ( ( dvd_dvd_real @ D @ D4 )
% 0.23/0.46 => ! [X5: real,K4: real] :
% 0.23/0.46 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
% 0.23/0.46 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(3)
% 0.23/0.46 thf(fact_3156_inf__period_I3_J,axiom,
% 0.23/0.46 ! [D: rat,D4: rat,T: rat] :
% 0.23/0.46 ( ( dvd_dvd_rat @ D @ D4 )
% 0.23/0.46 => ! [X5: rat,K4: rat] :
% 0.23/0.46 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
% 0.23/0.46 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(3)
% 0.23/0.46 thf(fact_3157_inf__period_I3_J,axiom,
% 0.23/0.46 ! [D: int,D4: int,T: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ D @ D4 )
% 0.23/0.46 => ! [X5: int,K4: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.23/0.46 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(3)
% 0.23/0.46 thf(fact_3158_inf__period_I4_J,axiom,
% 0.23/0.46 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 0.23/0.46 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 0.23/0.46 => ! [X5: code_integer,K4: code_integer] :
% 0.23/0.46 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(4)
% 0.23/0.46 thf(fact_3159_inf__period_I4_J,axiom,
% 0.23/0.46 ! [D: real,D4: real,T: real] :
% 0.23/0.46 ( ( dvd_dvd_real @ D @ D4 )
% 0.23/0.46 => ! [X5: real,K4: real] :
% 0.23/0.46 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(4)
% 0.23/0.46 thf(fact_3160_inf__period_I4_J,axiom,
% 0.23/0.46 ! [D: rat,D4: rat,T: rat] :
% 0.23/0.46 ( ( dvd_dvd_rat @ D @ D4 )
% 0.23/0.46 => ! [X5: rat,K4: rat] :
% 0.23/0.46 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(4)
% 0.23/0.46 thf(fact_3161_inf__period_I4_J,axiom,
% 0.23/0.46 ! [D: int,D4: int,T: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ D @ D4 )
% 0.23/0.46 => ! [X5: int,K4: int] :
% 0.23/0.46 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(4)
% 0.23/0.46 thf(fact_3162_unity__coeff__ex,axiom,
% 0.23/0.46 ! [P: code_integer > $o,L2: code_integer] :
% 0.23/0.46 ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X ) ) )
% 0.23/0.46 = ( ? [X: code_integer] :
% 0.23/0.46 ( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
% 0.23/0.46 & ( P @ X ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % unity_coeff_ex
% 0.23/0.46 thf(fact_3163_unity__coeff__ex,axiom,
% 0.23/0.46 ! [P: real > $o,L2: real] :
% 0.23/0.46 ( ( ? [X: real] : ( P @ ( times_times_real @ L2 @ X ) ) )
% 0.23/0.46 = ( ? [X: real] :
% 0.23/0.46 ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X @ zero_zero_real ) )
% 0.23/0.46 & ( P @ X ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % unity_coeff_ex
% 0.23/0.46 thf(fact_3164_unity__coeff__ex,axiom,
% 0.23/0.46 ! [P: rat > $o,L2: rat] :
% 0.23/0.46 ( ( ? [X: rat] : ( P @ ( times_times_rat @ L2 @ X ) ) )
% 0.23/0.46 = ( ? [X: rat] :
% 0.23/0.46 ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X @ zero_zero_rat ) )
% 0.23/0.46 & ( P @ X ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % unity_coeff_ex
% 0.23/0.46 thf(fact_3165_unity__coeff__ex,axiom,
% 0.23/0.46 ! [P: nat > $o,L2: nat] :
% 0.23/0.46 ( ( ? [X: nat] : ( P @ ( times_times_nat @ L2 @ X ) ) )
% 0.23/0.46 = ( ? [X: nat] :
% 0.23/0.46 ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X @ zero_zero_nat ) )
% 0.23/0.46 & ( P @ X ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % unity_coeff_ex
% 0.23/0.46 thf(fact_3166_unity__coeff__ex,axiom,
% 0.23/0.46 ! [P: int > $o,L2: int] :
% 0.23/0.46 ( ( ? [X: int] : ( P @ ( times_times_int @ L2 @ X ) ) )
% 0.23/0.46 = ( ? [X: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X @ zero_zero_int ) )
% 0.23/0.46 & ( P @ X ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % unity_coeff_ex
% 0.23/0.46 thf(fact_3167_unset__bit__nonnegative__int__iff,axiom,
% 0.23/0.46 ! [N: nat,K: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 0.23/0.46 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.23/0.46
% 0.23/0.46 % unset_bit_nonnegative_int_iff
% 0.23/0.46 thf(fact_3168_flip__bit__nonnegative__int__iff,axiom,
% 0.23/0.46 ! [N: nat,K: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 0.23/0.46 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.23/0.46
% 0.23/0.46 % flip_bit_nonnegative_int_iff
% 0.23/0.46 thf(fact_3169_set__bit__nonnegative__int__iff,axiom,
% 0.23/0.46 ! [N: nat,K: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 0.23/0.46 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.23/0.46
% 0.23/0.46 % set_bit_nonnegative_int_iff
% 0.23/0.46 thf(fact_3170_set__bit__negative__int__iff,axiom,
% 0.23/0.46 ! [N: nat,K: int] :
% 0.23/0.46 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 0.23/0.46 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % set_bit_negative_int_iff
% 0.23/0.46 thf(fact_3171_flip__bit__negative__int__iff,axiom,
% 0.23/0.46 ! [N: nat,K: int] :
% 0.23/0.46 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 0.23/0.46 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % flip_bit_negative_int_iff
% 0.23/0.46 thf(fact_3172_unset__bit__negative__int__iff,axiom,
% 0.23/0.46 ! [N: nat,K: int] :
% 0.23/0.46 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 0.23/0.46 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % unset_bit_negative_int_iff
% 0.23/0.46 thf(fact_3173_zle__add1__eq__le,axiom,
% 0.23/0.46 ! [W: int,Z: int] :
% 0.23/0.46 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.23/0.46 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.23/0.46
% 0.23/0.46 % zle_add1_eq_le
% 0.23/0.46 thf(fact_3174_zle__diff1__eq,axiom,
% 0.23/0.46 ! [W: int,Z: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 0.23/0.46 = ( ord_less_int @ W @ Z ) ) ).
% 0.23/0.46
% 0.23/0.46 % zle_diff1_eq
% 0.23/0.46 thf(fact_3175_mod__pos__pos__trivial,axiom,
% 0.23/0.46 ! [K: int,L2: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.46 => ( ( ord_less_int @ K @ L2 )
% 0.23/0.46 => ( ( modulo_modulo_int @ K @ L2 )
% 0.23/0.46 = K ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mod_pos_pos_trivial
% 0.23/0.46 thf(fact_3176_mod__neg__neg__trivial,axiom,
% 0.23/0.46 ! [K: int,L2: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.23/0.46 => ( ( ord_less_int @ L2 @ K )
% 0.23/0.46 => ( ( modulo_modulo_int @ K @ L2 )
% 0.23/0.46 = K ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mod_neg_neg_trivial
% 0.23/0.46 thf(fact_3177_le__imp__0__less,axiom,
% 0.23/0.46 ! [Z: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.23/0.46 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % le_imp_0_less
% 0.23/0.46 thf(fact_3178_split__zmod,axiom,
% 0.23/0.46 ! [P: int > $o,N: int,K: int] :
% 0.23/0.46 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 0.23/0.46 = ( ( ( K = zero_zero_int )
% 0.23/0.46 => ( P @ N ) )
% 0.23/0.46 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.23/0.46 => ! [I2: int,J3: int] :
% 0.23/0.46 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.23/0.46 & ( ord_less_int @ J3 @ K )
% 0.23/0.46 & ( N
% 0.23/0.46 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.23/0.46 => ( P @ J3 ) ) )
% 0.23/0.46 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.23/0.46 => ! [I2: int,J3: int] :
% 0.23/0.46 ( ( ( ord_less_int @ K @ J3 )
% 0.23/0.46 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.23/0.46 & ( N
% 0.23/0.46 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.23/0.46 => ( P @ J3 ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % split_zmod
% 0.23/0.46 thf(fact_3179_odd__less__0__iff,axiom,
% 0.23/0.46 ! [Z: int] :
% 0.23/0.46 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 0.23/0.46 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % odd_less_0_iff
% 0.23/0.46 thf(fact_3180_mod__pos__geq,axiom,
% 0.23/0.46 ! [L2: int,K: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.23/0.46 => ( ( ord_less_eq_int @ L2 @ K )
% 0.23/0.46 => ( ( modulo_modulo_int @ K @ L2 )
% 0.23/0.46 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mod_pos_geq
% 0.23/0.46 thf(fact_3181_q__pos__lemma,axiom,
% 0.23/0.46 ! [B: int,Q5: int,R4: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 0.23/0.46 => ( ( ord_less_int @ R4 @ B )
% 0.23/0.46 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.23/0.46 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % q_pos_lemma
% 0.23/0.46 thf(fact_3182_zmult__zless__mono2,axiom,
% 0.23/0.46 ! [I: int,J: int,K: int] :
% 0.23/0.46 ( ( ord_less_int @ I @ J )
% 0.23/0.46 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.23/0.46 => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zmult_zless_mono2
% 0.23/0.46 thf(fact_3183_int__mod__neg__eq,axiom,
% 0.23/0.46 ! [A: int,B2: int,Q2: int,R2: int] :
% 0.23/0.46 ( ( A
% 0.23/0.46 = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 0.23/0.46 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.23/0.46 => ( ( ord_less_int @ B2 @ R2 )
% 0.23/0.46 => ( ( modulo_modulo_int @ A @ B2 )
% 0.23/0.46 = R2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_mod_neg_eq
% 0.23/0.46 thf(fact_3184_int__mod__pos__eq,axiom,
% 0.23/0.46 ! [A: int,B2: int,Q2: int,R2: int] :
% 0.23/0.46 ( ( A
% 0.23/0.46 = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.23/0.46 => ( ( ord_less_int @ R2 @ B2 )
% 0.23/0.46 => ( ( modulo_modulo_int @ A @ B2 )
% 0.23/0.46 = R2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_mod_pos_eq
% 0.23/0.46 thf(fact_3185_pos__zmult__eq__1__iff,axiom,
% 0.23/0.46 ! [M: int,N: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ M )
% 0.23/0.46 => ( ( ( times_times_int @ M @ N )
% 0.23/0.46 = one_one_int )
% 0.23/0.46 = ( ( M = one_one_int )
% 0.23/0.46 & ( N = one_one_int ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pos_zmult_eq_1_iff
% 0.23/0.46 thf(fact_3186_plusinfinity,axiom,
% 0.23/0.46 ! [D: int,P4: int > $o,P: int > $o] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ D )
% 0.23/0.46 => ( ! [X3: int,K2: int] :
% 0.23/0.46 ( ( P4 @ X3 )
% 0.23/0.46 = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.23/0.46 => ( ? [Z3: int] :
% 0.23/0.46 ! [X3: int] :
% 0.23/0.46 ( ( ord_less_int @ Z3 @ X3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [X_12: int] : ( P4 @ X_12 )
% 0.23/0.46 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % plusinfinity
% 0.23/0.46 thf(fact_3187_zdiv__mono2__lemma,axiom,
% 0.23/0.46 ! [B2: int,Q2: int,R2: int,B: int,Q5: int,R4: int] :
% 0.23/0.46 ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 )
% 0.23/0.46 = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 0.23/0.46 => ( ( ord_less_int @ R4 @ B )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.23/0.46 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.23/0.46 => ( ( ord_less_eq_int @ B @ B2 )
% 0.23/0.46 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zdiv_mono2_lemma
% 0.23/0.46 thf(fact_3188_minusinfinity,axiom,
% 0.23/0.46 ! [D: int,P1: int > $o,P: int > $o] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ D )
% 0.23/0.46 => ( ! [X3: int,K2: int] :
% 0.23/0.46 ( ( P1 @ X3 )
% 0.23/0.46 = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.23/0.46 => ( ? [Z3: int] :
% 0.23/0.46 ! [X3: int] :
% 0.23/0.46 ( ( ord_less_int @ X3 @ Z3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P1 @ X3 ) ) )
% 0.23/0.46 => ( ? [X_12: int] : ( P1 @ X_12 )
% 0.23/0.46 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minusinfinity
% 0.23/0.46 thf(fact_3189_decr__mult__lemma,axiom,
% 0.23/0.46 ! [D: int,P: int > $o,K: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ D )
% 0.23/0.46 => ( ! [X3: int] :
% 0.23/0.46 ( ( P @ X3 )
% 0.23/0.46 => ( P @ ( minus_minus_int @ X3 @ D ) ) )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.46 => ! [X5: int] :
% 0.23/0.46 ( ( P @ X5 )
% 0.23/0.46 => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % decr_mult_lemma
% 0.23/0.46 thf(fact_3190_incr__mult__lemma,axiom,
% 0.23/0.46 ! [D: int,P: int > $o,K: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ D )
% 0.23/0.46 => ( ! [X3: int] :
% 0.23/0.46 ( ( P @ X3 )
% 0.23/0.46 => ( P @ ( plus_plus_int @ X3 @ D ) ) )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.46 => ! [X5: int] :
% 0.23/0.46 ( ( P @ X5 )
% 0.23/0.46 => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % incr_mult_lemma
% 0.23/0.46 thf(fact_3191_zdiv__mono2__neg__lemma,axiom,
% 0.23/0.46 ! [B2: int,Q2: int,R2: int,B: int,Q5: int,R4: int] :
% 0.23/0.46 ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 )
% 0.23/0.46 = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 0.23/0.46 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ zero_zero_int )
% 0.23/0.46 => ( ( ord_less_int @ R2 @ B2 )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.23/0.46 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.23/0.46 => ( ( ord_less_eq_int @ B @ B2 )
% 0.23/0.46 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zdiv_mono2_neg_lemma
% 0.23/0.46 thf(fact_3192_int__one__le__iff__zero__less,axiom,
% 0.23/0.46 ! [Z: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ one_one_int @ Z )
% 0.23/0.46 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_one_le_iff_zero_less
% 0.23/0.46 thf(fact_3193_unique__quotient__lemma,axiom,
% 0.23/0.46 ! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.23/0.46 => ( ( ord_less_int @ R4 @ B2 )
% 0.23/0.46 => ( ( ord_less_int @ R2 @ B2 )
% 0.23/0.46 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % unique_quotient_lemma
% 0.23/0.46 thf(fact_3194_unique__quotient__lemma__neg,axiom,
% 0.23/0.46 ! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 0.23/0.46 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.23/0.46 => ( ( ord_less_int @ B2 @ R2 )
% 0.23/0.46 => ( ( ord_less_int @ B2 @ R4 )
% 0.23/0.46 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % unique_quotient_lemma_neg
% 0.23/0.46 thf(fact_3195_mod__pos__neg__trivial,axiom,
% 0.23/0.46 ! [K: int,L2: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ K )
% 0.23/0.46 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 0.23/0.46 => ( ( modulo_modulo_int @ K @ L2 )
% 0.23/0.46 = ( plus_plus_int @ K @ L2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mod_pos_neg_trivial
% 0.23/0.46 thf(fact_3196_zdvd__reduce,axiom,
% 0.23/0.46 ! [K: int,N: int,M: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 0.23/0.46 = ( dvd_dvd_int @ K @ N ) ) ).
% 0.23/0.46
% 0.23/0.46 % zdvd_reduce
% 0.23/0.46 thf(fact_3197_zdvd__period,axiom,
% 0.23/0.46 ! [A: int,D: int,X4: int,T: int,C: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ A @ D )
% 0.23/0.46 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X4 @ T ) )
% 0.23/0.46 = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X4 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zdvd_period
% 0.23/0.46 thf(fact_3198_odd__nonzero,axiom,
% 0.23/0.46 ! [Z: int] :
% 0.23/0.46 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 0.23/0.46 != zero_zero_int ) ).
% 0.23/0.46
% 0.23/0.46 % odd_nonzero
% 0.23/0.46 thf(fact_3199_times__int__code_I2_J,axiom,
% 0.23/0.46 ! [L2: int] :
% 0.23/0.46 ( ( times_times_int @ zero_zero_int @ L2 )
% 0.23/0.46 = zero_zero_int ) ).
% 0.23/0.46
% 0.23/0.46 % times_int_code(2)
% 0.23/0.46 thf(fact_3200_times__int__code_I1_J,axiom,
% 0.23/0.46 ! [K: int] :
% 0.23/0.46 ( ( times_times_int @ K @ zero_zero_int )
% 0.23/0.46 = zero_zero_int ) ).
% 0.23/0.46
% 0.23/0.46 % times_int_code(1)
% 0.23/0.46 thf(fact_3201_zmod__eq__0D,axiom,
% 0.23/0.46 ! [M: int,D: int] :
% 0.23/0.46 ( ( ( modulo_modulo_int @ M @ D )
% 0.23/0.46 = zero_zero_int )
% 0.23/0.46 => ? [Q3: int] :
% 0.23/0.46 ( M
% 0.23/0.46 = ( times_times_int @ D @ Q3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zmod_eq_0D
% 0.23/0.46 thf(fact_3202_zdvd__mult__cancel,axiom,
% 0.23/0.46 ! [K: int,M: int,N: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 0.23/0.46 => ( ( K != zero_zero_int )
% 0.23/0.46 => ( dvd_dvd_int @ M @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zdvd_mult_cancel
% 0.23/0.46 thf(fact_3203_zdvd__mono,axiom,
% 0.23/0.46 ! [K: int,M: int,T: int] :
% 0.23/0.46 ( ( K != zero_zero_int )
% 0.23/0.46 => ( ( dvd_dvd_int @ M @ T )
% 0.23/0.46 = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zdvd_mono
% 0.23/0.46 thf(fact_3204_zmod__eq__0__iff,axiom,
% 0.23/0.46 ! [M: int,D: int] :
% 0.23/0.46 ( ( ( modulo_modulo_int @ M @ D )
% 0.23/0.46 = zero_zero_int )
% 0.23/0.46 = ( ? [Q4: int] :
% 0.23/0.46 ( M
% 0.23/0.46 = ( times_times_int @ D @ Q4 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zmod_eq_0_iff
% 0.23/0.46 thf(fact_3205_imult__is__0,axiom,
% 0.23/0.46 ! [M: extended_enat,N: extended_enat] :
% 0.23/0.46 ( ( ( times_7803423173614009249d_enat @ M @ N )
% 0.23/0.46 = zero_z5237406670263579293d_enat )
% 0.23/0.46 = ( ( M = zero_z5237406670263579293d_enat )
% 0.23/0.46 | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % imult_is_0
% 0.23/0.46 thf(fact_3206_signed__take__bit__mult,axiom,
% 0.23/0.46 ! [N: nat,K: int,L2: int] :
% 0.23/0.46 ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 0.23/0.46 = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % signed_take_bit_mult
% 0.23/0.46 thf(fact_3207_Euclidean__Division_Opos__mod__bound,axiom,
% 0.23/0.46 ! [L2: int,K: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.23/0.46 => ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % Euclidean_Division.pos_mod_bound
% 0.23/0.46 thf(fact_3208_neg__mod__bound,axiom,
% 0.23/0.46 ! [L2: int,K: int] :
% 0.23/0.46 ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.23/0.46 => ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_mod_bound
% 0.23/0.46 thf(fact_3209_Euclidean__Division_Opos__mod__sign,axiom,
% 0.23/0.46 ! [L2: int,K: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.23/0.46 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % Euclidean_Division.pos_mod_sign
% 0.23/0.46 thf(fact_3210_neg__mod__sign,axiom,
% 0.23/0.46 ! [L2: int,K: int] :
% 0.23/0.46 ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.23/0.46 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_mod_sign
% 0.23/0.46 thf(fact_3211_zmod__le__nonneg__dividend,axiom,
% 0.23/0.46 ! [M: int,K: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.23/0.46 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 0.23/0.46
% 0.23/0.46 % zmod_le_nonneg_dividend
% 0.23/0.46 thf(fact_3212_zmod__trivial__iff,axiom,
% 0.23/0.46 ! [I: int,K: int] :
% 0.23/0.46 ( ( ( modulo_modulo_int @ I @ K )
% 0.23/0.46 = I )
% 0.23/0.46 = ( ( K = zero_zero_int )
% 0.23/0.46 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.23/0.46 & ( ord_less_int @ I @ K ) )
% 0.23/0.46 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.23/0.46 & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zmod_trivial_iff
% 0.23/0.46 thf(fact_3213_mod__int__pos__iff,axiom,
% 0.23/0.46 ! [K: int,L2: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) )
% 0.23/0.46 = ( ( dvd_dvd_int @ L2 @ K )
% 0.23/0.46 | ( ( L2 = zero_zero_int )
% 0.23/0.46 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 0.23/0.46 | ( ord_less_int @ zero_zero_int @ L2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mod_int_pos_iff
% 0.23/0.46 thf(fact_3214_pos__mod__conj,axiom,
% 0.23/0.46 ! [B2: int,A: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) )
% 0.23/0.46 & ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pos_mod_conj
% 0.23/0.46 thf(fact_3215_neg__mod__conj,axiom,
% 0.23/0.46 ! [B2: int,A: int] :
% 0.23/0.46 ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.23/0.46 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ zero_zero_int )
% 0.23/0.46 & ( ord_less_int @ B2 @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_mod_conj
% 0.23/0.46 thf(fact_3216_zdvd__not__zless,axiom,
% 0.23/0.46 ! [M: int,N: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ M )
% 0.23/0.46 => ( ( ord_less_int @ M @ N )
% 0.23/0.46 => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zdvd_not_zless
% 0.23/0.46 thf(fact_3217_zdvd__imp__le,axiom,
% 0.23/0.46 ! [Z: int,N: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ Z @ N )
% 0.23/0.46 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.23/0.46 => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zdvd_imp_le
% 0.23/0.46 thf(fact_3218_less__int__code_I1_J,axiom,
% 0.23/0.46 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.23/0.46
% 0.23/0.46 % less_int_code(1)
% 0.23/0.46 thf(fact_3219_plus__int__code_I2_J,axiom,
% 0.23/0.46 ! [L2: int] :
% 0.23/0.46 ( ( plus_plus_int @ zero_zero_int @ L2 )
% 0.23/0.46 = L2 ) ).
% 0.23/0.46
% 0.23/0.46 % plus_int_code(2)
% 0.23/0.46 thf(fact_3220_plus__int__code_I1_J,axiom,
% 0.23/0.46 ! [K: int] :
% 0.23/0.46 ( ( plus_plus_int @ K @ zero_zero_int )
% 0.23/0.46 = K ) ).
% 0.23/0.46
% 0.23/0.46 % plus_int_code(1)
% 0.23/0.46 thf(fact_3221_minus__int__code_I1_J,axiom,
% 0.23/0.46 ! [K: int] :
% 0.23/0.46 ( ( minus_minus_int @ K @ zero_zero_int )
% 0.23/0.46 = K ) ).
% 0.23/0.46
% 0.23/0.46 % minus_int_code(1)
% 0.23/0.46 thf(fact_3222_zdvd__zdiffD,axiom,
% 0.23/0.46 ! [K: int,M: int,N: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 0.23/0.46 => ( ( dvd_dvd_int @ K @ N )
% 0.23/0.46 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zdvd_zdiffD
% 0.23/0.46 thf(fact_3223_signed__take__bit__add,axiom,
% 0.23/0.46 ! [N: nat,K: int,L2: int] :
% 0.23/0.46 ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 0.23/0.46 = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % signed_take_bit_add
% 0.23/0.46 thf(fact_3224_signed__take__bit__diff,axiom,
% 0.23/0.46 ! [N: nat,K: int,L2: int] :
% 0.23/0.46 ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 0.23/0.46 = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % signed_take_bit_diff
% 0.23/0.46 thf(fact_3225_zero__one__enat__neq_I1_J,axiom,
% 0.23/0.46 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 0.23/0.46
% 0.23/0.46 % zero_one_enat_neq(1)
% 0.23/0.46 thf(fact_3226_set__bit__greater__eq,axiom,
% 0.23/0.46 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 0.23/0.46
% 0.23/0.46 % set_bit_greater_eq
% 0.23/0.46 thf(fact_3227_unset__bit__less__eq,axiom,
% 0.23/0.46 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 0.23/0.46
% 0.23/0.46 % unset_bit_less_eq
% 0.23/0.46 thf(fact_3228_int__distrib_I4_J,axiom,
% 0.23/0.46 ! [W: int,Z1: int,Z22: int] :
% 0.23/0.46 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 0.23/0.46 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_distrib(4)
% 0.23/0.46 thf(fact_3229_int__distrib_I3_J,axiom,
% 0.23/0.46 ! [Z1: int,Z22: int,W: int] :
% 0.23/0.46 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 0.23/0.46 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_distrib(3)
% 0.23/0.46 thf(fact_3230_int__distrib_I2_J,axiom,
% 0.23/0.46 ! [W: int,Z1: int,Z22: int] :
% 0.23/0.46 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 0.23/0.46 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_distrib(2)
% 0.23/0.46 thf(fact_3231_int__distrib_I1_J,axiom,
% 0.23/0.46 ! [Z1: int,Z22: int,W: int] :
% 0.23/0.46 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 0.23/0.46 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_distrib(1)
% 0.23/0.46 thf(fact_3232_zless__imp__add1__zle,axiom,
% 0.23/0.46 ! [W: int,Z: int] :
% 0.23/0.46 ( ( ord_less_int @ W @ Z )
% 0.23/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 0.23/0.46
% 0.23/0.46 % zless_imp_add1_zle
% 0.23/0.46 thf(fact_3233_int__less__induct,axiom,
% 0.23/0.46 ! [I: int,K: int,P: int > $o] :
% 0.23/0.46 ( ( ord_less_int @ I @ K )
% 0.23/0.46 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.23/0.46 => ( ! [I3: int] :
% 0.23/0.46 ( ( ord_less_int @ I3 @ K )
% 0.23/0.46 => ( ( P @ I3 )
% 0.23/0.46 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.23/0.46 => ( P @ I ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_less_induct
% 0.23/0.46 thf(fact_3234_zless__add1__eq,axiom,
% 0.23/0.46 ! [W: int,Z: int] :
% 0.23/0.46 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.23/0.46 = ( ( ord_less_int @ W @ Z )
% 0.23/0.46 | ( W = Z ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % zless_add1_eq
% 0.23/0.46 thf(fact_3235_int__le__induct,axiom,
% 0.23/0.46 ! [I: int,K: int,P: int > $o] :
% 0.23/0.46 ( ( ord_less_eq_int @ I @ K )
% 0.23/0.46 => ( ( P @ K )
% 0.23/0.46 => ( ! [I3: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ I3 @ K )
% 0.23/0.46 => ( ( P @ I3 )
% 0.23/0.46 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.23/0.46 => ( P @ I ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_le_induct
% 0.23/0.46 thf(fact_3236_int__gr__induct,axiom,
% 0.23/0.46 ! [K: int,I: int,P: int > $o] :
% 0.23/0.46 ( ( ord_less_int @ K @ I )
% 0.23/0.46 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 0.23/0.46 => ( ! [I3: int] :
% 0.23/0.46 ( ( ord_less_int @ K @ I3 )
% 0.23/0.46 => ( ( P @ I3 )
% 0.23/0.46 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.23/0.46 => ( P @ I ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_gr_induct
% 0.23/0.46 thf(fact_3237_int__ge__induct,axiom,
% 0.23/0.46 ! [K: int,I: int,P: int > $o] :
% 0.23/0.46 ( ( ord_less_eq_int @ K @ I )
% 0.23/0.46 => ( ( P @ K )
% 0.23/0.46 => ( ! [I3: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ K @ I3 )
% 0.23/0.46 => ( ( P @ I3 )
% 0.23/0.46 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.23/0.46 => ( P @ I ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_ge_induct
% 0.23/0.46 thf(fact_3238_add1__zle__eq,axiom,
% 0.23/0.46 ! [W: int,Z: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 0.23/0.46 = ( ord_less_int @ W @ Z ) ) ).
% 0.23/0.46
% 0.23/0.46 % add1_zle_eq
% 0.23/0.46 thf(fact_3239_int__induct,axiom,
% 0.23/0.46 ! [P: int > $o,K: int,I: int] :
% 0.23/0.46 ( ( P @ K )
% 0.23/0.46 => ( ! [I3: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ K @ I3 )
% 0.23/0.46 => ( ( P @ I3 )
% 0.23/0.46 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.23/0.46 => ( ! [I3: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ I3 @ K )
% 0.23/0.46 => ( ( P @ I3 )
% 0.23/0.46 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.23/0.46 => ( P @ I ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % int_induct
% 0.23/0.46 thf(fact_3240_even__diff__iff,axiom,
% 0.23/0.46 ! [K: int,L2: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
% 0.23/0.46 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % even_diff_iff
% 0.23/0.46 thf(fact_3241_minf_I7_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.23/0.46 => ~ ( ord_less_real @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(7)
% 0.23/0.46 thf(fact_3242_minf_I7_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.23/0.46 => ~ ( ord_less_rat @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(7)
% 0.23/0.46 thf(fact_3243_minf_I7_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ X5 @ Z2 )
% 0.23/0.46 => ~ ( ord_less_num @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(7)
% 0.23/0.46 thf(fact_3244_minf_I7_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.23/0.46 => ~ ( ord_less_nat @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(7)
% 0.23/0.46 thf(fact_3245_minf_I7_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.23/0.46 => ~ ( ord_less_int @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(7)
% 0.23/0.46 thf(fact_3246_minf_I5_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.23/0.46 => ( ord_less_real @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(5)
% 0.23/0.46 thf(fact_3247_minf_I5_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.23/0.46 => ( ord_less_rat @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(5)
% 0.23/0.46 thf(fact_3248_minf_I5_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ X5 @ Z2 )
% 0.23/0.46 => ( ord_less_num @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(5)
% 0.23/0.46 thf(fact_3249_minf_I5_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.23/0.46 => ( ord_less_nat @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(5)
% 0.23/0.46 thf(fact_3250_minf_I5_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.23/0.46 => ( ord_less_int @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(5)
% 0.23/0.46 thf(fact_3251_minf_I4_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(4)
% 0.23/0.46 thf(fact_3252_minf_I4_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(4)
% 0.23/0.46 thf(fact_3253_minf_I4_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ X5 @ Z2 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(4)
% 0.23/0.46 thf(fact_3254_minf_I4_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(4)
% 0.23/0.46 thf(fact_3255_minf_I4_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(4)
% 0.23/0.46 thf(fact_3256_minf_I3_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(3)
% 0.23/0.46 thf(fact_3257_minf_I3_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(3)
% 0.23/0.46 thf(fact_3258_minf_I3_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ X5 @ Z2 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(3)
% 0.23/0.46 thf(fact_3259_minf_I3_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(3)
% 0.23/0.46 thf(fact_3260_minf_I3_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(3)
% 0.23/0.46 thf(fact_3261_minf_I2_J,axiom,
% 0.23/0.46 ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
% 0.23/0.46 ( ? [Z3: real] :
% 0.23/0.46 ! [X3: real] :
% 0.23/0.46 ( ( ord_less_real @ X3 @ Z3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: real] :
% 0.23/0.46 ! [X3: real] :
% 0.23/0.46 ( ( ord_less_real @ X3 @ Z3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(2)
% 0.23/0.46 thf(fact_3262_minf_I2_J,axiom,
% 0.23/0.46 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.23/0.46 ( ? [Z3: rat] :
% 0.23/0.46 ! [X3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X3 @ Z3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: rat] :
% 0.23/0.46 ! [X3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X3 @ Z3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(2)
% 0.23/0.46 thf(fact_3263_minf_I2_J,axiom,
% 0.23/0.46 ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
% 0.23/0.46 ( ? [Z3: num] :
% 0.23/0.46 ! [X3: num] :
% 0.23/0.46 ( ( ord_less_num @ X3 @ Z3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: num] :
% 0.23/0.46 ! [X3: num] :
% 0.23/0.46 ( ( ord_less_num @ X3 @ Z3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ X5 @ Z2 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(2)
% 0.23/0.46 thf(fact_3264_minf_I2_J,axiom,
% 0.23/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.23/0.46 ( ? [Z3: nat] :
% 0.23/0.46 ! [X3: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X3 @ Z3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: nat] :
% 0.23/0.46 ! [X3: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X3 @ Z3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(2)
% 0.23/0.46 thf(fact_3265_minf_I2_J,axiom,
% 0.23/0.46 ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
% 0.23/0.46 ( ? [Z3: int] :
% 0.23/0.46 ! [X3: int] :
% 0.23/0.46 ( ( ord_less_int @ X3 @ Z3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: int] :
% 0.23/0.46 ! [X3: int] :
% 0.23/0.46 ( ( ord_less_int @ X3 @ Z3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(2)
% 0.23/0.46 thf(fact_3266_minf_I1_J,axiom,
% 0.23/0.46 ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
% 0.23/0.46 ( ? [Z3: real] :
% 0.23/0.46 ! [X3: real] :
% 0.23/0.46 ( ( ord_less_real @ X3 @ Z3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: real] :
% 0.23/0.46 ! [X3: real] :
% 0.23/0.46 ( ( ord_less_real @ X3 @ Z3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(1)
% 0.23/0.46 thf(fact_3267_minf_I1_J,axiom,
% 0.23/0.46 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.23/0.46 ( ? [Z3: rat] :
% 0.23/0.46 ! [X3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X3 @ Z3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: rat] :
% 0.23/0.46 ! [X3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X3 @ Z3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(1)
% 0.23/0.46 thf(fact_3268_minf_I1_J,axiom,
% 0.23/0.46 ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
% 0.23/0.46 ( ? [Z3: num] :
% 0.23/0.46 ! [X3: num] :
% 0.23/0.46 ( ( ord_less_num @ X3 @ Z3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: num] :
% 0.23/0.46 ! [X3: num] :
% 0.23/0.46 ( ( ord_less_num @ X3 @ Z3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ X5 @ Z2 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(1)
% 0.23/0.46 thf(fact_3269_minf_I1_J,axiom,
% 0.23/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.23/0.46 ( ? [Z3: nat] :
% 0.23/0.46 ! [X3: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X3 @ Z3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: nat] :
% 0.23/0.46 ! [X3: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X3 @ Z3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(1)
% 0.23/0.46 thf(fact_3270_minf_I1_J,axiom,
% 0.23/0.46 ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
% 0.23/0.46 ( ? [Z3: int] :
% 0.23/0.46 ! [X3: int] :
% 0.23/0.46 ( ( ord_less_int @ X3 @ Z3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: int] :
% 0.23/0.46 ! [X3: int] :
% 0.23/0.46 ( ( ord_less_int @ X3 @ Z3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(1)
% 0.23/0.46 thf(fact_3271_pinf_I7_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.23/0.46 => ( ord_less_real @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(7)
% 0.23/0.46 thf(fact_3272_pinf_I7_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.23/0.46 => ( ord_less_rat @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(7)
% 0.23/0.46 thf(fact_3273_pinf_I7_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ Z2 @ X5 )
% 0.23/0.46 => ( ord_less_num @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(7)
% 0.23/0.46 thf(fact_3274_pinf_I7_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.23/0.46 => ( ord_less_nat @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(7)
% 0.23/0.46 thf(fact_3275_pinf_I7_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.23/0.46 => ( ord_less_int @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(7)
% 0.23/0.46 thf(fact_3276_pinf_I5_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.23/0.46 => ~ ( ord_less_real @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(5)
% 0.23/0.46 thf(fact_3277_pinf_I5_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.23/0.46 => ~ ( ord_less_rat @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(5)
% 0.23/0.46 thf(fact_3278_pinf_I5_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ Z2 @ X5 )
% 0.23/0.46 => ~ ( ord_less_num @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(5)
% 0.23/0.46 thf(fact_3279_pinf_I5_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.23/0.46 => ~ ( ord_less_nat @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(5)
% 0.23/0.46 thf(fact_3280_pinf_I5_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.23/0.46 => ~ ( ord_less_int @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(5)
% 0.23/0.46 thf(fact_3281_pinf_I4_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(4)
% 0.23/0.46 thf(fact_3282_pinf_I4_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(4)
% 0.23/0.46 thf(fact_3283_pinf_I4_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ Z2 @ X5 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(4)
% 0.23/0.46 thf(fact_3284_pinf_I4_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(4)
% 0.23/0.46 thf(fact_3285_pinf_I4_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(4)
% 0.23/0.46 thf(fact_3286_pinf_I3_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(3)
% 0.23/0.46 thf(fact_3287_pinf_I3_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(3)
% 0.23/0.46 thf(fact_3288_pinf_I3_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ Z2 @ X5 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(3)
% 0.23/0.46 thf(fact_3289_pinf_I3_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(3)
% 0.23/0.46 thf(fact_3290_pinf_I3_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.23/0.46 => ( X5 != T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(3)
% 0.23/0.46 thf(fact_3291_pinf_I2_J,axiom,
% 0.23/0.46 ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
% 0.23/0.46 ( ? [Z3: real] :
% 0.23/0.46 ! [X3: real] :
% 0.23/0.46 ( ( ord_less_real @ Z3 @ X3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: real] :
% 0.23/0.46 ! [X3: real] :
% 0.23/0.46 ( ( ord_less_real @ Z3 @ X3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(2)
% 0.23/0.46 thf(fact_3292_pinf_I2_J,axiom,
% 0.23/0.46 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.23/0.46 ( ? [Z3: rat] :
% 0.23/0.46 ! [X3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z3 @ X3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: rat] :
% 0.23/0.46 ! [X3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z3 @ X3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(2)
% 0.23/0.46 thf(fact_3293_pinf_I2_J,axiom,
% 0.23/0.46 ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
% 0.23/0.46 ( ? [Z3: num] :
% 0.23/0.46 ! [X3: num] :
% 0.23/0.46 ( ( ord_less_num @ Z3 @ X3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: num] :
% 0.23/0.46 ! [X3: num] :
% 0.23/0.46 ( ( ord_less_num @ Z3 @ X3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ Z2 @ X5 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(2)
% 0.23/0.46 thf(fact_3294_pinf_I2_J,axiom,
% 0.23/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.23/0.46 ( ? [Z3: nat] :
% 0.23/0.46 ! [X3: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z3 @ X3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: nat] :
% 0.23/0.46 ! [X3: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z3 @ X3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(2)
% 0.23/0.46 thf(fact_3295_pinf_I2_J,axiom,
% 0.23/0.46 ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
% 0.23/0.46 ( ? [Z3: int] :
% 0.23/0.46 ! [X3: int] :
% 0.23/0.46 ( ( ord_less_int @ Z3 @ X3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: int] :
% 0.23/0.46 ! [X3: int] :
% 0.23/0.46 ( ( ord_less_int @ Z3 @ X3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(2)
% 0.23/0.46 thf(fact_3296_pinf_I1_J,axiom,
% 0.23/0.46 ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
% 0.23/0.46 ( ? [Z3: real] :
% 0.23/0.46 ! [X3: real] :
% 0.23/0.46 ( ( ord_less_real @ Z3 @ X3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: real] :
% 0.23/0.46 ! [X3: real] :
% 0.23/0.46 ( ( ord_less_real @ Z3 @ X3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(1)
% 0.23/0.46 thf(fact_3297_pinf_I1_J,axiom,
% 0.23/0.46 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.23/0.46 ( ? [Z3: rat] :
% 0.23/0.46 ! [X3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z3 @ X3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: rat] :
% 0.23/0.46 ! [X3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z3 @ X3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(1)
% 0.23/0.46 thf(fact_3298_pinf_I1_J,axiom,
% 0.23/0.46 ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
% 0.23/0.46 ( ? [Z3: num] :
% 0.23/0.46 ! [X3: num] :
% 0.23/0.46 ( ( ord_less_num @ Z3 @ X3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: num] :
% 0.23/0.46 ! [X3: num] :
% 0.23/0.46 ( ( ord_less_num @ Z3 @ X3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ Z2 @ X5 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(1)
% 0.23/0.46 thf(fact_3299_pinf_I1_J,axiom,
% 0.23/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.23/0.46 ( ? [Z3: nat] :
% 0.23/0.46 ! [X3: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z3 @ X3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: nat] :
% 0.23/0.46 ! [X3: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z3 @ X3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(1)
% 0.23/0.46 thf(fact_3300_pinf_I1_J,axiom,
% 0.23/0.46 ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
% 0.23/0.46 ( ? [Z3: int] :
% 0.23/0.46 ! [X3: int] :
% 0.23/0.46 ( ( ord_less_int @ Z3 @ X3 )
% 0.23/0.46 => ( ( P @ X3 )
% 0.23/0.46 = ( P4 @ X3 ) ) )
% 0.23/0.46 => ( ? [Z3: int] :
% 0.23/0.46 ! [X3: int] :
% 0.23/0.46 ( ( ord_less_int @ Z3 @ X3 )
% 0.23/0.46 => ( ( Q @ X3 )
% 0.23/0.46 = ( Q6 @ X3 ) ) )
% 0.23/0.46 => ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.23/0.46 => ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P4 @ X5 )
% 0.23/0.46 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(1)
% 0.23/0.46 thf(fact_3301_vebt__buildup_Ocases,axiom,
% 0.23/0.46 ! [X4: nat] :
% 0.23/0.46 ( ( X4 != zero_zero_nat )
% 0.23/0.46 => ( ( X4
% 0.23/0.46 != ( suc @ zero_zero_nat ) )
% 0.23/0.46 => ~ ! [Va: nat] :
% 0.23/0.46 ( X4
% 0.23/0.46 != ( suc @ ( suc @ Va ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % vebt_buildup.cases
% 0.23/0.46 thf(fact_3302_Euclid__induct,axiom,
% 0.23/0.46 ! [P: nat > nat > $o,A: nat,B2: nat] :
% 0.23/0.46 ( ! [A5: nat,B5: nat] :
% 0.23/0.46 ( ( P @ A5 @ B5 )
% 0.23/0.46 = ( P @ B5 @ A5 ) )
% 0.23/0.46 => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
% 0.23/0.46 => ( ! [A5: nat,B5: nat] :
% 0.23/0.46 ( ( P @ A5 @ B5 )
% 0.23/0.46 => ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
% 0.23/0.46 => ( P @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % Euclid_induct
% 0.23/0.46 thf(fact_3303_minf_I8_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.23/0.46 => ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(8)
% 0.23/0.46 thf(fact_3304_minf_I8_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.23/0.46 => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(8)
% 0.23/0.46 thf(fact_3305_minf_I8_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ X5 @ Z2 )
% 0.23/0.46 => ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(8)
% 0.23/0.46 thf(fact_3306_minf_I8_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.23/0.46 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(8)
% 0.23/0.46 thf(fact_3307_minf_I8_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.23/0.46 => ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(8)
% 0.23/0.46 thf(fact_3308_minf_I6_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.23/0.46 => ( ord_less_eq_real @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(6)
% 0.23/0.46 thf(fact_3309_minf_I6_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.23/0.46 => ( ord_less_eq_rat @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(6)
% 0.23/0.46 thf(fact_3310_minf_I6_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ X5 @ Z2 )
% 0.23/0.46 => ( ord_less_eq_num @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(6)
% 0.23/0.46 thf(fact_3311_minf_I6_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.23/0.46 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(6)
% 0.23/0.46 thf(fact_3312_minf_I6_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.23/0.46 => ( ord_less_eq_int @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(6)
% 0.23/0.46 thf(fact_3313_pinf_I8_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.23/0.46 => ( ord_less_eq_real @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(8)
% 0.23/0.46 thf(fact_3314_pinf_I8_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.23/0.46 => ( ord_less_eq_rat @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(8)
% 0.23/0.46 thf(fact_3315_pinf_I8_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ Z2 @ X5 )
% 0.23/0.46 => ( ord_less_eq_num @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(8)
% 0.23/0.46 thf(fact_3316_pinf_I8_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.23/0.46 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(8)
% 0.23/0.46 thf(fact_3317_pinf_I8_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.23/0.46 => ( ord_less_eq_int @ T @ X5 ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(8)
% 0.23/0.46 thf(fact_3318_pinf_I6_J,axiom,
% 0.23/0.46 ! [T: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.23/0.46 => ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(6)
% 0.23/0.46 thf(fact_3319_pinf_I6_J,axiom,
% 0.23/0.46 ! [T: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.23/0.46 => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(6)
% 0.23/0.46 thf(fact_3320_pinf_I6_J,axiom,
% 0.23/0.46 ! [T: num] :
% 0.23/0.46 ? [Z2: num] :
% 0.23/0.46 ! [X5: num] :
% 0.23/0.46 ( ( ord_less_num @ Z2 @ X5 )
% 0.23/0.46 => ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(6)
% 0.23/0.46 thf(fact_3321_pinf_I6_J,axiom,
% 0.23/0.46 ! [T: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.23/0.46 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(6)
% 0.23/0.46 thf(fact_3322_pinf_I6_J,axiom,
% 0.23/0.46 ! [T: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.23/0.46 => ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(6)
% 0.23/0.46 thf(fact_3323_inf__period_I2_J,axiom,
% 0.23/0.46 ! [P: real > $o,D4: real,Q: real > $o] :
% 0.23/0.46 ( ! [X3: real,K2: real] :
% 0.23/0.46 ( ( P @ X3 )
% 0.23/0.46 = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ( ! [X3: real,K2: real] :
% 0.23/0.46 ( ( Q @ X3 )
% 0.23/0.46 = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ! [X5: real,K4: real] :
% 0.23/0.46 ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
% 0.23/0.46 | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(2)
% 0.23/0.46 thf(fact_3324_inf__period_I2_J,axiom,
% 0.23/0.46 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 0.23/0.46 ( ! [X3: rat,K2: rat] :
% 0.23/0.46 ( ( P @ X3 )
% 0.23/0.46 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ( ! [X3: rat,K2: rat] :
% 0.23/0.46 ( ( Q @ X3 )
% 0.23/0.46 = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ! [X5: rat,K4: rat] :
% 0.23/0.46 ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
% 0.23/0.46 | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(2)
% 0.23/0.46 thf(fact_3325_inf__period_I2_J,axiom,
% 0.23/0.46 ! [P: int > $o,D4: int,Q: int > $o] :
% 0.23/0.46 ( ! [X3: int,K2: int] :
% 0.23/0.46 ( ( P @ X3 )
% 0.23/0.46 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ( ! [X3: int,K2: int] :
% 0.23/0.46 ( ( Q @ X3 )
% 0.23/0.46 = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ! [X5: int,K4: int] :
% 0.23/0.46 ( ( ( P @ X5 )
% 0.23/0.46 | ( Q @ X5 ) )
% 0.23/0.46 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
% 0.23/0.46 | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(2)
% 0.23/0.46 thf(fact_3326_inf__period_I1_J,axiom,
% 0.23/0.46 ! [P: real > $o,D4: real,Q: real > $o] :
% 0.23/0.46 ( ! [X3: real,K2: real] :
% 0.23/0.46 ( ( P @ X3 )
% 0.23/0.46 = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ( ! [X3: real,K2: real] :
% 0.23/0.46 ( ( Q @ X3 )
% 0.23/0.46 = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ! [X5: real,K4: real] :
% 0.23/0.46 ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
% 0.23/0.46 & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(1)
% 0.23/0.46 thf(fact_3327_inf__period_I1_J,axiom,
% 0.23/0.46 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 0.23/0.46 ( ! [X3: rat,K2: rat] :
% 0.23/0.46 ( ( P @ X3 )
% 0.23/0.46 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ( ! [X3: rat,K2: rat] :
% 0.23/0.46 ( ( Q @ X3 )
% 0.23/0.46 = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ! [X5: rat,K4: rat] :
% 0.23/0.46 ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
% 0.23/0.46 & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(1)
% 0.23/0.46 thf(fact_3328_inf__period_I1_J,axiom,
% 0.23/0.46 ! [P: int > $o,D4: int,Q: int > $o] :
% 0.23/0.46 ( ! [X3: int,K2: int] :
% 0.23/0.46 ( ( P @ X3 )
% 0.23/0.46 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ( ! [X3: int,K2: int] :
% 0.23/0.46 ( ( Q @ X3 )
% 0.23/0.46 = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 0.23/0.46 => ! [X5: int,K4: int] :
% 0.23/0.46 ( ( ( P @ X5 )
% 0.23/0.46 & ( Q @ X5 ) )
% 0.23/0.46 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
% 0.23/0.46 & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % inf_period(1)
% 0.23/0.46 thf(fact_3329_dvd__productE,axiom,
% 0.23/0.46 ! [P2: nat,A: nat,B2: nat] :
% 0.23/0.46 ( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A @ B2 ) )
% 0.23/0.46 => ~ ! [X3: nat,Y4: nat] :
% 0.23/0.46 ( ( P2
% 0.23/0.46 = ( times_times_nat @ X3 @ Y4 ) )
% 0.23/0.46 => ( ( dvd_dvd_nat @ X3 @ A )
% 0.23/0.46 => ~ ( dvd_dvd_nat @ Y4 @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % dvd_productE
% 0.23/0.46 thf(fact_3330_dvd__productE,axiom,
% 0.23/0.46 ! [P2: int,A: int,B2: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ P2 @ ( times_times_int @ A @ B2 ) )
% 0.23/0.46 => ~ ! [X3: int,Y4: int] :
% 0.23/0.46 ( ( P2
% 0.23/0.46 = ( times_times_int @ X3 @ Y4 ) )
% 0.23/0.46 => ( ( dvd_dvd_int @ X3 @ A )
% 0.23/0.46 => ~ ( dvd_dvd_int @ Y4 @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % dvd_productE
% 0.23/0.46 thf(fact_3331_division__decomp,axiom,
% 0.23/0.46 ! [A: nat,B2: nat,C: nat] :
% 0.23/0.46 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.23/0.46 => ? [B6: nat,C4: nat] :
% 0.23/0.46 ( ( A
% 0.23/0.46 = ( times_times_nat @ B6 @ C4 ) )
% 0.23/0.46 & ( dvd_dvd_nat @ B6 @ B2 )
% 0.23/0.46 & ( dvd_dvd_nat @ C4 @ C ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % division_decomp
% 0.23/0.46 thf(fact_3332_division__decomp,axiom,
% 0.23/0.46 ! [A: int,B2: int,C: int] :
% 0.23/0.46 ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.23/0.46 => ? [B6: int,C4: int] :
% 0.23/0.46 ( ( A
% 0.23/0.46 = ( times_times_int @ B6 @ C4 ) )
% 0.23/0.46 & ( dvd_dvd_int @ B6 @ B2 )
% 0.23/0.46 & ( dvd_dvd_int @ C4 @ C ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % division_decomp
% 0.23/0.46 thf(fact_3333_dvd__pos__nat,axiom,
% 0.23/0.46 ! [N: nat,M: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.46 => ( ( dvd_dvd_nat @ M @ N )
% 0.23/0.46 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % dvd_pos_nat
% 0.23/0.46 thf(fact_3334_gcd__nat__induct,axiom,
% 0.23/0.46 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.23/0.46 ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 0.23/0.46 => ( ! [M4: nat,N4: nat] :
% 0.23/0.46 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.23/0.46 => ( ( P @ N4 @ ( modulo_modulo_nat @ M4 @ N4 ) )
% 0.23/0.46 => ( P @ M4 @ N4 ) ) )
% 0.23/0.46 => ( P @ M @ N ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % gcd_nat_induct
% 0.23/0.46 thf(fact_3335_pinf_I9_J,axiom,
% 0.23/0.46 ! [D: code_integer,S: code_integer] :
% 0.23/0.46 ? [Z2: code_integer] :
% 0.23/0.46 ! [X5: code_integer] :
% 0.23/0.46 ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
% 0.23/0.46 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 0.23/0.46 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(9)
% 0.23/0.46 thf(fact_3336_pinf_I9_J,axiom,
% 0.23/0.46 ! [D: real,S: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.23/0.46 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 0.23/0.46 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(9)
% 0.23/0.46 thf(fact_3337_pinf_I9_J,axiom,
% 0.23/0.46 ! [D: rat,S: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.23/0.46 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 0.23/0.46 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(9)
% 0.23/0.46 thf(fact_3338_pinf_I9_J,axiom,
% 0.23/0.46 ! [D: nat,S: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.23/0.46 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 0.23/0.46 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(9)
% 0.23/0.46 thf(fact_3339_pinf_I9_J,axiom,
% 0.23/0.46 ! [D: int,S: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.23/0.46 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 0.23/0.46 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(9)
% 0.23/0.46 thf(fact_3340_pinf_I10_J,axiom,
% 0.23/0.46 ! [D: code_integer,S: code_integer] :
% 0.23/0.46 ? [Z2: code_integer] :
% 0.23/0.46 ! [X5: code_integer] :
% 0.23/0.46 ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
% 0.23/0.46 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(10)
% 0.23/0.46 thf(fact_3341_pinf_I10_J,axiom,
% 0.23/0.46 ! [D: real,S: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.23/0.46 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(10)
% 0.23/0.46 thf(fact_3342_pinf_I10_J,axiom,
% 0.23/0.46 ! [D: rat,S: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.23/0.46 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(10)
% 0.23/0.46 thf(fact_3343_pinf_I10_J,axiom,
% 0.23/0.46 ! [D: nat,S: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.23/0.46 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(10)
% 0.23/0.46 thf(fact_3344_pinf_I10_J,axiom,
% 0.23/0.46 ! [D: int,S: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.23/0.46 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % pinf(10)
% 0.23/0.46 thf(fact_3345_minf_I9_J,axiom,
% 0.23/0.46 ! [D: code_integer,S: code_integer] :
% 0.23/0.46 ? [Z2: code_integer] :
% 0.23/0.46 ! [X5: code_integer] :
% 0.23/0.46 ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
% 0.23/0.46 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 0.23/0.46 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(9)
% 0.23/0.46 thf(fact_3346_minf_I9_J,axiom,
% 0.23/0.46 ! [D: real,S: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.23/0.46 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 0.23/0.46 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(9)
% 0.23/0.46 thf(fact_3347_minf_I9_J,axiom,
% 0.23/0.46 ! [D: rat,S: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.23/0.46 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 0.23/0.46 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(9)
% 0.23/0.46 thf(fact_3348_minf_I9_J,axiom,
% 0.23/0.46 ! [D: nat,S: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.23/0.46 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 0.23/0.46 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(9)
% 0.23/0.46 thf(fact_3349_minf_I9_J,axiom,
% 0.23/0.46 ! [D: int,S: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.23/0.46 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 0.23/0.46 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(9)
% 0.23/0.46 thf(fact_3350_minf_I10_J,axiom,
% 0.23/0.46 ! [D: code_integer,S: code_integer] :
% 0.23/0.46 ? [Z2: code_integer] :
% 0.23/0.46 ! [X5: code_integer] :
% 0.23/0.46 ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
% 0.23/0.46 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(10)
% 0.23/0.46 thf(fact_3351_minf_I10_J,axiom,
% 0.23/0.46 ! [D: real,S: real] :
% 0.23/0.46 ? [Z2: real] :
% 0.23/0.46 ! [X5: real] :
% 0.23/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.23/0.46 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(10)
% 0.23/0.46 thf(fact_3352_minf_I10_J,axiom,
% 0.23/0.46 ! [D: rat,S: rat] :
% 0.23/0.46 ? [Z2: rat] :
% 0.23/0.46 ! [X5: rat] :
% 0.23/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.23/0.46 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(10)
% 0.23/0.46 thf(fact_3353_minf_I10_J,axiom,
% 0.23/0.46 ! [D: nat,S: nat] :
% 0.23/0.46 ? [Z2: nat] :
% 0.23/0.46 ! [X5: nat] :
% 0.23/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.23/0.46 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(10)
% 0.23/0.46 thf(fact_3354_minf_I10_J,axiom,
% 0.23/0.46 ! [D: int,S: int] :
% 0.23/0.46 ? [Z2: int] :
% 0.23/0.46 ! [X5: int] :
% 0.23/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.23/0.46 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 0.23/0.46 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minf(10)
% 0.23/0.46 thf(fact_3355_bezout__lemma__nat,axiom,
% 0.23/0.46 ! [D: nat,A: nat,B2: nat,X4: nat,Y3: nat] :
% 0.23/0.46 ( ( dvd_dvd_nat @ D @ A )
% 0.23/0.46 => ( ( dvd_dvd_nat @ D @ B2 )
% 0.23/0.46 => ( ( ( ( times_times_nat @ A @ X4 )
% 0.23/0.46 = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D ) )
% 0.23/0.46 | ( ( times_times_nat @ B2 @ X4 )
% 0.23/0.46 = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
% 0.23/0.46 => ? [X3: nat,Y4: nat] :
% 0.23/0.46 ( ( dvd_dvd_nat @ D @ A )
% 0.23/0.46 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B2 ) )
% 0.23/0.46 & ( ( ( times_times_nat @ A @ X3 )
% 0.23/0.46 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ Y4 ) @ D ) )
% 0.23/0.46 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ X3 )
% 0.23/0.46 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % bezout_lemma_nat
% 0.23/0.46 thf(fact_3356_bezout__add__nat,axiom,
% 0.23/0.46 ! [A: nat,B2: nat] :
% 0.23/0.46 ? [D3: nat,X3: nat,Y4: nat] :
% 0.23/0.46 ( ( dvd_dvd_nat @ D3 @ A )
% 0.23/0.46 & ( dvd_dvd_nat @ D3 @ B2 )
% 0.23/0.46 & ( ( ( times_times_nat @ A @ X3 )
% 0.23/0.46 = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) )
% 0.23/0.46 | ( ( times_times_nat @ B2 @ X3 )
% 0.23/0.46 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % bezout_add_nat
% 0.23/0.46 thf(fact_3357_bezout__add__strong__nat,axiom,
% 0.23/0.46 ! [A: nat,B2: nat] :
% 0.23/0.46 ( ( A != zero_zero_nat )
% 0.23/0.46 => ? [D3: nat,X3: nat,Y4: nat] :
% 0.23/0.46 ( ( dvd_dvd_nat @ D3 @ A )
% 0.23/0.46 & ( dvd_dvd_nat @ D3 @ B2 )
% 0.23/0.46 & ( ( times_times_nat @ A @ X3 )
% 0.23/0.46 = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % bezout_add_strong_nat
% 0.23/0.46 thf(fact_3358_bezout1__nat,axiom,
% 0.23/0.46 ! [A: nat,B2: nat] :
% 0.23/0.46 ? [D3: nat,X3: nat,Y4: nat] :
% 0.23/0.46 ( ( dvd_dvd_nat @ D3 @ A )
% 0.23/0.46 & ( dvd_dvd_nat @ D3 @ B2 )
% 0.23/0.46 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B2 @ Y4 ) )
% 0.23/0.46 = D3 )
% 0.23/0.46 | ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.23/0.46 = D3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % bezout1_nat
% 0.23/0.46 thf(fact_3359_Bolzano,axiom,
% 0.23/0.46 ! [A: real,B2: real,P: real > real > $o] :
% 0.23/0.46 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.46 => ( ! [A5: real,B5: real,C3: real] :
% 0.23/0.46 ( ( P @ A5 @ B5 )
% 0.23/0.46 => ( ( P @ B5 @ C3 )
% 0.23/0.46 => ( ( ord_less_eq_real @ A5 @ B5 )
% 0.23/0.46 => ( ( ord_less_eq_real @ B5 @ C3 )
% 0.23/0.46 => ( P @ A5 @ C3 ) ) ) ) )
% 0.23/0.46 => ( ! [X3: real] :
% 0.23/0.46 ( ( ord_less_eq_real @ A @ X3 )
% 0.23/0.46 => ( ( ord_less_eq_real @ X3 @ B2 )
% 0.23/0.46 => ? [D5: real] :
% 0.23/0.46 ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.23/0.46 & ! [A5: real,B5: real] :
% 0.23/0.46 ( ( ( ord_less_eq_real @ A5 @ X3 )
% 0.23/0.46 & ( ord_less_eq_real @ X3 @ B5 )
% 0.23/0.46 & ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D5 ) )
% 0.23/0.46 => ( P @ A5 @ B5 ) ) ) ) )
% 0.23/0.46 => ( P @ A @ B2 ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % Bolzano
% 0.23/0.46 thf(fact_3360_mult__le__cancel__iff1,axiom,
% 0.23/0.46 ! [Z: real,X4: real,Y3: real] :
% 0.23/0.46 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.23/0.46 => ( ( ord_less_eq_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 0.23/0.46 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_le_cancel_iff1
% 0.23/0.46 thf(fact_3361_mult__le__cancel__iff1,axiom,
% 0.23/0.46 ! [Z: rat,X4: rat,Y3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.23/0.46 => ( ( ord_less_eq_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 0.23/0.46 = ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_le_cancel_iff1
% 0.23/0.46 thf(fact_3362_mult__le__cancel__iff1,axiom,
% 0.23/0.46 ! [Z: int,X4: int,Y3: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.23/0.46 => ( ( ord_less_eq_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 0.23/0.46 = ( ord_less_eq_int @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_le_cancel_iff1
% 0.23/0.46 thf(fact_3363_mult__le__cancel__iff2,axiom,
% 0.23/0.46 ! [Z: real,X4: real,Y3: real] :
% 0.23/0.46 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.23/0.46 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X4 ) @ ( times_times_real @ Z @ Y3 ) )
% 0.23/0.46 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_le_cancel_iff2
% 0.23/0.46 thf(fact_3364_mult__le__cancel__iff2,axiom,
% 0.23/0.46 ! [Z: rat,X4: rat,Y3: rat] :
% 0.23/0.46 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.23/0.46 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X4 ) @ ( times_times_rat @ Z @ Y3 ) )
% 0.23/0.46 = ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_le_cancel_iff2
% 0.23/0.46 thf(fact_3365_mult__le__cancel__iff2,axiom,
% 0.23/0.46 ! [Z: int,X4: int,Y3: int] :
% 0.23/0.46 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.23/0.46 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X4 ) @ ( times_times_int @ Z @ Y3 ) )
% 0.23/0.46 = ( ord_less_eq_int @ X4 @ Y3 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_le_cancel_iff2
% 0.23/0.46 thf(fact_3366_triangle__def,axiom,
% 0.23/0.46 ( nat_triangle
% 0.23/0.46 = ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % triangle_def
% 0.23/0.46 thf(fact_3367_signed__take__bit__rec,axiom,
% 0.23/0.46 ( bit_ri6519982836138164636nteger
% 0.23/0.46 = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % signed_take_bit_rec
% 0.23/0.46 thf(fact_3368_signed__take__bit__rec,axiom,
% 0.23/0.46 ( bit_ri631733984087533419it_int
% 0.23/0.46 = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % signed_take_bit_rec
% 0.23/0.46 thf(fact_3369_verit__minus__simplify_I4_J,axiom,
% 0.23/0.46 ! [B2: real] :
% 0.23/0.46 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B2 ) )
% 0.23/0.46 = B2 ) ).
% 0.23/0.46
% 0.23/0.46 % verit_minus_simplify(4)
% 0.23/0.46 thf(fact_3370_verit__minus__simplify_I4_J,axiom,
% 0.23/0.46 ! [B2: int] :
% 0.23/0.46 ( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.46 = B2 ) ).
% 0.23/0.46
% 0.23/0.46 % verit_minus_simplify(4)
% 0.23/0.46 thf(fact_3371_verit__minus__simplify_I4_J,axiom,
% 0.23/0.46 ! [B2: complex] :
% 0.23/0.46 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.23/0.46 = B2 ) ).
% 0.23/0.46
% 0.23/0.46 % verit_minus_simplify(4)
% 0.23/0.46 thf(fact_3372_verit__minus__simplify_I4_J,axiom,
% 0.23/0.46 ! [B2: code_integer] :
% 0.23/0.46 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.46 = B2 ) ).
% 0.23/0.46
% 0.23/0.46 % verit_minus_simplify(4)
% 0.23/0.46 thf(fact_3373_verit__minus__simplify_I4_J,axiom,
% 0.23/0.46 ! [B2: rat] :
% 0.23/0.46 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B2 ) )
% 0.23/0.46 = B2 ) ).
% 0.23/0.46
% 0.23/0.46 % verit_minus_simplify(4)
% 0.23/0.46 thf(fact_3374_add_Oinverse__inverse,axiom,
% 0.23/0.46 ! [A: real] :
% 0.23/0.46 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.23/0.46 = A ) ).
% 0.23/0.46
% 0.23/0.46 % add.inverse_inverse
% 0.23/0.46 thf(fact_3375_add_Oinverse__inverse,axiom,
% 0.23/0.46 ! [A: int] :
% 0.23/0.46 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.23/0.46 = A ) ).
% 0.23/0.46
% 0.23/0.46 % add.inverse_inverse
% 0.23/0.46 thf(fact_3376_add_Oinverse__inverse,axiom,
% 0.23/0.46 ! [A: complex] :
% 0.23/0.46 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.23/0.46 = A ) ).
% 0.23/0.46
% 0.23/0.46 % add.inverse_inverse
% 0.23/0.46 thf(fact_3377_add_Oinverse__inverse,axiom,
% 0.23/0.46 ! [A: code_integer] :
% 0.23/0.46 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.46 = A ) ).
% 0.23/0.46
% 0.23/0.46 % add.inverse_inverse
% 0.23/0.46 thf(fact_3378_add_Oinverse__inverse,axiom,
% 0.23/0.46 ! [A: rat] :
% 0.23/0.46 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 0.23/0.46 = A ) ).
% 0.23/0.46
% 0.23/0.46 % add.inverse_inverse
% 0.23/0.46 thf(fact_3379_neg__equal__iff__equal,axiom,
% 0.23/0.46 ! [A: real,B2: real] :
% 0.23/0.46 ( ( ( uminus_uminus_real @ A )
% 0.23/0.46 = ( uminus_uminus_real @ B2 ) )
% 0.23/0.46 = ( A = B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_iff_equal
% 0.23/0.46 thf(fact_3380_neg__equal__iff__equal,axiom,
% 0.23/0.46 ! [A: int,B2: int] :
% 0.23/0.46 ( ( ( uminus_uminus_int @ A )
% 0.23/0.46 = ( uminus_uminus_int @ B2 ) )
% 0.23/0.46 = ( A = B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_iff_equal
% 0.23/0.46 thf(fact_3381_neg__equal__iff__equal,axiom,
% 0.23/0.46 ! [A: complex,B2: complex] :
% 0.23/0.46 ( ( ( uminus1482373934393186551omplex @ A )
% 0.23/0.46 = ( uminus1482373934393186551omplex @ B2 ) )
% 0.23/0.46 = ( A = B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_iff_equal
% 0.23/0.46 thf(fact_3382_neg__equal__iff__equal,axiom,
% 0.23/0.46 ! [A: code_integer,B2: code_integer] :
% 0.23/0.46 ( ( ( uminus1351360451143612070nteger @ A )
% 0.23/0.46 = ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.46 = ( A = B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_iff_equal
% 0.23/0.46 thf(fact_3383_neg__equal__iff__equal,axiom,
% 0.23/0.46 ! [A: rat,B2: rat] :
% 0.23/0.46 ( ( ( uminus_uminus_rat @ A )
% 0.23/0.46 = ( uminus_uminus_rat @ B2 ) )
% 0.23/0.46 = ( A = B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_iff_equal
% 0.23/0.46 thf(fact_3384_neg__le__iff__le,axiom,
% 0.23/0.46 ! [B2: real,A: real] :
% 0.23/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
% 0.23/0.46 = ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_le_iff_le
% 0.23/0.46 thf(fact_3385_neg__le__iff__le,axiom,
% 0.23/0.46 ! [B2: code_integer,A: code_integer] :
% 0.23/0.46 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.46 = ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_le_iff_le
% 0.23/0.46 thf(fact_3386_neg__le__iff__le,axiom,
% 0.23/0.46 ! [B2: rat,A: rat] :
% 0.23/0.46 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
% 0.23/0.46 = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_le_iff_le
% 0.23/0.46 thf(fact_3387_neg__le__iff__le,axiom,
% 0.23/0.46 ! [B2: int,A: int] :
% 0.23/0.46 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
% 0.23/0.46 = ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_le_iff_le
% 0.23/0.46 thf(fact_3388_add_Oinverse__neutral,axiom,
% 0.23/0.46 ( ( uminus_uminus_real @ zero_zero_real )
% 0.23/0.46 = zero_zero_real ) ).
% 0.23/0.46
% 0.23/0.46 % add.inverse_neutral
% 0.23/0.46 thf(fact_3389_add_Oinverse__neutral,axiom,
% 0.23/0.46 ( ( uminus_uminus_int @ zero_zero_int )
% 0.23/0.46 = zero_zero_int ) ).
% 0.23/0.46
% 0.23/0.46 % add.inverse_neutral
% 0.23/0.46 thf(fact_3390_add_Oinverse__neutral,axiom,
% 0.23/0.46 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 0.23/0.46 = zero_zero_complex ) ).
% 0.23/0.46
% 0.23/0.46 % add.inverse_neutral
% 0.23/0.46 thf(fact_3391_add_Oinverse__neutral,axiom,
% 0.23/0.46 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.23/0.46 = zero_z3403309356797280102nteger ) ).
% 0.23/0.46
% 0.23/0.46 % add.inverse_neutral
% 0.23/0.46 thf(fact_3392_add_Oinverse__neutral,axiom,
% 0.23/0.46 ( ( uminus_uminus_rat @ zero_zero_rat )
% 0.23/0.46 = zero_zero_rat ) ).
% 0.23/0.46
% 0.23/0.46 % add.inverse_neutral
% 0.23/0.46 thf(fact_3393_neg__0__equal__iff__equal,axiom,
% 0.23/0.46 ! [A: real] :
% 0.23/0.46 ( ( zero_zero_real
% 0.23/0.46 = ( uminus_uminus_real @ A ) )
% 0.23/0.46 = ( zero_zero_real = A ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_0_equal_iff_equal
% 0.23/0.46 thf(fact_3394_neg__0__equal__iff__equal,axiom,
% 0.23/0.46 ! [A: int] :
% 0.23/0.46 ( ( zero_zero_int
% 0.23/0.46 = ( uminus_uminus_int @ A ) )
% 0.23/0.46 = ( zero_zero_int = A ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_0_equal_iff_equal
% 0.23/0.46 thf(fact_3395_neg__0__equal__iff__equal,axiom,
% 0.23/0.46 ! [A: complex] :
% 0.23/0.46 ( ( zero_zero_complex
% 0.23/0.46 = ( uminus1482373934393186551omplex @ A ) )
% 0.23/0.46 = ( zero_zero_complex = A ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_0_equal_iff_equal
% 0.23/0.46 thf(fact_3396_neg__0__equal__iff__equal,axiom,
% 0.23/0.46 ! [A: code_integer] :
% 0.23/0.46 ( ( zero_z3403309356797280102nteger
% 0.23/0.46 = ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.46 = ( zero_z3403309356797280102nteger = A ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_0_equal_iff_equal
% 0.23/0.46 thf(fact_3397_neg__0__equal__iff__equal,axiom,
% 0.23/0.46 ! [A: rat] :
% 0.23/0.46 ( ( zero_zero_rat
% 0.23/0.46 = ( uminus_uminus_rat @ A ) )
% 0.23/0.46 = ( zero_zero_rat = A ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_0_equal_iff_equal
% 0.23/0.46 thf(fact_3398_neg__equal__0__iff__equal,axiom,
% 0.23/0.46 ! [A: real] :
% 0.23/0.46 ( ( ( uminus_uminus_real @ A )
% 0.23/0.46 = zero_zero_real )
% 0.23/0.46 = ( A = zero_zero_real ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_0_iff_equal
% 0.23/0.46 thf(fact_3399_neg__equal__0__iff__equal,axiom,
% 0.23/0.46 ! [A: int] :
% 0.23/0.46 ( ( ( uminus_uminus_int @ A )
% 0.23/0.46 = zero_zero_int )
% 0.23/0.46 = ( A = zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_0_iff_equal
% 0.23/0.46 thf(fact_3400_neg__equal__0__iff__equal,axiom,
% 0.23/0.46 ! [A: complex] :
% 0.23/0.46 ( ( ( uminus1482373934393186551omplex @ A )
% 0.23/0.46 = zero_zero_complex )
% 0.23/0.46 = ( A = zero_zero_complex ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_0_iff_equal
% 0.23/0.46 thf(fact_3401_neg__equal__0__iff__equal,axiom,
% 0.23/0.46 ! [A: code_integer] :
% 0.23/0.46 ( ( ( uminus1351360451143612070nteger @ A )
% 0.23/0.46 = zero_z3403309356797280102nteger )
% 0.23/0.46 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_0_iff_equal
% 0.23/0.46 thf(fact_3402_neg__equal__0__iff__equal,axiom,
% 0.23/0.46 ! [A: rat] :
% 0.23/0.46 ( ( ( uminus_uminus_rat @ A )
% 0.23/0.46 = zero_zero_rat )
% 0.23/0.46 = ( A = zero_zero_rat ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_0_iff_equal
% 0.23/0.46 thf(fact_3403_equal__neg__zero,axiom,
% 0.23/0.46 ! [A: real] :
% 0.23/0.46 ( ( A
% 0.23/0.46 = ( uminus_uminus_real @ A ) )
% 0.23/0.46 = ( A = zero_zero_real ) ) ).
% 0.23/0.46
% 0.23/0.46 % equal_neg_zero
% 0.23/0.46 thf(fact_3404_equal__neg__zero,axiom,
% 0.23/0.46 ! [A: int] :
% 0.23/0.46 ( ( A
% 0.23/0.46 = ( uminus_uminus_int @ A ) )
% 0.23/0.46 = ( A = zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % equal_neg_zero
% 0.23/0.46 thf(fact_3405_equal__neg__zero,axiom,
% 0.23/0.46 ! [A: code_integer] :
% 0.23/0.46 ( ( A
% 0.23/0.46 = ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.46 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.46
% 0.23/0.46 % equal_neg_zero
% 0.23/0.46 thf(fact_3406_equal__neg__zero,axiom,
% 0.23/0.46 ! [A: rat] :
% 0.23/0.46 ( ( A
% 0.23/0.46 = ( uminus_uminus_rat @ A ) )
% 0.23/0.46 = ( A = zero_zero_rat ) ) ).
% 0.23/0.46
% 0.23/0.46 % equal_neg_zero
% 0.23/0.46 thf(fact_3407_neg__equal__zero,axiom,
% 0.23/0.46 ! [A: real] :
% 0.23/0.46 ( ( ( uminus_uminus_real @ A )
% 0.23/0.46 = A )
% 0.23/0.46 = ( A = zero_zero_real ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_zero
% 0.23/0.46 thf(fact_3408_neg__equal__zero,axiom,
% 0.23/0.46 ! [A: int] :
% 0.23/0.46 ( ( ( uminus_uminus_int @ A )
% 0.23/0.46 = A )
% 0.23/0.46 = ( A = zero_zero_int ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_zero
% 0.23/0.46 thf(fact_3409_neg__equal__zero,axiom,
% 0.23/0.46 ! [A: code_integer] :
% 0.23/0.46 ( ( ( uminus1351360451143612070nteger @ A )
% 0.23/0.46 = A )
% 0.23/0.46 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_zero
% 0.23/0.46 thf(fact_3410_neg__equal__zero,axiom,
% 0.23/0.46 ! [A: rat] :
% 0.23/0.46 ( ( ( uminus_uminus_rat @ A )
% 0.23/0.46 = A )
% 0.23/0.46 = ( A = zero_zero_rat ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_equal_zero
% 0.23/0.46 thf(fact_3411_neg__less__iff__less,axiom,
% 0.23/0.46 ! [B2: real,A: real] :
% 0.23/0.46 ( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
% 0.23/0.46 = ( ord_less_real @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_less_iff_less
% 0.23/0.46 thf(fact_3412_neg__less__iff__less,axiom,
% 0.23/0.46 ! [B2: int,A: int] :
% 0.23/0.46 ( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
% 0.23/0.46 = ( ord_less_int @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_less_iff_less
% 0.23/0.46 thf(fact_3413_neg__less__iff__less,axiom,
% 0.23/0.46 ! [B2: code_integer,A: code_integer] :
% 0.23/0.46 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.46 = ( ord_le6747313008572928689nteger @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_less_iff_less
% 0.23/0.46 thf(fact_3414_neg__less__iff__less,axiom,
% 0.23/0.46 ! [B2: rat,A: rat] :
% 0.23/0.46 ( ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
% 0.23/0.46 = ( ord_less_rat @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_less_iff_less
% 0.23/0.46 thf(fact_3415_neg__numeral__eq__iff,axiom,
% 0.23/0.46 ! [M: num,N: num] :
% 0.23/0.46 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.23/0.46 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.46 = ( M = N ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_numeral_eq_iff
% 0.23/0.46 thf(fact_3416_neg__numeral__eq__iff,axiom,
% 0.23/0.46 ! [M: num,N: num] :
% 0.23/0.46 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.23/0.46 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.46 = ( M = N ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_numeral_eq_iff
% 0.23/0.46 thf(fact_3417_neg__numeral__eq__iff,axiom,
% 0.23/0.46 ! [M: num,N: num] :
% 0.23/0.46 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.23/0.46 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.23/0.46 = ( M = N ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_numeral_eq_iff
% 0.23/0.46 thf(fact_3418_neg__numeral__eq__iff,axiom,
% 0.23/0.46 ! [M: num,N: num] :
% 0.23/0.46 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.23/0.46 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.46 = ( M = N ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_numeral_eq_iff
% 0.23/0.46 thf(fact_3419_neg__numeral__eq__iff,axiom,
% 0.23/0.46 ! [M: num,N: num] :
% 0.23/0.46 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.23/0.46 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.46 = ( M = N ) ) ).
% 0.23/0.46
% 0.23/0.46 % neg_numeral_eq_iff
% 0.23/0.46 thf(fact_3420_mult__minus__right,axiom,
% 0.23/0.46 ! [A: real,B2: real] :
% 0.23/0.46 ( ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) )
% 0.23/0.46 = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_minus_right
% 0.23/0.46 thf(fact_3421_mult__minus__right,axiom,
% 0.23/0.46 ! [A: int,B2: int] :
% 0.23/0.46 ( ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.46 = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_minus_right
% 0.23/0.46 thf(fact_3422_mult__minus__right,axiom,
% 0.23/0.46 ! [A: complex,B2: complex] :
% 0.23/0.46 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.23/0.46 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_minus_right
% 0.23/0.46 thf(fact_3423_mult__minus__right,axiom,
% 0.23/0.46 ! [A: code_integer,B2: code_integer] :
% 0.23/0.46 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.46 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_minus_right
% 0.23/0.46 thf(fact_3424_mult__minus__right,axiom,
% 0.23/0.46 ! [A: rat,B2: rat] :
% 0.23/0.46 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 0.23/0.46 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_minus_right
% 0.23/0.46 thf(fact_3425_minus__mult__minus,axiom,
% 0.23/0.46 ! [A: real,B2: real] :
% 0.23/0.46 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 0.23/0.46 = ( times_times_real @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minus_mult_minus
% 0.23/0.46 thf(fact_3426_minus__mult__minus,axiom,
% 0.23/0.46 ! [A: int,B2: int] :
% 0.23/0.46 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.46 = ( times_times_int @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minus_mult_minus
% 0.23/0.46 thf(fact_3427_minus__mult__minus,axiom,
% 0.23/0.46 ! [A: complex,B2: complex] :
% 0.23/0.46 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.23/0.46 = ( times_times_complex @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minus_mult_minus
% 0.23/0.46 thf(fact_3428_minus__mult__minus,axiom,
% 0.23/0.46 ! [A: code_integer,B2: code_integer] :
% 0.23/0.46 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.46 = ( times_3573771949741848930nteger @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minus_mult_minus
% 0.23/0.46 thf(fact_3429_minus__mult__minus,axiom,
% 0.23/0.46 ! [A: rat,B2: rat] :
% 0.23/0.46 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 0.23/0.46 = ( times_times_rat @ A @ B2 ) ) ).
% 0.23/0.46
% 0.23/0.46 % minus_mult_minus
% 0.23/0.46 thf(fact_3430_mult__minus__left,axiom,
% 0.23/0.46 ! [A: real,B2: real] :
% 0.23/0.46 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.23/0.46 = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_minus_left
% 0.23/0.46 thf(fact_3431_mult__minus__left,axiom,
% 0.23/0.46 ! [A: int,B2: int] :
% 0.23/0.46 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.46 = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_minus_left
% 0.23/0.46 thf(fact_3432_mult__minus__left,axiom,
% 0.23/0.46 ! [A: complex,B2: complex] :
% 0.23/0.46 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 0.23/0.46 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_minus_left
% 0.23/0.46 thf(fact_3433_mult__minus__left,axiom,
% 0.23/0.46 ! [A: code_integer,B2: code_integer] :
% 0.23/0.46 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.23/0.46 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_minus_left
% 0.23/0.46 thf(fact_3434_mult__minus__left,axiom,
% 0.23/0.46 ! [A: rat,B2: rat] :
% 0.23/0.46 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.23/0.46 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % mult_minus_left
% 0.23/0.46 thf(fact_3435_minus__add__distrib,axiom,
% 0.23/0.46 ! [A: real,B2: real] :
% 0.23/0.46 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 0.23/0.46 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minus_add_distrib
% 0.23/0.46 thf(fact_3436_minus__add__distrib,axiom,
% 0.23/0.46 ! [A: int,B2: int] :
% 0.23/0.46 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 0.23/0.46 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minus_add_distrib
% 0.23/0.46 thf(fact_3437_minus__add__distrib,axiom,
% 0.23/0.46 ! [A: complex,B2: complex] :
% 0.23/0.46 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 0.23/0.46 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minus_add_distrib
% 0.23/0.46 thf(fact_3438_minus__add__distrib,axiom,
% 0.23/0.46 ! [A: code_integer,B2: code_integer] :
% 0.23/0.46 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.23/0.46 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minus_add_distrib
% 0.23/0.46 thf(fact_3439_minus__add__distrib,axiom,
% 0.23/0.46 ! [A: rat,B2: rat] :
% 0.23/0.46 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 0.23/0.46 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 0.23/0.46
% 0.23/0.46 % minus_add_distrib
% 0.23/0.46 thf(fact_3440_minus__add__cancel,axiom,
% 0.23/0.46 ! [A: real,B2: real] :
% 0.23/0.46 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B2 ) )
% 0.23/0.46 = B2 ) ).
% 0.23/0.46
% 0.23/0.46 % minus_add_cancel
% 0.23/0.46 thf(fact_3441_minus__add__cancel,axiom,
% 0.23/0.46 ! [A: int,B2: int] :
% 0.23/0.46 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B2 ) )
% 0.23/0.46 = B2 ) ).
% 0.23/0.46
% 0.23/0.46 % minus_add_cancel
% 0.23/0.46 thf(fact_3442_minus__add__cancel,axiom,
% 0.23/0.46 ! [A: complex,B2: complex] :
% 0.23/0.46 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B2 ) )
% 0.23/0.46 = B2 ) ).
% 0.23/0.46
% 0.23/0.46 % minus_add_cancel
% 0.23/0.46 thf(fact_3443_minus__add__cancel,axiom,
% 0.23/0.46 ! [A: code_integer,B2: code_integer] :
% 0.23/0.46 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.23/0.46 = B2 ) ).
% 0.23/0.46
% 0.23/0.46 % minus_add_cancel
% 0.23/0.46 thf(fact_3444_minus__add__cancel,axiom,
% 0.23/0.46 ! [A: rat,B2: rat] :
% 0.23/0.46 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B2 ) )
% 0.23/0.46 = B2 ) ).
% 0.23/0.46
% 0.23/0.46 % minus_add_cancel
% 0.23/0.46 thf(fact_3445_add__minus__cancel,axiom,
% 0.23/0.46 ! [A: real,B2: real] :
% 0.23/0.46 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 ) )
% 0.23/0.47 = B2 ) ).
% 0.23/0.47
% 0.23/0.47 % add_minus_cancel
% 0.23/0.47 thf(fact_3446_add__minus__cancel,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 ) )
% 0.23/0.47 = B2 ) ).
% 0.23/0.47
% 0.23/0.47 % add_minus_cancel
% 0.23/0.47 thf(fact_3447_add__minus__cancel,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) )
% 0.23/0.47 = B2 ) ).
% 0.23/0.47
% 0.23/0.47 % add_minus_cancel
% 0.23/0.47 thf(fact_3448_add__minus__cancel,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) )
% 0.23/0.47 = B2 ) ).
% 0.23/0.47
% 0.23/0.47 % add_minus_cancel
% 0.23/0.47 thf(fact_3449_add__minus__cancel,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) )
% 0.23/0.47 = B2 ) ).
% 0.23/0.47
% 0.23/0.47 % add_minus_cancel
% 0.23/0.47 thf(fact_3450_minus__diff__eq,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) )
% 0.23/0.47 = ( minus_minus_real @ B2 @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_eq
% 0.23/0.47 thf(fact_3451_minus__diff__eq,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) )
% 0.23/0.47 = ( minus_minus_int @ B2 @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_eq
% 0.23/0.47 thf(fact_3452_minus__diff__eq,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) )
% 0.23/0.47 = ( minus_minus_complex @ B2 @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_eq
% 0.23/0.47 thf(fact_3453_minus__diff__eq,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 0.23/0.47 = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_eq
% 0.23/0.47 thf(fact_3454_minus__diff__eq,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) )
% 0.23/0.47 = ( minus_minus_rat @ B2 @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_eq
% 0.23/0.47 thf(fact_3455_div__minus__minus,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.47 = ( divide_divide_int @ A @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % div_minus_minus
% 0.23/0.47 thf(fact_3456_div__minus__minus,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.47 = ( divide6298287555418463151nteger @ A @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % div_minus_minus
% 0.23/0.47 thf(fact_3457_dvd__minus__iff,axiom,
% 0.23/0.47 ! [X4: real,Y3: real] :
% 0.23/0.47 ( ( dvd_dvd_real @ X4 @ ( uminus_uminus_real @ Y3 ) )
% 0.23/0.47 = ( dvd_dvd_real @ X4 @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_minus_iff
% 0.23/0.47 thf(fact_3458_dvd__minus__iff,axiom,
% 0.23/0.47 ! [X4: int,Y3: int] :
% 0.23/0.47 ( ( dvd_dvd_int @ X4 @ ( uminus_uminus_int @ Y3 ) )
% 0.23/0.47 = ( dvd_dvd_int @ X4 @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_minus_iff
% 0.23/0.47 thf(fact_3459_dvd__minus__iff,axiom,
% 0.23/0.47 ! [X4: complex,Y3: complex] :
% 0.23/0.47 ( ( dvd_dvd_complex @ X4 @ ( uminus1482373934393186551omplex @ Y3 ) )
% 0.23/0.47 = ( dvd_dvd_complex @ X4 @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_minus_iff
% 0.23/0.47 thf(fact_3460_dvd__minus__iff,axiom,
% 0.23/0.47 ! [X4: code_integer,Y3: code_integer] :
% 0.23/0.47 ( ( dvd_dvd_Code_integer @ X4 @ ( uminus1351360451143612070nteger @ Y3 ) )
% 0.23/0.47 = ( dvd_dvd_Code_integer @ X4 @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_minus_iff
% 0.23/0.47 thf(fact_3461_dvd__minus__iff,axiom,
% 0.23/0.47 ! [X4: rat,Y3: rat] :
% 0.23/0.47 ( ( dvd_dvd_rat @ X4 @ ( uminus_uminus_rat @ Y3 ) )
% 0.23/0.47 = ( dvd_dvd_rat @ X4 @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_minus_iff
% 0.23/0.47 thf(fact_3462_minus__dvd__iff,axiom,
% 0.23/0.47 ! [X4: real,Y3: real] :
% 0.23/0.47 ( ( dvd_dvd_real @ ( uminus_uminus_real @ X4 ) @ Y3 )
% 0.23/0.47 = ( dvd_dvd_real @ X4 @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_dvd_iff
% 0.23/0.47 thf(fact_3463_minus__dvd__iff,axiom,
% 0.23/0.47 ! [X4: int,Y3: int] :
% 0.23/0.47 ( ( dvd_dvd_int @ ( uminus_uminus_int @ X4 ) @ Y3 )
% 0.23/0.47 = ( dvd_dvd_int @ X4 @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_dvd_iff
% 0.23/0.47 thf(fact_3464_minus__dvd__iff,axiom,
% 0.23/0.47 ! [X4: complex,Y3: complex] :
% 0.23/0.47 ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X4 ) @ Y3 )
% 0.23/0.47 = ( dvd_dvd_complex @ X4 @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_dvd_iff
% 0.23/0.47 thf(fact_3465_minus__dvd__iff,axiom,
% 0.23/0.47 ! [X4: code_integer,Y3: code_integer] :
% 0.23/0.47 ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X4 ) @ Y3 )
% 0.23/0.47 = ( dvd_dvd_Code_integer @ X4 @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_dvd_iff
% 0.23/0.47 thf(fact_3466_minus__dvd__iff,axiom,
% 0.23/0.47 ! [X4: rat,Y3: rat] :
% 0.23/0.47 ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X4 ) @ Y3 )
% 0.23/0.47 = ( dvd_dvd_rat @ X4 @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_dvd_iff
% 0.23/0.47 thf(fact_3467_mod__minus__minus,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.47 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mod_minus_minus
% 0.23/0.47 thf(fact_3468_mod__minus__minus,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mod_minus_minus
% 0.23/0.47 thf(fact_3469_real__add__minus__iff,axiom,
% 0.23/0.47 ! [X4: real,A: real] :
% 0.23/0.47 ( ( ( plus_plus_real @ X4 @ ( uminus_uminus_real @ A ) )
% 0.23/0.47 = zero_zero_real )
% 0.23/0.47 = ( X4 = A ) ) ).
% 0.23/0.47
% 0.23/0.47 % real_add_minus_iff
% 0.23/0.47 thf(fact_3470_neg__less__eq__nonneg,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.23/0.47 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_eq_nonneg
% 0.23/0.47 thf(fact_3471_neg__less__eq__nonneg,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.23/0.47 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_eq_nonneg
% 0.23/0.47 thf(fact_3472_neg__less__eq__nonneg,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.23/0.47 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_eq_nonneg
% 0.23/0.47 thf(fact_3473_neg__less__eq__nonneg,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 0.23/0.47 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_eq_nonneg
% 0.23/0.47 thf(fact_3474_less__eq__neg__nonpos,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.23/0.47 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_eq_neg_nonpos
% 0.23/0.47 thf(fact_3475_less__eq__neg__nonpos,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.47 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_eq_neg_nonpos
% 0.23/0.47 thf(fact_3476_less__eq__neg__nonpos,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.23/0.47 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_eq_neg_nonpos
% 0.23/0.47 thf(fact_3477_less__eq__neg__nonpos,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 0.23/0.47 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_eq_neg_nonpos
% 0.23/0.47 thf(fact_3478_neg__le__0__iff__le,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.23/0.47 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_le_0_iff_le
% 0.23/0.47 thf(fact_3479_neg__le__0__iff__le,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.23/0.47 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_le_0_iff_le
% 0.23/0.47 thf(fact_3480_neg__le__0__iff__le,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.23/0.47 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_le_0_iff_le
% 0.23/0.47 thf(fact_3481_neg__le__0__iff__le,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.23/0.47 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_le_0_iff_le
% 0.23/0.47 thf(fact_3482_neg__0__le__iff__le,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.23/0.47 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_0_le_iff_le
% 0.23/0.47 thf(fact_3483_neg__0__le__iff__le,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.47 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_0_le_iff_le
% 0.23/0.47 thf(fact_3484_neg__0__le__iff__le,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.23/0.47 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_0_le_iff_le
% 0.23/0.47 thf(fact_3485_neg__0__le__iff__le,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.23/0.47 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_0_le_iff_le
% 0.23/0.47 thf(fact_3486_neg__less__0__iff__less,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.23/0.47 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_0_iff_less
% 0.23/0.47 thf(fact_3487_neg__less__0__iff__less,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.23/0.47 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_0_iff_less
% 0.23/0.47 thf(fact_3488_neg__less__0__iff__less,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.23/0.47 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_0_iff_less
% 0.23/0.47 thf(fact_3489_neg__less__0__iff__less,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.23/0.47 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_0_iff_less
% 0.23/0.47 thf(fact_3490_neg__0__less__iff__less,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.23/0.47 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_0_less_iff_less
% 0.23/0.47 thf(fact_3491_neg__0__less__iff__less,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.23/0.47 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_0_less_iff_less
% 0.23/0.47 thf(fact_3492_neg__0__less__iff__less,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.47 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_0_less_iff_less
% 0.23/0.47 thf(fact_3493_neg__0__less__iff__less,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.23/0.47 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_0_less_iff_less
% 0.23/0.47 thf(fact_3494_neg__less__pos,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.23/0.47 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_pos
% 0.23/0.47 thf(fact_3495_neg__less__pos,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 0.23/0.47 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_pos
% 0.23/0.47 thf(fact_3496_neg__less__pos,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.23/0.47 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_pos
% 0.23/0.47 thf(fact_3497_neg__less__pos,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.23/0.47 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_less_pos
% 0.23/0.47 thf(fact_3498_less__neg__neg,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.23/0.47 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_neg_neg
% 0.23/0.47 thf(fact_3499_less__neg__neg,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 0.23/0.47 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_neg_neg
% 0.23/0.47 thf(fact_3500_less__neg__neg,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.47 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_neg_neg
% 0.23/0.47 thf(fact_3501_less__neg__neg,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.23/0.47 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_neg_neg
% 0.23/0.47 thf(fact_3502_ab__left__minus,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.23/0.47 = zero_zero_real ) ).
% 0.23/0.47
% 0.23/0.47 % ab_left_minus
% 0.23/0.47 thf(fact_3503_ab__left__minus,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.23/0.47 = zero_zero_int ) ).
% 0.23/0.47
% 0.23/0.47 % ab_left_minus
% 0.23/0.47 thf(fact_3504_ab__left__minus,axiom,
% 0.23/0.47 ! [A: complex] :
% 0.23/0.47 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.23/0.47 = zero_zero_complex ) ).
% 0.23/0.47
% 0.23/0.47 % ab_left_minus
% 0.23/0.47 thf(fact_3505_ab__left__minus,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.23/0.47 = zero_z3403309356797280102nteger ) ).
% 0.23/0.47
% 0.23/0.47 % ab_left_minus
% 0.23/0.47 thf(fact_3506_ab__left__minus,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.23/0.47 = zero_zero_rat ) ).
% 0.23/0.47
% 0.23/0.47 % ab_left_minus
% 0.23/0.47 thf(fact_3507_add_Oright__inverse,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.23/0.47 = zero_zero_real ) ).
% 0.23/0.47
% 0.23/0.47 % add.right_inverse
% 0.23/0.47 thf(fact_3508_add_Oright__inverse,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 0.23/0.47 = zero_zero_int ) ).
% 0.23/0.47
% 0.23/0.47 % add.right_inverse
% 0.23/0.47 thf(fact_3509_add_Oright__inverse,axiom,
% 0.23/0.47 ! [A: complex] :
% 0.23/0.47 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 0.23/0.47 = zero_zero_complex ) ).
% 0.23/0.47
% 0.23/0.47 % add.right_inverse
% 0.23/0.47 thf(fact_3510_add_Oright__inverse,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.47 = zero_z3403309356797280102nteger ) ).
% 0.23/0.47
% 0.23/0.47 % add.right_inverse
% 0.23/0.47 thf(fact_3511_add_Oright__inverse,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.23/0.47 = zero_zero_rat ) ).
% 0.23/0.47
% 0.23/0.47 % add.right_inverse
% 0.23/0.47 thf(fact_3512_diff__0,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( minus_minus_real @ zero_zero_real @ A )
% 0.23/0.47 = ( uminus_uminus_real @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_0
% 0.23/0.47 thf(fact_3513_diff__0,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( minus_minus_int @ zero_zero_int @ A )
% 0.23/0.47 = ( uminus_uminus_int @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_0
% 0.23/0.47 thf(fact_3514_diff__0,axiom,
% 0.23/0.47 ! [A: complex] :
% 0.23/0.47 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_0
% 0.23/0.47 thf(fact_3515_diff__0,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_0
% 0.23/0.47 thf(fact_3516_diff__0,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 0.23/0.47 = ( uminus_uminus_rat @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_0
% 0.23/0.47 thf(fact_3517_verit__minus__simplify_I3_J,axiom,
% 0.23/0.47 ! [B2: real] :
% 0.23/0.47 ( ( minus_minus_real @ zero_zero_real @ B2 )
% 0.23/0.47 = ( uminus_uminus_real @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_minus_simplify(3)
% 0.23/0.47 thf(fact_3518_verit__minus__simplify_I3_J,axiom,
% 0.23/0.47 ! [B2: int] :
% 0.23/0.47 ( ( minus_minus_int @ zero_zero_int @ B2 )
% 0.23/0.47 = ( uminus_uminus_int @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_minus_simplify(3)
% 0.23/0.47 thf(fact_3519_verit__minus__simplify_I3_J,axiom,
% 0.23/0.47 ! [B2: complex] :
% 0.23/0.47 ( ( minus_minus_complex @ zero_zero_complex @ B2 )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_minus_simplify(3)
% 0.23/0.47 thf(fact_3520_verit__minus__simplify_I3_J,axiom,
% 0.23/0.47 ! [B2: code_integer] :
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_minus_simplify(3)
% 0.23/0.47 thf(fact_3521_verit__minus__simplify_I3_J,axiom,
% 0.23/0.47 ! [B2: rat] :
% 0.23/0.47 ( ( minus_minus_rat @ zero_zero_rat @ B2 )
% 0.23/0.47 = ( uminus_uminus_rat @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_minus_simplify(3)
% 0.23/0.47 thf(fact_3522_add__neg__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.47 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_simps(3)
% 0.23/0.47 thf(fact_3523_add__neg__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.47 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_simps(3)
% 0.23/0.47 thf(fact_3524_add__neg__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_simps(3)
% 0.23/0.47 thf(fact_3525_add__neg__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_simps(3)
% 0.23/0.47 thf(fact_3526_add__neg__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.47 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_simps(3)
% 0.23/0.47 thf(fact_3527_mult__minus1,axiom,
% 0.23/0.47 ! [Z: real] :
% 0.23/0.47 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 0.23/0.47 = ( uminus_uminus_real @ Z ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_minus1
% 0.23/0.47 thf(fact_3528_mult__minus1,axiom,
% 0.23/0.47 ! [Z: int] :
% 0.23/0.47 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 0.23/0.47 = ( uminus_uminus_int @ Z ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_minus1
% 0.23/0.47 thf(fact_3529_mult__minus1,axiom,
% 0.23/0.47 ! [Z: complex] :
% 0.23/0.47 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_minus1
% 0.23/0.47 thf(fact_3530_mult__minus1,axiom,
% 0.23/0.47 ! [Z: code_integer] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_minus1
% 0.23/0.47 thf(fact_3531_mult__minus1,axiom,
% 0.23/0.47 ! [Z: rat] :
% 0.23/0.47 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 0.23/0.47 = ( uminus_uminus_rat @ Z ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_minus1
% 0.23/0.47 thf(fact_3532_mult__minus1__right,axiom,
% 0.23/0.47 ! [Z: real] :
% 0.23/0.47 ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.47 = ( uminus_uminus_real @ Z ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_minus1_right
% 0.23/0.47 thf(fact_3533_mult__minus1__right,axiom,
% 0.23/0.47 ! [Z: int] :
% 0.23/0.47 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 = ( uminus_uminus_int @ Z ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_minus1_right
% 0.23/0.47 thf(fact_3534_mult__minus1__right,axiom,
% 0.23/0.47 ! [Z: complex] :
% 0.23/0.47 ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_minus1_right
% 0.23/0.47 thf(fact_3535_mult__minus1__right,axiom,
% 0.23/0.47 ! [Z: code_integer] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_minus1_right
% 0.23/0.47 thf(fact_3536_mult__minus1__right,axiom,
% 0.23/0.47 ! [Z: rat] :
% 0.23/0.47 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.47 = ( uminus_uminus_rat @ Z ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_minus1_right
% 0.23/0.47 thf(fact_3537_uminus__add__conv__diff,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.23/0.47 = ( minus_minus_real @ B2 @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % uminus_add_conv_diff
% 0.23/0.47 thf(fact_3538_uminus__add__conv__diff,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.47 = ( minus_minus_int @ B2 @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % uminus_add_conv_diff
% 0.23/0.47 thf(fact_3539_uminus__add__conv__diff,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 0.23/0.47 = ( minus_minus_complex @ B2 @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % uminus_add_conv_diff
% 0.23/0.47 thf(fact_3540_uminus__add__conv__diff,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.23/0.47 = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % uminus_add_conv_diff
% 0.23/0.47 thf(fact_3541_uminus__add__conv__diff,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.23/0.47 = ( minus_minus_rat @ B2 @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % uminus_add_conv_diff
% 0.23/0.47 thf(fact_3542_diff__minus__eq__add,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B2 ) )
% 0.23/0.47 = ( plus_plus_real @ A @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_minus_eq_add
% 0.23/0.47 thf(fact_3543_diff__minus__eq__add,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.47 = ( plus_plus_int @ A @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_minus_eq_add
% 0.23/0.47 thf(fact_3544_diff__minus__eq__add,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.23/0.47 = ( plus_plus_complex @ A @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_minus_eq_add
% 0.23/0.47 thf(fact_3545_diff__minus__eq__add,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.47 = ( plus_p5714425477246183910nteger @ A @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_minus_eq_add
% 0.23/0.47 thf(fact_3546_diff__minus__eq__add,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 0.23/0.47 = ( plus_plus_rat @ A @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_minus_eq_add
% 0.23/0.47 thf(fact_3547_div__minus1__right,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 = ( uminus_uminus_int @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % div_minus1_right
% 0.23/0.47 thf(fact_3548_div__minus1__right,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % div_minus1_right
% 0.23/0.47 thf(fact_3549_divide__minus1,axiom,
% 0.23/0.47 ! [X4: real] :
% 0.23/0.47 ( ( divide_divide_real @ X4 @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.47 = ( uminus_uminus_real @ X4 ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_minus1
% 0.23/0.47 thf(fact_3550_divide__minus1,axiom,
% 0.23/0.47 ! [X4: complex] :
% 0.23/0.47 ( ( divide1717551699836669952omplex @ X4 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ X4 ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_minus1
% 0.23/0.47 thf(fact_3551_divide__minus1,axiom,
% 0.23/0.47 ! [X4: rat] :
% 0.23/0.47 ( ( divide_divide_rat @ X4 @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.47 = ( uminus_uminus_rat @ X4 ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_minus1
% 0.23/0.47 thf(fact_3552_minus__mod__self1,axiom,
% 0.23/0.47 ! [B2: int,A: int] :
% 0.23/0.47 ( ( modulo_modulo_int @ ( minus_minus_int @ B2 @ A ) @ B2 )
% 0.23/0.47 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_mod_self1
% 0.23/0.47 thf(fact_3553_minus__mod__self1,axiom,
% 0.23/0.47 ! [B2: code_integer,A: code_integer] :
% 0.23/0.47 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B2 @ A ) @ B2 )
% 0.23/0.47 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_mod_self1
% 0.23/0.47 thf(fact_3554_signed__take__bit__of__minus__1,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % signed_take_bit_of_minus_1
% 0.23/0.47 thf(fact_3555_signed__take__bit__of__minus__1,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % signed_take_bit_of_minus_1
% 0.23/0.47 thf(fact_3556_dbl__simps_I1_J,axiom,
% 0.23/0.47 ! [K: num] :
% 0.23/0.47 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.23/0.47 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dbl_simps(1)
% 0.23/0.47 thf(fact_3557_dbl__simps_I1_J,axiom,
% 0.23/0.47 ! [K: num] :
% 0.23/0.47 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.47 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dbl_simps(1)
% 0.23/0.47 thf(fact_3558_dbl__simps_I1_J,axiom,
% 0.23/0.47 ! [K: num] :
% 0.23/0.47 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dbl_simps(1)
% 0.23/0.47 thf(fact_3559_dbl__simps_I1_J,axiom,
% 0.23/0.47 ! [K: num] :
% 0.23/0.47 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dbl_simps(1)
% 0.23/0.47 thf(fact_3560_dbl__simps_I1_J,axiom,
% 0.23/0.47 ! [K: num] :
% 0.23/0.47 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.23/0.47 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dbl_simps(1)
% 0.23/0.47 thf(fact_3561_triangle__Suc,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( nat_triangle @ ( suc @ N ) )
% 0.23/0.47 = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % triangle_Suc
% 0.23/0.47 thf(fact_3562_add__neg__numeral__special_I8_J,axiom,
% 0.23/0.47 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.23/0.47 = zero_zero_real ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(8)
% 0.23/0.47 thf(fact_3563_add__neg__numeral__special_I8_J,axiom,
% 0.23/0.47 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.23/0.47 = zero_zero_int ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(8)
% 0.23/0.47 thf(fact_3564_add__neg__numeral__special_I8_J,axiom,
% 0.23/0.47 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.23/0.47 = zero_zero_complex ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(8)
% 0.23/0.47 thf(fact_3565_add__neg__numeral__special_I8_J,axiom,
% 0.23/0.47 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.23/0.47 = zero_z3403309356797280102nteger ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(8)
% 0.23/0.47 thf(fact_3566_add__neg__numeral__special_I8_J,axiom,
% 0.23/0.47 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.23/0.47 = zero_zero_rat ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(8)
% 0.23/0.47 thf(fact_3567_add__neg__numeral__special_I7_J,axiom,
% 0.23/0.47 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.47 = zero_zero_real ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(7)
% 0.23/0.47 thf(fact_3568_add__neg__numeral__special_I7_J,axiom,
% 0.23/0.47 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 = zero_zero_int ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(7)
% 0.23/0.47 thf(fact_3569_add__neg__numeral__special_I7_J,axiom,
% 0.23/0.47 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.47 = zero_zero_complex ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(7)
% 0.23/0.47 thf(fact_3570_add__neg__numeral__special_I7_J,axiom,
% 0.23/0.47 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.47 = zero_z3403309356797280102nteger ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(7)
% 0.23/0.47 thf(fact_3571_add__neg__numeral__special_I7_J,axiom,
% 0.23/0.47 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.47 = zero_zero_rat ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(7)
% 0.23/0.47 thf(fact_3572_diff__numeral__special_I12_J,axiom,
% 0.23/0.47 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.47 = zero_zero_real ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(12)
% 0.23/0.47 thf(fact_3573_diff__numeral__special_I12_J,axiom,
% 0.23/0.47 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 = zero_zero_int ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(12)
% 0.23/0.47 thf(fact_3574_diff__numeral__special_I12_J,axiom,
% 0.23/0.47 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.47 = zero_zero_complex ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(12)
% 0.23/0.47 thf(fact_3575_diff__numeral__special_I12_J,axiom,
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.47 = zero_z3403309356797280102nteger ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(12)
% 0.23/0.47 thf(fact_3576_diff__numeral__special_I12_J,axiom,
% 0.23/0.47 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.47 = zero_zero_rat ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(12)
% 0.23/0.47 thf(fact_3577_neg__one__eq__numeral__iff,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( ( uminus_uminus_real @ one_one_real )
% 0.23/0.47 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.47 = ( N = one ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_eq_numeral_iff
% 0.23/0.47 thf(fact_3578_neg__one__eq__numeral__iff,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( ( uminus_uminus_int @ one_one_int )
% 0.23/0.47 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.47 = ( N = one ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_eq_numeral_iff
% 0.23/0.47 thf(fact_3579_neg__one__eq__numeral__iff,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.23/0.47 = ( N = one ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_eq_numeral_iff
% 0.23/0.47 thf(fact_3580_neg__one__eq__numeral__iff,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.47 = ( N = one ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_eq_numeral_iff
% 0.23/0.47 thf(fact_3581_neg__one__eq__numeral__iff,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( ( uminus_uminus_rat @ one_one_rat )
% 0.23/0.47 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.47 = ( N = one ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_eq_numeral_iff
% 0.23/0.47 thf(fact_3582_numeral__eq__neg__one__iff,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 0.23/0.47 = ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.47 = ( N = one ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_eq_neg_one_iff
% 0.23/0.47 thf(fact_3583_numeral__eq__neg__one__iff,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 0.23/0.47 = ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 = ( N = one ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_eq_neg_one_iff
% 0.23/0.47 thf(fact_3584_numeral__eq__neg__one__iff,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.47 = ( N = one ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_eq_neg_one_iff
% 0.23/0.47 thf(fact_3585_numeral__eq__neg__one__iff,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.47 = ( N = one ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_eq_neg_one_iff
% 0.23/0.47 thf(fact_3586_numeral__eq__neg__one__iff,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 0.23/0.47 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.47 = ( N = one ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_eq_neg_one_iff
% 0.23/0.47 thf(fact_3587_minus__one__mult__self,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 0.23/0.47 = one_one_real ) ).
% 0.23/0.47
% 0.23/0.47 % minus_one_mult_self
% 0.23/0.47 thf(fact_3588_minus__one__mult__self,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 0.23/0.47 = one_one_int ) ).
% 0.23/0.47
% 0.23/0.47 % minus_one_mult_self
% 0.23/0.47 thf(fact_3589_minus__one__mult__self,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 0.23/0.47 = one_one_complex ) ).
% 0.23/0.47
% 0.23/0.47 % minus_one_mult_self
% 0.23/0.47 thf(fact_3590_minus__one__mult__self,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 0.23/0.47 = one_one_Code_integer ) ).
% 0.23/0.47
% 0.23/0.47 % minus_one_mult_self
% 0.23/0.47 thf(fact_3591_minus__one__mult__self,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 0.23/0.47 = one_one_rat ) ).
% 0.23/0.47
% 0.23/0.47 % minus_one_mult_self
% 0.23/0.47 thf(fact_3592_left__minus__one__mult__self,axiom,
% 0.23/0.47 ! [N: nat,A: real] :
% 0.23/0.47 ( ( 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 ) )
% 0.23/0.47 = A ) ).
% 0.23/0.47
% 0.23/0.47 % left_minus_one_mult_self
% 0.23/0.47 thf(fact_3593_left__minus__one__mult__self,axiom,
% 0.23/0.47 ! [N: nat,A: int] :
% 0.23/0.47 ( ( 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 ) )
% 0.23/0.47 = A ) ).
% 0.23/0.47
% 0.23/0.47 % left_minus_one_mult_self
% 0.23/0.47 thf(fact_3594_left__minus__one__mult__self,axiom,
% 0.23/0.47 ! [N: nat,A: complex] :
% 0.23/0.47 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 0.23/0.47 = A ) ).
% 0.23/0.47
% 0.23/0.47 % left_minus_one_mult_self
% 0.23/0.47 thf(fact_3595_left__minus__one__mult__self,axiom,
% 0.23/0.47 ! [N: nat,A: code_integer] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 0.23/0.47 = A ) ).
% 0.23/0.47
% 0.23/0.47 % left_minus_one_mult_self
% 0.23/0.47 thf(fact_3596_left__minus__one__mult__self,axiom,
% 0.23/0.47 ! [N: nat,A: rat] :
% 0.23/0.47 ( ( 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 ) )
% 0.23/0.47 = A ) ).
% 0.23/0.47
% 0.23/0.47 % left_minus_one_mult_self
% 0.23/0.47 thf(fact_3597_mod__minus1__right,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 = zero_zero_int ) ).
% 0.23/0.47
% 0.23/0.47 % mod_minus1_right
% 0.23/0.47 thf(fact_3598_mod__minus1__right,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.47 = zero_z3403309356797280102nteger ) ).
% 0.23/0.47
% 0.23/0.47 % mod_minus1_right
% 0.23/0.47 thf(fact_3599_semiring__norm_I168_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: real] :
% 0.23/0.47 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(168)
% 0.23/0.47 thf(fact_3600_semiring__norm_I168_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: int] :
% 0.23/0.47 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(168)
% 0.23/0.47 thf(fact_3601_semiring__norm_I168_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: complex] :
% 0.23/0.47 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(168)
% 0.23/0.47 thf(fact_3602_semiring__norm_I168_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: code_integer] :
% 0.23/0.47 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(168)
% 0.23/0.47 thf(fact_3603_semiring__norm_I168_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: rat] :
% 0.23/0.47 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(168)
% 0.23/0.47 thf(fact_3604_diff__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.23/0.47 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_simps(3)
% 0.23/0.47 thf(fact_3605_diff__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.47 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_simps(3)
% 0.23/0.47 thf(fact_3606_diff__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_simps(3)
% 0.23/0.47 thf(fact_3607_diff__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_simps(3)
% 0.23/0.47 thf(fact_3608_diff__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.23/0.47 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_simps(3)
% 0.23/0.47 thf(fact_3609_diff__numeral__simps_I2_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.47 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_simps(2)
% 0.23/0.47 thf(fact_3610_diff__numeral__simps_I2_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.47 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_simps(2)
% 0.23/0.47 thf(fact_3611_diff__numeral__simps_I2_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.23/0.47 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_simps(2)
% 0.23/0.47 thf(fact_3612_diff__numeral__simps_I2_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.47 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_simps(2)
% 0.23/0.47 thf(fact_3613_diff__numeral__simps_I2_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.47 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_simps(2)
% 0.23/0.47 thf(fact_3614_semiring__norm_I172_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: real] :
% 0.23/0.47 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(172)
% 0.23/0.47 thf(fact_3615_semiring__norm_I172_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: int] :
% 0.23/0.47 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(172)
% 0.23/0.47 thf(fact_3616_semiring__norm_I172_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: complex] :
% 0.23/0.47 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(172)
% 0.23/0.47 thf(fact_3617_semiring__norm_I172_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: code_integer] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(172)
% 0.23/0.47 thf(fact_3618_semiring__norm_I172_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: rat] :
% 0.23/0.47 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(172)
% 0.23/0.47 thf(fact_3619_semiring__norm_I171_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: real] :
% 0.23/0.47 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(171)
% 0.23/0.47 thf(fact_3620_semiring__norm_I171_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: int] :
% 0.23/0.47 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(171)
% 0.23/0.47 thf(fact_3621_semiring__norm_I171_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: complex] :
% 0.23/0.47 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(171)
% 0.23/0.47 thf(fact_3622_semiring__norm_I171_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: code_integer] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(171)
% 0.23/0.47 thf(fact_3623_semiring__norm_I171_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: rat] :
% 0.23/0.47 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.23/0.47 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(171)
% 0.23/0.47 thf(fact_3624_semiring__norm_I170_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: real] :
% 0.23/0.47 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
% 0.23/0.47 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(170)
% 0.23/0.47 thf(fact_3625_semiring__norm_I170_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: int] :
% 0.23/0.47 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
% 0.23/0.47 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(170)
% 0.23/0.47 thf(fact_3626_semiring__norm_I170_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: complex] :
% 0.23/0.47 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y3 ) )
% 0.23/0.47 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(170)
% 0.23/0.47 thf(fact_3627_semiring__norm_I170_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: code_integer] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y3 ) )
% 0.23/0.47 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(170)
% 0.23/0.47 thf(fact_3628_semiring__norm_I170_J,axiom,
% 0.23/0.47 ! [V: num,W: num,Y3: rat] :
% 0.23/0.47 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y3 ) )
% 0.23/0.47 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.23/0.47
% 0.23/0.47 % semiring_norm(170)
% 0.23/0.47 thf(fact_3629_mult__neg__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.47 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(3)
% 0.23/0.47 thf(fact_3630_mult__neg__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.47 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(3)
% 0.23/0.47 thf(fact_3631_mult__neg__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(3)
% 0.23/0.47 thf(fact_3632_mult__neg__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(3)
% 0.23/0.47 thf(fact_3633_mult__neg__numeral__simps_I3_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.47 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(3)
% 0.23/0.47 thf(fact_3634_mult__neg__numeral__simps_I2_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.23/0.47 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(2)
% 0.23/0.47 thf(fact_3635_mult__neg__numeral__simps_I2_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.47 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(2)
% 0.23/0.47 thf(fact_3636_mult__neg__numeral__simps_I2_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(2)
% 0.23/0.47 thf(fact_3637_mult__neg__numeral__simps_I2_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(2)
% 0.23/0.47 thf(fact_3638_mult__neg__numeral__simps_I2_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.23/0.47 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(2)
% 0.23/0.47 thf(fact_3639_mult__neg__numeral__simps_I1_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.47 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(1)
% 0.23/0.47 thf(fact_3640_mult__neg__numeral__simps_I1_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.47 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(1)
% 0.23/0.47 thf(fact_3641_mult__neg__numeral__simps_I1_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.23/0.47 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(1)
% 0.23/0.47 thf(fact_3642_mult__neg__numeral__simps_I1_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.47 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(1)
% 0.23/0.47 thf(fact_3643_mult__neg__numeral__simps_I1_J,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.47 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_neg_numeral_simps(1)
% 0.23/0.47 thf(fact_3644_neg__numeral__le__iff,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.47 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_iff
% 0.23/0.47 thf(fact_3645_neg__numeral__le__iff,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.47 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_iff
% 0.23/0.47 thf(fact_3646_neg__numeral__le__iff,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.47 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_iff
% 0.23/0.47 thf(fact_3647_neg__numeral__le__iff,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.47 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_iff
% 0.23/0.47 thf(fact_3648_neg__numeral__less__iff,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.47 = ( ord_less_num @ N @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_iff
% 0.23/0.47 thf(fact_3649_neg__numeral__less__iff,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.47 = ( ord_less_num @ N @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_iff
% 0.23/0.47 thf(fact_3650_neg__numeral__less__iff,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.47 = ( ord_less_num @ N @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_iff
% 0.23/0.47 thf(fact_3651_neg__numeral__less__iff,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.47 = ( ord_less_num @ N @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_iff
% 0.23/0.47 thf(fact_3652_not__neg__one__le__neg__numeral__iff,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 0.23/0.47 = ( M != one ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_neg_one_le_neg_numeral_iff
% 0.23/0.47 thf(fact_3653_not__neg__one__le__neg__numeral__iff,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 0.23/0.47 = ( M != one ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_neg_one_le_neg_numeral_iff
% 0.23/0.47 thf(fact_3654_not__neg__one__le__neg__numeral__iff,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 0.23/0.47 = ( M != one ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_neg_one_le_neg_numeral_iff
% 0.23/0.47 thf(fact_3655_not__neg__one__le__neg__numeral__iff,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 0.23/0.47 = ( M != one ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_neg_one_le_neg_numeral_iff
% 0.23/0.47 thf(fact_3656_divide__le__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [B2: real,W: num,A: real] :
% 0.23/0.47 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.23/0.47 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_le_eq_numeral1(2)
% 0.23/0.47 thf(fact_3657_divide__le__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [B2: rat,W: num,A: rat] :
% 0.23/0.47 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.23/0.47 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_le_eq_numeral1(2)
% 0.23/0.47 thf(fact_3658_le__divide__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [A: real,B2: real,W: num] :
% 0.23/0.47 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.23/0.47 = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_divide_eq_numeral1(2)
% 0.23/0.47 thf(fact_3659_le__divide__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [A: rat,B2: rat,W: num] :
% 0.23/0.47 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.23/0.47 = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_divide_eq_numeral1(2)
% 0.23/0.47 thf(fact_3660_divide__eq__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [B2: real,W: num,A: real] :
% 0.23/0.47 ( ( ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.47 = A )
% 0.23/0.47 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.23/0.47 != zero_zero_real )
% 0.23/0.47 => ( B2
% 0.23/0.47 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 0.23/0.47 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.23/0.47 = zero_zero_real )
% 0.23/0.47 => ( A = zero_zero_real ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_eq_eq_numeral1(2)
% 0.23/0.47 thf(fact_3661_divide__eq__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [B2: complex,W: num,A: complex] :
% 0.23/0.47 ( ( ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.23/0.47 = A )
% 0.23/0.47 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.23/0.47 != zero_zero_complex )
% 0.23/0.47 => ( B2
% 0.23/0.47 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 0.23/0.47 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.23/0.47 = zero_zero_complex )
% 0.23/0.47 => ( A = zero_zero_complex ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_eq_eq_numeral1(2)
% 0.23/0.47 thf(fact_3662_divide__eq__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [B2: rat,W: num,A: rat] :
% 0.23/0.47 ( ( ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.23/0.47 = A )
% 0.23/0.47 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.23/0.47 != zero_zero_rat )
% 0.23/0.47 => ( B2
% 0.23/0.47 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 0.23/0.47 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.23/0.47 = zero_zero_rat )
% 0.23/0.47 => ( A = zero_zero_rat ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_eq_eq_numeral1(2)
% 0.23/0.47 thf(fact_3663_eq__divide__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [A: real,B2: real,W: num] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.23/0.47 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.23/0.47 != zero_zero_real )
% 0.23/0.47 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.47 = B2 ) )
% 0.23/0.47 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.23/0.47 = zero_zero_real )
% 0.23/0.47 => ( A = zero_zero_real ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % eq_divide_eq_numeral1(2)
% 0.23/0.47 thf(fact_3664_eq__divide__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [A: complex,B2: complex,W: num] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.23/0.47 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.23/0.47 != zero_zero_complex )
% 0.23/0.47 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.23/0.47 = B2 ) )
% 0.23/0.47 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.23/0.47 = zero_zero_complex )
% 0.23/0.47 => ( A = zero_zero_complex ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % eq_divide_eq_numeral1(2)
% 0.23/0.47 thf(fact_3665_eq__divide__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [A: rat,B2: rat,W: num] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.23/0.47 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.23/0.47 != zero_zero_rat )
% 0.23/0.47 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.23/0.47 = B2 ) )
% 0.23/0.47 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.23/0.47 = zero_zero_rat )
% 0.23/0.47 => ( A = zero_zero_rat ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % eq_divide_eq_numeral1(2)
% 0.23/0.47 thf(fact_3666_neg__numeral__less__neg__one__iff,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.47 = ( M != one ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_neg_one_iff
% 0.23/0.47 thf(fact_3667_neg__numeral__less__neg__one__iff,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 = ( M != one ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_neg_one_iff
% 0.23/0.47 thf(fact_3668_neg__numeral__less__neg__one__iff,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.47 = ( M != one ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_neg_one_iff
% 0.23/0.47 thf(fact_3669_neg__numeral__less__neg__one__iff,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.47 = ( M != one ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_neg_one_iff
% 0.23/0.47 thf(fact_3670_divide__less__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [B2: real,W: num,A: real] :
% 0.23/0.47 ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.23/0.47 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_less_eq_numeral1(2)
% 0.23/0.47 thf(fact_3671_divide__less__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [B2: rat,W: num,A: rat] :
% 0.23/0.47 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.23/0.47 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_less_eq_numeral1(2)
% 0.23/0.47 thf(fact_3672_less__divide__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [A: real,B2: real,W: num] :
% 0.23/0.47 ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.23/0.47 = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_divide_eq_numeral1(2)
% 0.23/0.47 thf(fact_3673_less__divide__eq__numeral1_I2_J,axiom,
% 0.23/0.47 ! [A: rat,B2: rat,W: num] :
% 0.23/0.47 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.23/0.47 = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_divide_eq_numeral1(2)
% 0.23/0.47 thf(fact_3674_power2__minus,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.47 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % power2_minus
% 0.23/0.47 thf(fact_3675_power2__minus,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.47 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % power2_minus
% 0.23/0.47 thf(fact_3676_power2__minus,axiom,
% 0.23/0.47 ! [A: complex] :
% 0.23/0.47 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.47 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % power2_minus
% 0.23/0.47 thf(fact_3677_power2__minus,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.47 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % power2_minus
% 0.23/0.47 thf(fact_3678_power2__minus,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.47 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % power2_minus
% 0.23/0.47 thf(fact_3679_add__neg__numeral__special_I9_J,axiom,
% 0.23/0.47 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.47 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(9)
% 0.23/0.47 thf(fact_3680_add__neg__numeral__special_I9_J,axiom,
% 0.23/0.47 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(9)
% 0.23/0.47 thf(fact_3681_add__neg__numeral__special_I9_J,axiom,
% 0.23/0.47 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(9)
% 0.23/0.47 thf(fact_3682_add__neg__numeral__special_I9_J,axiom,
% 0.23/0.47 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(9)
% 0.23/0.47 thf(fact_3683_add__neg__numeral__special_I9_J,axiom,
% 0.23/0.47 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.47 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_neg_numeral_special(9)
% 0.23/0.47 thf(fact_3684_diff__numeral__special_I11_J,axiom,
% 0.23/0.47 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.47 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(11)
% 0.23/0.47 thf(fact_3685_diff__numeral__special_I11_J,axiom,
% 0.23/0.47 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(11)
% 0.23/0.47 thf(fact_3686_diff__numeral__special_I11_J,axiom,
% 0.23/0.47 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.47 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(11)
% 0.23/0.47 thf(fact_3687_diff__numeral__special_I11_J,axiom,
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.47 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(11)
% 0.23/0.47 thf(fact_3688_diff__numeral__special_I11_J,axiom,
% 0.23/0.47 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.47 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(11)
% 0.23/0.47 thf(fact_3689_diff__numeral__special_I10_J,axiom,
% 0.23/0.47 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.23/0.47 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(10)
% 0.23/0.47 thf(fact_3690_diff__numeral__special_I10_J,axiom,
% 0.23/0.47 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.23/0.47 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(10)
% 0.23/0.47 thf(fact_3691_diff__numeral__special_I10_J,axiom,
% 0.23/0.47 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(10)
% 0.23/0.47 thf(fact_3692_diff__numeral__special_I10_J,axiom,
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(10)
% 0.23/0.47 thf(fact_3693_diff__numeral__special_I10_J,axiom,
% 0.23/0.47 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.23/0.47 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(10)
% 0.23/0.47 thf(fact_3694_minus__1__div__2__eq,axiom,
% 0.23/0.47 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.47 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_1_div_2_eq
% 0.23/0.47 thf(fact_3695_minus__1__div__2__eq,axiom,
% 0.23/0.47 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_1_div_2_eq
% 0.23/0.47 thf(fact_3696_minus__1__mod__2__eq,axiom,
% 0.23/0.47 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.47 = one_one_int ) ).
% 0.23/0.47
% 0.23/0.47 % minus_1_mod_2_eq
% 0.23/0.47 thf(fact_3697_minus__1__mod__2__eq,axiom,
% 0.23/0.47 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.47 = one_one_Code_integer ) ).
% 0.23/0.47
% 0.23/0.47 % minus_1_mod_2_eq
% 0.23/0.47 thf(fact_3698_bits__minus__1__mod__2__eq,axiom,
% 0.23/0.47 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.47 = one_one_int ) ).
% 0.23/0.47
% 0.23/0.47 % bits_minus_1_mod_2_eq
% 0.23/0.47 thf(fact_3699_bits__minus__1__mod__2__eq,axiom,
% 0.23/0.47 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.47 = one_one_Code_integer ) ).
% 0.23/0.47
% 0.23/0.47 % bits_minus_1_mod_2_eq
% 0.23/0.47 thf(fact_3700_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.23/0.47 ! [A: real,N: nat] :
% 0.23/0.47 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.47 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % Power.ring_1_class.power_minus_even
% 0.23/0.47 thf(fact_3701_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.23/0.47 ! [A: int,N: nat] :
% 0.23/0.47 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.47 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % Power.ring_1_class.power_minus_even
% 0.23/0.47 thf(fact_3702_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.23/0.47 ! [A: complex,N: nat] :
% 0.23/0.47 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.47 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % Power.ring_1_class.power_minus_even
% 0.23/0.47 thf(fact_3703_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.23/0.47 ! [A: code_integer,N: nat] :
% 0.23/0.47 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.47 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % Power.ring_1_class.power_minus_even
% 0.23/0.47 thf(fact_3704_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.23/0.47 ! [A: rat,N: nat] :
% 0.23/0.47 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.47 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % Power.ring_1_class.power_minus_even
% 0.23/0.47 thf(fact_3705_power__minus__odd,axiom,
% 0.23/0.47 ! [N: nat,A: real] :
% 0.23/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.23/0.47 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % power_minus_odd
% 0.23/0.47 thf(fact_3706_power__minus__odd,axiom,
% 0.23/0.47 ! [N: nat,A: int] :
% 0.23/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.23/0.47 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % power_minus_odd
% 0.23/0.47 thf(fact_3707_power__minus__odd,axiom,
% 0.23/0.47 ! [N: nat,A: complex] :
% 0.23/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % power_minus_odd
% 0.23/0.47 thf(fact_3708_power__minus__odd,axiom,
% 0.23/0.47 ! [N: nat,A: code_integer] :
% 0.23/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % power_minus_odd
% 0.23/0.47 thf(fact_3709_power__minus__odd,axiom,
% 0.23/0.47 ! [N: nat,A: rat] :
% 0.23/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.23/0.47 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % power_minus_odd
% 0.23/0.47 thf(fact_3710_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.23/0.47 ! [N: nat,A: real] :
% 0.23/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.23/0.47 = ( power_power_real @ A @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % Parity.ring_1_class.power_minus_even
% 0.23/0.47 thf(fact_3711_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.23/0.47 ! [N: nat,A: int] :
% 0.23/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.23/0.47 = ( power_power_int @ A @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % Parity.ring_1_class.power_minus_even
% 0.23/0.47 thf(fact_3712_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.23/0.47 ! [N: nat,A: complex] :
% 0.23/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.23/0.47 = ( power_power_complex @ A @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % Parity.ring_1_class.power_minus_even
% 0.23/0.47 thf(fact_3713_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.23/0.47 ! [N: nat,A: code_integer] :
% 0.23/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.23/0.47 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % Parity.ring_1_class.power_minus_even
% 0.23/0.47 thf(fact_3714_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.23/0.47 ! [N: nat,A: rat] :
% 0.23/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.23/0.47 = ( power_power_rat @ A @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % Parity.ring_1_class.power_minus_even
% 0.23/0.47 thf(fact_3715_diff__numeral__special_I3_J,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.47 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(3)
% 0.23/0.47 thf(fact_3716_diff__numeral__special_I3_J,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.47 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(3)
% 0.23/0.47 thf(fact_3717_diff__numeral__special_I3_J,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.23/0.47 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(3)
% 0.23/0.47 thf(fact_3718_diff__numeral__special_I3_J,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.47 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(3)
% 0.23/0.47 thf(fact_3719_diff__numeral__special_I3_J,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.47 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(3)
% 0.23/0.47 thf(fact_3720_diff__numeral__special_I4_J,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 0.23/0.47 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(4)
% 0.23/0.47 thf(fact_3721_diff__numeral__special_I4_J,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 0.23/0.47 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(4)
% 0.23/0.47 thf(fact_3722_diff__numeral__special_I4_J,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(4)
% 0.23/0.47 thf(fact_3723_diff__numeral__special_I4_J,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(4)
% 0.23/0.47 thf(fact_3724_diff__numeral__special_I4_J,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 0.23/0.47 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_numeral_special(4)
% 0.23/0.47 thf(fact_3725_signed__take__bit__Suc__minus__bit0,axiom,
% 0.23/0.47 ! [N: nat,K: num] :
% 0.23/0.47 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.23/0.47 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % signed_take_bit_Suc_minus_bit0
% 0.23/0.47 thf(fact_3726_dbl__simps_I4_J,axiom,
% 0.23/0.47 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.47 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dbl_simps(4)
% 0.23/0.47 thf(fact_3727_dbl__simps_I4_J,axiom,
% 0.23/0.47 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dbl_simps(4)
% 0.23/0.47 thf(fact_3728_dbl__simps_I4_J,axiom,
% 0.23/0.47 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dbl_simps(4)
% 0.23/0.47 thf(fact_3729_dbl__simps_I4_J,axiom,
% 0.23/0.47 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dbl_simps(4)
% 0.23/0.47 thf(fact_3730_dbl__simps_I4_J,axiom,
% 0.23/0.47 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.47 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dbl_simps(4)
% 0.23/0.47 thf(fact_3731_power__minus1__even,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.47 = one_one_real ) ).
% 0.23/0.47
% 0.23/0.47 % power_minus1_even
% 0.23/0.47 thf(fact_3732_power__minus1__even,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.47 = one_one_int ) ).
% 0.23/0.47
% 0.23/0.47 % power_minus1_even
% 0.23/0.47 thf(fact_3733_power__minus1__even,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.47 = one_one_complex ) ).
% 0.23/0.47
% 0.23/0.47 % power_minus1_even
% 0.23/0.47 thf(fact_3734_power__minus1__even,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.47 = one_one_Code_integer ) ).
% 0.23/0.47
% 0.23/0.47 % power_minus1_even
% 0.23/0.47 thf(fact_3735_power__minus1__even,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.47 = one_one_rat ) ).
% 0.23/0.47
% 0.23/0.47 % power_minus1_even
% 0.23/0.47 thf(fact_3736_neg__one__odd__power,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.23/0.47 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_odd_power
% 0.23/0.47 thf(fact_3737_neg__one__odd__power,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.23/0.47 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_odd_power
% 0.23/0.47 thf(fact_3738_neg__one__odd__power,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_odd_power
% 0.23/0.47 thf(fact_3739_neg__one__odd__power,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_odd_power
% 0.23/0.47 thf(fact_3740_neg__one__odd__power,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.23/0.47 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_odd_power
% 0.23/0.47 thf(fact_3741_neg__one__even__power,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.23/0.47 = one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_even_power
% 0.23/0.47 thf(fact_3742_neg__one__even__power,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.23/0.47 = one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_even_power
% 0.23/0.47 thf(fact_3743_neg__one__even__power,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.23/0.47 = one_one_complex ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_even_power
% 0.23/0.47 thf(fact_3744_neg__one__even__power,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.23/0.47 = one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_even_power
% 0.23/0.47 thf(fact_3745_neg__one__even__power,axiom,
% 0.23/0.47 ! [N: nat] :
% 0.23/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.47 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.23/0.47 = one_one_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_even_power
% 0.23/0.47 thf(fact_3746_signed__take__bit__0,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % signed_take_bit_0
% 0.23/0.47 thf(fact_3747_signed__take__bit__0,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 0.23/0.47 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % signed_take_bit_0
% 0.23/0.47 thf(fact_3748_verit__negate__coefficient_I3_J,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( A = B2 )
% 0.23/0.47 => ( ( uminus_uminus_real @ A )
% 0.23/0.47 = ( uminus_uminus_real @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_negate_coefficient(3)
% 0.23/0.47 thf(fact_3749_verit__negate__coefficient_I3_J,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( A = B2 )
% 0.23/0.47 => ( ( uminus_uminus_int @ A )
% 0.23/0.47 = ( uminus_uminus_int @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_negate_coefficient(3)
% 0.23/0.47 thf(fact_3750_verit__negate__coefficient_I3_J,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( A = B2 )
% 0.23/0.47 => ( ( uminus1351360451143612070nteger @ A )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_negate_coefficient(3)
% 0.23/0.47 thf(fact_3751_verit__negate__coefficient_I3_J,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( A = B2 )
% 0.23/0.47 => ( ( uminus_uminus_rat @ A )
% 0.23/0.47 = ( uminus_uminus_rat @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_negate_coefficient(3)
% 0.23/0.47 thf(fact_3752_equation__minus__iff,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus_uminus_real @ B2 ) )
% 0.23/0.47 = ( B2
% 0.23/0.47 = ( uminus_uminus_real @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % equation_minus_iff
% 0.23/0.47 thf(fact_3753_equation__minus__iff,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus_uminus_int @ B2 ) )
% 0.23/0.47 = ( B2
% 0.23/0.47 = ( uminus_uminus_int @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % equation_minus_iff
% 0.23/0.47 thf(fact_3754_equation__minus__iff,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus1482373934393186551omplex @ B2 ) )
% 0.23/0.47 = ( B2
% 0.23/0.47 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % equation_minus_iff
% 0.23/0.47 thf(fact_3755_equation__minus__iff,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.47 = ( B2
% 0.23/0.47 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % equation_minus_iff
% 0.23/0.47 thf(fact_3756_equation__minus__iff,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus_uminus_rat @ B2 ) )
% 0.23/0.47 = ( B2
% 0.23/0.47 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % equation_minus_iff
% 0.23/0.47 thf(fact_3757_minus__equation__iff,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ( uminus_uminus_real @ A )
% 0.23/0.47 = B2 )
% 0.23/0.47 = ( ( uminus_uminus_real @ B2 )
% 0.23/0.47 = A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_equation_iff
% 0.23/0.47 thf(fact_3758_minus__equation__iff,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( ( uminus_uminus_int @ A )
% 0.23/0.47 = B2 )
% 0.23/0.47 = ( ( uminus_uminus_int @ B2 )
% 0.23/0.47 = A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_equation_iff
% 0.23/0.47 thf(fact_3759_minus__equation__iff,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( ( uminus1482373934393186551omplex @ A )
% 0.23/0.47 = B2 )
% 0.23/0.47 = ( ( uminus1482373934393186551omplex @ B2 )
% 0.23/0.47 = A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_equation_iff
% 0.23/0.47 thf(fact_3760_minus__equation__iff,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( ( uminus1351360451143612070nteger @ A )
% 0.23/0.47 = B2 )
% 0.23/0.47 = ( ( uminus1351360451143612070nteger @ B2 )
% 0.23/0.47 = A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_equation_iff
% 0.23/0.47 thf(fact_3761_minus__equation__iff,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ( uminus_uminus_rat @ A )
% 0.23/0.47 = B2 )
% 0.23/0.47 = ( ( uminus_uminus_rat @ B2 )
% 0.23/0.47 = A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_equation_iff
% 0.23/0.47 thf(fact_3762_le__imp__neg__le,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.47 => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_imp_neg_le
% 0.23/0.47 thf(fact_3763_le__imp__neg__le,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 0.23/0.47 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_imp_neg_le
% 0.23/0.47 thf(fact_3764_le__imp__neg__le,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.47 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_imp_neg_le
% 0.23/0.47 thf(fact_3765_le__imp__neg__le,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( ord_less_eq_int @ A @ B2 )
% 0.23/0.47 => ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_imp_neg_le
% 0.23/0.47 thf(fact_3766_minus__le__iff,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.23/0.47 = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_le_iff
% 0.23/0.47 thf(fact_3767_minus__le__iff,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.23/0.47 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_le_iff
% 0.23/0.47 thf(fact_3768_minus__le__iff,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.23/0.47 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_le_iff
% 0.23/0.47 thf(fact_3769_minus__le__iff,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.47 = ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_le_iff
% 0.23/0.47 thf(fact_3770_le__minus__iff,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B2 ) )
% 0.23/0.47 = ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_iff
% 0.23/0.47 thf(fact_3771_le__minus__iff,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.47 = ( ord_le3102999989581377725nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_iff
% 0.23/0.47 thf(fact_3772_le__minus__iff,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 0.23/0.47 = ( ord_less_eq_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_iff
% 0.23/0.47 thf(fact_3773_le__minus__iff,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.47 = ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_iff
% 0.23/0.47 thf(fact_3774_minus__less__iff,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.23/0.47 = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_less_iff
% 0.23/0.47 thf(fact_3775_minus__less__iff,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.47 = ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_less_iff
% 0.23/0.47 thf(fact_3776_minus__less__iff,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.23/0.47 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_less_iff
% 0.23/0.47 thf(fact_3777_minus__less__iff,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.23/0.47 = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_less_iff
% 0.23/0.47 thf(fact_3778_less__minus__iff,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B2 ) )
% 0.23/0.47 = ( ord_less_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_iff
% 0.23/0.47 thf(fact_3779_less__minus__iff,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.47 = ( ord_less_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_iff
% 0.23/0.47 thf(fact_3780_less__minus__iff,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.47 = ( ord_le6747313008572928689nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_iff
% 0.23/0.47 thf(fact_3781_less__minus__iff,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 0.23/0.47 = ( ord_less_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_iff
% 0.23/0.47 thf(fact_3782_verit__negate__coefficient_I2_J,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ord_less_real @ A @ B2 )
% 0.23/0.47 => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_negate_coefficient(2)
% 0.23/0.47 thf(fact_3783_verit__negate__coefficient_I2_J,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( ord_less_int @ A @ B2 )
% 0.23/0.47 => ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_negate_coefficient(2)
% 0.23/0.47 thf(fact_3784_verit__negate__coefficient_I2_J,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 0.23/0.47 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_negate_coefficient(2)
% 0.23/0.47 thf(fact_3785_verit__negate__coefficient_I2_J,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.47 => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % verit_negate_coefficient(2)
% 0.23/0.47 thf(fact_3786_neg__numeral__neq__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.23/0.47 != ( numeral_numeral_real @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_neq_numeral
% 0.23/0.47 thf(fact_3787_neg__numeral__neq__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.23/0.47 != ( numeral_numeral_int @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_neq_numeral
% 0.23/0.47 thf(fact_3788_neg__numeral__neq__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.23/0.47 != ( numera6690914467698888265omplex @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_neq_numeral
% 0.23/0.47 thf(fact_3789_neg__numeral__neq__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.23/0.47 != ( numera6620942414471956472nteger @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_neq_numeral
% 0.23/0.47 thf(fact_3790_neg__numeral__neq__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.23/0.47 != ( numeral_numeral_rat @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_neq_numeral
% 0.23/0.47 thf(fact_3791_numeral__neq__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( numeral_numeral_real @ M )
% 0.23/0.47 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_neq_neg_numeral
% 0.23/0.47 thf(fact_3792_numeral__neq__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( numeral_numeral_int @ M )
% 0.23/0.47 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_neq_neg_numeral
% 0.23/0.47 thf(fact_3793_numeral__neq__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( numera6690914467698888265omplex @ M )
% 0.23/0.47 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_neq_neg_numeral
% 0.23/0.47 thf(fact_3794_numeral__neq__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( numera6620942414471956472nteger @ M )
% 0.23/0.47 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_neq_neg_numeral
% 0.23/0.47 thf(fact_3795_numeral__neq__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ( ( numeral_numeral_rat @ M )
% 0.23/0.47 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_neq_neg_numeral
% 0.23/0.47 thf(fact_3796_minus__mult__commute,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.23/0.47 = ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_mult_commute
% 0.23/0.47 thf(fact_3797_minus__mult__commute,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.47 = ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_mult_commute
% 0.23/0.47 thf(fact_3798_minus__mult__commute,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 0.23/0.47 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_mult_commute
% 0.23/0.47 thf(fact_3799_minus__mult__commute,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.23/0.47 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_mult_commute
% 0.23/0.47 thf(fact_3800_minus__mult__commute,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.23/0.47 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_mult_commute
% 0.23/0.47 thf(fact_3801_square__eq__iff,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ( times_times_real @ A @ A )
% 0.23/0.47 = ( times_times_real @ B2 @ B2 ) )
% 0.23/0.47 = ( ( A = B2 )
% 0.23/0.47 | ( A
% 0.23/0.47 = ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % square_eq_iff
% 0.23/0.47 thf(fact_3802_square__eq__iff,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( ( times_times_int @ A @ A )
% 0.23/0.47 = ( times_times_int @ B2 @ B2 ) )
% 0.23/0.47 = ( ( A = B2 )
% 0.23/0.47 | ( A
% 0.23/0.47 = ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % square_eq_iff
% 0.23/0.47 thf(fact_3803_square__eq__iff,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( ( times_times_complex @ A @ A )
% 0.23/0.47 = ( times_times_complex @ B2 @ B2 ) )
% 0.23/0.47 = ( ( A = B2 )
% 0.23/0.47 | ( A
% 0.23/0.47 = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % square_eq_iff
% 0.23/0.47 thf(fact_3804_square__eq__iff,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( ( times_3573771949741848930nteger @ A @ A )
% 0.23/0.47 = ( times_3573771949741848930nteger @ B2 @ B2 ) )
% 0.23/0.47 = ( ( A = B2 )
% 0.23/0.47 | ( A
% 0.23/0.47 = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % square_eq_iff
% 0.23/0.47 thf(fact_3805_square__eq__iff,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ( times_times_rat @ A @ A )
% 0.23/0.47 = ( times_times_rat @ B2 @ B2 ) )
% 0.23/0.47 = ( ( A = B2 )
% 0.23/0.47 | ( A
% 0.23/0.47 = ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % square_eq_iff
% 0.23/0.47 thf(fact_3806_one__neq__neg__one,axiom,
% 0.23/0.47 ( one_one_real
% 0.23/0.47 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % one_neq_neg_one
% 0.23/0.47 thf(fact_3807_one__neq__neg__one,axiom,
% 0.23/0.47 ( one_one_int
% 0.23/0.47 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % one_neq_neg_one
% 0.23/0.47 thf(fact_3808_one__neq__neg__one,axiom,
% 0.23/0.47 ( one_one_complex
% 0.23/0.47 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.47
% 0.23/0.47 % one_neq_neg_one
% 0.23/0.47 thf(fact_3809_one__neq__neg__one,axiom,
% 0.23/0.47 ( one_one_Code_integer
% 0.23/0.47 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % one_neq_neg_one
% 0.23/0.47 thf(fact_3810_one__neq__neg__one,axiom,
% 0.23/0.47 ( one_one_rat
% 0.23/0.47 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % one_neq_neg_one
% 0.23/0.47 thf(fact_3811_add_Oinverse__distrib__swap,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 0.23/0.47 = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add.inverse_distrib_swap
% 0.23/0.47 thf(fact_3812_add_Oinverse__distrib__swap,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 0.23/0.47 = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add.inverse_distrib_swap
% 0.23/0.47 thf(fact_3813_add_Oinverse__distrib__swap,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 0.23/0.47 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add.inverse_distrib_swap
% 0.23/0.47 thf(fact_3814_add_Oinverse__distrib__swap,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.23/0.47 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add.inverse_distrib_swap
% 0.23/0.47 thf(fact_3815_add_Oinverse__distrib__swap,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 0.23/0.47 = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add.inverse_distrib_swap
% 0.23/0.47 thf(fact_3816_group__cancel_Oneg1,axiom,
% 0.23/0.47 ! [A3: real,K: real,A: real] :
% 0.23/0.47 ( ( A3
% 0.23/0.47 = ( plus_plus_real @ K @ A ) )
% 0.23/0.47 => ( ( uminus_uminus_real @ A3 )
% 0.23/0.47 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % group_cancel.neg1
% 0.23/0.47 thf(fact_3817_group__cancel_Oneg1,axiom,
% 0.23/0.47 ! [A3: int,K: int,A: int] :
% 0.23/0.47 ( ( A3
% 0.23/0.47 = ( plus_plus_int @ K @ A ) )
% 0.23/0.47 => ( ( uminus_uminus_int @ A3 )
% 0.23/0.47 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % group_cancel.neg1
% 0.23/0.47 thf(fact_3818_group__cancel_Oneg1,axiom,
% 0.23/0.47 ! [A3: complex,K: complex,A: complex] :
% 0.23/0.47 ( ( A3
% 0.23/0.47 = ( plus_plus_complex @ K @ A ) )
% 0.23/0.47 => ( ( uminus1482373934393186551omplex @ A3 )
% 0.23/0.47 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % group_cancel.neg1
% 0.23/0.47 thf(fact_3819_group__cancel_Oneg1,axiom,
% 0.23/0.47 ! [A3: code_integer,K: code_integer,A: code_integer] :
% 0.23/0.47 ( ( A3
% 0.23/0.47 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 0.23/0.47 => ( ( uminus1351360451143612070nteger @ A3 )
% 0.23/0.47 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % group_cancel.neg1
% 0.23/0.47 thf(fact_3820_group__cancel_Oneg1,axiom,
% 0.23/0.47 ! [A3: rat,K: rat,A: rat] :
% 0.23/0.47 ( ( A3
% 0.23/0.47 = ( plus_plus_rat @ K @ A ) )
% 0.23/0.47 => ( ( uminus_uminus_rat @ A3 )
% 0.23/0.47 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % group_cancel.neg1
% 0.23/0.47 thf(fact_3821_is__num__normalize_I8_J,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 0.23/0.47 = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % is_num_normalize(8)
% 0.23/0.47 thf(fact_3822_is__num__normalize_I8_J,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 0.23/0.47 = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % is_num_normalize(8)
% 0.23/0.47 thf(fact_3823_is__num__normalize_I8_J,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 0.23/0.47 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % is_num_normalize(8)
% 0.23/0.47 thf(fact_3824_is__num__normalize_I8_J,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.23/0.47 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % is_num_normalize(8)
% 0.23/0.47 thf(fact_3825_is__num__normalize_I8_J,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 0.23/0.47 = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % is_num_normalize(8)
% 0.23/0.47 thf(fact_3826_minus__diff__commute,axiom,
% 0.23/0.47 ! [B2: real,A: real] :
% 0.23/0.47 ( ( minus_minus_real @ ( uminus_uminus_real @ B2 ) @ A )
% 0.23/0.47 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_commute
% 0.23/0.47 thf(fact_3827_minus__diff__commute,axiom,
% 0.23/0.47 ! [B2: int,A: int] :
% 0.23/0.47 ( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A )
% 0.23/0.47 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_commute
% 0.23/0.47 thf(fact_3828_minus__diff__commute,axiom,
% 0.23/0.47 ! [B2: complex,A: complex] :
% 0.23/0.47 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ A )
% 0.23/0.47 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_commute
% 0.23/0.47 thf(fact_3829_minus__diff__commute,axiom,
% 0.23/0.47 ! [B2: code_integer,A: code_integer] :
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A )
% 0.23/0.47 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_commute
% 0.23/0.47 thf(fact_3830_minus__diff__commute,axiom,
% 0.23/0.47 ! [B2: rat,A: rat] :
% 0.23/0.47 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B2 ) @ A )
% 0.23/0.47 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_commute
% 0.23/0.47 thf(fact_3831_minus__diff__minus,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 0.23/0.47 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_minus
% 0.23/0.47 thf(fact_3832_minus__diff__minus,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.47 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_minus
% 0.23/0.47 thf(fact_3833_minus__diff__minus,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_minus
% 0.23/0.47 thf(fact_3834_minus__diff__minus,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_minus
% 0.23/0.47 thf(fact_3835_minus__diff__minus,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 0.23/0.47 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_diff_minus
% 0.23/0.47 thf(fact_3836_div__minus__right,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.47 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % div_minus_right
% 0.23/0.47 thf(fact_3837_div__minus__right,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.47 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % div_minus_right
% 0.23/0.47 thf(fact_3838_minus__divide__right,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 0.23/0.47 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_right
% 0.23/0.47 thf(fact_3839_minus__divide__right,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.23/0.47 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_right
% 0.23/0.47 thf(fact_3840_minus__divide__right,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.23/0.47 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_right
% 0.23/0.47 thf(fact_3841_minus__divide__divide,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 0.23/0.47 = ( divide_divide_real @ A @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_divide
% 0.23/0.47 thf(fact_3842_minus__divide__divide,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.23/0.47 = ( divide1717551699836669952omplex @ A @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_divide
% 0.23/0.47 thf(fact_3843_minus__divide__divide,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 0.23/0.47 = ( divide_divide_rat @ A @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_divide
% 0.23/0.47 thf(fact_3844_minus__divide__left,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 0.23/0.47 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_left
% 0.23/0.47 thf(fact_3845_minus__divide__left,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.23/0.47 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_left
% 0.23/0.47 thf(fact_3846_minus__divide__left,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.23/0.47 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_left
% 0.23/0.47 thf(fact_3847_mod__minus__eq,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) @ B2 )
% 0.23/0.47 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mod_minus_eq
% 0.23/0.47 thf(fact_3848_mod__minus__eq,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ B2 )
% 0.23/0.47 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mod_minus_eq
% 0.23/0.47 thf(fact_3849_mod__minus__cong,axiom,
% 0.23/0.47 ! [A: int,B2: int,A2: int] :
% 0.23/0.47 ( ( ( modulo_modulo_int @ A @ B2 )
% 0.23/0.47 = ( modulo_modulo_int @ A2 @ B2 ) )
% 0.23/0.47 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.47 = ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mod_minus_cong
% 0.23/0.47 thf(fact_3850_mod__minus__cong,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer,A2: code_integer] :
% 0.23/0.47 ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.23/0.47 = ( modulo364778990260209775nteger @ A2 @ B2 ) )
% 0.23/0.47 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.23/0.47 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mod_minus_cong
% 0.23/0.47 thf(fact_3851_mod__minus__right,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.47 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mod_minus_right
% 0.23/0.47 thf(fact_3852_mod__minus__right,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % mod_minus_right
% 0.23/0.47 thf(fact_3853_signed__take__bit__minus,axiom,
% 0.23/0.47 ! [N: nat,K: int] :
% 0.23/0.47 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 0.23/0.47 = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % signed_take_bit_minus
% 0.23/0.47 thf(fact_3854_neg__numeral__le__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_numeral
% 0.23/0.47 thf(fact_3855_neg__numeral__le__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_numeral
% 0.23/0.47 thf(fact_3856_neg__numeral__le__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_numeral
% 0.23/0.47 thf(fact_3857_neg__numeral__le__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_numeral
% 0.23/0.47 thf(fact_3858_not__numeral__le__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_le_neg_numeral
% 0.23/0.47 thf(fact_3859_not__numeral__le__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_le_neg_numeral
% 0.23/0.47 thf(fact_3860_not__numeral__le__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_le_neg_numeral
% 0.23/0.47 thf(fact_3861_not__numeral__le__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_le_neg_numeral
% 0.23/0.47 thf(fact_3862_zero__neq__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( zero_zero_real
% 0.23/0.47 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % zero_neq_neg_numeral
% 0.23/0.47 thf(fact_3863_zero__neq__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( zero_zero_int
% 0.23/0.47 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % zero_neq_neg_numeral
% 0.23/0.47 thf(fact_3864_zero__neq__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( zero_zero_complex
% 0.23/0.47 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % zero_neq_neg_numeral
% 0.23/0.47 thf(fact_3865_zero__neq__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( zero_z3403309356797280102nteger
% 0.23/0.47 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % zero_neq_neg_numeral
% 0.23/0.47 thf(fact_3866_zero__neq__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( zero_zero_rat
% 0.23/0.47 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % zero_neq_neg_numeral
% 0.23/0.47 thf(fact_3867_not__numeral__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_less_neg_numeral
% 0.23/0.47 thf(fact_3868_not__numeral__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_less_neg_numeral
% 0.23/0.47 thf(fact_3869_not__numeral__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_less_neg_numeral
% 0.23/0.47 thf(fact_3870_not__numeral__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] :
% 0.23/0.47 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_less_neg_numeral
% 0.23/0.47 thf(fact_3871_neg__numeral__less__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_numeral
% 0.23/0.47 thf(fact_3872_neg__numeral__less__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_numeral
% 0.23/0.47 thf(fact_3873_neg__numeral__less__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_numeral
% 0.23/0.47 thf(fact_3874_neg__numeral__less__numeral,axiom,
% 0.23/0.47 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_numeral
% 0.23/0.47 thf(fact_3875_le__minus__one__simps_I2_J,axiom,
% 0.23/0.47 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(2)
% 0.23/0.47 thf(fact_3876_le__minus__one__simps_I2_J,axiom,
% 0.23/0.47 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(2)
% 0.23/0.47 thf(fact_3877_le__minus__one__simps_I2_J,axiom,
% 0.23/0.47 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(2)
% 0.23/0.47 thf(fact_3878_le__minus__one__simps_I2_J,axiom,
% 0.23/0.47 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(2)
% 0.23/0.47 thf(fact_3879_le__minus__one__simps_I4_J,axiom,
% 0.23/0.47 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(4)
% 0.23/0.47 thf(fact_3880_le__minus__one__simps_I4_J,axiom,
% 0.23/0.47 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(4)
% 0.23/0.47 thf(fact_3881_le__minus__one__simps_I4_J,axiom,
% 0.23/0.47 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(4)
% 0.23/0.47 thf(fact_3882_le__minus__one__simps_I4_J,axiom,
% 0.23/0.47 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(4)
% 0.23/0.47 thf(fact_3883_zero__neq__neg__one,axiom,
% 0.23/0.47 ( zero_zero_real
% 0.23/0.47 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % zero_neq_neg_one
% 0.23/0.47 thf(fact_3884_zero__neq__neg__one,axiom,
% 0.23/0.47 ( zero_zero_int
% 0.23/0.47 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % zero_neq_neg_one
% 0.23/0.47 thf(fact_3885_zero__neq__neg__one,axiom,
% 0.23/0.47 ( zero_zero_complex
% 0.23/0.47 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.47
% 0.23/0.47 % zero_neq_neg_one
% 0.23/0.47 thf(fact_3886_zero__neq__neg__one,axiom,
% 0.23/0.47 ( zero_z3403309356797280102nteger
% 0.23/0.47 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % zero_neq_neg_one
% 0.23/0.47 thf(fact_3887_zero__neq__neg__one,axiom,
% 0.23/0.47 ( zero_zero_rat
% 0.23/0.47 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % zero_neq_neg_one
% 0.23/0.47 thf(fact_3888_neg__eq__iff__add__eq__0,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ( uminus_uminus_real @ A )
% 0.23/0.47 = B2 )
% 0.23/0.47 = ( ( plus_plus_real @ A @ B2 )
% 0.23/0.47 = zero_zero_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_eq_iff_add_eq_0
% 0.23/0.47 thf(fact_3889_neg__eq__iff__add__eq__0,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( ( uminus_uminus_int @ A )
% 0.23/0.47 = B2 )
% 0.23/0.47 = ( ( plus_plus_int @ A @ B2 )
% 0.23/0.47 = zero_zero_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_eq_iff_add_eq_0
% 0.23/0.47 thf(fact_3890_neg__eq__iff__add__eq__0,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( ( uminus1482373934393186551omplex @ A )
% 0.23/0.47 = B2 )
% 0.23/0.47 = ( ( plus_plus_complex @ A @ B2 )
% 0.23/0.47 = zero_zero_complex ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_eq_iff_add_eq_0
% 0.23/0.47 thf(fact_3891_neg__eq__iff__add__eq__0,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( ( uminus1351360451143612070nteger @ A )
% 0.23/0.47 = B2 )
% 0.23/0.47 = ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 0.23/0.47 = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_eq_iff_add_eq_0
% 0.23/0.47 thf(fact_3892_neg__eq__iff__add__eq__0,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ( uminus_uminus_rat @ A )
% 0.23/0.47 = B2 )
% 0.23/0.47 = ( ( plus_plus_rat @ A @ B2 )
% 0.23/0.47 = zero_zero_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_eq_iff_add_eq_0
% 0.23/0.47 thf(fact_3893_eq__neg__iff__add__eq__0,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus_uminus_real @ B2 ) )
% 0.23/0.47 = ( ( plus_plus_real @ A @ B2 )
% 0.23/0.47 = zero_zero_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % eq_neg_iff_add_eq_0
% 0.23/0.47 thf(fact_3894_eq__neg__iff__add__eq__0,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus_uminus_int @ B2 ) )
% 0.23/0.47 = ( ( plus_plus_int @ A @ B2 )
% 0.23/0.47 = zero_zero_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % eq_neg_iff_add_eq_0
% 0.23/0.47 thf(fact_3895_eq__neg__iff__add__eq__0,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus1482373934393186551omplex @ B2 ) )
% 0.23/0.47 = ( ( plus_plus_complex @ A @ B2 )
% 0.23/0.47 = zero_zero_complex ) ) ).
% 0.23/0.47
% 0.23/0.47 % eq_neg_iff_add_eq_0
% 0.23/0.47 thf(fact_3896_eq__neg__iff__add__eq__0,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.47 = ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 0.23/0.47 = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.47
% 0.23/0.47 % eq_neg_iff_add_eq_0
% 0.23/0.47 thf(fact_3897_eq__neg__iff__add__eq__0,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus_uminus_rat @ B2 ) )
% 0.23/0.47 = ( ( plus_plus_rat @ A @ B2 )
% 0.23/0.47 = zero_zero_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % eq_neg_iff_add_eq_0
% 0.23/0.47 thf(fact_3898_add_Oinverse__unique,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ( plus_plus_real @ A @ B2 )
% 0.23/0.47 = zero_zero_real )
% 0.23/0.47 => ( ( uminus_uminus_real @ A )
% 0.23/0.47 = B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % add.inverse_unique
% 0.23/0.47 thf(fact_3899_add_Oinverse__unique,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( ( plus_plus_int @ A @ B2 )
% 0.23/0.47 = zero_zero_int )
% 0.23/0.47 => ( ( uminus_uminus_int @ A )
% 0.23/0.47 = B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % add.inverse_unique
% 0.23/0.47 thf(fact_3900_add_Oinverse__unique,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( ( plus_plus_complex @ A @ B2 )
% 0.23/0.47 = zero_zero_complex )
% 0.23/0.47 => ( ( uminus1482373934393186551omplex @ A )
% 0.23/0.47 = B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % add.inverse_unique
% 0.23/0.47 thf(fact_3901_add_Oinverse__unique,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 0.23/0.47 = zero_z3403309356797280102nteger )
% 0.23/0.47 => ( ( uminus1351360451143612070nteger @ A )
% 0.23/0.47 = B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % add.inverse_unique
% 0.23/0.47 thf(fact_3902_add_Oinverse__unique,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ( plus_plus_rat @ A @ B2 )
% 0.23/0.47 = zero_zero_rat )
% 0.23/0.47 => ( ( uminus_uminus_rat @ A )
% 0.23/0.47 = B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % add.inverse_unique
% 0.23/0.47 thf(fact_3903_ab__group__add__class_Oab__left__minus,axiom,
% 0.23/0.47 ! [A: real] :
% 0.23/0.47 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.23/0.47 = zero_zero_real ) ).
% 0.23/0.47
% 0.23/0.47 % ab_group_add_class.ab_left_minus
% 0.23/0.47 thf(fact_3904_ab__group__add__class_Oab__left__minus,axiom,
% 0.23/0.47 ! [A: int] :
% 0.23/0.47 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.23/0.47 = zero_zero_int ) ).
% 0.23/0.47
% 0.23/0.47 % ab_group_add_class.ab_left_minus
% 0.23/0.47 thf(fact_3905_ab__group__add__class_Oab__left__minus,axiom,
% 0.23/0.47 ! [A: complex] :
% 0.23/0.47 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.23/0.47 = zero_zero_complex ) ).
% 0.23/0.47
% 0.23/0.47 % ab_group_add_class.ab_left_minus
% 0.23/0.47 thf(fact_3906_ab__group__add__class_Oab__left__minus,axiom,
% 0.23/0.47 ! [A: code_integer] :
% 0.23/0.47 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.23/0.47 = zero_z3403309356797280102nteger ) ).
% 0.23/0.47
% 0.23/0.47 % ab_group_add_class.ab_left_minus
% 0.23/0.47 thf(fact_3907_ab__group__add__class_Oab__left__minus,axiom,
% 0.23/0.47 ! [A: rat] :
% 0.23/0.47 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.23/0.47 = zero_zero_rat ) ).
% 0.23/0.47
% 0.23/0.47 % ab_group_add_class.ab_left_minus
% 0.23/0.47 thf(fact_3908_add__eq__0__iff,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ( plus_plus_real @ A @ B2 )
% 0.23/0.47 = zero_zero_real )
% 0.23/0.47 = ( B2
% 0.23/0.47 = ( uminus_uminus_real @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_eq_0_iff
% 0.23/0.47 thf(fact_3909_add__eq__0__iff,axiom,
% 0.23/0.47 ! [A: int,B2: int] :
% 0.23/0.47 ( ( ( plus_plus_int @ A @ B2 )
% 0.23/0.47 = zero_zero_int )
% 0.23/0.47 = ( B2
% 0.23/0.47 = ( uminus_uminus_int @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_eq_0_iff
% 0.23/0.47 thf(fact_3910_add__eq__0__iff,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( ( plus_plus_complex @ A @ B2 )
% 0.23/0.47 = zero_zero_complex )
% 0.23/0.47 = ( B2
% 0.23/0.47 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_eq_0_iff
% 0.23/0.47 thf(fact_3911_add__eq__0__iff,axiom,
% 0.23/0.47 ! [A: code_integer,B2: code_integer] :
% 0.23/0.47 ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 0.23/0.47 = zero_z3403309356797280102nteger )
% 0.23/0.47 = ( B2
% 0.23/0.47 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_eq_0_iff
% 0.23/0.47 thf(fact_3912_add__eq__0__iff,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ( plus_plus_rat @ A @ B2 )
% 0.23/0.47 = zero_zero_rat )
% 0.23/0.47 = ( B2
% 0.23/0.47 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % add_eq_0_iff
% 0.23/0.47 thf(fact_3913_less__minus__one__simps_I2_J,axiom,
% 0.23/0.47 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(2)
% 0.23/0.47 thf(fact_3914_less__minus__one__simps_I2_J,axiom,
% 0.23/0.47 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(2)
% 0.23/0.47 thf(fact_3915_less__minus__one__simps_I2_J,axiom,
% 0.23/0.47 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(2)
% 0.23/0.47 thf(fact_3916_less__minus__one__simps_I2_J,axiom,
% 0.23/0.47 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(2)
% 0.23/0.47 thf(fact_3917_less__minus__one__simps_I4_J,axiom,
% 0.23/0.47 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(4)
% 0.23/0.47 thf(fact_3918_less__minus__one__simps_I4_J,axiom,
% 0.23/0.47 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(4)
% 0.23/0.47 thf(fact_3919_less__minus__one__simps_I4_J,axiom,
% 0.23/0.47 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(4)
% 0.23/0.47 thf(fact_3920_less__minus__one__simps_I4_J,axiom,
% 0.23/0.47 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(4)
% 0.23/0.47 thf(fact_3921_numeral__times__minus__swap,axiom,
% 0.23/0.47 ! [W: num,X4: real] :
% 0.23/0.47 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X4 ) )
% 0.23/0.47 = ( times_times_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_times_minus_swap
% 0.23/0.47 thf(fact_3922_numeral__times__minus__swap,axiom,
% 0.23/0.47 ! [W: num,X4: int] :
% 0.23/0.47 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X4 ) )
% 0.23/0.47 = ( times_times_int @ X4 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_times_minus_swap
% 0.23/0.47 thf(fact_3923_numeral__times__minus__swap,axiom,
% 0.23/0.47 ! [W: num,X4: complex] :
% 0.23/0.47 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X4 ) )
% 0.23/0.47 = ( times_times_complex @ X4 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_times_minus_swap
% 0.23/0.47 thf(fact_3924_numeral__times__minus__swap,axiom,
% 0.23/0.47 ! [W: num,X4: code_integer] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X4 ) )
% 0.23/0.47 = ( times_3573771949741848930nteger @ X4 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_times_minus_swap
% 0.23/0.47 thf(fact_3925_numeral__times__minus__swap,axiom,
% 0.23/0.47 ! [W: num,X4: rat] :
% 0.23/0.47 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X4 ) )
% 0.23/0.47 = ( times_times_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_times_minus_swap
% 0.23/0.47 thf(fact_3926_nonzero__minus__divide__right,axiom,
% 0.23/0.47 ! [B2: real,A: real] :
% 0.23/0.47 ( ( B2 != zero_zero_real )
% 0.23/0.47 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 0.23/0.47 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_minus_divide_right
% 0.23/0.47 thf(fact_3927_nonzero__minus__divide__right,axiom,
% 0.23/0.47 ! [B2: complex,A: complex] :
% 0.23/0.47 ( ( B2 != zero_zero_complex )
% 0.23/0.47 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.23/0.47 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_minus_divide_right
% 0.23/0.47 thf(fact_3928_nonzero__minus__divide__right,axiom,
% 0.23/0.47 ! [B2: rat,A: rat] :
% 0.23/0.47 ( ( B2 != zero_zero_rat )
% 0.23/0.47 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.23/0.47 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_minus_divide_right
% 0.23/0.47 thf(fact_3929_nonzero__minus__divide__divide,axiom,
% 0.23/0.47 ! [B2: real,A: real] :
% 0.23/0.47 ( ( B2 != zero_zero_real )
% 0.23/0.47 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 0.23/0.47 = ( divide_divide_real @ A @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_minus_divide_divide
% 0.23/0.47 thf(fact_3930_nonzero__minus__divide__divide,axiom,
% 0.23/0.47 ! [B2: complex,A: complex] :
% 0.23/0.47 ( ( B2 != zero_zero_complex )
% 0.23/0.47 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.23/0.47 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_minus_divide_divide
% 0.23/0.47 thf(fact_3931_nonzero__minus__divide__divide,axiom,
% 0.23/0.47 ! [B2: rat,A: rat] :
% 0.23/0.47 ( ( B2 != zero_zero_rat )
% 0.23/0.47 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 0.23/0.47 = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_minus_divide_divide
% 0.23/0.47 thf(fact_3932_one__neq__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( one_one_real
% 0.23/0.47 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % one_neq_neg_numeral
% 0.23/0.47 thf(fact_3933_one__neq__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( one_one_int
% 0.23/0.47 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % one_neq_neg_numeral
% 0.23/0.47 thf(fact_3934_one__neq__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( one_one_complex
% 0.23/0.47 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % one_neq_neg_numeral
% 0.23/0.47 thf(fact_3935_one__neq__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( one_one_Code_integer
% 0.23/0.47 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % one_neq_neg_numeral
% 0.23/0.47 thf(fact_3936_one__neq__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( one_one_rat
% 0.23/0.47 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % one_neq_neg_numeral
% 0.23/0.47 thf(fact_3937_numeral__neq__neg__one,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( numeral_numeral_real @ N )
% 0.23/0.47 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_neq_neg_one
% 0.23/0.47 thf(fact_3938_numeral__neq__neg__one,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( numeral_numeral_int @ N )
% 0.23/0.47 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_neq_neg_one
% 0.23/0.47 thf(fact_3939_numeral__neq__neg__one,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( numera6690914467698888265omplex @ N )
% 0.23/0.47 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_neq_neg_one
% 0.23/0.47 thf(fact_3940_numeral__neq__neg__one,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( numera6620942414471956472nteger @ N )
% 0.23/0.47 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_neq_neg_one
% 0.23/0.47 thf(fact_3941_numeral__neq__neg__one,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ( ( numeral_numeral_rat @ N )
% 0.23/0.47 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % numeral_neq_neg_one
% 0.23/0.47 thf(fact_3942_square__eq__1__iff,axiom,
% 0.23/0.47 ! [X4: real] :
% 0.23/0.47 ( ( ( times_times_real @ X4 @ X4 )
% 0.23/0.47 = one_one_real )
% 0.23/0.47 = ( ( X4 = one_one_real )
% 0.23/0.47 | ( X4
% 0.23/0.47 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % square_eq_1_iff
% 0.23/0.47 thf(fact_3943_square__eq__1__iff,axiom,
% 0.23/0.47 ! [X4: int] :
% 0.23/0.47 ( ( ( times_times_int @ X4 @ X4 )
% 0.23/0.47 = one_one_int )
% 0.23/0.47 = ( ( X4 = one_one_int )
% 0.23/0.47 | ( X4
% 0.23/0.47 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % square_eq_1_iff
% 0.23/0.47 thf(fact_3944_square__eq__1__iff,axiom,
% 0.23/0.47 ! [X4: complex] :
% 0.23/0.47 ( ( ( times_times_complex @ X4 @ X4 )
% 0.23/0.47 = one_one_complex )
% 0.23/0.47 = ( ( X4 = one_one_complex )
% 0.23/0.47 | ( X4
% 0.23/0.47 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % square_eq_1_iff
% 0.23/0.47 thf(fact_3945_square__eq__1__iff,axiom,
% 0.23/0.47 ! [X4: code_integer] :
% 0.23/0.47 ( ( ( times_3573771949741848930nteger @ X4 @ X4 )
% 0.23/0.47 = one_one_Code_integer )
% 0.23/0.47 = ( ( X4 = one_one_Code_integer )
% 0.23/0.47 | ( X4
% 0.23/0.47 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % square_eq_1_iff
% 0.23/0.47 thf(fact_3946_square__eq__1__iff,axiom,
% 0.23/0.47 ! [X4: rat] :
% 0.23/0.47 ( ( ( times_times_rat @ X4 @ X4 )
% 0.23/0.47 = one_one_rat )
% 0.23/0.47 = ( ( X4 = one_one_rat )
% 0.23/0.47 | ( X4
% 0.23/0.47 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % square_eq_1_iff
% 0.23/0.47 thf(fact_3947_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.23/0.47 ( minus_minus_real
% 0.23/0.47 = ( ^ [A4: real,B3: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.23/0.47 thf(fact_3948_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.23/0.47 ( minus_minus_int
% 0.23/0.47 = ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.23/0.47 thf(fact_3949_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.23/0.47 ( minus_minus_complex
% 0.23/0.47 = ( ^ [A4: complex,B3: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.23/0.47 thf(fact_3950_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.23/0.47 ( minus_8373710615458151222nteger
% 0.23/0.47 = ( ^ [A4: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.23/0.47 thf(fact_3951_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.23/0.47 ( minus_minus_rat
% 0.23/0.47 = ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.23/0.47 thf(fact_3952_diff__conv__add__uminus,axiom,
% 0.23/0.47 ( minus_minus_real
% 0.23/0.47 = ( ^ [A4: real,B3: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_conv_add_uminus
% 0.23/0.47 thf(fact_3953_diff__conv__add__uminus,axiom,
% 0.23/0.47 ( minus_minus_int
% 0.23/0.47 = ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_conv_add_uminus
% 0.23/0.47 thf(fact_3954_diff__conv__add__uminus,axiom,
% 0.23/0.47 ( minus_minus_complex
% 0.23/0.47 = ( ^ [A4: complex,B3: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_conv_add_uminus
% 0.23/0.47 thf(fact_3955_diff__conv__add__uminus,axiom,
% 0.23/0.47 ( minus_8373710615458151222nteger
% 0.23/0.47 = ( ^ [A4: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_conv_add_uminus
% 0.23/0.47 thf(fact_3956_diff__conv__add__uminus,axiom,
% 0.23/0.47 ( minus_minus_rat
% 0.23/0.47 = ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % diff_conv_add_uminus
% 0.23/0.47 thf(fact_3957_group__cancel_Osub2,axiom,
% 0.23/0.47 ! [B4: real,K: real,B2: real,A: real] :
% 0.23/0.47 ( ( B4
% 0.23/0.47 = ( plus_plus_real @ K @ B2 ) )
% 0.23/0.47 => ( ( minus_minus_real @ A @ B4 )
% 0.23/0.47 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % group_cancel.sub2
% 0.23/0.47 thf(fact_3958_group__cancel_Osub2,axiom,
% 0.23/0.47 ! [B4: int,K: int,B2: int,A: int] :
% 0.23/0.47 ( ( B4
% 0.23/0.47 = ( plus_plus_int @ K @ B2 ) )
% 0.23/0.47 => ( ( minus_minus_int @ A @ B4 )
% 0.23/0.47 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % group_cancel.sub2
% 0.23/0.47 thf(fact_3959_group__cancel_Osub2,axiom,
% 0.23/0.47 ! [B4: complex,K: complex,B2: complex,A: complex] :
% 0.23/0.47 ( ( B4
% 0.23/0.47 = ( plus_plus_complex @ K @ B2 ) )
% 0.23/0.47 => ( ( minus_minus_complex @ A @ B4 )
% 0.23/0.47 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % group_cancel.sub2
% 0.23/0.47 thf(fact_3960_group__cancel_Osub2,axiom,
% 0.23/0.47 ! [B4: code_integer,K: code_integer,B2: code_integer,A: code_integer] :
% 0.23/0.47 ( ( B4
% 0.23/0.47 = ( plus_p5714425477246183910nteger @ K @ B2 ) )
% 0.23/0.47 => ( ( minus_8373710615458151222nteger @ A @ B4 )
% 0.23/0.47 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % group_cancel.sub2
% 0.23/0.47 thf(fact_3961_group__cancel_Osub2,axiom,
% 0.23/0.47 ! [B4: rat,K: rat,B2: rat,A: rat] :
% 0.23/0.47 ( ( B4
% 0.23/0.47 = ( plus_plus_rat @ K @ B2 ) )
% 0.23/0.47 => ( ( minus_minus_rat @ A @ B4 )
% 0.23/0.47 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % group_cancel.sub2
% 0.23/0.47 thf(fact_3962_dvd__div__neg,axiom,
% 0.23/0.47 ! [B2: real,A: real] :
% 0.23/0.47 ( ( dvd_dvd_real @ B2 @ A )
% 0.23/0.47 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) )
% 0.23/0.47 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_div_neg
% 0.23/0.47 thf(fact_3963_dvd__div__neg,axiom,
% 0.23/0.47 ! [B2: int,A: int] :
% 0.23/0.47 ( ( dvd_dvd_int @ B2 @ A )
% 0.23/0.47 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.47 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_div_neg
% 0.23/0.47 thf(fact_3964_dvd__div__neg,axiom,
% 0.23/0.47 ! [B2: complex,A: complex] :
% 0.23/0.47 ( ( dvd_dvd_complex @ B2 @ A )
% 0.23/0.47 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_div_neg
% 0.23/0.47 thf(fact_3965_dvd__div__neg,axiom,
% 0.23/0.47 ! [B2: code_integer,A: code_integer] :
% 0.23/0.47 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.23/0.47 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_div_neg
% 0.23/0.47 thf(fact_3966_dvd__div__neg,axiom,
% 0.23/0.47 ! [B2: rat,A: rat] :
% 0.23/0.47 ( ( dvd_dvd_rat @ B2 @ A )
% 0.23/0.47 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 0.23/0.47 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_div_neg
% 0.23/0.47 thf(fact_3967_dvd__neg__div,axiom,
% 0.23/0.47 ! [B2: real,A: real] :
% 0.23/0.47 ( ( dvd_dvd_real @ B2 @ A )
% 0.23/0.47 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.23/0.47 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_neg_div
% 0.23/0.47 thf(fact_3968_dvd__neg__div,axiom,
% 0.23/0.47 ! [B2: int,A: int] :
% 0.23/0.47 ( ( dvd_dvd_int @ B2 @ A )
% 0.23/0.47 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.47 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_neg_div
% 0.23/0.47 thf(fact_3969_dvd__neg__div,axiom,
% 0.23/0.47 ! [B2: complex,A: complex] :
% 0.23/0.47 ( ( dvd_dvd_complex @ B2 @ A )
% 0.23/0.47 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_neg_div
% 0.23/0.47 thf(fact_3970_dvd__neg__div,axiom,
% 0.23/0.47 ! [B2: code_integer,A: code_integer] :
% 0.23/0.47 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.23/0.47 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_neg_div
% 0.23/0.47 thf(fact_3971_dvd__neg__div,axiom,
% 0.23/0.47 ! [B2: rat,A: rat] :
% 0.23/0.47 ( ( dvd_dvd_rat @ B2 @ A )
% 0.23/0.47 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.23/0.47 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % dvd_neg_div
% 0.23/0.47 thf(fact_3972_real__minus__mult__self__le,axiom,
% 0.23/0.47 ! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X4 @ X4 ) ) ).
% 0.23/0.47
% 0.23/0.47 % real_minus_mult_self_le
% 0.23/0.47 thf(fact_3973_pos__zmult__eq__1__iff__lemma,axiom,
% 0.23/0.47 ! [M: int,N: int] :
% 0.23/0.47 ( ( ( times_times_int @ M @ N )
% 0.23/0.47 = one_one_int )
% 0.23/0.47 => ( ( M = one_one_int )
% 0.23/0.47 | ( M
% 0.23/0.47 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % pos_zmult_eq_1_iff_lemma
% 0.23/0.47 thf(fact_3974_zmult__eq__1__iff,axiom,
% 0.23/0.47 ! [M: int,N: int] :
% 0.23/0.47 ( ( ( times_times_int @ M @ N )
% 0.23/0.47 = one_one_int )
% 0.23/0.47 = ( ( ( M = one_one_int )
% 0.23/0.47 & ( N = one_one_int ) )
% 0.23/0.47 | ( ( M
% 0.23/0.47 = ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.47 & ( N
% 0.23/0.47 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % zmult_eq_1_iff
% 0.23/0.47 thf(fact_3975_minus__int__code_I2_J,axiom,
% 0.23/0.47 ! [L2: int] :
% 0.23/0.47 ( ( minus_minus_int @ zero_zero_int @ L2 )
% 0.23/0.47 = ( uminus_uminus_int @ L2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_int_code(2)
% 0.23/0.47 thf(fact_3976_zmod__zminus2__not__zero,axiom,
% 0.23/0.47 ! [K: int,L2: int] :
% 0.23/0.47 ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L2 ) )
% 0.23/0.47 != zero_zero_int )
% 0.23/0.47 => ( ( modulo_modulo_int @ K @ L2 )
% 0.23/0.47 != zero_zero_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % zmod_zminus2_not_zero
% 0.23/0.47 thf(fact_3977_zmod__zminus1__not__zero,axiom,
% 0.23/0.47 ! [K: int,L2: int] :
% 0.23/0.47 ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
% 0.23/0.47 != zero_zero_int )
% 0.23/0.47 => ( ( modulo_modulo_int @ K @ L2 )
% 0.23/0.47 != zero_zero_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % zmod_zminus1_not_zero
% 0.23/0.47 thf(fact_3978_minus__real__def,axiom,
% 0.23/0.47 ( minus_minus_real
% 0.23/0.47 = ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_real_def
% 0.23/0.47 thf(fact_3979_not__zero__le__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_zero_le_neg_numeral
% 0.23/0.47 thf(fact_3980_not__zero__le__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_zero_le_neg_numeral
% 0.23/0.47 thf(fact_3981_not__zero__le__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_zero_le_neg_numeral
% 0.23/0.47 thf(fact_3982_not__zero__le__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_zero_le_neg_numeral
% 0.23/0.47 thf(fact_3983_neg__numeral__le__zero,axiom,
% 0.23/0.47 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_zero
% 0.23/0.47 thf(fact_3984_neg__numeral__le__zero,axiom,
% 0.23/0.47 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_zero
% 0.23/0.47 thf(fact_3985_neg__numeral__le__zero,axiom,
% 0.23/0.47 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_zero
% 0.23/0.47 thf(fact_3986_neg__numeral__le__zero,axiom,
% 0.23/0.47 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_zero
% 0.23/0.47 thf(fact_3987_not__zero__less__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_zero_less_neg_numeral
% 0.23/0.47 thf(fact_3988_not__zero__less__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_zero_less_neg_numeral
% 0.23/0.47 thf(fact_3989_not__zero__less__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_zero_less_neg_numeral
% 0.23/0.47 thf(fact_3990_not__zero__less__neg__numeral,axiom,
% 0.23/0.47 ! [N: num] :
% 0.23/0.47 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_zero_less_neg_numeral
% 0.23/0.47 thf(fact_3991_neg__numeral__less__zero,axiom,
% 0.23/0.47 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_zero
% 0.23/0.47 thf(fact_3992_neg__numeral__less__zero,axiom,
% 0.23/0.47 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_zero
% 0.23/0.47 thf(fact_3993_neg__numeral__less__zero,axiom,
% 0.23/0.47 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_zero
% 0.23/0.47 thf(fact_3994_neg__numeral__less__zero,axiom,
% 0.23/0.47 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_zero
% 0.23/0.47 thf(fact_3995_le__minus__one__simps_I3_J,axiom,
% 0.23/0.47 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(3)
% 0.23/0.47 thf(fact_3996_le__minus__one__simps_I3_J,axiom,
% 0.23/0.47 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(3)
% 0.23/0.47 thf(fact_3997_le__minus__one__simps_I3_J,axiom,
% 0.23/0.47 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(3)
% 0.23/0.47 thf(fact_3998_le__minus__one__simps_I3_J,axiom,
% 0.23/0.47 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(3)
% 0.23/0.47 thf(fact_3999_le__minus__one__simps_I1_J,axiom,
% 0.23/0.47 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(1)
% 0.23/0.47 thf(fact_4000_le__minus__one__simps_I1_J,axiom,
% 0.23/0.47 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(1)
% 0.23/0.47 thf(fact_4001_le__minus__one__simps_I1_J,axiom,
% 0.23/0.47 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(1)
% 0.23/0.47 thf(fact_4002_le__minus__one__simps_I1_J,axiom,
% 0.23/0.47 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.23/0.47
% 0.23/0.47 % le_minus_one_simps(1)
% 0.23/0.47 thf(fact_4003_less__minus__one__simps_I3_J,axiom,
% 0.23/0.47 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(3)
% 0.23/0.47 thf(fact_4004_less__minus__one__simps_I3_J,axiom,
% 0.23/0.47 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(3)
% 0.23/0.47 thf(fact_4005_less__minus__one__simps_I3_J,axiom,
% 0.23/0.47 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(3)
% 0.23/0.47 thf(fact_4006_less__minus__one__simps_I3_J,axiom,
% 0.23/0.47 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(3)
% 0.23/0.47 thf(fact_4007_less__minus__one__simps_I1_J,axiom,
% 0.23/0.47 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(1)
% 0.23/0.47 thf(fact_4008_less__minus__one__simps_I1_J,axiom,
% 0.23/0.47 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(1)
% 0.23/0.47 thf(fact_4009_less__minus__one__simps_I1_J,axiom,
% 0.23/0.47 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(1)
% 0.23/0.47 thf(fact_4010_less__minus__one__simps_I1_J,axiom,
% 0.23/0.47 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.23/0.47
% 0.23/0.47 % less_minus_one_simps(1)
% 0.23/0.47 thf(fact_4011_not__one__le__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_one_le_neg_numeral
% 0.23/0.47 thf(fact_4012_not__one__le__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_one_le_neg_numeral
% 0.23/0.47 thf(fact_4013_not__one__le__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_one_le_neg_numeral
% 0.23/0.47 thf(fact_4014_not__one__le__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_one_le_neg_numeral
% 0.23/0.47 thf(fact_4015_not__numeral__le__neg__one,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_le_neg_one
% 0.23/0.47 thf(fact_4016_not__numeral__le__neg__one,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_le_neg_one
% 0.23/0.47 thf(fact_4017_not__numeral__le__neg__one,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_le_neg_one
% 0.23/0.47 thf(fact_4018_not__numeral__le__neg__one,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_le_neg_one
% 0.23/0.47 thf(fact_4019_neg__numeral__le__neg__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_neg_one
% 0.23/0.47 thf(fact_4020_neg__numeral__le__neg__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_neg_one
% 0.23/0.47 thf(fact_4021_neg__numeral__le__neg__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_neg_one
% 0.23/0.47 thf(fact_4022_neg__numeral__le__neg__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_neg_one
% 0.23/0.47 thf(fact_4023_neg__one__le__numeral,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_le_numeral
% 0.23/0.47 thf(fact_4024_neg__one__le__numeral,axiom,
% 0.23/0.47 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_le_numeral
% 0.23/0.47 thf(fact_4025_neg__one__le__numeral,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_le_numeral
% 0.23/0.47 thf(fact_4026_neg__one__le__numeral,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_le_numeral
% 0.23/0.47 thf(fact_4027_neg__numeral__le__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_one
% 0.23/0.47 thf(fact_4028_neg__numeral__le__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_one
% 0.23/0.47 thf(fact_4029_neg__numeral__le__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_one
% 0.23/0.47 thf(fact_4030_neg__numeral__le__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_le_one
% 0.23/0.47 thf(fact_4031_not__neg__one__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_neg_one_less_neg_numeral
% 0.23/0.47 thf(fact_4032_not__neg__one__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_neg_one_less_neg_numeral
% 0.23/0.47 thf(fact_4033_not__neg__one__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_neg_one_less_neg_numeral
% 0.23/0.47 thf(fact_4034_not__neg__one__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_neg_one_less_neg_numeral
% 0.23/0.47 thf(fact_4035_not__one__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_one_less_neg_numeral
% 0.23/0.47 thf(fact_4036_not__one__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_one_less_neg_numeral
% 0.23/0.47 thf(fact_4037_not__one__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_one_less_neg_numeral
% 0.23/0.47 thf(fact_4038_not__one__less__neg__numeral,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_one_less_neg_numeral
% 0.23/0.47 thf(fact_4039_not__numeral__less__neg__one,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_less_neg_one
% 0.23/0.47 thf(fact_4040_not__numeral__less__neg__one,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_less_neg_one
% 0.23/0.47 thf(fact_4041_not__numeral__less__neg__one,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_less_neg_one
% 0.23/0.47 thf(fact_4042_not__numeral__less__neg__one,axiom,
% 0.23/0.47 ! [M: num] :
% 0.23/0.47 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.47
% 0.23/0.47 % not_numeral_less_neg_one
% 0.23/0.47 thf(fact_4043_neg__one__less__numeral,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_less_numeral
% 0.23/0.47 thf(fact_4044_neg__one__less__numeral,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_less_numeral
% 0.23/0.47 thf(fact_4045_neg__one__less__numeral,axiom,
% 0.23/0.47 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_less_numeral
% 0.23/0.47 thf(fact_4046_neg__one__less__numeral,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.23/0.47
% 0.23/0.47 % neg_one_less_numeral
% 0.23/0.47 thf(fact_4047_neg__numeral__less__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_one
% 0.23/0.47 thf(fact_4048_neg__numeral__less__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_one
% 0.23/0.47 thf(fact_4049_neg__numeral__less__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_one
% 0.23/0.47 thf(fact_4050_neg__numeral__less__one,axiom,
% 0.23/0.47 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.23/0.47
% 0.23/0.47 % neg_numeral_less_one
% 0.23/0.47 thf(fact_4051_eq__minus__divide__eq,axiom,
% 0.23/0.47 ! [A: real,B2: real,C: real] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.23/0.47 = ( ( ( C != zero_zero_real )
% 0.23/0.47 => ( ( times_times_real @ A @ C )
% 0.23/0.47 = ( uminus_uminus_real @ B2 ) ) )
% 0.23/0.47 & ( ( C = zero_zero_real )
% 0.23/0.47 => ( A = zero_zero_real ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % eq_minus_divide_eq
% 0.23/0.47 thf(fact_4052_eq__minus__divide__eq,axiom,
% 0.23/0.47 ! [A: complex,B2: complex,C: complex] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) ) )
% 0.23/0.47 = ( ( ( C != zero_zero_complex )
% 0.23/0.47 => ( ( times_times_complex @ A @ C )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ B2 ) ) )
% 0.23/0.47 & ( ( C = zero_zero_complex )
% 0.23/0.47 => ( A = zero_zero_complex ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % eq_minus_divide_eq
% 0.23/0.47 thf(fact_4053_eq__minus__divide__eq,axiom,
% 0.23/0.47 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.47 ( ( A
% 0.23/0.47 = ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.23/0.47 = ( ( ( C != zero_zero_rat )
% 0.23/0.47 => ( ( times_times_rat @ A @ C )
% 0.23/0.47 = ( uminus_uminus_rat @ B2 ) ) )
% 0.23/0.47 & ( ( C = zero_zero_rat )
% 0.23/0.47 => ( A = zero_zero_rat ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % eq_minus_divide_eq
% 0.23/0.47 thf(fact_4054_minus__divide__eq__eq,axiom,
% 0.23/0.47 ! [B2: real,C: real,A: real] :
% 0.23/0.47 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) )
% 0.23/0.47 = A )
% 0.23/0.47 = ( ( ( C != zero_zero_real )
% 0.23/0.47 => ( ( uminus_uminus_real @ B2 )
% 0.23/0.47 = ( times_times_real @ A @ C ) ) )
% 0.23/0.47 & ( ( C = zero_zero_real )
% 0.23/0.47 => ( A = zero_zero_real ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_eq_eq
% 0.23/0.47 thf(fact_4055_minus__divide__eq__eq,axiom,
% 0.23/0.47 ! [B2: complex,C: complex,A: complex] :
% 0.23/0.47 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.23/0.47 = A )
% 0.23/0.47 = ( ( ( C != zero_zero_complex )
% 0.23/0.47 => ( ( uminus1482373934393186551omplex @ B2 )
% 0.23/0.47 = ( times_times_complex @ A @ C ) ) )
% 0.23/0.47 & ( ( C = zero_zero_complex )
% 0.23/0.47 => ( A = zero_zero_complex ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_eq_eq
% 0.23/0.47 thf(fact_4056_minus__divide__eq__eq,axiom,
% 0.23/0.47 ! [B2: rat,C: rat,A: rat] :
% 0.23/0.47 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) )
% 0.23/0.47 = A )
% 0.23/0.47 = ( ( ( C != zero_zero_rat )
% 0.23/0.47 => ( ( uminus_uminus_rat @ B2 )
% 0.23/0.47 = ( times_times_rat @ A @ C ) ) )
% 0.23/0.47 & ( ( C = zero_zero_rat )
% 0.23/0.47 => ( A = zero_zero_rat ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % minus_divide_eq_eq
% 0.23/0.47 thf(fact_4057_nonzero__neg__divide__eq__eq,axiom,
% 0.23/0.47 ! [B2: real,A: real,C: real] :
% 0.23/0.47 ( ( B2 != zero_zero_real )
% 0.23/0.47 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 0.23/0.47 = C )
% 0.23/0.47 = ( ( uminus_uminus_real @ A )
% 0.23/0.47 = ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_neg_divide_eq_eq
% 0.23/0.47 thf(fact_4058_nonzero__neg__divide__eq__eq,axiom,
% 0.23/0.47 ! [B2: complex,A: complex,C: complex] :
% 0.23/0.47 ( ( B2 != zero_zero_complex )
% 0.23/0.47 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.23/0.47 = C )
% 0.23/0.47 = ( ( uminus1482373934393186551omplex @ A )
% 0.23/0.47 = ( times_times_complex @ C @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_neg_divide_eq_eq
% 0.23/0.47 thf(fact_4059_nonzero__neg__divide__eq__eq,axiom,
% 0.23/0.47 ! [B2: rat,A: rat,C: rat] :
% 0.23/0.47 ( ( B2 != zero_zero_rat )
% 0.23/0.47 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.23/0.47 = C )
% 0.23/0.47 = ( ( uminus_uminus_rat @ A )
% 0.23/0.47 = ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_neg_divide_eq_eq
% 0.23/0.47 thf(fact_4060_nonzero__neg__divide__eq__eq2,axiom,
% 0.23/0.47 ! [B2: real,C: real,A: real] :
% 0.23/0.47 ( ( B2 != zero_zero_real )
% 0.23/0.47 => ( ( C
% 0.23/0.47 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) )
% 0.23/0.47 = ( ( times_times_real @ C @ B2 )
% 0.23/0.47 = ( uminus_uminus_real @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_neg_divide_eq_eq2
% 0.23/0.47 thf(fact_4061_nonzero__neg__divide__eq__eq2,axiom,
% 0.23/0.47 ! [B2: complex,C: complex,A: complex] :
% 0.23/0.47 ( ( B2 != zero_zero_complex )
% 0.23/0.47 => ( ( C
% 0.23/0.47 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.23/0.47 = ( ( times_times_complex @ C @ B2 )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_neg_divide_eq_eq2
% 0.23/0.47 thf(fact_4062_nonzero__neg__divide__eq__eq2,axiom,
% 0.23/0.47 ! [B2: rat,C: rat,A: rat] :
% 0.23/0.47 ( ( B2 != zero_zero_rat )
% 0.23/0.47 => ( ( C
% 0.23/0.47 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) )
% 0.23/0.47 = ( ( times_times_rat @ C @ B2 )
% 0.23/0.47 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % nonzero_neg_divide_eq_eq2
% 0.23/0.47 thf(fact_4063_mult__1s__ring__1_I2_J,axiom,
% 0.23/0.47 ! [B2: real] :
% 0.23/0.47 ( ( times_times_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 0.23/0.47 = ( uminus_uminus_real @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_1s_ring_1(2)
% 0.23/0.47 thf(fact_4064_mult__1s__ring__1_I2_J,axiom,
% 0.23/0.47 ! [B2: int] :
% 0.23/0.47 ( ( times_times_int @ B2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 0.23/0.47 = ( uminus_uminus_int @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_1s_ring_1(2)
% 0.23/0.47 thf(fact_4065_mult__1s__ring__1_I2_J,axiom,
% 0.23/0.47 ! [B2: complex] :
% 0.23/0.47 ( ( times_times_complex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_1s_ring_1(2)
% 0.23/0.47 thf(fact_4066_mult__1s__ring__1_I2_J,axiom,
% 0.23/0.47 ! [B2: code_integer] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ B2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_1s_ring_1(2)
% 0.23/0.47 thf(fact_4067_mult__1s__ring__1_I2_J,axiom,
% 0.23/0.47 ! [B2: rat] :
% 0.23/0.47 ( ( times_times_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 0.23/0.47 = ( uminus_uminus_rat @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_1s_ring_1(2)
% 0.23/0.47 thf(fact_4068_mult__1s__ring__1_I1_J,axiom,
% 0.23/0.47 ! [B2: real] :
% 0.23/0.47 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B2 )
% 0.23/0.47 = ( uminus_uminus_real @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_1s_ring_1(1)
% 0.23/0.47 thf(fact_4069_mult__1s__ring__1_I1_J,axiom,
% 0.23/0.47 ! [B2: int] :
% 0.23/0.47 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B2 )
% 0.23/0.47 = ( uminus_uminus_int @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_1s_ring_1(1)
% 0.23/0.47 thf(fact_4070_mult__1s__ring__1_I1_J,axiom,
% 0.23/0.47 ! [B2: complex] :
% 0.23/0.47 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B2 )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_1s_ring_1(1)
% 0.23/0.47 thf(fact_4071_mult__1s__ring__1_I1_J,axiom,
% 0.23/0.47 ! [B2: code_integer] :
% 0.23/0.47 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B2 )
% 0.23/0.47 = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_1s_ring_1(1)
% 0.23/0.47 thf(fact_4072_mult__1s__ring__1_I1_J,axiom,
% 0.23/0.47 ! [B2: rat] :
% 0.23/0.47 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B2 )
% 0.23/0.47 = ( uminus_uminus_rat @ B2 ) ) ).
% 0.23/0.47
% 0.23/0.47 % mult_1s_ring_1(1)
% 0.23/0.47 thf(fact_4073_divide__eq__minus__1__iff,axiom,
% 0.23/0.47 ! [A: real,B2: real] :
% 0.23/0.47 ( ( ( divide_divide_real @ A @ B2 )
% 0.23/0.47 = ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.47 = ( ( B2 != zero_zero_real )
% 0.23/0.47 & ( A
% 0.23/0.47 = ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_eq_minus_1_iff
% 0.23/0.47 thf(fact_4074_divide__eq__minus__1__iff,axiom,
% 0.23/0.47 ! [A: complex,B2: complex] :
% 0.23/0.47 ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 0.23/0.47 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.47 = ( ( B2 != zero_zero_complex )
% 0.23/0.47 & ( A
% 0.23/0.47 = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_eq_minus_1_iff
% 0.23/0.47 thf(fact_4075_divide__eq__minus__1__iff,axiom,
% 0.23/0.47 ! [A: rat,B2: rat] :
% 0.23/0.47 ( ( ( divide_divide_rat @ A @ B2 )
% 0.23/0.47 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.47 = ( ( B2 != zero_zero_rat )
% 0.23/0.47 & ( A
% 0.23/0.47 = ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 % divide_eq_minus_1_iff
% 0.23/0.47 thf(fact_4076_uminus__numeral__One,axiom,
% 0.23/0.47 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 0.23/0.47 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.47
% 0.23/0.47 % uminus_numeral_One
% 0.23/0.47 thf(fact_4077_uminus__numeral__One,axiom,
% 0.23/0.47 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 0.23/0.47 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % uminus_numeral_One
% 0.23/0.48 thf(fact_4078_uminus__numeral__One,axiom,
% 0.23/0.48 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.48
% 0.23/0.48 % uminus_numeral_One
% 0.23/0.48 thf(fact_4079_uminus__numeral__One,axiom,
% 0.23/0.48 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 0.23/0.48 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.48
% 0.23/0.48 % uminus_numeral_One
% 0.23/0.48 thf(fact_4080_uminus__numeral__One,axiom,
% 0.23/0.48 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 0.23/0.48 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.48
% 0.23/0.48 % uminus_numeral_One
% 0.23/0.48 thf(fact_4081_power__minus,axiom,
% 0.23/0.48 ! [A: real,N: nat] :
% 0.23/0.48 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.23/0.48 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus
% 0.23/0.48 thf(fact_4082_power__minus,axiom,
% 0.23/0.48 ! [A: int,N: nat] :
% 0.23/0.48 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.23/0.48 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus
% 0.23/0.48 thf(fact_4083_power__minus,axiom,
% 0.23/0.48 ! [A: complex,N: nat] :
% 0.23/0.48 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.23/0.48 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus
% 0.23/0.48 thf(fact_4084_power__minus,axiom,
% 0.23/0.48 ! [A: code_integer,N: nat] :
% 0.23/0.48 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.23/0.48 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus
% 0.23/0.48 thf(fact_4085_power__minus,axiom,
% 0.23/0.48 ! [A: rat,N: nat] :
% 0.23/0.48 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.23/0.48 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus
% 0.23/0.48 thf(fact_4086_power__minus__Bit0,axiom,
% 0.23/0.48 ! [X4: real,K: num] :
% 0.23/0.48 ( ( power_power_real @ ( uminus_uminus_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.23/0.48 = ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus_Bit0
% 0.23/0.48 thf(fact_4087_power__minus__Bit0,axiom,
% 0.23/0.48 ! [X4: int,K: num] :
% 0.23/0.48 ( ( power_power_int @ ( uminus_uminus_int @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.23/0.48 = ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus_Bit0
% 0.23/0.48 thf(fact_4088_power__minus__Bit0,axiom,
% 0.23/0.48 ! [X4: complex,K: num] :
% 0.23/0.48 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.23/0.48 = ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus_Bit0
% 0.23/0.48 thf(fact_4089_power__minus__Bit0,axiom,
% 0.23/0.48 ! [X4: code_integer,K: num] :
% 0.23/0.48 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.23/0.48 = ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus_Bit0
% 0.23/0.48 thf(fact_4090_power__minus__Bit0,axiom,
% 0.23/0.48 ! [X4: rat,K: num] :
% 0.23/0.48 ( ( power_power_rat @ ( uminus_uminus_rat @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.23/0.48 = ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus_Bit0
% 0.23/0.48 thf(fact_4091_real__add__less__0__iff,axiom,
% 0.23/0.48 ! [X4: real,Y3: real] :
% 0.23/0.48 ( ( ord_less_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
% 0.23/0.48 = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X4 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_add_less_0_iff
% 0.23/0.48 thf(fact_4092_real__0__less__add__iff,axiom,
% 0.23/0.48 ! [X4: real,Y3: real] :
% 0.23/0.48 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.48 = ( ord_less_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_0_less_add_iff
% 0.23/0.48 thf(fact_4093_real__0__le__add__iff,axiom,
% 0.23/0.48 ! [X4: real,Y3: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.48 = ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_0_le_add_iff
% 0.23/0.48 thf(fact_4094_real__add__le__0__iff,axiom,
% 0.23/0.48 ! [X4: real,Y3: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
% 0.23/0.48 = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X4 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_add_le_0_iff
% 0.23/0.48 thf(fact_4095_zmod__zminus2__eq__if,axiom,
% 0.23/0.48 ! [A: int,B2: int] :
% 0.23/0.48 ( ( ( ( modulo_modulo_int @ A @ B2 )
% 0.23/0.48 = zero_zero_int )
% 0.23/0.48 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.48 = zero_zero_int ) )
% 0.23/0.48 & ( ( ( modulo_modulo_int @ A @ B2 )
% 0.23/0.48 != zero_zero_int )
% 0.23/0.48 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.48 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % zmod_zminus2_eq_if
% 0.23/0.48 thf(fact_4096_zmod__zminus1__eq__if,axiom,
% 0.23/0.48 ! [A: int,B2: int] :
% 0.23/0.48 ( ( ( ( modulo_modulo_int @ A @ B2 )
% 0.23/0.48 = zero_zero_int )
% 0.23/0.48 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.48 = zero_zero_int ) )
% 0.23/0.48 & ( ( ( modulo_modulo_int @ A @ B2 )
% 0.23/0.48 != zero_zero_int )
% 0.23/0.48 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.48 = ( minus_minus_int @ B2 @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % zmod_zminus1_eq_if
% 0.23/0.48 thf(fact_4097_pos__minus__divide__less__eq,axiom,
% 0.23/0.48 ! [C: real,B2: real,A: real] :
% 0.23/0.48 ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % pos_minus_divide_less_eq
% 0.23/0.48 thf(fact_4098_pos__minus__divide__less__eq,axiom,
% 0.23/0.48 ! [C: rat,B2: rat,A: rat] :
% 0.23/0.48 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % pos_minus_divide_less_eq
% 0.23/0.48 thf(fact_4099_pos__less__minus__divide__eq,axiom,
% 0.23/0.48 ! [C: real,A: real,B2: real] :
% 0.23/0.48 ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.23/0.48 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % pos_less_minus_divide_eq
% 0.23/0.48 thf(fact_4100_pos__less__minus__divide__eq,axiom,
% 0.23/0.48 ! [C: rat,A: rat,B2: rat] :
% 0.23/0.48 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.23/0.48 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % pos_less_minus_divide_eq
% 0.23/0.48 thf(fact_4101_neg__minus__divide__less__eq,axiom,
% 0.23/0.48 ! [C: real,B2: real,A: real] :
% 0.23/0.48 ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_minus_divide_less_eq
% 0.23/0.48 thf(fact_4102_neg__minus__divide__less__eq,axiom,
% 0.23/0.48 ! [C: rat,B2: rat,A: rat] :
% 0.23/0.48 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_minus_divide_less_eq
% 0.23/0.48 thf(fact_4103_neg__less__minus__divide__eq,axiom,
% 0.23/0.48 ! [C: real,A: real,B2: real] :
% 0.23/0.48 ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.23/0.48 = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_less_minus_divide_eq
% 0.23/0.48 thf(fact_4104_neg__less__minus__divide__eq,axiom,
% 0.23/0.48 ! [C: rat,A: rat,B2: rat] :
% 0.23/0.48 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.23/0.48 = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_less_minus_divide_eq
% 0.23/0.48 thf(fact_4105_minus__divide__less__eq,axiom,
% 0.23/0.48 ! [B2: real,C: real,A: real] :
% 0.23/0.48 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_divide_less_eq
% 0.23/0.48 thf(fact_4106_minus__divide__less__eq,axiom,
% 0.23/0.48 ! [B2: rat,C: rat,A: rat] :
% 0.23/0.48 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_divide_less_eq
% 0.23/0.48 thf(fact_4107_less__minus__divide__eq,axiom,
% 0.23/0.48 ! [A: real,B2: real,C: real] :
% 0.23/0.48 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.23/0.48 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % less_minus_divide_eq
% 0.23/0.48 thf(fact_4108_less__minus__divide__eq,axiom,
% 0.23/0.48 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.48 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.23/0.48 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % less_minus_divide_eq
% 0.23/0.48 thf(fact_4109_divide__eq__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [B2: real,C: real,W: num] :
% 0.23/0.48 ( ( ( divide_divide_real @ B2 @ C )
% 0.23/0.48 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.48 = ( ( ( C != zero_zero_real )
% 0.23/0.48 => ( B2
% 0.23/0.48 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.23/0.48 & ( ( C = zero_zero_real )
% 0.23/0.48 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.23/0.48 = zero_zero_real ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % divide_eq_eq_numeral(2)
% 0.23/0.48 thf(fact_4110_divide__eq__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [B2: complex,C: complex,W: num] :
% 0.23/0.48 ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.23/0.48 = ( ( ( C != zero_zero_complex )
% 0.23/0.48 => ( B2
% 0.23/0.48 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 0.23/0.48 & ( ( C = zero_zero_complex )
% 0.23/0.48 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.23/0.48 = zero_zero_complex ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % divide_eq_eq_numeral(2)
% 0.23/0.48 thf(fact_4111_divide__eq__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [B2: rat,C: rat,W: num] :
% 0.23/0.48 ( ( ( divide_divide_rat @ B2 @ C )
% 0.23/0.48 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.23/0.48 = ( ( ( C != zero_zero_rat )
% 0.23/0.48 => ( B2
% 0.23/0.48 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.23/0.48 & ( ( C = zero_zero_rat )
% 0.23/0.48 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.23/0.48 = zero_zero_rat ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % divide_eq_eq_numeral(2)
% 0.23/0.48 thf(fact_4112_eq__divide__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [W: num,B2: real,C: real] :
% 0.23/0.48 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.23/0.48 = ( divide_divide_real @ B2 @ C ) )
% 0.23/0.48 = ( ( ( C != zero_zero_real )
% 0.23/0.48 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 0.23/0.48 = B2 ) )
% 0.23/0.48 & ( ( C = zero_zero_real )
% 0.23/0.48 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.23/0.48 = zero_zero_real ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % eq_divide_eq_numeral(2)
% 0.23/0.48 thf(fact_4113_eq__divide__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [W: num,B2: complex,C: complex] :
% 0.23/0.48 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.23/0.48 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.23/0.48 = ( ( ( C != zero_zero_complex )
% 0.23/0.48 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 0.23/0.48 = B2 ) )
% 0.23/0.48 & ( ( C = zero_zero_complex )
% 0.23/0.48 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.23/0.48 = zero_zero_complex ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % eq_divide_eq_numeral(2)
% 0.23/0.48 thf(fact_4114_eq__divide__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [W: num,B2: rat,C: rat] :
% 0.23/0.48 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.23/0.48 = ( divide_divide_rat @ B2 @ C ) )
% 0.23/0.48 = ( ( ( C != zero_zero_rat )
% 0.23/0.48 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 0.23/0.48 = B2 ) )
% 0.23/0.48 & ( ( C = zero_zero_rat )
% 0.23/0.48 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.23/0.48 = zero_zero_rat ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % eq_divide_eq_numeral(2)
% 0.23/0.48 thf(fact_4115_add__divide__eq__if__simps_I3_J,axiom,
% 0.23/0.48 ! [Z: real,A: real,B2: real] :
% 0.23/0.48 ( ( ( Z = zero_zero_real )
% 0.23/0.48 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 0.23/0.48 = B2 ) )
% 0.23/0.48 & ( ( Z != zero_zero_real )
% 0.23/0.48 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 0.23/0.48 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_divide_eq_if_simps(3)
% 0.23/0.48 thf(fact_4116_add__divide__eq__if__simps_I3_J,axiom,
% 0.23/0.48 ! [Z: complex,A: complex,B2: complex] :
% 0.23/0.48 ( ( ( Z = zero_zero_complex )
% 0.23/0.48 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 0.23/0.48 = B2 ) )
% 0.23/0.48 & ( ( Z != zero_zero_complex )
% 0.23/0.48 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 0.23/0.48 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_divide_eq_if_simps(3)
% 0.23/0.48 thf(fact_4117_add__divide__eq__if__simps_I3_J,axiom,
% 0.23/0.48 ! [Z: rat,A: rat,B2: rat] :
% 0.23/0.48 ( ( ( Z = zero_zero_rat )
% 0.23/0.48 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 0.23/0.48 = B2 ) )
% 0.23/0.48 & ( ( Z != zero_zero_rat )
% 0.23/0.48 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 0.23/0.48 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_divide_eq_if_simps(3)
% 0.23/0.48 thf(fact_4118_minus__divide__add__eq__iff,axiom,
% 0.23/0.48 ! [Z: real,X4: real,Y3: real] :
% 0.23/0.48 ( ( Z != zero_zero_real )
% 0.23/0.48 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X4 @ Z ) ) @ Y3 )
% 0.23/0.48 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_divide_add_eq_iff
% 0.23/0.48 thf(fact_4119_minus__divide__add__eq__iff,axiom,
% 0.23/0.48 ! [Z: complex,X4: complex,Y3: complex] :
% 0.23/0.48 ( ( Z != zero_zero_complex )
% 0.23/0.48 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X4 @ Z ) ) @ Y3 )
% 0.23/0.48 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_divide_add_eq_iff
% 0.23/0.48 thf(fact_4120_minus__divide__add__eq__iff,axiom,
% 0.23/0.48 ! [Z: rat,X4: rat,Y3: rat] :
% 0.23/0.48 ( ( Z != zero_zero_rat )
% 0.23/0.48 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X4 @ Z ) ) @ Y3 )
% 0.23/0.48 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X4 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_divide_add_eq_iff
% 0.23/0.48 thf(fact_4121_add__divide__eq__if__simps_I6_J,axiom,
% 0.23/0.48 ! [Z: real,A: real,B2: real] :
% 0.23/0.48 ( ( ( Z = zero_zero_real )
% 0.23/0.48 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 0.23/0.48 = ( uminus_uminus_real @ B2 ) ) )
% 0.23/0.48 & ( ( Z != zero_zero_real )
% 0.23/0.48 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 0.23/0.48 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_divide_eq_if_simps(6)
% 0.23/0.48 thf(fact_4122_add__divide__eq__if__simps_I6_J,axiom,
% 0.23/0.48 ! [Z: complex,A: complex,B2: complex] :
% 0.23/0.48 ( ( ( Z = zero_zero_complex )
% 0.23/0.48 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ B2 ) ) )
% 0.23/0.48 & ( ( Z != zero_zero_complex )
% 0.23/0.48 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 0.23/0.48 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_divide_eq_if_simps(6)
% 0.23/0.48 thf(fact_4123_add__divide__eq__if__simps_I6_J,axiom,
% 0.23/0.48 ! [Z: rat,A: rat,B2: rat] :
% 0.23/0.48 ( ( ( Z = zero_zero_rat )
% 0.23/0.48 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 0.23/0.48 = ( uminus_uminus_rat @ B2 ) ) )
% 0.23/0.48 & ( ( Z != zero_zero_rat )
% 0.23/0.48 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 0.23/0.48 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_divide_eq_if_simps(6)
% 0.23/0.48 thf(fact_4124_add__divide__eq__if__simps_I5_J,axiom,
% 0.23/0.48 ! [Z: real,A: real,B2: real] :
% 0.23/0.48 ( ( ( Z = zero_zero_real )
% 0.23/0.48 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 0.23/0.48 = ( uminus_uminus_real @ B2 ) ) )
% 0.23/0.48 & ( ( Z != zero_zero_real )
% 0.23/0.48 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 0.23/0.48 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_divide_eq_if_simps(5)
% 0.23/0.48 thf(fact_4125_add__divide__eq__if__simps_I5_J,axiom,
% 0.23/0.48 ! [Z: complex,A: complex,B2: complex] :
% 0.23/0.48 ( ( ( Z = zero_zero_complex )
% 0.23/0.48 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ B2 ) ) )
% 0.23/0.48 & ( ( Z != zero_zero_complex )
% 0.23/0.48 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 0.23/0.48 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_divide_eq_if_simps(5)
% 0.23/0.48 thf(fact_4126_add__divide__eq__if__simps_I5_J,axiom,
% 0.23/0.48 ! [Z: rat,A: rat,B2: rat] :
% 0.23/0.48 ( ( ( Z = zero_zero_rat )
% 0.23/0.48 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 0.23/0.48 = ( uminus_uminus_rat @ B2 ) ) )
% 0.23/0.48 & ( ( Z != zero_zero_rat )
% 0.23/0.48 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 0.23/0.48 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_divide_eq_if_simps(5)
% 0.23/0.48 thf(fact_4127_minus__divide__diff__eq__iff,axiom,
% 0.23/0.48 ! [Z: real,X4: real,Y3: real] :
% 0.23/0.48 ( ( Z != zero_zero_real )
% 0.23/0.48 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X4 @ Z ) ) @ Y3 )
% 0.23/0.48 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_divide_diff_eq_iff
% 0.23/0.48 thf(fact_4128_minus__divide__diff__eq__iff,axiom,
% 0.23/0.48 ! [Z: complex,X4: complex,Y3: complex] :
% 0.23/0.48 ( ( Z != zero_zero_complex )
% 0.23/0.48 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X4 @ Z ) ) @ Y3 )
% 0.23/0.48 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_divide_diff_eq_iff
% 0.23/0.48 thf(fact_4129_minus__divide__diff__eq__iff,axiom,
% 0.23/0.48 ! [Z: rat,X4: rat,Y3: rat] :
% 0.23/0.48 ( ( Z != zero_zero_rat )
% 0.23/0.48 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X4 @ Z ) ) @ Y3 )
% 0.23/0.48 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X4 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_divide_diff_eq_iff
% 0.23/0.48 thf(fact_4130_even__minus,axiom,
% 0.23/0.48 ! [A: int] :
% 0.23/0.48 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 0.23/0.48 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.23/0.48
% 0.23/0.48 % even_minus
% 0.23/0.48 thf(fact_4131_even__minus,axiom,
% 0.23/0.48 ! [A: code_integer] :
% 0.23/0.48 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.48 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.23/0.48
% 0.23/0.48 % even_minus
% 0.23/0.48 thf(fact_4132_power2__eq__iff,axiom,
% 0.23/0.48 ! [X4: real,Y3: real] :
% 0.23/0.48 ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.48 = ( ( X4 = Y3 )
% 0.23/0.48 | ( X4
% 0.23/0.48 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power2_eq_iff
% 0.23/0.48 thf(fact_4133_power2__eq__iff,axiom,
% 0.23/0.48 ! [X4: int,Y3: int] :
% 0.23/0.48 ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.48 = ( ( X4 = Y3 )
% 0.23/0.48 | ( X4
% 0.23/0.48 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power2_eq_iff
% 0.23/0.48 thf(fact_4134_power2__eq__iff,axiom,
% 0.23/0.48 ! [X4: complex,Y3: complex] :
% 0.23/0.48 ( ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.48 = ( ( X4 = Y3 )
% 0.23/0.48 | ( X4
% 0.23/0.48 = ( uminus1482373934393186551omplex @ Y3 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power2_eq_iff
% 0.23/0.48 thf(fact_4135_power2__eq__iff,axiom,
% 0.23/0.48 ! [X4: code_integer,Y3: code_integer] :
% 0.23/0.48 ( ( ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.48 = ( ( X4 = Y3 )
% 0.23/0.48 | ( X4
% 0.23/0.48 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power2_eq_iff
% 0.23/0.48 thf(fact_4136_power2__eq__iff,axiom,
% 0.23/0.48 ! [X4: rat,Y3: rat] :
% 0.23/0.48 ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.48 = ( ( X4 = Y3 )
% 0.23/0.48 | ( X4
% 0.23/0.48 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power2_eq_iff
% 0.23/0.48 thf(fact_4137_verit__less__mono__div__int2,axiom,
% 0.23/0.48 ! [A3: int,B4: int,N: int] :
% 0.23/0.48 ( ( ord_less_eq_int @ A3 @ B4 )
% 0.23/0.48 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 0.23/0.48 => ( ord_less_eq_int @ ( divide_divide_int @ B4 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % verit_less_mono_div_int2
% 0.23/0.48 thf(fact_4138_div__eq__minus1,axiom,
% 0.23/0.48 ! [B2: int] :
% 0.23/0.48 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.48 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
% 0.23/0.48 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % div_eq_minus1
% 0.23/0.48 thf(fact_4139_pos__minus__divide__le__eq,axiom,
% 0.23/0.48 ! [C: real,B2: real,A: real] :
% 0.23/0.48 ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % pos_minus_divide_le_eq
% 0.23/0.48 thf(fact_4140_pos__minus__divide__le__eq,axiom,
% 0.23/0.48 ! [C: rat,B2: rat,A: rat] :
% 0.23/0.48 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % pos_minus_divide_le_eq
% 0.23/0.48 thf(fact_4141_pos__le__minus__divide__eq,axiom,
% 0.23/0.48 ! [C: real,A: real,B2: real] :
% 0.23/0.48 ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.23/0.48 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % pos_le_minus_divide_eq
% 0.23/0.48 thf(fact_4142_pos__le__minus__divide__eq,axiom,
% 0.23/0.48 ! [C: rat,A: rat,B2: rat] :
% 0.23/0.48 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.23/0.48 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % pos_le_minus_divide_eq
% 0.23/0.48 thf(fact_4143_neg__minus__divide__le__eq,axiom,
% 0.23/0.48 ! [C: real,B2: real,A: real] :
% 0.23/0.48 ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_minus_divide_le_eq
% 0.23/0.48 thf(fact_4144_neg__minus__divide__le__eq,axiom,
% 0.23/0.48 ! [C: rat,B2: rat,A: rat] :
% 0.23/0.48 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_minus_divide_le_eq
% 0.23/0.48 thf(fact_4145_neg__le__minus__divide__eq,axiom,
% 0.23/0.48 ! [C: real,A: real,B2: real] :
% 0.23/0.48 ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.23/0.48 = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_le_minus_divide_eq
% 0.23/0.48 thf(fact_4146_neg__le__minus__divide__eq,axiom,
% 0.23/0.48 ! [C: rat,A: rat,B2: rat] :
% 0.23/0.48 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.23/0.48 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_le_minus_divide_eq
% 0.23/0.48 thf(fact_4147_minus__divide__le__eq,axiom,
% 0.23/0.48 ! [B2: real,C: real,A: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_divide_le_eq
% 0.23/0.48 thf(fact_4148_minus__divide__le__eq,axiom,
% 0.23/0.48 ! [B2: rat,C: rat,A: rat] :
% 0.23/0.48 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.23/0.48 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_divide_le_eq
% 0.23/0.48 thf(fact_4149_le__minus__divide__eq,axiom,
% 0.23/0.48 ! [A: real,B2: real,C: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.23/0.48 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % le_minus_divide_eq
% 0.23/0.48 thf(fact_4150_le__minus__divide__eq,axiom,
% 0.23/0.48 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.48 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.23/0.48 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % le_minus_divide_eq
% 0.23/0.48 thf(fact_4151_less__divide__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [W: num,B2: real,C: real] :
% 0.23/0.48 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
% 0.23/0.48 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % less_divide_eq_numeral(2)
% 0.23/0.48 thf(fact_4152_less__divide__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [W: num,B2: rat,C: rat] :
% 0.23/0.48 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.23/0.48 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % less_divide_eq_numeral(2)
% 0.23/0.48 thf(fact_4153_divide__less__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [B2: real,C: real,W: num] :
% 0.23/0.48 ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.48 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % divide_less_eq_numeral(2)
% 0.23/0.48 thf(fact_4154_divide__less__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [B2: rat,C: rat,W: num] :
% 0.23/0.48 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.23/0.48 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % divide_less_eq_numeral(2)
% 0.23/0.48 thf(fact_4155_power2__eq__1__iff,axiom,
% 0.23/0.48 ! [A: real] :
% 0.23/0.48 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = one_one_real )
% 0.23/0.48 = ( ( A = one_one_real )
% 0.23/0.48 | ( A
% 0.23/0.48 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power2_eq_1_iff
% 0.23/0.48 thf(fact_4156_power2__eq__1__iff,axiom,
% 0.23/0.48 ! [A: int] :
% 0.23/0.48 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = one_one_int )
% 0.23/0.48 = ( ( A = one_one_int )
% 0.23/0.48 | ( A
% 0.23/0.48 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power2_eq_1_iff
% 0.23/0.48 thf(fact_4157_power2__eq__1__iff,axiom,
% 0.23/0.48 ! [A: complex] :
% 0.23/0.48 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = one_one_complex )
% 0.23/0.48 = ( ( A = one_one_complex )
% 0.23/0.48 | ( A
% 0.23/0.48 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power2_eq_1_iff
% 0.23/0.48 thf(fact_4158_power2__eq__1__iff,axiom,
% 0.23/0.48 ! [A: code_integer] :
% 0.23/0.48 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = one_one_Code_integer )
% 0.23/0.48 = ( ( A = one_one_Code_integer )
% 0.23/0.48 | ( A
% 0.23/0.48 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power2_eq_1_iff
% 0.23/0.48 thf(fact_4159_power2__eq__1__iff,axiom,
% 0.23/0.48 ! [A: rat] :
% 0.23/0.48 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = one_one_rat )
% 0.23/0.48 = ( ( A = one_one_rat )
% 0.23/0.48 | ( A
% 0.23/0.48 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power2_eq_1_iff
% 0.23/0.48 thf(fact_4160_uminus__power__if,axiom,
% 0.23/0.48 ! [N: nat,A: real] :
% 0.23/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.23/0.48 = ( power_power_real @ A @ N ) ) )
% 0.23/0.48 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.23/0.48 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % uminus_power_if
% 0.23/0.48 thf(fact_4161_uminus__power__if,axiom,
% 0.23/0.48 ! [N: nat,A: int] :
% 0.23/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.23/0.48 = ( power_power_int @ A @ N ) ) )
% 0.23/0.48 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.23/0.48 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % uminus_power_if
% 0.23/0.48 thf(fact_4162_uminus__power__if,axiom,
% 0.23/0.48 ! [N: nat,A: complex] :
% 0.23/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.23/0.48 = ( power_power_complex @ A @ N ) ) )
% 0.23/0.48 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % uminus_power_if
% 0.23/0.48 thf(fact_4163_uminus__power__if,axiom,
% 0.23/0.48 ! [N: nat,A: code_integer] :
% 0.23/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.23/0.48 = ( power_8256067586552552935nteger @ A @ N ) ) )
% 0.23/0.48 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.23/0.48 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % uminus_power_if
% 0.23/0.48 thf(fact_4164_uminus__power__if,axiom,
% 0.23/0.48 ! [N: nat,A: rat] :
% 0.23/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.23/0.48 = ( power_power_rat @ A @ N ) ) )
% 0.23/0.48 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.23/0.48 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % uminus_power_if
% 0.23/0.48 thf(fact_4165_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.23/0.48 ! [K: nat,N: nat] :
% 0.23/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.48 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 0.23/0.48 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_one_power_add_eq_neg_one_power_diff
% 0.23/0.48 thf(fact_4166_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.23/0.48 ! [K: nat,N: nat] :
% 0.23/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.48 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 0.23/0.48 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_one_power_add_eq_neg_one_power_diff
% 0.23/0.48 thf(fact_4167_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.23/0.48 ! [K: nat,N: nat] :
% 0.23/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.48 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 0.23/0.48 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_one_power_add_eq_neg_one_power_diff
% 0.23/0.48 thf(fact_4168_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.23/0.48 ! [K: nat,N: nat] :
% 0.23/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.48 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 0.23/0.48 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_one_power_add_eq_neg_one_power_diff
% 0.23/0.48 thf(fact_4169_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.23/0.48 ! [K: nat,N: nat] :
% 0.23/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.48 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 0.23/0.48 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_one_power_add_eq_neg_one_power_diff
% 0.23/0.48 thf(fact_4170_realpow__square__minus__le,axiom,
% 0.23/0.48 ! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % realpow_square_minus_le
% 0.23/0.48 thf(fact_4171_minus__mod__int__eq,axiom,
% 0.23/0.48 ! [L2: int,K: int] :
% 0.23/0.48 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 0.23/0.48 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
% 0.23/0.48 = ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_mod_int_eq
% 0.23/0.48 thf(fact_4172_zmod__minus1,axiom,
% 0.23/0.48 ! [B2: int] :
% 0.23/0.48 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.48 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
% 0.23/0.48 = ( minus_minus_int @ B2 @ one_one_int ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % zmod_minus1
% 0.23/0.48 thf(fact_4173_zdiv__zminus1__eq__if,axiom,
% 0.23/0.48 ! [B2: int,A: int] :
% 0.23/0.48 ( ( B2 != zero_zero_int )
% 0.23/0.48 => ( ( ( ( modulo_modulo_int @ A @ B2 )
% 0.23/0.48 = zero_zero_int )
% 0.23/0.48 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.48 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) )
% 0.23/0.48 & ( ( ( modulo_modulo_int @ A @ B2 )
% 0.23/0.48 != zero_zero_int )
% 0.23/0.48 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.48 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) @ one_one_int ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % zdiv_zminus1_eq_if
% 0.23/0.48 thf(fact_4174_zdiv__zminus2__eq__if,axiom,
% 0.23/0.48 ! [B2: int,A: int] :
% 0.23/0.48 ( ( B2 != zero_zero_int )
% 0.23/0.48 => ( ( ( ( modulo_modulo_int @ A @ B2 )
% 0.23/0.48 = zero_zero_int )
% 0.23/0.48 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.48 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) )
% 0.23/0.48 & ( ( ( modulo_modulo_int @ A @ B2 )
% 0.23/0.48 != zero_zero_int )
% 0.23/0.48 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.23/0.48 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) @ one_one_int ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % zdiv_zminus2_eq_if
% 0.23/0.48 thf(fact_4175_le__divide__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [W: num,B2: real,C: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
% 0.23/0.48 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % le_divide_eq_numeral(2)
% 0.23/0.48 thf(fact_4176_le__divide__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [W: num,B2: rat,C: rat] :
% 0.23/0.48 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.23/0.48 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % le_divide_eq_numeral(2)
% 0.23/0.48 thf(fact_4177_divide__le__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [B2: real,C: real,W: num] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.48 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.48 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 0.23/0.48 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.23/0.48 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % divide_le_eq_numeral(2)
% 0.23/0.48 thf(fact_4178_divide__le__eq__numeral_I2_J,axiom,
% 0.23/0.48 ! [B2: rat,C: rat,W: num] :
% 0.23/0.48 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.23/0.48 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.23/0.48 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 0.23/0.48 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.23/0.48 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % divide_le_eq_numeral(2)
% 0.23/0.48 thf(fact_4179_square__le__1,axiom,
% 0.23/0.48 ! [X4: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.48 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.48 => ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % square_le_1
% 0.23/0.48 thf(fact_4180_square__le__1,axiom,
% 0.23/0.48 ! [X4: code_integer] :
% 0.23/0.48 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X4 )
% 0.23/0.48 => ( ( ord_le3102999989581377725nteger @ X4 @ one_one_Code_integer )
% 0.23/0.48 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % square_le_1
% 0.23/0.48 thf(fact_4181_square__le__1,axiom,
% 0.23/0.48 ! [X4: rat] :
% 0.23/0.48 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X4 )
% 0.23/0.48 => ( ( ord_less_eq_rat @ X4 @ one_one_rat )
% 0.23/0.48 => ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % square_le_1
% 0.23/0.48 thf(fact_4182_square__le__1,axiom,
% 0.23/0.48 ! [X4: int] :
% 0.23/0.48 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X4 )
% 0.23/0.48 => ( ( ord_less_eq_int @ X4 @ one_one_int )
% 0.23/0.48 => ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % square_le_1
% 0.23/0.48 thf(fact_4183_minus__power__mult__self,axiom,
% 0.23/0.48 ! [A: real,N: nat] :
% 0.23/0.48 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.23/0.48 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_power_mult_self
% 0.23/0.48 thf(fact_4184_minus__power__mult__self,axiom,
% 0.23/0.48 ! [A: int,N: nat] :
% 0.23/0.48 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.23/0.48 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_power_mult_self
% 0.23/0.48 thf(fact_4185_minus__power__mult__self,axiom,
% 0.23/0.48 ! [A: complex,N: nat] :
% 0.23/0.48 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 0.23/0.48 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_power_mult_self
% 0.23/0.48 thf(fact_4186_minus__power__mult__self,axiom,
% 0.23/0.48 ! [A: code_integer,N: nat] :
% 0.23/0.48 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.23/0.48 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_power_mult_self
% 0.23/0.48 thf(fact_4187_minus__power__mult__self,axiom,
% 0.23/0.48 ! [A: rat,N: nat] :
% 0.23/0.48 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.23/0.48 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_power_mult_self
% 0.23/0.48 thf(fact_4188_minus__one__power__iff,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.23/0.48 = one_one_real ) )
% 0.23/0.48 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.23/0.48 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_one_power_iff
% 0.23/0.48 thf(fact_4189_minus__one__power__iff,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.23/0.48 = one_one_int ) )
% 0.23/0.48 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.23/0.48 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_one_power_iff
% 0.23/0.48 thf(fact_4190_minus__one__power__iff,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.23/0.48 = one_one_complex ) )
% 0.23/0.48 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_one_power_iff
% 0.23/0.48 thf(fact_4191_minus__one__power__iff,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.23/0.48 = one_one_Code_integer ) )
% 0.23/0.48 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.23/0.48 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_one_power_iff
% 0.23/0.48 thf(fact_4192_minus__one__power__iff,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.23/0.48 = one_one_rat ) )
% 0.23/0.48 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.48 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.23/0.48 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_one_power_iff
% 0.23/0.48 thf(fact_4193_minus__1__div__exp__eq__int,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % minus_1_div_exp_eq_int
% 0.23/0.48 thf(fact_4194_div__pos__neg__trivial,axiom,
% 0.23/0.48 ! [K: int,L2: int] :
% 0.23/0.48 ( ( ord_less_int @ zero_zero_int @ K )
% 0.23/0.48 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 0.23/0.48 => ( ( divide_divide_int @ K @ L2 )
% 0.23/0.48 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % div_pos_neg_trivial
% 0.23/0.48 thf(fact_4195_signed__take__bit__int__less__eq__self__iff,axiom,
% 0.23/0.48 ! [N: nat,K: int] :
% 0.23/0.48 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.23/0.48 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_int_less_eq_self_iff
% 0.23/0.48 thf(fact_4196_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 0.23/0.48 ! [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 ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_int_greater_eq_minus_exp
% 0.23/0.48 thf(fact_4197_signed__take__bit__int__greater__self__iff,axiom,
% 0.23/0.48 ! [K: int,N: nat] :
% 0.23/0.48 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.23/0.48 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_int_greater_self_iff
% 0.23/0.48 thf(fact_4198_power__minus1__odd,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.48 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus1_odd
% 0.23/0.48 thf(fact_4199_power__minus1__odd,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.48 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus1_odd
% 0.23/0.48 thf(fact_4200_power__minus1__odd,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus1_odd
% 0.23/0.48 thf(fact_4201_power__minus1__odd,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.48 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus1_odd
% 0.23/0.48 thf(fact_4202_power__minus1__odd,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.48 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus1_odd
% 0.23/0.48 thf(fact_4203_int__bit__induct,axiom,
% 0.23/0.48 ! [P: int > $o,K: int] :
% 0.23/0.48 ( ( P @ zero_zero_int )
% 0.23/0.48 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.48 => ( ! [K2: int] :
% 0.23/0.48 ( ( P @ K2 )
% 0.23/0.48 => ( ( K2 != zero_zero_int )
% 0.23/0.48 => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.48 => ( ! [K2: int] :
% 0.23/0.48 ( ( P @ K2 )
% 0.23/0.48 => ( ( K2
% 0.23/0.48 != ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.48 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 0.23/0.48 => ( P @ K ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % int_bit_induct
% 0.23/0.48 thf(fact_4204_signed__take__bit__int__eq__self,axiom,
% 0.23/0.48 ! [N: nat,K: int] :
% 0.23/0.48 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.23/0.48 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.23/0.48 = K ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_int_eq_self
% 0.23/0.48 thf(fact_4205_signed__take__bit__int__eq__self__iff,axiom,
% 0.23/0.48 ! [N: nat,K: int] :
% 0.23/0.48 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.23/0.48 = K )
% 0.23/0.48 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.23/0.48 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_int_eq_self_iff
% 0.23/0.48 thf(fact_4206_signed__take__bit__int__greater__eq,axiom,
% 0.23/0.48 ! [K: int,N: nat] :
% 0.23/0.48 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.48 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_int_greater_eq
% 0.23/0.48 thf(fact_4207_mult__less__iff1,axiom,
% 0.23/0.48 ! [Z: real,X4: real,Y3: real] :
% 0.23/0.48 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.23/0.48 => ( ( ord_less_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 0.23/0.48 = ( ord_less_real @ X4 @ Y3 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % mult_less_iff1
% 0.23/0.48 thf(fact_4208_mult__less__iff1,axiom,
% 0.23/0.48 ! [Z: rat,X4: rat,Y3: rat] :
% 0.23/0.48 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.23/0.48 => ( ( ord_less_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 0.23/0.48 = ( ord_less_rat @ X4 @ Y3 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % mult_less_iff1
% 0.23/0.48 thf(fact_4209_mult__less__iff1,axiom,
% 0.23/0.48 ! [Z: int,X4: int,Y3: int] :
% 0.23/0.48 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.23/0.48 => ( ( ord_less_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 0.23/0.48 = ( ord_less_int @ X4 @ Y3 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % mult_less_iff1
% 0.23/0.48 thf(fact_4210_compl__le__compl__iff,axiom,
% 0.23/0.48 ! [X4: set_nat,Y3: set_nat] :
% 0.23/0.48 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ ( uminus5710092332889474511et_nat @ Y3 ) )
% 0.23/0.48 = ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ).
% 0.23/0.48
% 0.23/0.48 % compl_le_compl_iff
% 0.23/0.48 thf(fact_4211_signed__take__bit__Suc__minus__bit1,axiom,
% 0.23/0.48 ! [N: nat,K: num] :
% 0.23/0.48 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.23/0.48 = ( 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 ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_Suc_minus_bit1
% 0.23/0.48 thf(fact_4212_concat__bit__Suc,axiom,
% 0.23/0.48 ! [N: nat,K: int,L2: int] :
% 0.23/0.48 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L2 )
% 0.23/0.48 = ( 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 ) ) ) @ L2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % concat_bit_Suc
% 0.23/0.48 thf(fact_4213_flip__bit__0,axiom,
% 0.23/0.48 ! [A: code_integer] :
% 0.23/0.48 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 0.23/0.48 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % flip_bit_0
% 0.23/0.48 thf(fact_4214_flip__bit__0,axiom,
% 0.23/0.48 ! [A: int] :
% 0.23/0.48 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 0.23/0.48 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % flip_bit_0
% 0.23/0.48 thf(fact_4215_flip__bit__0,axiom,
% 0.23/0.48 ! [A: nat] :
% 0.23/0.48 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 0.23/0.48 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % flip_bit_0
% 0.23/0.48 thf(fact_4216_set__decode__0,axiom,
% 0.23/0.48 ! [X4: nat] :
% 0.23/0.48 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X4 ) )
% 0.23/0.48 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % set_decode_0
% 0.23/0.48 thf(fact_4217_semiring__norm_I90_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( ( bit1 @ M )
% 0.23/0.48 = ( bit1 @ N ) )
% 0.23/0.48 = ( M = N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(90)
% 0.23/0.48 thf(fact_4218_verit__eq__simplify_I9_J,axiom,
% 0.23/0.48 ! [X32: num,Y32: num] :
% 0.23/0.48 ( ( ( bit1 @ X32 )
% 0.23/0.48 = ( bit1 @ Y32 ) )
% 0.23/0.48 = ( X32 = Y32 ) ) ).
% 0.23/0.48
% 0.23/0.48 % verit_eq_simplify(9)
% 0.23/0.48 thf(fact_4219_semiring__norm_I88_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( bit0 @ M )
% 0.23/0.48 != ( bit1 @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(88)
% 0.23/0.48 thf(fact_4220_semiring__norm_I89_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( bit1 @ M )
% 0.23/0.48 != ( bit0 @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(89)
% 0.23/0.48 thf(fact_4221_semiring__norm_I84_J,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( one
% 0.23/0.48 != ( bit1 @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(84)
% 0.23/0.48 thf(fact_4222_semiring__norm_I86_J,axiom,
% 0.23/0.48 ! [M: num] :
% 0.23/0.48 ( ( bit1 @ M )
% 0.23/0.48 != one ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(86)
% 0.23/0.48 thf(fact_4223_of__bool__less__eq__iff,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.23/0.48 = ( P
% 0.23/0.48 => Q ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_eq_iff
% 0.23/0.48 thf(fact_4224_of__bool__less__eq__iff,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.23/0.48 = ( P
% 0.23/0.48 => Q ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_eq_iff
% 0.23/0.48 thf(fact_4225_of__bool__less__eq__iff,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.23/0.48 = ( P
% 0.23/0.48 => Q ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_eq_iff
% 0.23/0.48 thf(fact_4226_of__bool__less__eq__iff,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.23/0.48 = ( P
% 0.23/0.48 => Q ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_eq_iff
% 0.23/0.48 thf(fact_4227_of__bool__eq__0__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ( zero_n3304061248610475627l_real @ P )
% 0.23/0.48 = zero_zero_real )
% 0.23/0.48 = ( ~ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_0_iff
% 0.23/0.48 thf(fact_4228_of__bool__eq__0__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.23/0.48 = zero_zero_rat )
% 0.23/0.48 = ( ~ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_0_iff
% 0.23/0.48 thf(fact_4229_of__bool__eq__0__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.23/0.48 = zero_zero_nat )
% 0.23/0.48 = ( ~ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_0_iff
% 0.23/0.48 thf(fact_4230_of__bool__eq__0__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.23/0.48 = zero_zero_int )
% 0.23/0.48 = ( ~ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_0_iff
% 0.23/0.48 thf(fact_4231_of__bool__eq__0__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ( zero_n356916108424825756nteger @ P )
% 0.23/0.48 = zero_z3403309356797280102nteger )
% 0.23/0.48 = ( ~ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_0_iff
% 0.23/0.48 thf(fact_4232_of__bool__eq_I1_J,axiom,
% 0.23/0.48 ( ( zero_n3304061248610475627l_real @ $false )
% 0.23/0.48 = zero_zero_real ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq(1)
% 0.23/0.48 thf(fact_4233_of__bool__eq_I1_J,axiom,
% 0.23/0.48 ( ( zero_n2052037380579107095ol_rat @ $false )
% 0.23/0.48 = zero_zero_rat ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq(1)
% 0.23/0.48 thf(fact_4234_of__bool__eq_I1_J,axiom,
% 0.23/0.48 ( ( zero_n2687167440665602831ol_nat @ $false )
% 0.23/0.48 = zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq(1)
% 0.23/0.48 thf(fact_4235_of__bool__eq_I1_J,axiom,
% 0.23/0.48 ( ( zero_n2684676970156552555ol_int @ $false )
% 0.23/0.48 = zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq(1)
% 0.23/0.48 thf(fact_4236_of__bool__eq_I1_J,axiom,
% 0.23/0.48 ( ( zero_n356916108424825756nteger @ $false )
% 0.23/0.48 = zero_z3403309356797280102nteger ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq(1)
% 0.23/0.48 thf(fact_4237_of__bool__less__iff,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 0.23/0.48 = ( ~ P
% 0.23/0.48 & Q ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_iff
% 0.23/0.48 thf(fact_4238_of__bool__less__iff,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.23/0.48 = ( ~ P
% 0.23/0.48 & Q ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_iff
% 0.23/0.48 thf(fact_4239_of__bool__less__iff,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.23/0.48 = ( ~ P
% 0.23/0.48 & Q ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_iff
% 0.23/0.48 thf(fact_4240_of__bool__less__iff,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.23/0.48 = ( ~ P
% 0.23/0.48 & Q ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_iff
% 0.23/0.48 thf(fact_4241_of__bool__less__iff,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.23/0.48 = ( ~ P
% 0.23/0.48 & Q ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_iff
% 0.23/0.48 thf(fact_4242_of__bool__eq_I2_J,axiom,
% 0.23/0.48 ( ( zero_n1201886186963655149omplex @ $true )
% 0.23/0.48 = one_one_complex ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq(2)
% 0.23/0.48 thf(fact_4243_of__bool__eq_I2_J,axiom,
% 0.23/0.48 ( ( zero_n3304061248610475627l_real @ $true )
% 0.23/0.48 = one_one_real ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq(2)
% 0.23/0.48 thf(fact_4244_of__bool__eq_I2_J,axiom,
% 0.23/0.48 ( ( zero_n2052037380579107095ol_rat @ $true )
% 0.23/0.48 = one_one_rat ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq(2)
% 0.23/0.48 thf(fact_4245_of__bool__eq_I2_J,axiom,
% 0.23/0.48 ( ( zero_n2687167440665602831ol_nat @ $true )
% 0.23/0.48 = one_one_nat ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq(2)
% 0.23/0.48 thf(fact_4246_of__bool__eq_I2_J,axiom,
% 0.23/0.48 ( ( zero_n2684676970156552555ol_int @ $true )
% 0.23/0.48 = one_one_int ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq(2)
% 0.23/0.48 thf(fact_4247_of__bool__eq_I2_J,axiom,
% 0.23/0.48 ( ( zero_n356916108424825756nteger @ $true )
% 0.23/0.48 = one_one_Code_integer ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq(2)
% 0.23/0.48 thf(fact_4248_of__bool__eq__1__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ( zero_n1201886186963655149omplex @ P )
% 0.23/0.48 = one_one_complex )
% 0.23/0.48 = P ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_1_iff
% 0.23/0.48 thf(fact_4249_of__bool__eq__1__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ( zero_n3304061248610475627l_real @ P )
% 0.23/0.48 = one_one_real )
% 0.23/0.48 = P ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_1_iff
% 0.23/0.48 thf(fact_4250_of__bool__eq__1__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.23/0.48 = one_one_rat )
% 0.23/0.48 = P ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_1_iff
% 0.23/0.48 thf(fact_4251_of__bool__eq__1__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.23/0.48 = one_one_nat )
% 0.23/0.48 = P ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_1_iff
% 0.23/0.48 thf(fact_4252_of__bool__eq__1__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.23/0.48 = one_one_int )
% 0.23/0.48 = P ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_1_iff
% 0.23/0.48 thf(fact_4253_of__bool__eq__1__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ( zero_n356916108424825756nteger @ P )
% 0.23/0.48 = one_one_Code_integer )
% 0.23/0.48 = P ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_1_iff
% 0.23/0.48 thf(fact_4254_concat__bit__0,axiom,
% 0.23/0.48 ! [K: int,L2: int] :
% 0.23/0.48 ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
% 0.23/0.48 = L2 ) ).
% 0.23/0.48
% 0.23/0.48 % concat_bit_0
% 0.23/0.48 thf(fact_4255_semiring__norm_I73_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.48 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(73)
% 0.23/0.48 thf(fact_4256_semiring__norm_I80_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.48 = ( ord_less_num @ M @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(80)
% 0.23/0.48 thf(fact_4257_zero__less__of__bool__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.23/0.48 = P ) ).
% 0.23/0.48
% 0.23/0.48 % zero_less_of_bool_iff
% 0.23/0.48 thf(fact_4258_zero__less__of__bool__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.23/0.48 = P ) ).
% 0.23/0.48
% 0.23/0.48 % zero_less_of_bool_iff
% 0.23/0.48 thf(fact_4259_zero__less__of__bool__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.23/0.48 = P ) ).
% 0.23/0.48
% 0.23/0.48 % zero_less_of_bool_iff
% 0.23/0.48 thf(fact_4260_zero__less__of__bool__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.23/0.48 = P ) ).
% 0.23/0.48
% 0.23/0.48 % zero_less_of_bool_iff
% 0.23/0.48 thf(fact_4261_zero__less__of__bool__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 0.23/0.48 = P ) ).
% 0.23/0.48
% 0.23/0.48 % zero_less_of_bool_iff
% 0.23/0.48 thf(fact_4262_of__bool__less__one__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 0.23/0.48 = ( ~ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_one_iff
% 0.23/0.48 thf(fact_4263_of__bool__less__one__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 0.23/0.48 = ( ~ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_one_iff
% 0.23/0.48 thf(fact_4264_of__bool__less__one__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 0.23/0.48 = ( ~ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_one_iff
% 0.23/0.48 thf(fact_4265_of__bool__less__one__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 0.23/0.48 = ( ~ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_one_iff
% 0.23/0.48 thf(fact_4266_of__bool__less__one__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 0.23/0.48 = ( ~ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_one_iff
% 0.23/0.48 thf(fact_4267_of__bool__not__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( zero_n1201886186963655149omplex @ ~ P )
% 0.23/0.48 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_not_iff
% 0.23/0.48 thf(fact_4268_of__bool__not__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( zero_n3304061248610475627l_real @ ~ P )
% 0.23/0.48 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_not_iff
% 0.23/0.48 thf(fact_4269_of__bool__not__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 0.23/0.48 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_not_iff
% 0.23/0.48 thf(fact_4270_of__bool__not__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 0.23/0.48 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_not_iff
% 0.23/0.48 thf(fact_4271_of__bool__not__iff,axiom,
% 0.23/0.48 ! [P: $o] :
% 0.23/0.48 ( ( zero_n356916108424825756nteger @ ~ P )
% 0.23/0.48 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_not_iff
% 0.23/0.48 thf(fact_4272_Suc__0__mod__eq,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.23/0.48 = ( zero_n2687167440665602831ol_nat
% 0.23/0.48 @ ( N
% 0.23/0.48 != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % Suc_0_mod_eq
% 0.23/0.48 thf(fact_4273_semiring__norm_I9_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.23/0.48 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(9)
% 0.23/0.48 thf(fact_4274_semiring__norm_I7_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.48 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(7)
% 0.23/0.48 thf(fact_4275_semiring__norm_I15_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.23/0.48 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(15)
% 0.23/0.48 thf(fact_4276_semiring__norm_I14_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.48 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(14)
% 0.23/0.48 thf(fact_4277_semiring__norm_I72_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.48 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(72)
% 0.23/0.48 thf(fact_4278_semiring__norm_I81_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.23/0.48 = ( ord_less_num @ M @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(81)
% 0.23/0.48 thf(fact_4279_semiring__norm_I70_J,axiom,
% 0.23/0.48 ! [M: num] :
% 0.23/0.48 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(70)
% 0.23/0.48 thf(fact_4280_semiring__norm_I77_J,axiom,
% 0.23/0.48 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(77)
% 0.23/0.48 thf(fact_4281_concat__bit__nonnegative__iff,axiom,
% 0.23/0.48 ! [N: nat,K: int,L2: int] :
% 0.23/0.48 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L2 ) )
% 0.23/0.48 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).
% 0.23/0.48
% 0.23/0.48 % concat_bit_nonnegative_iff
% 0.23/0.48 thf(fact_4282_concat__bit__negative__iff,axiom,
% 0.23/0.48 ! [N: nat,K: int,L2: int] :
% 0.23/0.48 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L2 ) @ zero_zero_int )
% 0.23/0.48 = ( ord_less_int @ L2 @ zero_zero_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % concat_bit_negative_iff
% 0.23/0.48 thf(fact_4283_zdiv__numeral__Bit1,axiom,
% 0.23/0.48 ! [V: num,W: num] :
% 0.23/0.48 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.23/0.48 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % zdiv_numeral_Bit1
% 0.23/0.48 thf(fact_4284_semiring__norm_I3_J,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 0.23/0.48 = ( bit1 @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(3)
% 0.23/0.48 thf(fact_4285_semiring__norm_I4_J,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 0.23/0.48 = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(4)
% 0.23/0.48 thf(fact_4286_semiring__norm_I5_J,axiom,
% 0.23/0.48 ! [M: num] :
% 0.23/0.48 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 0.23/0.48 = ( bit1 @ M ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(5)
% 0.23/0.48 thf(fact_4287_semiring__norm_I8_J,axiom,
% 0.23/0.48 ! [M: num] :
% 0.23/0.48 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 0.23/0.48 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(8)
% 0.23/0.48 thf(fact_4288_semiring__norm_I10_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.48 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(10)
% 0.23/0.48 thf(fact_4289_semiring__norm_I16_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.48 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(16)
% 0.23/0.48 thf(fact_4290_semiring__norm_I74_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.23/0.48 = ( ord_less_num @ M @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(74)
% 0.23/0.48 thf(fact_4291_semiring__norm_I79_J,axiom,
% 0.23/0.48 ! [M: num,N: num] :
% 0.23/0.48 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.48 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % semiring_norm(79)
% 0.23/0.48 thf(fact_4292_odd__of__bool__self,axiom,
% 0.23/0.48 ! [P2: $o] :
% 0.23/0.48 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P2 ) ) )
% 0.23/0.48 = P2 ) ).
% 0.23/0.48
% 0.23/0.48 % odd_of_bool_self
% 0.23/0.48 thf(fact_4293_odd__of__bool__self,axiom,
% 0.23/0.48 ! [P2: $o] :
% 0.23/0.48 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P2 ) ) )
% 0.23/0.48 = P2 ) ).
% 0.23/0.48
% 0.23/0.48 % odd_of_bool_self
% 0.23/0.48 thf(fact_4294_odd__of__bool__self,axiom,
% 0.23/0.48 ! [P2: $o] :
% 0.23/0.48 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P2 ) ) )
% 0.23/0.48 = P2 ) ).
% 0.23/0.48
% 0.23/0.48 % odd_of_bool_self
% 0.23/0.48 thf(fact_4295_of__bool__half__eq__0,axiom,
% 0.23/0.48 ! [B2: $o] :
% 0.23/0.48 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_half_eq_0
% 0.23/0.48 thf(fact_4296_of__bool__half__eq__0,axiom,
% 0.23/0.48 ! [B2: $o] :
% 0.23/0.48 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.48 = zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_half_eq_0
% 0.23/0.48 thf(fact_4297_of__bool__half__eq__0,axiom,
% 0.23/0.48 ! [B2: $o] :
% 0.23/0.48 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.48 = zero_z3403309356797280102nteger ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_half_eq_0
% 0.23/0.48 thf(fact_4298_Suc__div__eq__add3__div__numeral,axiom,
% 0.23/0.48 ! [M: nat,V: num] :
% 0.23/0.48 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.23/0.48 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % Suc_div_eq_add3_div_numeral
% 0.23/0.48 thf(fact_4299_div__Suc__eq__div__add3,axiom,
% 0.23/0.48 ! [M: nat,N: nat] :
% 0.23/0.48 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.23/0.48 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % div_Suc_eq_div_add3
% 0.23/0.48 thf(fact_4300_mod__Suc__eq__mod__add3,axiom,
% 0.23/0.48 ! [M: nat,N: nat] :
% 0.23/0.48 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.23/0.48 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % mod_Suc_eq_mod_add3
% 0.23/0.48 thf(fact_4301_Suc__mod__eq__add3__mod__numeral,axiom,
% 0.23/0.48 ! [M: nat,V: num] :
% 0.23/0.48 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.23/0.48 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % Suc_mod_eq_add3_mod_numeral
% 0.23/0.48 thf(fact_4302_set__decode__Suc,axiom,
% 0.23/0.48 ! [N: nat,X4: nat] :
% 0.23/0.48 ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X4 ) )
% 0.23/0.48 = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % set_decode_Suc
% 0.23/0.48 thf(fact_4303_bits__1__div__exp,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % bits_1_div_exp
% 0.23/0.48 thf(fact_4304_bits__1__div__exp,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % bits_1_div_exp
% 0.23/0.48 thf(fact_4305_bits__1__div__exp,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % bits_1_div_exp
% 0.23/0.48 thf(fact_4306_one__div__2__pow__eq,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % one_div_2_pow_eq
% 0.23/0.48 thf(fact_4307_one__div__2__pow__eq,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % one_div_2_pow_eq
% 0.23/0.48 thf(fact_4308_one__div__2__pow__eq,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % one_div_2_pow_eq
% 0.23/0.48 thf(fact_4309_zmod__numeral__Bit1,axiom,
% 0.23/0.48 ! [V: num,W: num] :
% 0.23/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.23/0.48 = ( 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 ) ) ).
% 0.23/0.48
% 0.23/0.48 % zmod_numeral_Bit1
% 0.23/0.48 thf(fact_4310_one__mod__2__pow__eq,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % one_mod_2_pow_eq
% 0.23/0.48 thf(fact_4311_one__mod__2__pow__eq,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % one_mod_2_pow_eq
% 0.23/0.48 thf(fact_4312_one__mod__2__pow__eq,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % one_mod_2_pow_eq
% 0.23/0.48 thf(fact_4313_signed__take__bit__Suc__bit1,axiom,
% 0.23/0.48 ! [N: nat,K: num] :
% 0.23/0.48 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.23/0.48 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_Suc_bit1
% 0.23/0.48 thf(fact_4314_of__bool__eq__iff,axiom,
% 0.23/0.48 ! [P2: $o,Q2: $o] :
% 0.23/0.48 ( ( ( zero_n2687167440665602831ol_nat @ P2 )
% 0.23/0.48 = ( zero_n2687167440665602831ol_nat @ Q2 ) )
% 0.23/0.48 = ( P2 = Q2 ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_iff
% 0.23/0.48 thf(fact_4315_of__bool__eq__iff,axiom,
% 0.23/0.48 ! [P2: $o,Q2: $o] :
% 0.23/0.48 ( ( ( zero_n2684676970156552555ol_int @ P2 )
% 0.23/0.48 = ( zero_n2684676970156552555ol_int @ Q2 ) )
% 0.23/0.48 = ( P2 = Q2 ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_iff
% 0.23/0.48 thf(fact_4316_of__bool__eq__iff,axiom,
% 0.23/0.48 ! [P2: $o,Q2: $o] :
% 0.23/0.48 ( ( ( zero_n356916108424825756nteger @ P2 )
% 0.23/0.48 = ( zero_n356916108424825756nteger @ Q2 ) )
% 0.23/0.48 = ( P2 = Q2 ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_eq_iff
% 0.23/0.48 thf(fact_4317_of__bool__conj,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( zero_n3304061248610475627l_real
% 0.23/0.48 @ ( P
% 0.23/0.48 & Q ) )
% 0.23/0.48 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_conj
% 0.23/0.48 thf(fact_4318_of__bool__conj,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( zero_n2052037380579107095ol_rat
% 0.23/0.48 @ ( P
% 0.23/0.48 & Q ) )
% 0.23/0.48 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_conj
% 0.23/0.48 thf(fact_4319_of__bool__conj,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( zero_n2687167440665602831ol_nat
% 0.23/0.48 @ ( P
% 0.23/0.48 & Q ) )
% 0.23/0.48 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_conj
% 0.23/0.48 thf(fact_4320_of__bool__conj,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( zero_n2684676970156552555ol_int
% 0.23/0.48 @ ( P
% 0.23/0.48 & Q ) )
% 0.23/0.48 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_conj
% 0.23/0.48 thf(fact_4321_of__bool__conj,axiom,
% 0.23/0.48 ! [P: $o,Q: $o] :
% 0.23/0.48 ( ( zero_n356916108424825756nteger
% 0.23/0.48 @ ( P
% 0.23/0.48 & Q ) )
% 0.23/0.48 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_conj
% 0.23/0.48 thf(fact_4322_verit__eq__simplify_I14_J,axiom,
% 0.23/0.48 ! [X2: num,X32: num] :
% 0.23/0.48 ( ( bit0 @ X2 )
% 0.23/0.48 != ( bit1 @ X32 ) ) ).
% 0.23/0.48
% 0.23/0.48 % verit_eq_simplify(14)
% 0.23/0.48 thf(fact_4323_verit__eq__simplify_I12_J,axiom,
% 0.23/0.48 ! [X32: num] :
% 0.23/0.48 ( one
% 0.23/0.48 != ( bit1 @ X32 ) ) ).
% 0.23/0.48
% 0.23/0.48 % verit_eq_simplify(12)
% 0.23/0.48 thf(fact_4324_concat__bit__assoc,axiom,
% 0.23/0.48 ! [N: nat,K: int,M: nat,L2: int,R2: int] :
% 0.23/0.48 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L2 @ R2 ) )
% 0.23/0.48 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L2 ) @ R2 ) ) ).
% 0.23/0.48
% 0.23/0.48 % concat_bit_assoc
% 0.23/0.48 thf(fact_4325_subset__decode__imp__le,axiom,
% 0.23/0.48 ! [M: nat,N: nat] :
% 0.23/0.48 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 0.23/0.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % subset_decode_imp_le
% 0.23/0.48 thf(fact_4326_zero__less__eq__of__bool,axiom,
% 0.23/0.48 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % zero_less_eq_of_bool
% 0.23/0.48 thf(fact_4327_zero__less__eq__of__bool,axiom,
% 0.23/0.48 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % zero_less_eq_of_bool
% 0.23/0.48 thf(fact_4328_zero__less__eq__of__bool,axiom,
% 0.23/0.48 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % zero_less_eq_of_bool
% 0.23/0.48 thf(fact_4329_zero__less__eq__of__bool,axiom,
% 0.23/0.48 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % zero_less_eq_of_bool
% 0.23/0.48 thf(fact_4330_zero__less__eq__of__bool,axiom,
% 0.23/0.48 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 0.23/0.48
% 0.23/0.48 % zero_less_eq_of_bool
% 0.23/0.48 thf(fact_4331_of__bool__less__eq__one,axiom,
% 0.23/0.48 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_eq_one
% 0.23/0.48 thf(fact_4332_of__bool__less__eq__one,axiom,
% 0.23/0.48 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_eq_one
% 0.23/0.48 thf(fact_4333_of__bool__less__eq__one,axiom,
% 0.23/0.48 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_eq_one
% 0.23/0.48 thf(fact_4334_of__bool__less__eq__one,axiom,
% 0.23/0.48 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_eq_one
% 0.23/0.48 thf(fact_4335_of__bool__less__eq__one,axiom,
% 0.23/0.48 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_less_eq_one
% 0.23/0.48 thf(fact_4336_split__of__bool__asm,axiom,
% 0.23/0.48 ! [P: complex > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
% 0.23/0.48 = ( ~ ( ( P2
% 0.23/0.48 & ~ ( P @ one_one_complex ) )
% 0.23/0.48 | ( ~ P2
% 0.23/0.48 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool_asm
% 0.23/0.48 thf(fact_4337_split__of__bool__asm,axiom,
% 0.23/0.48 ! [P: real > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
% 0.23/0.48 = ( ~ ( ( P2
% 0.23/0.48 & ~ ( P @ one_one_real ) )
% 0.23/0.48 | ( ~ P2
% 0.23/0.48 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool_asm
% 0.23/0.48 thf(fact_4338_split__of__bool__asm,axiom,
% 0.23/0.48 ! [P: rat > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
% 0.23/0.48 = ( ~ ( ( P2
% 0.23/0.48 & ~ ( P @ one_one_rat ) )
% 0.23/0.48 | ( ~ P2
% 0.23/0.48 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool_asm
% 0.23/0.48 thf(fact_4339_split__of__bool__asm,axiom,
% 0.23/0.48 ! [P: nat > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
% 0.23/0.48 = ( ~ ( ( P2
% 0.23/0.48 & ~ ( P @ one_one_nat ) )
% 0.23/0.48 | ( ~ P2
% 0.23/0.48 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool_asm
% 0.23/0.48 thf(fact_4340_split__of__bool__asm,axiom,
% 0.23/0.48 ! [P: int > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
% 0.23/0.48 = ( ~ ( ( P2
% 0.23/0.48 & ~ ( P @ one_one_int ) )
% 0.23/0.48 | ( ~ P2
% 0.23/0.48 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool_asm
% 0.23/0.48 thf(fact_4341_split__of__bool__asm,axiom,
% 0.23/0.48 ! [P: code_integer > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
% 0.23/0.48 = ( ~ ( ( P2
% 0.23/0.48 & ~ ( P @ one_one_Code_integer ) )
% 0.23/0.48 | ( ~ P2
% 0.23/0.48 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool_asm
% 0.23/0.48 thf(fact_4342_split__of__bool,axiom,
% 0.23/0.48 ! [P: complex > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
% 0.23/0.48 = ( ( P2
% 0.23/0.48 => ( P @ one_one_complex ) )
% 0.23/0.48 & ( ~ P2
% 0.23/0.48 => ( P @ zero_zero_complex ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool
% 0.23/0.48 thf(fact_4343_split__of__bool,axiom,
% 0.23/0.48 ! [P: real > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
% 0.23/0.48 = ( ( P2
% 0.23/0.48 => ( P @ one_one_real ) )
% 0.23/0.48 & ( ~ P2
% 0.23/0.48 => ( P @ zero_zero_real ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool
% 0.23/0.48 thf(fact_4344_split__of__bool,axiom,
% 0.23/0.48 ! [P: rat > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
% 0.23/0.48 = ( ( P2
% 0.23/0.48 => ( P @ one_one_rat ) )
% 0.23/0.48 & ( ~ P2
% 0.23/0.48 => ( P @ zero_zero_rat ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool
% 0.23/0.48 thf(fact_4345_split__of__bool,axiom,
% 0.23/0.48 ! [P: nat > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
% 0.23/0.48 = ( ( P2
% 0.23/0.48 => ( P @ one_one_nat ) )
% 0.23/0.48 & ( ~ P2
% 0.23/0.48 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool
% 0.23/0.48 thf(fact_4346_split__of__bool,axiom,
% 0.23/0.48 ! [P: int > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
% 0.23/0.48 = ( ( P2
% 0.23/0.48 => ( P @ one_one_int ) )
% 0.23/0.48 & ( ~ P2
% 0.23/0.48 => ( P @ zero_zero_int ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool
% 0.23/0.48 thf(fact_4347_split__of__bool,axiom,
% 0.23/0.48 ! [P: code_integer > $o,P2: $o] :
% 0.23/0.48 ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
% 0.23/0.48 = ( ( P2
% 0.23/0.48 => ( P @ one_one_Code_integer ) )
% 0.23/0.48 & ( ~ P2
% 0.23/0.48 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % split_of_bool
% 0.23/0.48 thf(fact_4348_of__bool__def,axiom,
% 0.23/0.48 ( zero_n1201886186963655149omplex
% 0.23/0.48 = ( ^ [P3: $o] : ( if_complex @ P3 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_def
% 0.23/0.48 thf(fact_4349_of__bool__def,axiom,
% 0.23/0.48 ( zero_n3304061248610475627l_real
% 0.23/0.48 = ( ^ [P3: $o] : ( if_real @ P3 @ one_one_real @ zero_zero_real ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_def
% 0.23/0.48 thf(fact_4350_of__bool__def,axiom,
% 0.23/0.48 ( zero_n2052037380579107095ol_rat
% 0.23/0.48 = ( ^ [P3: $o] : ( if_rat @ P3 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_def
% 0.23/0.48 thf(fact_4351_of__bool__def,axiom,
% 0.23/0.48 ( zero_n2687167440665602831ol_nat
% 0.23/0.48 = ( ^ [P3: $o] : ( if_nat @ P3 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_def
% 0.23/0.48 thf(fact_4352_of__bool__def,axiom,
% 0.23/0.48 ( zero_n2684676970156552555ol_int
% 0.23/0.48 = ( ^ [P3: $o] : ( if_int @ P3 @ one_one_int @ zero_zero_int ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_def
% 0.23/0.48 thf(fact_4353_of__bool__def,axiom,
% 0.23/0.48 ( zero_n356916108424825756nteger
% 0.23/0.48 = ( ^ [P3: $o] : ( if_Code_integer @ P3 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_def
% 0.23/0.48 thf(fact_4354_num_Oexhaust,axiom,
% 0.23/0.48 ! [Y3: num] :
% 0.23/0.48 ( ( Y3 != one )
% 0.23/0.48 => ( ! [X22: num] :
% 0.23/0.48 ( Y3
% 0.23/0.48 != ( bit0 @ X22 ) )
% 0.23/0.48 => ~ ! [X33: num] :
% 0.23/0.48 ( Y3
% 0.23/0.48 != ( bit1 @ X33 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % num.exhaust
% 0.23/0.48 thf(fact_4355_numeral__Bit1,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 0.23/0.48 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 0.23/0.48
% 0.23/0.48 % numeral_Bit1
% 0.23/0.48 thf(fact_4356_numeral__Bit1,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 0.23/0.48 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 0.23/0.48
% 0.23/0.48 % numeral_Bit1
% 0.23/0.48 thf(fact_4357_numeral__Bit1,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 0.23/0.48 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 0.23/0.48
% 0.23/0.48 % numeral_Bit1
% 0.23/0.48 thf(fact_4358_numeral__Bit1,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.23/0.48 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 0.23/0.48
% 0.23/0.48 % numeral_Bit1
% 0.23/0.48 thf(fact_4359_numeral__Bit1,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 0.23/0.48 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % numeral_Bit1
% 0.23/0.48 thf(fact_4360_eval__nat__numeral_I3_J,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.23/0.48 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % eval_nat_numeral(3)
% 0.23/0.48 thf(fact_4361_cong__exp__iff__simps_I10_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num,N: num] :
% 0.23/0.48 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(10)
% 0.23/0.48 thf(fact_4362_cong__exp__iff__simps_I10_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num,N: num] :
% 0.23/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(10)
% 0.23/0.48 thf(fact_4363_cong__exp__iff__simps_I10_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num,N: num] :
% 0.23/0.48 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(10)
% 0.23/0.48 thf(fact_4364_cong__exp__iff__simps_I12_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num,N: num] :
% 0.23/0.48 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(12)
% 0.23/0.48 thf(fact_4365_cong__exp__iff__simps_I12_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num,N: num] :
% 0.23/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(12)
% 0.23/0.48 thf(fact_4366_cong__exp__iff__simps_I12_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num,N: num] :
% 0.23/0.48 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(12)
% 0.23/0.48 thf(fact_4367_cong__exp__iff__simps_I13_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num,N: num] :
% 0.23/0.48 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.23/0.48 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.23/0.48 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(13)
% 0.23/0.48 thf(fact_4368_cong__exp__iff__simps_I13_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num,N: num] :
% 0.23/0.48 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.23/0.48 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.23/0.48 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(13)
% 0.23/0.48 thf(fact_4369_cong__exp__iff__simps_I13_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num,N: num] :
% 0.23/0.48 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.23/0.48 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.23/0.48 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(13)
% 0.23/0.48 thf(fact_4370_power__minus__Bit1,axiom,
% 0.23/0.48 ! [X4: real,K: num] :
% 0.23/0.48 ( ( power_power_real @ ( uminus_uminus_real @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.23/0.48 = ( uminus_uminus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus_Bit1
% 0.23/0.48 thf(fact_4371_power__minus__Bit1,axiom,
% 0.23/0.48 ! [X4: int,K: num] :
% 0.23/0.48 ( ( power_power_int @ ( uminus_uminus_int @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.23/0.48 = ( uminus_uminus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus_Bit1
% 0.23/0.48 thf(fact_4372_power__minus__Bit1,axiom,
% 0.23/0.48 ! [X4: complex,K: num] :
% 0.23/0.48 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus_Bit1
% 0.23/0.48 thf(fact_4373_power__minus__Bit1,axiom,
% 0.23/0.48 ! [X4: code_integer,K: num] :
% 0.23/0.48 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.23/0.48 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus_Bit1
% 0.23/0.48 thf(fact_4374_power__minus__Bit1,axiom,
% 0.23/0.48 ! [X4: rat,K: num] :
% 0.23/0.48 ( ( power_power_rat @ ( uminus_uminus_rat @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.23/0.48 = ( uminus_uminus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % power_minus_Bit1
% 0.23/0.48 thf(fact_4375_numeral__Bit1__div__2,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = ( numeral_numeral_nat @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % numeral_Bit1_div_2
% 0.23/0.48 thf(fact_4376_numeral__Bit1__div__2,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.48 = ( numeral_numeral_int @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % numeral_Bit1_div_2
% 0.23/0.48 thf(fact_4377_odd__numeral,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % odd_numeral
% 0.23/0.48 thf(fact_4378_odd__numeral,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % odd_numeral
% 0.23/0.48 thf(fact_4379_odd__numeral,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % odd_numeral
% 0.23/0.48 thf(fact_4380_cong__exp__iff__simps_I3_J,axiom,
% 0.23/0.48 ! [N: num,Q2: num] :
% 0.23/0.48 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 != zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(3)
% 0.23/0.48 thf(fact_4381_cong__exp__iff__simps_I3_J,axiom,
% 0.23/0.48 ! [N: num,Q2: num] :
% 0.23/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 != zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(3)
% 0.23/0.48 thf(fact_4382_cong__exp__iff__simps_I3_J,axiom,
% 0.23/0.48 ! [N: num,Q2: num] :
% 0.23/0.48 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 != zero_z3403309356797280102nteger ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(3)
% 0.23/0.48 thf(fact_4383_power3__eq__cube,axiom,
% 0.23/0.48 ! [A: complex] :
% 0.23/0.48 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.23/0.48 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 0.23/0.48
% 0.23/0.48 % power3_eq_cube
% 0.23/0.48 thf(fact_4384_power3__eq__cube,axiom,
% 0.23/0.48 ! [A: real] :
% 0.23/0.48 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.23/0.48 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 0.23/0.48
% 0.23/0.48 % power3_eq_cube
% 0.23/0.48 thf(fact_4385_power3__eq__cube,axiom,
% 0.23/0.48 ! [A: rat] :
% 0.23/0.48 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.23/0.48 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 0.23/0.48
% 0.23/0.48 % power3_eq_cube
% 0.23/0.48 thf(fact_4386_power3__eq__cube,axiom,
% 0.23/0.48 ! [A: nat] :
% 0.23/0.48 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.23/0.48 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 0.23/0.48
% 0.23/0.48 % power3_eq_cube
% 0.23/0.48 thf(fact_4387_power3__eq__cube,axiom,
% 0.23/0.48 ! [A: int] :
% 0.23/0.48 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.23/0.48 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 0.23/0.48
% 0.23/0.48 % power3_eq_cube
% 0.23/0.48 thf(fact_4388_numeral__3__eq__3,axiom,
% 0.23/0.48 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 0.23/0.48 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % numeral_3_eq_3
% 0.23/0.48 thf(fact_4389_Suc3__eq__add__3,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 0.23/0.48 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % Suc3_eq_add_3
% 0.23/0.48 thf(fact_4390_of__bool__odd__eq__mod__2,axiom,
% 0.23/0.48 ! [A: nat] :
% 0.23/0.48 ( ( zero_n2687167440665602831ol_nat
% 0.23/0.48 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.23/0.48 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_odd_eq_mod_2
% 0.23/0.48 thf(fact_4391_of__bool__odd__eq__mod__2,axiom,
% 0.23/0.48 ! [A: int] :
% 0.23/0.48 ( ( zero_n2684676970156552555ol_int
% 0.23/0.48 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.23/0.48 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_odd_eq_mod_2
% 0.23/0.48 thf(fact_4392_of__bool__odd__eq__mod__2,axiom,
% 0.23/0.48 ! [A: code_integer] :
% 0.23/0.48 ( ( zero_n356916108424825756nteger
% 0.23/0.48 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.23/0.48 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_bool_odd_eq_mod_2
% 0.23/0.48 thf(fact_4393_cong__exp__iff__simps_I7_J,axiom,
% 0.23/0.48 ! [Q2: num,N: num] :
% 0.23/0.48 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.23/0.48 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.23/0.48 = zero_zero_nat ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(7)
% 0.23/0.48 thf(fact_4394_cong__exp__iff__simps_I7_J,axiom,
% 0.23/0.48 ! [Q2: num,N: num] :
% 0.23/0.48 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.23/0.48 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 0.23/0.48 = zero_zero_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(7)
% 0.23/0.48 thf(fact_4395_cong__exp__iff__simps_I7_J,axiom,
% 0.23/0.48 ! [Q2: num,N: num] :
% 0.23/0.48 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.23/0.48 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.23/0.48 = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(7)
% 0.23/0.48 thf(fact_4396_cong__exp__iff__simps_I11_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num] :
% 0.23/0.48 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.23/0.48 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.23/0.48 = zero_zero_nat ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(11)
% 0.23/0.48 thf(fact_4397_cong__exp__iff__simps_I11_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num] :
% 0.23/0.48 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.23/0.48 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.23/0.48 = zero_zero_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(11)
% 0.23/0.48 thf(fact_4398_cong__exp__iff__simps_I11_J,axiom,
% 0.23/0.48 ! [M: num,Q2: num] :
% 0.23/0.48 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.23/0.48 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.23/0.48 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.23/0.48 = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.48
% 0.23/0.48 % cong_exp_iff_simps(11)
% 0.23/0.48 thf(fact_4399_Suc__div__eq__add3__div,axiom,
% 0.23/0.48 ! [M: nat,N: nat] :
% 0.23/0.48 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.23/0.48 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % Suc_div_eq_add3_div
% 0.23/0.48 thf(fact_4400_Suc__mod__eq__add3__mod,axiom,
% 0.23/0.48 ! [M: nat,N: nat] :
% 0.23/0.48 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.23/0.48 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % Suc_mod_eq_add3_mod
% 0.23/0.48 thf(fact_4401_bits__induct,axiom,
% 0.23/0.48 ! [P: nat > $o,A: nat] :
% 0.23/0.48 ( ! [A5: nat] :
% 0.23/0.48 ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = A5 )
% 0.23/0.48 => ( P @ A5 ) )
% 0.23/0.48 => ( ! [A5: nat,B5: $o] :
% 0.23/0.48 ( ( P @ A5 )
% 0.23/0.48 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 = A5 )
% 0.23/0.48 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 0.23/0.48 => ( P @ A ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % bits_induct
% 0.23/0.48 thf(fact_4402_bits__induct,axiom,
% 0.23/0.48 ! [P: int > $o,A: int] :
% 0.23/0.48 ( ! [A5: int] :
% 0.23/0.48 ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.48 = A5 )
% 0.23/0.48 => ( P @ A5 ) )
% 0.23/0.48 => ( ! [A5: int,B5: $o] :
% 0.23/0.48 ( ( P @ A5 )
% 0.23/0.48 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.48 = A5 )
% 0.23/0.48 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 0.23/0.48 => ( P @ A ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % bits_induct
% 0.23/0.48 thf(fact_4403_bits__induct,axiom,
% 0.23/0.48 ! [P: code_integer > $o,A: code_integer] :
% 0.23/0.48 ( ! [A5: code_integer] :
% 0.23/0.48 ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.48 = A5 )
% 0.23/0.48 => ( P @ A5 ) )
% 0.23/0.48 => ( ! [A5: code_integer,B5: $o] :
% 0.23/0.48 ( ( P @ A5 )
% 0.23/0.48 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.48 = A5 )
% 0.23/0.48 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 0.23/0.48 => ( P @ A ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % bits_induct
% 0.23/0.48 thf(fact_4404_compl__le__swap2,axiom,
% 0.23/0.48 ! [Y3: set_nat,X4: set_nat] :
% 0.23/0.48 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ X4 )
% 0.23/0.48 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ Y3 ) ) ).
% 0.23/0.48
% 0.23/0.48 % compl_le_swap2
% 0.23/0.48 thf(fact_4405_compl__le__swap1,axiom,
% 0.23/0.48 ! [Y3: set_nat,X4: set_nat] :
% 0.23/0.48 ( ( ord_less_eq_set_nat @ Y3 @ ( uminus5710092332889474511et_nat @ X4 ) )
% 0.23/0.48 => ( ord_less_eq_set_nat @ X4 @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % compl_le_swap1
% 0.23/0.48 thf(fact_4406_compl__mono,axiom,
% 0.23/0.48 ! [X4: set_nat,Y3: set_nat] :
% 0.23/0.48 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 0.23/0.48 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ ( uminus5710092332889474511et_nat @ X4 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % compl_mono
% 0.23/0.48 thf(fact_4407_exp__mod__exp,axiom,
% 0.23/0.48 ! [M: nat,N: nat] :
% 0.23/0.48 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % exp_mod_exp
% 0.23/0.48 thf(fact_4408_exp__mod__exp,axiom,
% 0.23/0.48 ! [M: nat,N: nat] :
% 0.23/0.48 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % exp_mod_exp
% 0.23/0.48 thf(fact_4409_exp__mod__exp,axiom,
% 0.23/0.48 ! [M: nat,N: nat] :
% 0.23/0.48 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % exp_mod_exp
% 0.23/0.48 thf(fact_4410_exp__div__exp__eq,axiom,
% 0.23/0.48 ! [M: nat,N: nat] :
% 0.23/0.48 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( times_times_nat
% 0.23/0.48 @ ( zero_n2687167440665602831ol_nat
% 0.23/0.48 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.23/0.48 != zero_zero_nat )
% 0.23/0.48 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.23/0.48 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % exp_div_exp_eq
% 0.23/0.48 thf(fact_4411_exp__div__exp__eq,axiom,
% 0.23/0.48 ! [M: nat,N: nat] :
% 0.23/0.48 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( times_times_int
% 0.23/0.48 @ ( zero_n2684676970156552555ol_int
% 0.23/0.48 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.23/0.48 != zero_zero_int )
% 0.23/0.48 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.23/0.48 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % exp_div_exp_eq
% 0.23/0.48 thf(fact_4412_exp__div__exp__eq,axiom,
% 0.23/0.48 ! [M: nat,N: nat] :
% 0.23/0.48 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.48 = ( times_3573771949741848930nteger
% 0.23/0.48 @ ( zero_n356916108424825756nteger
% 0.23/0.48 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 0.23/0.48 != zero_z3403309356797280102nteger )
% 0.23/0.48 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.23/0.48 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % exp_div_exp_eq
% 0.23/0.48 thf(fact_4413_odd__mod__4__div__2,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.23/0.48 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.23/0.48 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % odd_mod_4_div_2
% 0.23/0.48 thf(fact_4414_mod__exhaust__less__4,axiom,
% 0.23/0.48 ! [M: nat] :
% 0.23/0.48 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.23/0.48 = zero_zero_nat )
% 0.23/0.48 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.23/0.48 = one_one_nat )
% 0.23/0.48 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.23/0.48 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.48 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.23/0.48 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % mod_exhaust_less_4
% 0.23/0.48 thf(fact_4415_signed__take__bit__numeral__minus__bit1,axiom,
% 0.23/0.48 ! [L2: num,K: num] :
% 0.23/0.48 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.23/0.48 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_numeral_minus_bit1
% 0.23/0.48 thf(fact_4416_dbl__dec__simps_I4_J,axiom,
% 0.23/0.48 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.48 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(4)
% 0.23/0.48 thf(fact_4417_dbl__dec__simps_I4_J,axiom,
% 0.23/0.48 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.48 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(4)
% 0.23/0.48 thf(fact_4418_dbl__dec__simps_I4_J,axiom,
% 0.23/0.48 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(4)
% 0.23/0.48 thf(fact_4419_dbl__dec__simps_I4_J,axiom,
% 0.23/0.48 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.48 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(4)
% 0.23/0.48 thf(fact_4420_dbl__dec__simps_I4_J,axiom,
% 0.23/0.48 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.48 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(4)
% 0.23/0.48 thf(fact_4421_signed__take__bit__numeral__bit1,axiom,
% 0.23/0.48 ! [L2: num,K: num] :
% 0.23/0.48 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.23/0.48 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_numeral_bit1
% 0.23/0.48 thf(fact_4422_dbl__inc__simps_I3_J,axiom,
% 0.23/0.48 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 0.23/0.48 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(3)
% 0.23/0.48 thf(fact_4423_dbl__inc__simps_I3_J,axiom,
% 0.23/0.48 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 0.23/0.48 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(3)
% 0.23/0.48 thf(fact_4424_dbl__inc__simps_I3_J,axiom,
% 0.23/0.48 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 0.23/0.48 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(3)
% 0.23/0.48 thf(fact_4425_dbl__inc__simps_I3_J,axiom,
% 0.23/0.48 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 0.23/0.48 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(3)
% 0.23/0.48 thf(fact_4426_add__scale__eq__noteq,axiom,
% 0.23/0.48 ! [R2: real,A: real,B2: real,C: real,D: real] :
% 0.23/0.48 ( ( R2 != zero_zero_real )
% 0.23/0.48 => ( ( ( A = B2 )
% 0.23/0.48 & ( C != D ) )
% 0.23/0.48 => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 0.23/0.48 != ( plus_plus_real @ B2 @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_scale_eq_noteq
% 0.23/0.48 thf(fact_4427_add__scale__eq__noteq,axiom,
% 0.23/0.48 ! [R2: rat,A: rat,B2: rat,C: rat,D: rat] :
% 0.23/0.48 ( ( R2 != zero_zero_rat )
% 0.23/0.48 => ( ( ( A = B2 )
% 0.23/0.48 & ( C != D ) )
% 0.23/0.48 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 0.23/0.48 != ( plus_plus_rat @ B2 @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_scale_eq_noteq
% 0.23/0.48 thf(fact_4428_add__scale__eq__noteq,axiom,
% 0.23/0.48 ! [R2: nat,A: nat,B2: nat,C: nat,D: nat] :
% 0.23/0.48 ( ( R2 != zero_zero_nat )
% 0.23/0.48 => ( ( ( A = B2 )
% 0.23/0.48 & ( C != D ) )
% 0.23/0.48 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 0.23/0.48 != ( plus_plus_nat @ B2 @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_scale_eq_noteq
% 0.23/0.48 thf(fact_4429_add__scale__eq__noteq,axiom,
% 0.23/0.48 ! [R2: int,A: int,B2: int,C: int,D: int] :
% 0.23/0.48 ( ( R2 != zero_zero_int )
% 0.23/0.48 => ( ( ( A = B2 )
% 0.23/0.48 & ( C != D ) )
% 0.23/0.48 => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 0.23/0.48 != ( plus_plus_int @ B2 @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_scale_eq_noteq
% 0.23/0.48 thf(fact_4430_num_Osize__gen_I3_J,axiom,
% 0.23/0.48 ! [X32: num] :
% 0.23/0.48 ( ( size_num @ ( bit1 @ X32 ) )
% 0.23/0.48 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % num.size_gen(3)
% 0.23/0.48 thf(fact_4431_dbl__dec__simps_I3_J,axiom,
% 0.23/0.48 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 0.23/0.48 = one_one_complex ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(3)
% 0.23/0.48 thf(fact_4432_dbl__dec__simps_I3_J,axiom,
% 0.23/0.48 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 0.23/0.48 = one_one_real ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(3)
% 0.23/0.48 thf(fact_4433_dbl__dec__simps_I3_J,axiom,
% 0.23/0.48 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 0.23/0.48 = one_one_rat ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(3)
% 0.23/0.48 thf(fact_4434_dbl__dec__simps_I3_J,axiom,
% 0.23/0.48 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 0.23/0.48 = one_one_int ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(3)
% 0.23/0.48 thf(fact_4435_pred__numeral__simps_I1_J,axiom,
% 0.23/0.48 ( ( pred_numeral @ one )
% 0.23/0.48 = zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % pred_numeral_simps(1)
% 0.23/0.48 thf(fact_4436_eq__numeral__Suc,axiom,
% 0.23/0.48 ! [K: num,N: nat] :
% 0.23/0.48 ( ( ( numeral_numeral_nat @ K )
% 0.23/0.48 = ( suc @ N ) )
% 0.23/0.48 = ( ( pred_numeral @ K )
% 0.23/0.48 = N ) ) ).
% 0.23/0.48
% 0.23/0.48 % eq_numeral_Suc
% 0.23/0.48 thf(fact_4437_Suc__eq__numeral,axiom,
% 0.23/0.48 ! [N: nat,K: num] :
% 0.23/0.48 ( ( ( suc @ N )
% 0.23/0.48 = ( numeral_numeral_nat @ K ) )
% 0.23/0.48 = ( N
% 0.23/0.48 = ( pred_numeral @ K ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % Suc_eq_numeral
% 0.23/0.48 thf(fact_4438_dbl__inc__simps_I2_J,axiom,
% 0.23/0.48 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 0.23/0.48 = one_one_complex ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(2)
% 0.23/0.48 thf(fact_4439_dbl__inc__simps_I2_J,axiom,
% 0.23/0.48 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 0.23/0.48 = one_one_real ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(2)
% 0.23/0.48 thf(fact_4440_dbl__inc__simps_I2_J,axiom,
% 0.23/0.48 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 0.23/0.48 = one_one_rat ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(2)
% 0.23/0.48 thf(fact_4441_dbl__inc__simps_I2_J,axiom,
% 0.23/0.48 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 0.23/0.48 = one_one_int ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(2)
% 0.23/0.48 thf(fact_4442_dbl__inc__simps_I4_J,axiom,
% 0.23/0.48 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.48 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(4)
% 0.23/0.48 thf(fact_4443_dbl__inc__simps_I4_J,axiom,
% 0.23/0.48 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.48 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(4)
% 0.23/0.48 thf(fact_4444_dbl__inc__simps_I4_J,axiom,
% 0.23/0.48 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(4)
% 0.23/0.48 thf(fact_4445_dbl__inc__simps_I4_J,axiom,
% 0.23/0.48 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.48 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(4)
% 0.23/0.48 thf(fact_4446_dbl__inc__simps_I4_J,axiom,
% 0.23/0.48 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.48 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(4)
% 0.23/0.48 thf(fact_4447_dbl__inc__simps_I5_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.23/0.48 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(5)
% 0.23/0.48 thf(fact_4448_dbl__inc__simps_I5_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 0.23/0.48 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(5)
% 0.23/0.48 thf(fact_4449_dbl__inc__simps_I5_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 0.23/0.48 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(5)
% 0.23/0.48 thf(fact_4450_dbl__inc__simps_I5_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 0.23/0.48 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(5)
% 0.23/0.48 thf(fact_4451_less__numeral__Suc,axiom,
% 0.23/0.48 ! [K: num,N: nat] :
% 0.23/0.48 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.23/0.48 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % less_numeral_Suc
% 0.23/0.48 thf(fact_4452_less__Suc__numeral,axiom,
% 0.23/0.48 ! [N: nat,K: num] :
% 0.23/0.48 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.23/0.48 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % less_Suc_numeral
% 0.23/0.48 thf(fact_4453_pred__numeral__simps_I3_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( pred_numeral @ ( bit1 @ K ) )
% 0.23/0.48 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % pred_numeral_simps(3)
% 0.23/0.48 thf(fact_4454_le__numeral__Suc,axiom,
% 0.23/0.48 ! [K: num,N: nat] :
% 0.23/0.48 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.23/0.48 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % le_numeral_Suc
% 0.23/0.48 thf(fact_4455_le__Suc__numeral,axiom,
% 0.23/0.48 ! [N: nat,K: num] :
% 0.23/0.48 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.23/0.48 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % le_Suc_numeral
% 0.23/0.48 thf(fact_4456_diff__Suc__numeral,axiom,
% 0.23/0.48 ! [N: nat,K: num] :
% 0.23/0.48 ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.23/0.48 = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % diff_Suc_numeral
% 0.23/0.48 thf(fact_4457_diff__numeral__Suc,axiom,
% 0.23/0.48 ! [K: num,N: nat] :
% 0.23/0.48 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.23/0.48 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % diff_numeral_Suc
% 0.23/0.48 thf(fact_4458_dbl__dec__simps_I2_J,axiom,
% 0.23/0.48 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 0.23/0.48 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(2)
% 0.23/0.48 thf(fact_4459_dbl__dec__simps_I2_J,axiom,
% 0.23/0.48 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.23/0.48 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(2)
% 0.23/0.48 thf(fact_4460_dbl__dec__simps_I2_J,axiom,
% 0.23/0.48 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(2)
% 0.23/0.48 thf(fact_4461_dbl__dec__simps_I2_J,axiom,
% 0.23/0.48 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.23/0.48 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(2)
% 0.23/0.48 thf(fact_4462_dbl__dec__simps_I2_J,axiom,
% 0.23/0.48 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 0.23/0.48 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(2)
% 0.23/0.48 thf(fact_4463_dbl__dec__simps_I1_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.23/0.48 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(1)
% 0.23/0.48 thf(fact_4464_dbl__dec__simps_I1_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.48 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(1)
% 0.23/0.48 thf(fact_4465_dbl__dec__simps_I1_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(1)
% 0.23/0.48 thf(fact_4466_dbl__dec__simps_I1_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.23/0.48 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(1)
% 0.23/0.48 thf(fact_4467_dbl__dec__simps_I1_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.23/0.48 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_simps(1)
% 0.23/0.48 thf(fact_4468_dbl__inc__simps_I1_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.23/0.48 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(1)
% 0.23/0.48 thf(fact_4469_dbl__inc__simps_I1_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.48 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(1)
% 0.23/0.48 thf(fact_4470_dbl__inc__simps_I1_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.23/0.48 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(1)
% 0.23/0.48 thf(fact_4471_dbl__inc__simps_I1_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.23/0.48 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(1)
% 0.23/0.48 thf(fact_4472_dbl__inc__simps_I1_J,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.23/0.48 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_simps(1)
% 0.23/0.48 thf(fact_4473_signed__take__bit__numeral__bit0,axiom,
% 0.23/0.48 ! [L2: num,K: num] :
% 0.23/0.48 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.23/0.48 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_numeral_bit0
% 0.23/0.48 thf(fact_4474_signed__take__bit__numeral__minus__bit0,axiom,
% 0.23/0.48 ! [L2: num,K: num] :
% 0.23/0.48 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.23/0.48 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % signed_take_bit_numeral_minus_bit0
% 0.23/0.48 thf(fact_4475_numeral__eq__Suc,axiom,
% 0.23/0.48 ( numeral_numeral_nat
% 0.23/0.48 = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % numeral_eq_Suc
% 0.23/0.48 thf(fact_4476_pred__numeral__def,axiom,
% 0.23/0.48 ( pred_numeral
% 0.23/0.48 = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % pred_numeral_def
% 0.23/0.48 thf(fact_4477_dbl__inc__def,axiom,
% 0.23/0.48 ( neg_nu8557863876264182079omplex
% 0.23/0.48 = ( ^ [X: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_def
% 0.23/0.48 thf(fact_4478_dbl__inc__def,axiom,
% 0.23/0.48 ( neg_nu8295874005876285629c_real
% 0.23/0.48 = ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_def
% 0.23/0.48 thf(fact_4479_dbl__inc__def,axiom,
% 0.23/0.48 ( neg_nu5219082963157363817nc_rat
% 0.23/0.48 = ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_def
% 0.23/0.48 thf(fact_4480_dbl__inc__def,axiom,
% 0.23/0.48 ( neg_nu5851722552734809277nc_int
% 0.23/0.48 = ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_inc_def
% 0.23/0.48 thf(fact_4481_num_Osize__gen_I1_J,axiom,
% 0.23/0.48 ( ( size_num @ one )
% 0.23/0.48 = zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % num.size_gen(1)
% 0.23/0.48 thf(fact_4482_dbl__dec__def,axiom,
% 0.23/0.48 ( neg_nu6511756317524482435omplex
% 0.23/0.48 = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_def
% 0.23/0.48 thf(fact_4483_dbl__dec__def,axiom,
% 0.23/0.48 ( neg_nu6075765906172075777c_real
% 0.23/0.48 = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_def
% 0.23/0.48 thf(fact_4484_dbl__dec__def,axiom,
% 0.23/0.48 ( neg_nu3179335615603231917ec_rat
% 0.23/0.48 = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_def
% 0.23/0.48 thf(fact_4485_dbl__dec__def,axiom,
% 0.23/0.48 ( neg_nu3811975205180677377ec_int
% 0.23/0.48 = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % dbl_dec_def
% 0.23/0.48 thf(fact_4486_add__0__iff,axiom,
% 0.23/0.48 ! [B2: real,A: real] :
% 0.23/0.48 ( ( B2
% 0.23/0.48 = ( plus_plus_real @ B2 @ A ) )
% 0.23/0.48 = ( A = zero_zero_real ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_0_iff
% 0.23/0.48 thf(fact_4487_add__0__iff,axiom,
% 0.23/0.48 ! [B2: rat,A: rat] :
% 0.23/0.48 ( ( B2
% 0.23/0.48 = ( plus_plus_rat @ B2 @ A ) )
% 0.23/0.48 = ( A = zero_zero_rat ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_0_iff
% 0.23/0.48 thf(fact_4488_add__0__iff,axiom,
% 0.23/0.48 ! [B2: nat,A: nat] :
% 0.23/0.48 ( ( B2
% 0.23/0.48 = ( plus_plus_nat @ B2 @ A ) )
% 0.23/0.48 = ( A = zero_zero_nat ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_0_iff
% 0.23/0.48 thf(fact_4489_add__0__iff,axiom,
% 0.23/0.48 ! [B2: int,A: int] :
% 0.23/0.48 ( ( B2
% 0.23/0.48 = ( plus_plus_int @ B2 @ A ) )
% 0.23/0.48 = ( A = zero_zero_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % add_0_iff
% 0.23/0.48 thf(fact_4490_crossproduct__noteq,axiom,
% 0.23/0.48 ! [A: real,B2: real,C: real,D: real] :
% 0.23/0.48 ( ( ( A != B2 )
% 0.23/0.48 & ( C != D ) )
% 0.23/0.48 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) )
% 0.23/0.48 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % crossproduct_noteq
% 0.23/0.48 thf(fact_4491_crossproduct__noteq,axiom,
% 0.23/0.48 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.23/0.48 ( ( ( A != B2 )
% 0.23/0.48 & ( C != D ) )
% 0.23/0.48 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) )
% 0.23/0.48 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % crossproduct_noteq
% 0.23/0.48 thf(fact_4492_crossproduct__noteq,axiom,
% 0.23/0.48 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.23/0.48 ( ( ( A != B2 )
% 0.23/0.48 & ( C != D ) )
% 0.23/0.48 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) )
% 0.23/0.48 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % crossproduct_noteq
% 0.23/0.48 thf(fact_4493_crossproduct__noteq,axiom,
% 0.23/0.48 ! [A: int,B2: int,C: int,D: int] :
% 0.23/0.48 ( ( ( A != B2 )
% 0.23/0.48 & ( C != D ) )
% 0.23/0.48 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) )
% 0.23/0.48 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % crossproduct_noteq
% 0.23/0.48 thf(fact_4494_crossproduct__eq,axiom,
% 0.23/0.48 ! [W: real,Y3: real,X4: real,Z: real] :
% 0.23/0.48 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X4 @ Z ) )
% 0.23/0.48 = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X4 @ Y3 ) ) )
% 0.23/0.48 = ( ( W = X4 )
% 0.23/0.48 | ( Y3 = Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % crossproduct_eq
% 0.23/0.48 thf(fact_4495_crossproduct__eq,axiom,
% 0.23/0.48 ! [W: rat,Y3: rat,X4: rat,Z: rat] :
% 0.23/0.48 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y3 ) @ ( times_times_rat @ X4 @ Z ) )
% 0.23/0.48 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X4 @ Y3 ) ) )
% 0.23/0.48 = ( ( W = X4 )
% 0.23/0.48 | ( Y3 = Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % crossproduct_eq
% 0.23/0.48 thf(fact_4496_crossproduct__eq,axiom,
% 0.23/0.48 ! [W: nat,Y3: nat,X4: nat,Z: nat] :
% 0.23/0.48 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X4 @ Z ) )
% 0.23/0.48 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X4 @ Y3 ) ) )
% 0.23/0.48 = ( ( W = X4 )
% 0.23/0.48 | ( Y3 = Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % crossproduct_eq
% 0.23/0.48 thf(fact_4497_crossproduct__eq,axiom,
% 0.23/0.48 ! [W: int,Y3: int,X4: int,Z: int] :
% 0.23/0.48 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X4 @ Z ) )
% 0.23/0.48 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X4 @ Y3 ) ) )
% 0.23/0.48 = ( ( W = X4 )
% 0.23/0.48 | ( Y3 = Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % crossproduct_eq
% 0.23/0.48 thf(fact_4498_eq__diff__eq_H,axiom,
% 0.23/0.48 ! [X4: real,Y3: real,Z: real] :
% 0.23/0.48 ( ( X4
% 0.23/0.48 = ( minus_minus_real @ Y3 @ Z ) )
% 0.23/0.48 = ( Y3
% 0.23/0.48 = ( plus_plus_real @ X4 @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % eq_diff_eq'
% 0.23/0.48 thf(fact_4499_num_Osize__gen_I2_J,axiom,
% 0.23/0.48 ! [X2: num] :
% 0.23/0.48 ( ( size_num @ ( bit0 @ X2 ) )
% 0.23/0.48 = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % num.size_gen(2)
% 0.23/0.48 thf(fact_4500_mask__numeral,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
% 0.23/0.48 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % mask_numeral
% 0.23/0.48 thf(fact_4501_mask__numeral,axiom,
% 0.23/0.48 ! [N: num] :
% 0.23/0.48 ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
% 0.23/0.48 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % mask_numeral
% 0.23/0.48 thf(fact_4502_take__bit__rec,axiom,
% 0.23/0.48 ( bit_se1745604003318907178nteger
% 0.23/0.48 = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_rec
% 0.23/0.48 thf(fact_4503_take__bit__rec,axiom,
% 0.23/0.48 ( bit_se2923211474154528505it_int
% 0.23/0.48 = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_rec
% 0.23/0.48 thf(fact_4504_take__bit__rec,axiom,
% 0.23/0.48 ( bit_se2925701944663578781it_nat
% 0.23/0.48 = ( ^ [N2: nat,A4: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_rec
% 0.23/0.48 thf(fact_4505_and__int__unfold,axiom,
% 0.23/0.48 ( bit_se725231765392027082nd_int
% 0.23/0.48 = ( ^ [K3: int,L: int] :
% 0.23/0.48 ( if_int
% 0.23/0.48 @ ( ( K3 = zero_zero_int )
% 0.23/0.48 | ( L = zero_zero_int ) )
% 0.23/0.48 @ zero_zero_int
% 0.23/0.48 @ ( if_int
% 0.23/0.48 @ ( K3
% 0.23/0.48 = ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.48 @ L
% 0.23/0.48 @ ( if_int
% 0.23/0.48 @ ( L
% 0.23/0.48 = ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.48 @ K3
% 0.23/0.48 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % and_int_unfold
% 0.23/0.48 thf(fact_4506_exp__lower__Taylor__quadratic,axiom,
% 0.23/0.48 ! [X4: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.48 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( divide_divide_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X4 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % exp_lower_Taylor_quadratic
% 0.23/0.48 thf(fact_4507_sqrt__sum__squares__half__less,axiom,
% 0.23/0.48 ! [X4: real,U: real,Y3: real] :
% 0.23/0.48 ( ( ord_less_real @ X4 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.48 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.48 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.48 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.48 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % sqrt_sum_squares_half_less
% 0.23/0.48 thf(fact_4508_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.48 ! [A: int,X4: num,N: nat] :
% 0.23/0.48 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) )
% 0.23/0.48 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_less_neg_numeral_power_cancel_iff
% 0.23/0.48 thf(fact_4509_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.48 ! [A: int,X4: num,N: nat] :
% 0.23/0.48 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) )
% 0.23/0.48 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_less_neg_numeral_power_cancel_iff
% 0.23/0.48 thf(fact_4510_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.48 ! [A: int,X4: num,N: nat] :
% 0.23/0.48 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) )
% 0.23/0.48 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_less_neg_numeral_power_cancel_iff
% 0.23/0.48 thf(fact_4511_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.48 ! [A: int,X4: num,N: nat] :
% 0.23/0.48 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) )
% 0.23/0.48 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_less_neg_numeral_power_cancel_iff
% 0.23/0.48 thf(fact_4512_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.23/0.48 ! [X4: num,N: nat,A: int] :
% 0.23/0.48 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.23/0.48 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_numeral_power_less_of_int_cancel_iff
% 0.23/0.48 thf(fact_4513_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.23/0.48 ! [X4: num,N: nat,A: int] :
% 0.23/0.48 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.23/0.48 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_numeral_power_less_of_int_cancel_iff
% 0.23/0.48 thf(fact_4514_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.23/0.48 ! [X4: num,N: nat,A: int] :
% 0.23/0.48 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.23/0.48 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_numeral_power_less_of_int_cancel_iff
% 0.23/0.48 thf(fact_4515_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.23/0.48 ! [X4: num,N: nat,A: int] :
% 0.23/0.48 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.23/0.48 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 0.23/0.48
% 0.23/0.48 % neg_numeral_power_less_of_int_cancel_iff
% 0.23/0.48 thf(fact_4516_and_Oright__idem,axiom,
% 0.23/0.48 ! [A: int,B2: int] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B2 ) @ B2 )
% 0.23/0.48 = ( bit_se725231765392027082nd_int @ A @ B2 ) ) ).
% 0.23/0.48
% 0.23/0.48 % and.right_idem
% 0.23/0.48 thf(fact_4517_and_Oright__idem,axiom,
% 0.23/0.48 ! [A: nat,B2: nat] :
% 0.23/0.48 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B2 ) @ B2 )
% 0.23/0.48 = ( bit_se727722235901077358nd_nat @ A @ B2 ) ) ).
% 0.23/0.48
% 0.23/0.48 % and.right_idem
% 0.23/0.48 thf(fact_4518_and_Oleft__idem,axiom,
% 0.23/0.48 ! [A: int,B2: int] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
% 0.23/0.48 = ( bit_se725231765392027082nd_int @ A @ B2 ) ) ).
% 0.23/0.48
% 0.23/0.48 % and.left_idem
% 0.23/0.48 thf(fact_4519_and_Oleft__idem,axiom,
% 0.23/0.48 ! [A: nat,B2: nat] :
% 0.23/0.48 ( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
% 0.23/0.48 = ( bit_se727722235901077358nd_nat @ A @ B2 ) ) ).
% 0.23/0.48
% 0.23/0.48 % and.left_idem
% 0.23/0.48 thf(fact_4520_and_Oidem,axiom,
% 0.23/0.48 ! [A: int] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ A @ A )
% 0.23/0.48 = A ) ).
% 0.23/0.48
% 0.23/0.48 % and.idem
% 0.23/0.48 thf(fact_4521_and_Oidem,axiom,
% 0.23/0.48 ! [A: nat] :
% 0.23/0.48 ( ( bit_se727722235901077358nd_nat @ A @ A )
% 0.23/0.48 = A ) ).
% 0.23/0.48
% 0.23/0.48 % and.idem
% 0.23/0.48 thf(fact_4522_real__sqrt__eq__iff,axiom,
% 0.23/0.48 ! [X4: real,Y3: real] :
% 0.23/0.48 ( ( ( sqrt @ X4 )
% 0.23/0.48 = ( sqrt @ Y3 ) )
% 0.23/0.48 = ( X4 = Y3 ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_eq_iff
% 0.23/0.48 thf(fact_4523_mask__nat__positive__iff,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.23/0.48 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % mask_nat_positive_iff
% 0.23/0.48 thf(fact_4524_take__bit__of__0,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 0.23/0.48 = zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_of_0
% 0.23/0.48 thf(fact_4525_take__bit__of__0,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 0.23/0.48 = zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_of_0
% 0.23/0.48 thf(fact_4526_bit_Oconj__zero__right,axiom,
% 0.23/0.48 ! [X4: int] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ X4 @ zero_zero_int )
% 0.23/0.48 = zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % bit.conj_zero_right
% 0.23/0.48 thf(fact_4527_bit_Oconj__zero__left,axiom,
% 0.23/0.48 ! [X4: int] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X4 )
% 0.23/0.48 = zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % bit.conj_zero_left
% 0.23/0.48 thf(fact_4528_zero__and__eq,axiom,
% 0.23/0.48 ! [A: int] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 0.23/0.48 = zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % zero_and_eq
% 0.23/0.48 thf(fact_4529_zero__and__eq,axiom,
% 0.23/0.48 ! [A: nat] :
% 0.23/0.48 ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 0.23/0.48 = zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % zero_and_eq
% 0.23/0.48 thf(fact_4530_and__zero__eq,axiom,
% 0.23/0.48 ! [A: int] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 0.23/0.48 = zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % and_zero_eq
% 0.23/0.48 thf(fact_4531_and__zero__eq,axiom,
% 0.23/0.48 ! [A: nat] :
% 0.23/0.48 ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 0.23/0.48 = zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % and_zero_eq
% 0.23/0.48 thf(fact_4532_real__sqrt__zero,axiom,
% 0.23/0.48 ( ( sqrt @ zero_zero_real )
% 0.23/0.48 = zero_zero_real ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_zero
% 0.23/0.48 thf(fact_4533_real__sqrt__eq__zero__cancel__iff,axiom,
% 0.23/0.48 ! [X4: real] :
% 0.23/0.48 ( ( ( sqrt @ X4 )
% 0.23/0.48 = zero_zero_real )
% 0.23/0.48 = ( X4 = zero_zero_real ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_eq_zero_cancel_iff
% 0.23/0.48 thf(fact_4534_real__sqrt__less__iff,axiom,
% 0.23/0.48 ! [X4: real,Y3: real] :
% 0.23/0.48 ( ( ord_less_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) )
% 0.23/0.48 = ( ord_less_real @ X4 @ Y3 ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_less_iff
% 0.23/0.48 thf(fact_4535_real__sqrt__le__iff,axiom,
% 0.23/0.48 ! [X4: real,Y3: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) )
% 0.23/0.48 = ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_le_iff
% 0.23/0.48 thf(fact_4536_real__sqrt__one,axiom,
% 0.23/0.48 ( ( sqrt @ one_one_real )
% 0.23/0.48 = one_one_real ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_one
% 0.23/0.48 thf(fact_4537_real__sqrt__eq__1__iff,axiom,
% 0.23/0.48 ! [X4: real] :
% 0.23/0.48 ( ( ( sqrt @ X4 )
% 0.23/0.48 = one_one_real )
% 0.23/0.48 = ( X4 = one_one_real ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_eq_1_iff
% 0.23/0.48 thf(fact_4538_take__bit__and,axiom,
% 0.23/0.48 ! [N: nat,A: int,B2: int] :
% 0.23/0.48 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
% 0.23/0.48 = ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B2 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_and
% 0.23/0.48 thf(fact_4539_take__bit__and,axiom,
% 0.23/0.48 ! [N: nat,A: nat,B2: nat] :
% 0.23/0.48 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
% 0.23/0.48 = ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B2 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_and
% 0.23/0.48 thf(fact_4540_concat__bit__of__zero__2,axiom,
% 0.23/0.48 ! [N: nat,K: int] :
% 0.23/0.48 ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 0.23/0.48 = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.23/0.48
% 0.23/0.48 % concat_bit_of_zero_2
% 0.23/0.48 thf(fact_4541_take__bit__0,axiom,
% 0.23/0.48 ! [A: int] :
% 0.23/0.48 ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 0.23/0.48 = zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_0
% 0.23/0.48 thf(fact_4542_take__bit__0,axiom,
% 0.23/0.48 ! [A: nat] :
% 0.23/0.48 ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 0.23/0.48 = zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_0
% 0.23/0.48 thf(fact_4543_take__bit__Suc__1,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
% 0.23/0.48 = one_one_int ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_Suc_1
% 0.23/0.48 thf(fact_4544_take__bit__Suc__1,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
% 0.23/0.48 = one_one_nat ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_Suc_1
% 0.23/0.48 thf(fact_4545_and_Oleft__neutral,axiom,
% 0.23/0.48 ! [A: code_integer] :
% 0.23/0.48 ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
% 0.23/0.48 = A ) ).
% 0.23/0.48
% 0.23/0.48 % and.left_neutral
% 0.23/0.48 thf(fact_4546_and_Oleft__neutral,axiom,
% 0.23/0.48 ! [A: int] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
% 0.23/0.48 = A ) ).
% 0.23/0.48
% 0.23/0.48 % and.left_neutral
% 0.23/0.48 thf(fact_4547_and_Oright__neutral,axiom,
% 0.23/0.48 ! [A: code_integer] :
% 0.23/0.48 ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.48 = A ) ).
% 0.23/0.48
% 0.23/0.48 % and.right_neutral
% 0.23/0.48 thf(fact_4548_and_Oright__neutral,axiom,
% 0.23/0.48 ! [A: int] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.48 = A ) ).
% 0.23/0.48
% 0.23/0.48 % and.right_neutral
% 0.23/0.48 thf(fact_4549_bit_Oconj__one__right,axiom,
% 0.23/0.48 ! [X4: code_integer] :
% 0.23/0.48 ( ( bit_se3949692690581998587nteger @ X4 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.48 = X4 ) ).
% 0.23/0.48
% 0.23/0.48 % bit.conj_one_right
% 0.23/0.48 thf(fact_4550_bit_Oconj__one__right,axiom,
% 0.23/0.48 ! [X4: int] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ X4 @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.48 = X4 ) ).
% 0.23/0.48
% 0.23/0.48 % bit.conj_one_right
% 0.23/0.48 thf(fact_4551_of__int__eq__numeral__iff,axiom,
% 0.23/0.48 ! [Z: int,N: num] :
% 0.23/0.48 ( ( ( ring_17405671764205052669omplex @ Z )
% 0.23/0.48 = ( numera6690914467698888265omplex @ N ) )
% 0.23/0.48 = ( Z
% 0.23/0.48 = ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_numeral_iff
% 0.23/0.48 thf(fact_4552_of__int__eq__numeral__iff,axiom,
% 0.23/0.48 ! [Z: int,N: num] :
% 0.23/0.48 ( ( ( ring_1_of_int_real @ Z )
% 0.23/0.48 = ( numeral_numeral_real @ N ) )
% 0.23/0.48 = ( Z
% 0.23/0.48 = ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_numeral_iff
% 0.23/0.48 thf(fact_4553_of__int__eq__numeral__iff,axiom,
% 0.23/0.48 ! [Z: int,N: num] :
% 0.23/0.48 ( ( ( ring_1_of_int_rat @ Z )
% 0.23/0.48 = ( numeral_numeral_rat @ N ) )
% 0.23/0.48 = ( Z
% 0.23/0.48 = ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_numeral_iff
% 0.23/0.48 thf(fact_4554_of__int__eq__numeral__iff,axiom,
% 0.23/0.48 ! [Z: int,N: num] :
% 0.23/0.48 ( ( ( ring_1_of_int_int @ Z )
% 0.23/0.48 = ( numeral_numeral_int @ N ) )
% 0.23/0.48 = ( Z
% 0.23/0.48 = ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_numeral_iff
% 0.23/0.48 thf(fact_4555_of__int__numeral,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 0.23/0.48 = ( numera6690914467698888265omplex @ K ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_numeral
% 0.23/0.48 thf(fact_4556_of__int__numeral,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 0.23/0.48 = ( numeral_numeral_real @ K ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_numeral
% 0.23/0.48 thf(fact_4557_of__int__numeral,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 0.23/0.48 = ( numeral_numeral_rat @ K ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_numeral
% 0.23/0.48 thf(fact_4558_of__int__numeral,axiom,
% 0.23/0.48 ! [K: num] :
% 0.23/0.48 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 0.23/0.48 = ( numeral_numeral_int @ K ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_numeral
% 0.23/0.48 thf(fact_4559_of__int__le__iff,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.23/0.48 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_le_iff
% 0.23/0.48 thf(fact_4560_of__int__le__iff,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.23/0.48 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_le_iff
% 0.23/0.48 thf(fact_4561_of__int__le__iff,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.23/0.48 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_le_iff
% 0.23/0.48 thf(fact_4562_take__bit__numeral__1,axiom,
% 0.23/0.48 ! [L2: num] :
% 0.23/0.48 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ one_one_int )
% 0.23/0.48 = one_one_int ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_numeral_1
% 0.23/0.48 thf(fact_4563_take__bit__numeral__1,axiom,
% 0.23/0.48 ! [L2: num] :
% 0.23/0.48 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ one_one_nat )
% 0.23/0.48 = one_one_nat ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_numeral_1
% 0.23/0.48 thf(fact_4564_of__int__less__iff,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.23/0.48 = ( ord_less_int @ W @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_less_iff
% 0.23/0.48 thf(fact_4565_of__int__less__iff,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.23/0.48 = ( ord_less_int @ W @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_less_iff
% 0.23/0.48 thf(fact_4566_of__int__less__iff,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.23/0.48 = ( ord_less_int @ W @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_less_iff
% 0.23/0.48 thf(fact_4567_of__int__1,axiom,
% 0.23/0.48 ( ( ring_17405671764205052669omplex @ one_one_int )
% 0.23/0.48 = one_one_complex ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_1
% 0.23/0.48 thf(fact_4568_of__int__1,axiom,
% 0.23/0.48 ( ( ring_1_of_int_int @ one_one_int )
% 0.23/0.48 = one_one_int ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_1
% 0.23/0.48 thf(fact_4569_of__int__1,axiom,
% 0.23/0.48 ( ( ring_1_of_int_real @ one_one_int )
% 0.23/0.48 = one_one_real ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_1
% 0.23/0.48 thf(fact_4570_of__int__1,axiom,
% 0.23/0.48 ( ( ring_1_of_int_rat @ one_one_int )
% 0.23/0.48 = one_one_rat ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_1
% 0.23/0.48 thf(fact_4571_of__int__eq__1__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ( ring_17405671764205052669omplex @ Z )
% 0.23/0.48 = one_one_complex )
% 0.23/0.48 = ( Z = one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_1_iff
% 0.23/0.48 thf(fact_4572_of__int__eq__1__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ( ring_1_of_int_int @ Z )
% 0.23/0.48 = one_one_int )
% 0.23/0.48 = ( Z = one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_1_iff
% 0.23/0.48 thf(fact_4573_of__int__eq__1__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ( ring_1_of_int_real @ Z )
% 0.23/0.48 = one_one_real )
% 0.23/0.48 = ( Z = one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_1_iff
% 0.23/0.48 thf(fact_4574_of__int__eq__1__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ( ring_1_of_int_rat @ Z )
% 0.23/0.48 = one_one_rat )
% 0.23/0.48 = ( Z = one_one_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_1_iff
% 0.23/0.48 thf(fact_4575_of__int__mult,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 0.23/0.48 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_mult
% 0.23/0.48 thf(fact_4576_of__int__mult,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 0.23/0.48 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_mult
% 0.23/0.48 thf(fact_4577_of__int__mult,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 0.23/0.48 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_mult
% 0.23/0.48 thf(fact_4578_of__int__add,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 0.23/0.48 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_add
% 0.23/0.48 thf(fact_4579_of__int__add,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 0.23/0.48 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_add
% 0.23/0.48 thf(fact_4580_of__int__add,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 0.23/0.48 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_add
% 0.23/0.48 thf(fact_4581_real__sqrt__lt__0__iff,axiom,
% 0.23/0.48 ! [X4: real] :
% 0.23/0.48 ( ( ord_less_real @ ( sqrt @ X4 ) @ zero_zero_real )
% 0.23/0.48 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_lt_0_iff
% 0.23/0.48 thf(fact_4582_real__sqrt__gt__0__iff,axiom,
% 0.23/0.48 ! [Y3: real] :
% 0.23/0.48 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 0.23/0.48 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_gt_0_iff
% 0.23/0.48 thf(fact_4583_real__sqrt__ge__0__iff,axiom,
% 0.23/0.48 ! [Y3: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 0.23/0.48 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_ge_0_iff
% 0.23/0.48 thf(fact_4584_real__sqrt__le__0__iff,axiom,
% 0.23/0.48 ! [X4: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ zero_zero_real )
% 0.23/0.48 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_le_0_iff
% 0.23/0.48 thf(fact_4585_mask__eq__0__iff,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( ( bit_se2002935070580805687sk_nat @ N )
% 0.23/0.48 = zero_zero_nat )
% 0.23/0.48 = ( N = zero_zero_nat ) ) ).
% 0.23/0.48
% 0.23/0.48 % mask_eq_0_iff
% 0.23/0.48 thf(fact_4586_mask__eq__0__iff,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( ( bit_se2000444600071755411sk_int @ N )
% 0.23/0.48 = zero_zero_int )
% 0.23/0.48 = ( N = zero_zero_nat ) ) ).
% 0.23/0.48
% 0.23/0.48 % mask_eq_0_iff
% 0.23/0.48 thf(fact_4587_mask__0,axiom,
% 0.23/0.48 ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
% 0.23/0.48 = zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % mask_0
% 0.23/0.48 thf(fact_4588_mask__0,axiom,
% 0.23/0.48 ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
% 0.23/0.48 = zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % mask_0
% 0.23/0.48 thf(fact_4589_of__int__diff,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 0.23/0.48 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_diff
% 0.23/0.48 thf(fact_4590_of__int__diff,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 0.23/0.48 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_diff
% 0.23/0.48 thf(fact_4591_of__int__diff,axiom,
% 0.23/0.48 ! [W: int,Z: int] :
% 0.23/0.48 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 0.23/0.48 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_diff
% 0.23/0.48 thf(fact_4592_real__sqrt__lt__1__iff,axiom,
% 0.23/0.48 ! [X4: real] :
% 0.23/0.48 ( ( ord_less_real @ ( sqrt @ X4 ) @ one_one_real )
% 0.23/0.48 = ( ord_less_real @ X4 @ one_one_real ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_lt_1_iff
% 0.23/0.48 thf(fact_4593_real__sqrt__gt__1__iff,axiom,
% 0.23/0.48 ! [Y3: real] :
% 0.23/0.48 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
% 0.23/0.48 = ( ord_less_real @ one_one_real @ Y3 ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_gt_1_iff
% 0.23/0.48 thf(fact_4594_real__sqrt__le__1__iff,axiom,
% 0.23/0.48 ! [X4: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ one_one_real )
% 0.23/0.48 = ( ord_less_eq_real @ X4 @ one_one_real ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_le_1_iff
% 0.23/0.48 thf(fact_4595_real__sqrt__ge__1__iff,axiom,
% 0.23/0.48 ! [Y3: real] :
% 0.23/0.48 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
% 0.23/0.48 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_ge_1_iff
% 0.23/0.48 thf(fact_4596_of__int__power,axiom,
% 0.23/0.48 ! [Z: int,N: nat] :
% 0.23/0.48 ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
% 0.23/0.48 = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_power
% 0.23/0.48 thf(fact_4597_of__int__power,axiom,
% 0.23/0.48 ! [Z: int,N: nat] :
% 0.23/0.48 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 0.23/0.48 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_power
% 0.23/0.48 thf(fact_4598_of__int__power,axiom,
% 0.23/0.48 ! [Z: int,N: nat] :
% 0.23/0.48 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 0.23/0.48 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_power
% 0.23/0.48 thf(fact_4599_of__int__power,axiom,
% 0.23/0.48 ! [Z: int,N: nat] :
% 0.23/0.48 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
% 0.23/0.48 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_power
% 0.23/0.48 thf(fact_4600_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.23/0.48 ! [B2: int,W: nat,X4: int] :
% 0.23/0.48 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W )
% 0.23/0.48 = ( ring_1_of_int_rat @ X4 ) )
% 0.23/0.48 = ( ( power_power_int @ B2 @ W )
% 0.23/0.48 = X4 ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_of_int_power_cancel_iff
% 0.23/0.48 thf(fact_4601_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.23/0.48 ! [B2: int,W: nat,X4: int] :
% 0.23/0.48 ( ( ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W )
% 0.23/0.48 = ( ring_1_of_int_real @ X4 ) )
% 0.23/0.48 = ( ( power_power_int @ B2 @ W )
% 0.23/0.48 = X4 ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_of_int_power_cancel_iff
% 0.23/0.48 thf(fact_4602_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.23/0.48 ! [B2: int,W: nat,X4: int] :
% 0.23/0.48 ( ( ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W )
% 0.23/0.48 = ( ring_1_of_int_int @ X4 ) )
% 0.23/0.48 = ( ( power_power_int @ B2 @ W )
% 0.23/0.48 = X4 ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_of_int_power_cancel_iff
% 0.23/0.48 thf(fact_4603_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.23/0.48 ! [B2: int,W: nat,X4: int] :
% 0.23/0.48 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B2 ) @ W )
% 0.23/0.48 = ( ring_17405671764205052669omplex @ X4 ) )
% 0.23/0.48 = ( ( power_power_int @ B2 @ W )
% 0.23/0.48 = X4 ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_eq_of_int_power_cancel_iff
% 0.23/0.48 thf(fact_4604_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.48 ! [X4: int,B2: int,W: nat] :
% 0.23/0.48 ( ( ( ring_1_of_int_rat @ X4 )
% 0.23/0.48 = ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
% 0.23/0.48 = ( X4
% 0.23/0.48 = ( power_power_int @ B2 @ W ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_power_eq_of_int_cancel_iff
% 0.23/0.48 thf(fact_4605_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.48 ! [X4: int,B2: int,W: nat] :
% 0.23/0.48 ( ( ( ring_1_of_int_real @ X4 )
% 0.23/0.48 = ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
% 0.23/0.48 = ( X4
% 0.23/0.48 = ( power_power_int @ B2 @ W ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_power_eq_of_int_cancel_iff
% 0.23/0.48 thf(fact_4606_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.48 ! [X4: int,B2: int,W: nat] :
% 0.23/0.48 ( ( ( ring_1_of_int_int @ X4 )
% 0.23/0.48 = ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
% 0.23/0.48 = ( X4
% 0.23/0.48 = ( power_power_int @ B2 @ W ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_power_eq_of_int_cancel_iff
% 0.23/0.48 thf(fact_4607_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.48 ! [X4: int,B2: int,W: nat] :
% 0.23/0.48 ( ( ( ring_17405671764205052669omplex @ X4 )
% 0.23/0.48 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B2 ) @ W ) )
% 0.23/0.48 = ( X4
% 0.23/0.48 = ( power_power_int @ B2 @ W ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_power_eq_of_int_cancel_iff
% 0.23/0.48 thf(fact_4608_and__nonnegative__int__iff,axiom,
% 0.23/0.48 ! [K: int,L2: int] :
% 0.23/0.48 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 0.23/0.48 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.48 | ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % and_nonnegative_int_iff
% 0.23/0.48 thf(fact_4609_and__negative__int__iff,axiom,
% 0.23/0.48 ! [K: int,L2: int] :
% 0.23/0.48 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ zero_zero_int )
% 0.23/0.48 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.23/0.48 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % and_negative_int_iff
% 0.23/0.48 thf(fact_4610_take__bit__of__1__eq__0__iff,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.23/0.48 = zero_zero_int )
% 0.23/0.48 = ( N = zero_zero_nat ) ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_of_1_eq_0_iff
% 0.23/0.48 thf(fact_4611_take__bit__of__1__eq__0__iff,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.23/0.48 = zero_zero_nat )
% 0.23/0.48 = ( N = zero_zero_nat ) ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_of_1_eq_0_iff
% 0.23/0.48 thf(fact_4612_and__numerals_I2_J,axiom,
% 0.23/0.48 ! [Y3: num] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 0.23/0.48 = one_one_int ) ).
% 0.23/0.48
% 0.23/0.48 % and_numerals(2)
% 0.23/0.48 thf(fact_4613_and__numerals_I2_J,axiom,
% 0.23/0.48 ! [Y3: num] :
% 0.23/0.48 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.23/0.48 = one_one_nat ) ).
% 0.23/0.48
% 0.23/0.48 % and_numerals(2)
% 0.23/0.48 thf(fact_4614_and__numerals_I8_J,axiom,
% 0.23/0.48 ! [X4: num] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ one_one_int )
% 0.23/0.48 = one_one_int ) ).
% 0.23/0.48
% 0.23/0.48 % and_numerals(8)
% 0.23/0.48 thf(fact_4615_and__numerals_I8_J,axiom,
% 0.23/0.48 ! [X4: num] :
% 0.23/0.48 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ one_one_nat )
% 0.23/0.48 = one_one_nat ) ).
% 0.23/0.48
% 0.23/0.48 % and_numerals(8)
% 0.23/0.48 thf(fact_4616_real__sqrt__four,axiom,
% 0.23/0.48 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.23/0.48 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % real_sqrt_four
% 0.23/0.48 thf(fact_4617_mask__Suc__0,axiom,
% 0.23/0.48 ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
% 0.23/0.48 = one_one_nat ) ).
% 0.23/0.48
% 0.23/0.48 % mask_Suc_0
% 0.23/0.48 thf(fact_4618_mask__Suc__0,axiom,
% 0.23/0.48 ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
% 0.23/0.48 = one_one_int ) ).
% 0.23/0.48
% 0.23/0.48 % mask_Suc_0
% 0.23/0.48 thf(fact_4619_take__bit__minus__one__eq__mask,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.48 = ( bit_se2119862282449309892nteger @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_minus_one_eq_mask
% 0.23/0.48 thf(fact_4620_take__bit__minus__one__eq__mask,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.48 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_minus_one_eq_mask
% 0.23/0.48 thf(fact_4621_take__bit__of__Suc__0,axiom,
% 0.23/0.48 ! [N: nat] :
% 0.23/0.48 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.23/0.48 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % take_bit_of_Suc_0
% 0.23/0.48 thf(fact_4622_and__numerals_I1_J,axiom,
% 0.23/0.48 ! [Y3: num] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 0.23/0.48 = zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % and_numerals(1)
% 0.23/0.48 thf(fact_4623_and__numerals_I1_J,axiom,
% 0.23/0.48 ! [Y3: num] :
% 0.23/0.48 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.23/0.48 = zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % and_numerals(1)
% 0.23/0.48 thf(fact_4624_and__numerals_I5_J,axiom,
% 0.23/0.48 ! [X4: num] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ one_one_int )
% 0.23/0.48 = zero_zero_int ) ).
% 0.23/0.48
% 0.23/0.48 % and_numerals(5)
% 0.23/0.48 thf(fact_4625_and__numerals_I5_J,axiom,
% 0.23/0.48 ! [X4: num] :
% 0.23/0.48 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ one_one_nat )
% 0.23/0.48 = zero_zero_nat ) ).
% 0.23/0.48
% 0.23/0.48 % and_numerals(5)
% 0.23/0.48 thf(fact_4626_and__numerals_I3_J,axiom,
% 0.23/0.48 ! [X4: num,Y3: num] :
% 0.23/0.48 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 0.23/0.48 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % and_numerals(3)
% 0.23/0.48 thf(fact_4627_and__numerals_I3_J,axiom,
% 0.23/0.48 ! [X4: num,Y3: num] :
% 0.23/0.48 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.23/0.48 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % and_numerals(3)
% 0.23/0.48 thf(fact_4628_of__int__le__0__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.23/0.48 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_le_0_iff
% 0.23/0.48 thf(fact_4629_of__int__le__0__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.23/0.48 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_le_0_iff
% 0.23/0.48 thf(fact_4630_of__int__le__0__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.23/0.48 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_le_0_iff
% 0.23/0.48 thf(fact_4631_of__int__0__le__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.23/0.48 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_0_le_iff
% 0.23/0.48 thf(fact_4632_of__int__0__le__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.23/0.48 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_0_le_iff
% 0.23/0.48 thf(fact_4633_of__int__0__le__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.23/0.48 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_0_le_iff
% 0.23/0.48 thf(fact_4634_of__int__0__less__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.23/0.48 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_0_less_iff
% 0.23/0.48 thf(fact_4635_of__int__0__less__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.23/0.48 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_0_less_iff
% 0.23/0.48 thf(fact_4636_of__int__0__less__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.23/0.48 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_0_less_iff
% 0.23/0.48 thf(fact_4637_of__int__less__0__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.23/0.48 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_less_0_iff
% 0.23/0.48 thf(fact_4638_of__int__less__0__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.23/0.48 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_less_0_iff
% 0.23/0.48 thf(fact_4639_of__int__less__0__iff,axiom,
% 0.23/0.48 ! [Z: int] :
% 0.23/0.48 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.23/0.48 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_less_0_iff
% 0.23/0.48 thf(fact_4640_of__int__le__numeral__iff,axiom,
% 0.23/0.48 ! [Z: int,N: num] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.23/0.48 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_le_numeral_iff
% 0.23/0.48 thf(fact_4641_of__int__le__numeral__iff,axiom,
% 0.23/0.48 ! [Z: int,N: num] :
% 0.23/0.48 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.23/0.48 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_le_numeral_iff
% 0.23/0.48 thf(fact_4642_of__int__le__numeral__iff,axiom,
% 0.23/0.48 ! [Z: int,N: num] :
% 0.23/0.48 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.48 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_le_numeral_iff
% 0.23/0.48 thf(fact_4643_of__int__numeral__le__iff,axiom,
% 0.23/0.48 ! [N: num,Z: int] :
% 0.23/0.48 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.23/0.48 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.23/0.48
% 0.23/0.48 % of_int_numeral_le_iff
% 0.23/0.48 thf(fact_4644_of__int__numeral__le__iff,axiom,
% 0.23/0.48 ! [N: num,Z: int] :
% 0.23/0.48 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.23/0.48 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_numeral_le_iff
% 0.23/0.49 thf(fact_4645_of__int__numeral__le__iff,axiom,
% 0.23/0.49 ! [N: num,Z: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.23/0.49 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_numeral_le_iff
% 0.23/0.49 thf(fact_4646_of__int__less__numeral__iff,axiom,
% 0.23/0.49 ! [Z: int,N: num] :
% 0.23/0.49 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.23/0.49 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_numeral_iff
% 0.23/0.49 thf(fact_4647_of__int__less__numeral__iff,axiom,
% 0.23/0.49 ! [Z: int,N: num] :
% 0.23/0.49 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.23/0.49 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_numeral_iff
% 0.23/0.49 thf(fact_4648_of__int__less__numeral__iff,axiom,
% 0.23/0.49 ! [Z: int,N: num] :
% 0.23/0.49 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.49 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_numeral_iff
% 0.23/0.49 thf(fact_4649_of__int__numeral__less__iff,axiom,
% 0.23/0.49 ! [N: num,Z: int] :
% 0.23/0.49 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.23/0.49 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_numeral_less_iff
% 0.23/0.49 thf(fact_4650_of__int__numeral__less__iff,axiom,
% 0.23/0.49 ! [N: num,Z: int] :
% 0.23/0.49 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.23/0.49 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_numeral_less_iff
% 0.23/0.49 thf(fact_4651_of__int__numeral__less__iff,axiom,
% 0.23/0.49 ! [N: num,Z: int] :
% 0.23/0.49 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.23/0.49 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_numeral_less_iff
% 0.23/0.49 thf(fact_4652_of__int__le__1__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.23/0.49 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_1_iff
% 0.23/0.49 thf(fact_4653_of__int__le__1__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.23/0.49 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_1_iff
% 0.23/0.49 thf(fact_4654_of__int__le__1__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.23/0.49 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_1_iff
% 0.23/0.49 thf(fact_4655_of__int__1__le__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.23/0.49 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_1_le_iff
% 0.23/0.49 thf(fact_4656_of__int__1__le__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.23/0.49 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_1_le_iff
% 0.23/0.49 thf(fact_4657_of__int__1__le__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.23/0.49 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_1_le_iff
% 0.23/0.49 thf(fact_4658_of__int__less__1__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.23/0.49 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_1_iff
% 0.23/0.49 thf(fact_4659_of__int__less__1__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.23/0.49 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_1_iff
% 0.23/0.49 thf(fact_4660_of__int__less__1__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.23/0.49 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_1_iff
% 0.23/0.49 thf(fact_4661_of__int__1__less__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.23/0.49 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_1_less_iff
% 0.23/0.49 thf(fact_4662_of__int__1__less__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.23/0.49 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_1_less_iff
% 0.23/0.49 thf(fact_4663_of__int__1__less__iff,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.23/0.49 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_1_less_iff
% 0.23/0.49 thf(fact_4664_take__bit__of__1,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
% 0.23/0.49 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_of_1
% 0.23/0.49 thf(fact_4665_take__bit__of__1,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.23/0.49 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_of_1
% 0.23/0.49 thf(fact_4666_take__bit__of__1,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.23/0.49 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_of_1
% 0.23/0.49 thf(fact_4667_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,Y3: int] :
% 0.23/0.49 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N )
% 0.23/0.49 = ( ring_17405671764205052669omplex @ Y3 ) )
% 0.23/0.49 = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 0.23/0.49 = Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_power_eq_of_int_cancel_iff
% 0.23/0.49 thf(fact_4668_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,Y3: int] :
% 0.23/0.49 ( ( ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N )
% 0.23/0.49 = ( ring_1_of_int_real @ Y3 ) )
% 0.23/0.49 = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 0.23/0.49 = Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_power_eq_of_int_cancel_iff
% 0.23/0.49 thf(fact_4669_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,Y3: int] :
% 0.23/0.49 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N )
% 0.23/0.49 = ( ring_1_of_int_rat @ Y3 ) )
% 0.23/0.49 = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 0.23/0.49 = Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_power_eq_of_int_cancel_iff
% 0.23/0.49 thf(fact_4670_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,Y3: int] :
% 0.23/0.49 ( ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 0.23/0.49 = ( ring_1_of_int_int @ Y3 ) )
% 0.23/0.49 = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 0.23/0.49 = Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_power_eq_of_int_cancel_iff
% 0.23/0.49 thf(fact_4671_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [Y3: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 0.23/0.49 = ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N ) )
% 0.23/0.49 = ( Y3
% 0.23/0.49 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_eq_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4672_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [Y3: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ( ring_1_of_int_real @ Y3 )
% 0.23/0.49 = ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 0.23/0.49 = ( Y3
% 0.23/0.49 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_eq_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4673_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [Y3: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ( ring_1_of_int_rat @ Y3 )
% 0.23/0.49 = ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 0.23/0.49 = ( Y3
% 0.23/0.49 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_eq_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4674_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [Y3: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ( ring_1_of_int_int @ Y3 )
% 0.23/0.49 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
% 0.23/0.49 = ( Y3
% 0.23/0.49 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_eq_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4675_of__int__le__of__int__power__cancel__iff,axiom,
% 0.23/0.49 ! [B2: int,W: nat,X4: int] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) @ ( ring_1_of_int_real @ X4 ) )
% 0.23/0.49 = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_of_int_power_cancel_iff
% 0.23/0.49 thf(fact_4676_of__int__le__of__int__power__cancel__iff,axiom,
% 0.23/0.49 ! [B2: int,W: nat,X4: int] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) @ ( ring_1_of_int_rat @ X4 ) )
% 0.23/0.49 = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_of_int_power_cancel_iff
% 0.23/0.49 thf(fact_4677_of__int__le__of__int__power__cancel__iff,axiom,
% 0.23/0.49 ! [B2: int,W: nat,X4: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) @ ( ring_1_of_int_int @ X4 ) )
% 0.23/0.49 = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_of_int_power_cancel_iff
% 0.23/0.49 thf(fact_4678_of__int__power__le__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: int,B2: int,W: nat] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
% 0.23/0.49 = ( ord_less_eq_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_power_le_of_int_cancel_iff
% 0.23/0.49 thf(fact_4679_of__int__power__le__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: int,B2: int,W: nat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
% 0.23/0.49 = ( ord_less_eq_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_power_le_of_int_cancel_iff
% 0.23/0.49 thf(fact_4680_of__int__power__le__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: int,B2: int,W: nat] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X4 ) @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
% 0.23/0.49 = ( ord_less_eq_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_power_le_of_int_cancel_iff
% 0.23/0.49 thf(fact_4681_of__int__less__of__int__power__cancel__iff,axiom,
% 0.23/0.49 ! [B2: int,W: nat,X4: int] :
% 0.23/0.49 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) @ ( ring_1_of_int_real @ X4 ) )
% 0.23/0.49 = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_of_int_power_cancel_iff
% 0.23/0.49 thf(fact_4682_of__int__less__of__int__power__cancel__iff,axiom,
% 0.23/0.49 ! [B2: int,W: nat,X4: int] :
% 0.23/0.49 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) @ ( ring_1_of_int_rat @ X4 ) )
% 0.23/0.49 = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_of_int_power_cancel_iff
% 0.23/0.49 thf(fact_4683_of__int__less__of__int__power__cancel__iff,axiom,
% 0.23/0.49 ! [B2: int,W: nat,X4: int] :
% 0.23/0.49 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) @ ( ring_1_of_int_int @ X4 ) )
% 0.23/0.49 = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_of_int_power_cancel_iff
% 0.23/0.49 thf(fact_4684_of__int__power__less__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: int,B2: int,W: nat] :
% 0.23/0.49 ( ( ord_less_real @ ( ring_1_of_int_real @ X4 ) @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
% 0.23/0.49 = ( ord_less_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_power_less_of_int_cancel_iff
% 0.23/0.49 thf(fact_4685_of__int__power__less__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: int,B2: int,W: nat] :
% 0.23/0.49 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X4 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
% 0.23/0.49 = ( ord_less_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_power_less_of_int_cancel_iff
% 0.23/0.49 thf(fact_4686_of__int__power__less__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: int,B2: int,W: nat] :
% 0.23/0.49 ( ( ord_less_int @ ( ring_1_of_int_int @ X4 ) @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
% 0.23/0.49 = ( ord_less_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_power_less_of_int_cancel_iff
% 0.23/0.49 thf(fact_4687_and__minus__numerals_I6_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.23/0.49 = one_one_int ) ).
% 0.23/0.49
% 0.23/0.49 % and_minus_numerals(6)
% 0.23/0.49 thf(fact_4688_and__minus__numerals_I2_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.23/0.49 = one_one_int ) ).
% 0.23/0.49
% 0.23/0.49 % and_minus_numerals(2)
% 0.23/0.49 thf(fact_4689_and__numerals_I4_J,axiom,
% 0.23/0.49 ! [X4: num,Y3: num] :
% 0.23/0.49 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 0.23/0.49 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_numerals(4)
% 0.23/0.49 thf(fact_4690_and__numerals_I4_J,axiom,
% 0.23/0.49 ! [X4: num,Y3: num] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.23/0.49 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_numerals(4)
% 0.23/0.49 thf(fact_4691_and__numerals_I6_J,axiom,
% 0.23/0.49 ! [X4: num,Y3: num] :
% 0.23/0.49 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 0.23/0.49 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_numerals(6)
% 0.23/0.49 thf(fact_4692_and__numerals_I6_J,axiom,
% 0.23/0.49 ! [X4: num,Y3: num] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.23/0.49 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_numerals(6)
% 0.23/0.49 thf(fact_4693_even__take__bit__eq,axiom,
% 0.23/0.49 ! [N: nat,A: code_integer] :
% 0.23/0.49 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 0.23/0.49 = ( ( N = zero_zero_nat )
% 0.23/0.49 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % even_take_bit_eq
% 0.23/0.49 thf(fact_4694_even__take__bit__eq,axiom,
% 0.23/0.49 ! [N: nat,A: int] :
% 0.23/0.49 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.23/0.49 = ( ( N = zero_zero_nat )
% 0.23/0.49 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % even_take_bit_eq
% 0.23/0.49 thf(fact_4695_even__take__bit__eq,axiom,
% 0.23/0.49 ! [N: nat,A: nat] :
% 0.23/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.23/0.49 = ( ( N = zero_zero_nat )
% 0.23/0.49 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % even_take_bit_eq
% 0.23/0.49 thf(fact_4696_and__minus__numerals_I5_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.23/0.49 = zero_zero_int ) ).
% 0.23/0.49
% 0.23/0.49 % and_minus_numerals(5)
% 0.23/0.49 thf(fact_4697_and__minus__numerals_I1_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.23/0.49 = zero_zero_int ) ).
% 0.23/0.49
% 0.23/0.49 % and_minus_numerals(1)
% 0.23/0.49 thf(fact_4698_take__bit__Suc__0,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 0.23/0.49 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc_0
% 0.23/0.49 thf(fact_4699_take__bit__Suc__0,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 0.23/0.49 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc_0
% 0.23/0.49 thf(fact_4700_take__bit__Suc__0,axiom,
% 0.23/0.49 ! [A: nat] :
% 0.23/0.49 ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 0.23/0.49 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc_0
% 0.23/0.49 thf(fact_4701_real__sqrt__pow2__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ( power_power_real @ ( sqrt @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.49 = X4 )
% 0.23/0.49 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_pow2_iff
% 0.23/0.49 thf(fact_4702_real__sqrt__pow2,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( power_power_real @ ( sqrt @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.49 = X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_pow2
% 0.23/0.49 thf(fact_4703_numeral__power__le__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,A: int] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.23/0.49 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_power_le_of_int_cancel_iff
% 0.23/0.49 thf(fact_4704_numeral__power__le__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,A: int] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.23/0.49 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_power_le_of_int_cancel_iff
% 0.23/0.49 thf(fact_4705_numeral__power__le__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,A: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.23/0.49 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_power_le_of_int_cancel_iff
% 0.23/0.49 thf(fact_4706_of__int__le__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [A: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 0.23/0.49 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4707_of__int__le__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [A: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 0.23/0.49 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4708_of__int__le__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [A: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
% 0.23/0.49 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4709_numeral__power__less__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,A: int] :
% 0.23/0.49 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.23/0.49 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_power_less_of_int_cancel_iff
% 0.23/0.49 thf(fact_4710_numeral__power__less__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,A: int] :
% 0.23/0.49 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.23/0.49 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_power_less_of_int_cancel_iff
% 0.23/0.49 thf(fact_4711_numeral__power__less__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,A: int] :
% 0.23/0.49 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.23/0.49 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_power_less_of_int_cancel_iff
% 0.23/0.49 thf(fact_4712_of__int__less__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [A: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 0.23/0.49 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4713_of__int__less__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [A: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 0.23/0.49 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4714_of__int__less__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [A: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
% 0.23/0.49 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_less_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4715_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 0.23/0.49 ! [X4: real,Y3: real,Xa: real,Ya: real] :
% 0.23/0.49 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( 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 ) ) )
% 0.23/0.49 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( 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 ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_sum_squares_mult_squared_eq
% 0.23/0.49 thf(fact_4716_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [Y3: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ( ring_1_of_int_real @ Y3 )
% 0.23/0.49 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) )
% 0.23/0.49 = ( Y3
% 0.23/0.49 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_eq_neg_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4717_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [Y3: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ( ring_1_of_int_int @ Y3 )
% 0.23/0.49 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) )
% 0.23/0.49 = ( Y3
% 0.23/0.49 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_eq_neg_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4718_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [Y3: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 0.23/0.49 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X4 ) ) @ N ) )
% 0.23/0.49 = ( Y3
% 0.23/0.49 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_eq_neg_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4719_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [Y3: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ( ring_18347121197199848620nteger @ Y3 )
% 0.23/0.49 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) )
% 0.23/0.49 = ( Y3
% 0.23/0.49 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_eq_neg_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4720_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [Y3: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ( ring_1_of_int_rat @ Y3 )
% 0.23/0.49 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) )
% 0.23/0.49 = ( Y3
% 0.23/0.49 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_eq_neg_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4721_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,Y3: int] :
% 0.23/0.49 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N )
% 0.23/0.49 = ( ring_1_of_int_real @ Y3 ) )
% 0.23/0.49 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 0.23/0.49 = Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % neg_numeral_power_eq_of_int_cancel_iff
% 0.23/0.49 thf(fact_4722_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,Y3: int] :
% 0.23/0.49 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 0.23/0.49 = ( ring_1_of_int_int @ Y3 ) )
% 0.23/0.49 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 0.23/0.49 = Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % neg_numeral_power_eq_of_int_cancel_iff
% 0.23/0.49 thf(fact_4723_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,Y3: int] :
% 0.23/0.49 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X4 ) ) @ N )
% 0.23/0.49 = ( ring_17405671764205052669omplex @ Y3 ) )
% 0.23/0.49 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 0.23/0.49 = Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % neg_numeral_power_eq_of_int_cancel_iff
% 0.23/0.49 thf(fact_4724_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,Y3: int] :
% 0.23/0.49 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N )
% 0.23/0.49 = ( ring_18347121197199848620nteger @ Y3 ) )
% 0.23/0.49 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 0.23/0.49 = Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % neg_numeral_power_eq_of_int_cancel_iff
% 0.23/0.49 thf(fact_4725_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,Y3: int] :
% 0.23/0.49 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N )
% 0.23/0.49 = ( ring_1_of_int_rat @ Y3 ) )
% 0.23/0.49 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 0.23/0.49 = Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % neg_numeral_power_eq_of_int_cancel_iff
% 0.23/0.49 thf(fact_4726_and__numerals_I7_J,axiom,
% 0.23/0.49 ! [X4: num,Y3: num] :
% 0.23/0.49 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 0.23/0.49 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_numerals(7)
% 0.23/0.49 thf(fact_4727_and__numerals_I7_J,axiom,
% 0.23/0.49 ! [X4: num,Y3: num] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.23/0.49 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_numerals(7)
% 0.23/0.49 thf(fact_4728_take__bit__of__exp,axiom,
% 0.23/0.49 ! [M: nat,N: nat] :
% 0.23/0.49 ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.49 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_of_exp
% 0.23/0.49 thf(fact_4729_take__bit__of__exp,axiom,
% 0.23/0.49 ! [M: nat,N: nat] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.49 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_of_exp
% 0.23/0.49 thf(fact_4730_take__bit__of__exp,axiom,
% 0.23/0.49 ! [M: nat,N: nat] :
% 0.23/0.49 ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.49 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_of_exp
% 0.23/0.49 thf(fact_4731_take__bit__of__2,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.49 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_of_2
% 0.23/0.49 thf(fact_4732_take__bit__of__2,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.49 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_of_2
% 0.23/0.49 thf(fact_4733_take__bit__of__2,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.49 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_of_2
% 0.23/0.49 thf(fact_4734_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,A: int] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.23/0.49 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % neg_numeral_power_le_of_int_cancel_iff
% 0.23/0.49 thf(fact_4735_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,A: int] :
% 0.23/0.49 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.23/0.49 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % neg_numeral_power_le_of_int_cancel_iff
% 0.23/0.49 thf(fact_4736_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,A: int] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.23/0.49 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % neg_numeral_power_le_of_int_cancel_iff
% 0.23/0.49 thf(fact_4737_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.23/0.49 ! [X4: num,N: nat,A: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.23/0.49 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % neg_numeral_power_le_of_int_cancel_iff
% 0.23/0.49 thf(fact_4738_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [A: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) )
% 0.23/0.49 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_neg_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4739_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [A: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) )
% 0.23/0.49 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_neg_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4740_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [A: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) )
% 0.23/0.49 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_neg_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4741_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.23/0.49 ! [A: int,X4: num,N: nat] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) )
% 0.23/0.49 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_le_neg_numeral_power_cancel_iff
% 0.23/0.49 thf(fact_4742_take__bit__eq__mask,axiom,
% 0.23/0.49 ( bit_se2923211474154528505it_int
% 0.23/0.49 = ( ^ [N2: nat,A4: int] : ( bit_se725231765392027082nd_int @ A4 @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_eq_mask
% 0.23/0.49 thf(fact_4743_take__bit__eq__mask,axiom,
% 0.23/0.49 ( bit_se2925701944663578781it_nat
% 0.23/0.49 = ( ^ [N2: nat,A4: nat] : ( bit_se727722235901077358nd_nat @ A4 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_eq_mask
% 0.23/0.49 thf(fact_4744_and_Oleft__commute,axiom,
% 0.23/0.49 ! [B2: int,A: int,C: int] :
% 0.23/0.49 ( ( bit_se725231765392027082nd_int @ B2 @ ( bit_se725231765392027082nd_int @ A @ C ) )
% 0.23/0.49 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B2 @ C ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and.left_commute
% 0.23/0.49 thf(fact_4745_and_Oleft__commute,axiom,
% 0.23/0.49 ! [B2: nat,A: nat,C: nat] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ B2 @ ( bit_se727722235901077358nd_nat @ A @ C ) )
% 0.23/0.49 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B2 @ C ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and.left_commute
% 0.23/0.49 thf(fact_4746_and_Ocommute,axiom,
% 0.23/0.49 ( bit_se725231765392027082nd_int
% 0.23/0.49 = ( ^ [A4: int,B3: int] : ( bit_se725231765392027082nd_int @ B3 @ A4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and.commute
% 0.23/0.49 thf(fact_4747_and_Ocommute,axiom,
% 0.23/0.49 ( bit_se727722235901077358nd_nat
% 0.23/0.49 = ( ^ [A4: nat,B3: nat] : ( bit_se727722235901077358nd_nat @ B3 @ A4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and.commute
% 0.23/0.49 thf(fact_4748_take__bit__of__int,axiom,
% 0.23/0.49 ! [N: nat,K: int] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
% 0.23/0.49 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_of_int
% 0.23/0.49 thf(fact_4749_of__int__mask__eq,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.23/0.49 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_mask_eq
% 0.23/0.49 thf(fact_4750_and_Oassoc,axiom,
% 0.23/0.49 ! [A: int,B2: int,C: int] :
% 0.23/0.49 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B2 ) @ C )
% 0.23/0.49 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B2 @ C ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and.assoc
% 0.23/0.49 thf(fact_4751_and_Oassoc,axiom,
% 0.23/0.49 ! [A: nat,B2: nat,C: nat] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B2 ) @ C )
% 0.23/0.49 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B2 @ C ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and.assoc
% 0.23/0.49 thf(fact_4752_of__int__and__eq,axiom,
% 0.23/0.49 ! [K: int,L2: int] :
% 0.23/0.49 ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 0.23/0.49 = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_and_eq
% 0.23/0.49 thf(fact_4753_take__bit__add,axiom,
% 0.23/0.49 ! [N: nat,A: int,B2: int] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B2 ) ) )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_add
% 0.23/0.49 thf(fact_4754_take__bit__add,axiom,
% 0.23/0.49 ! [N: nat,A: nat,B2: nat] :
% 0.23/0.49 ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B2 ) ) )
% 0.23/0.49 = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_add
% 0.23/0.49 thf(fact_4755_take__bit__tightened,axiom,
% 0.23/0.49 ! [N: nat,A: int,B2: int,M: nat] :
% 0.23/0.49 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ N @ B2 ) )
% 0.23/0.49 => ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ M @ A )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ M @ B2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_tightened
% 0.23/0.49 thf(fact_4756_take__bit__tightened,axiom,
% 0.23/0.49 ! [N: nat,A: nat,B2: nat,M: nat] :
% 0.23/0.49 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.23/0.49 = ( bit_se2925701944663578781it_nat @ N @ B2 ) )
% 0.23/0.49 => ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.49 => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 0.23/0.49 = ( bit_se2925701944663578781it_nat @ M @ B2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_tightened
% 0.23/0.49 thf(fact_4757_take__bit__nat__less__eq__self,axiom,
% 0.23/0.49 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_nat_less_eq_self
% 0.23/0.49 thf(fact_4758_take__bit__tightened__less__eq__nat,axiom,
% 0.23/0.49 ! [M: nat,N: nat,Q2: nat] :
% 0.23/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.49 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_tightened_less_eq_nat
% 0.23/0.49 thf(fact_4759_real__sqrt__less__mono,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.49 => ( ord_less_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_less_mono
% 0.23/0.49 thf(fact_4760_real__sqrt__le__mono,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.49 => ( ord_less_eq_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_le_mono
% 0.23/0.49 thf(fact_4761_mult__of__int__commute,axiom,
% 0.23/0.49 ! [X4: int,Y3: real] :
% 0.23/0.49 ( ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ Y3 )
% 0.23/0.49 = ( times_times_real @ Y3 @ ( ring_1_of_int_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % mult_of_int_commute
% 0.23/0.49 thf(fact_4762_mult__of__int__commute,axiom,
% 0.23/0.49 ! [X4: int,Y3: rat] :
% 0.23/0.49 ( ( times_times_rat @ ( ring_1_of_int_rat @ X4 ) @ Y3 )
% 0.23/0.49 = ( times_times_rat @ Y3 @ ( ring_1_of_int_rat @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % mult_of_int_commute
% 0.23/0.49 thf(fact_4763_mult__of__int__commute,axiom,
% 0.23/0.49 ! [X4: int,Y3: int] :
% 0.23/0.49 ( ( times_times_int @ ( ring_1_of_int_int @ X4 ) @ Y3 )
% 0.23/0.49 = ( times_times_int @ Y3 @ ( ring_1_of_int_int @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % mult_of_int_commute
% 0.23/0.49 thf(fact_4764_real__sqrt__divide,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( sqrt @ ( divide_divide_real @ X4 @ Y3 ) )
% 0.23/0.49 = ( divide_divide_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_divide
% 0.23/0.49 thf(fact_4765_real__sqrt__mult,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( sqrt @ ( times_times_real @ X4 @ Y3 ) )
% 0.23/0.49 = ( times_times_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_mult
% 0.23/0.49 thf(fact_4766_real__sqrt__power,axiom,
% 0.23/0.49 ! [X4: real,K: nat] :
% 0.23/0.49 ( ( sqrt @ ( power_power_real @ X4 @ K ) )
% 0.23/0.49 = ( power_power_real @ ( sqrt @ X4 ) @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_power
% 0.23/0.49 thf(fact_4767_real__sqrt__minus,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( sqrt @ ( uminus_uminus_real @ X4 ) )
% 0.23/0.49 = ( uminus_uminus_real @ ( sqrt @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_minus
% 0.23/0.49 thf(fact_4768_take__bit__minus,axiom,
% 0.23/0.49 ! [N: nat,K: int] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_minus
% 0.23/0.49 thf(fact_4769_take__bit__mult,axiom,
% 0.23/0.49 ! [N: nat,K: int,L2: int] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_mult
% 0.23/0.49 thf(fact_4770_take__bit__diff,axiom,
% 0.23/0.49 ! [N: nat,K: int,L2: int] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_diff
% 0.23/0.49 thf(fact_4771_concat__bit__take__bit__eq,axiom,
% 0.23/0.49 ! [N: nat,B2: int] :
% 0.23/0.49 ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B2 ) )
% 0.23/0.49 = ( bit_concat_bit @ N @ B2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % concat_bit_take_bit_eq
% 0.23/0.49 thf(fact_4772_concat__bit__eq__iff,axiom,
% 0.23/0.49 ! [N: nat,K: int,L2: int,R2: int,S: int] :
% 0.23/0.49 ( ( ( bit_concat_bit @ N @ K @ L2 )
% 0.23/0.49 = ( bit_concat_bit @ N @ R2 @ S ) )
% 0.23/0.49 = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ N @ R2 ) )
% 0.23/0.49 & ( L2 = S ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % concat_bit_eq_iff
% 0.23/0.49 thf(fact_4773_less__eq__mask,axiom,
% 0.23/0.49 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % less_eq_mask
% 0.23/0.49 thf(fact_4774_take__bit__eq__mask__iff,axiom,
% 0.23/0.49 ! [N: nat,K: int] :
% 0.23/0.49 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.23/0.49 = ( bit_se2000444600071755411sk_int @ N ) )
% 0.23/0.49 = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.23/0.49 = zero_zero_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_eq_mask_iff
% 0.23/0.49 thf(fact_4775_and__eq__minus__1__iff,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] :
% 0.23/0.49 ( ( ( bit_se3949692690581998587nteger @ A @ B2 )
% 0.23/0.49 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.49 = ( ( A
% 0.23/0.49 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.49 & ( B2
% 0.23/0.49 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_eq_minus_1_iff
% 0.23/0.49 thf(fact_4776_and__eq__minus__1__iff,axiom,
% 0.23/0.49 ! [A: int,B2: int] :
% 0.23/0.49 ( ( ( bit_se725231765392027082nd_int @ A @ B2 )
% 0.23/0.49 = ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.49 = ( ( A
% 0.23/0.49 = ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.49 & ( B2
% 0.23/0.49 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_eq_minus_1_iff
% 0.23/0.49 thf(fact_4777_real__sqrt__gt__zero,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_gt_zero
% 0.23/0.49 thf(fact_4778_real__sqrt__ge__zero,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_ge_zero
% 0.23/0.49 thf(fact_4779_real__sqrt__eq__zero__cancel,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ( sqrt @ X4 )
% 0.23/0.49 = zero_zero_real )
% 0.23/0.49 => ( X4 = zero_zero_real ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_eq_zero_cancel
% 0.23/0.49 thf(fact_4780_real__sqrt__ge__one,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ one_one_real @ X4 )
% 0.23/0.49 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_ge_one
% 0.23/0.49 thf(fact_4781_take__bit__tightened__less__eq__int,axiom,
% 0.23/0.49 ! [M: nat,N: nat,K: int] :
% 0.23/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.49 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_tightened_less_eq_int
% 0.23/0.49 thf(fact_4782_AND__upper2_H,axiom,
% 0.23/0.49 ! [Y3: int,Z: int,X4: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.49 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.23/0.49 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ Z ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % AND_upper2'
% 0.23/0.49 thf(fact_4783_AND__upper1_H,axiom,
% 0.23/0.49 ! [Y3: int,Z: int,Ya: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.49 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.23/0.49 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % AND_upper1'
% 0.23/0.49 thf(fact_4784_AND__upper2,axiom,
% 0.23/0.49 ! [Y3: int,X4: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.49 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % AND_upper2
% 0.23/0.49 thf(fact_4785_AND__upper1,axiom,
% 0.23/0.49 ! [X4: int,Y3: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.49 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % AND_upper1
% 0.23/0.49 thf(fact_4786_AND__lower,axiom,
% 0.23/0.49 ! [X4: int,Y3: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.49 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % AND_lower
% 0.23/0.49 thf(fact_4787_take__bit__int__less__eq__self__iff,axiom,
% 0.23/0.49 ! [N: nat,K: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.23/0.49 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_int_less_eq_self_iff
% 0.23/0.49 thf(fact_4788_take__bit__nonnegative,axiom,
% 0.23/0.49 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_nonnegative
% 0.23/0.49 thf(fact_4789_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 0.23/0.49 ! [N: nat,A: int,B2: int] :
% 0.23/0.49 ( ( ( bit_ri631733984087533419it_int @ N @ A )
% 0.23/0.49 = ( bit_ri631733984087533419it_int @ N @ B2 ) )
% 0.23/0.49 = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % signed_take_bit_eq_iff_take_bit_eq
% 0.23/0.49 thf(fact_4790_not__take__bit__negative,axiom,
% 0.23/0.49 ! [N: nat,K: int] :
% 0.23/0.49 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 0.23/0.49
% 0.23/0.49 % not_take_bit_negative
% 0.23/0.49 thf(fact_4791_take__bit__int__greater__self__iff,axiom,
% 0.23/0.49 ! [K: int,N: nat] :
% 0.23/0.49 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.23/0.49 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_int_greater_self_iff
% 0.23/0.49 thf(fact_4792_signed__take__bit__take__bit,axiom,
% 0.23/0.49 ! [M: nat,N: nat,A: int] :
% 0.23/0.49 ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.23/0.49 = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % signed_take_bit_take_bit
% 0.23/0.49 thf(fact_4793_take__bit__unset__bit__eq,axiom,
% 0.23/0.49 ! [N: nat,M: nat,A: int] :
% 0.23/0.49 ( ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.23/0.49 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.23/0.49 = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_unset_bit_eq
% 0.23/0.49 thf(fact_4794_take__bit__unset__bit__eq,axiom,
% 0.23/0.49 ! [N: nat,M: nat,A: nat] :
% 0.23/0.49 ( ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.23/0.49 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.23/0.49 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.23/0.49 = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_unset_bit_eq
% 0.23/0.49 thf(fact_4795_take__bit__set__bit__eq,axiom,
% 0.23/0.49 ! [N: nat,M: nat,A: int] :
% 0.23/0.49 ( ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.23/0.49 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.23/0.49 = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_set_bit_eq
% 0.23/0.49 thf(fact_4796_take__bit__set__bit__eq,axiom,
% 0.23/0.49 ! [N: nat,M: nat,A: nat] :
% 0.23/0.49 ( ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.23/0.49 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.23/0.49 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.23/0.49 = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_set_bit_eq
% 0.23/0.49 thf(fact_4797_take__bit__flip__bit__eq,axiom,
% 0.23/0.49 ! [N: nat,M: nat,A: int] :
% 0.23/0.49 ( ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.23/0.49 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.23/0.49 = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_flip_bit_eq
% 0.23/0.49 thf(fact_4798_take__bit__flip__bit__eq,axiom,
% 0.23/0.49 ! [N: nat,M: nat,A: nat] :
% 0.23/0.49 ( ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.23/0.49 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.23/0.49 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.23/0.49 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.23/0.49 = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_flip_bit_eq
% 0.23/0.49 thf(fact_4799_mask__nonnegative__int,axiom,
% 0.23/0.49 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % mask_nonnegative_int
% 0.23/0.49 thf(fact_4800_not__mask__negative__int,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 0.23/0.49
% 0.23/0.49 % not_mask_negative_int
% 0.23/0.49 thf(fact_4801_real__div__sqrt,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( divide_divide_real @ X4 @ ( sqrt @ X4 ) )
% 0.23/0.49 = ( sqrt @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_div_sqrt
% 0.23/0.49 thf(fact_4802_sqrt__add__le__add__sqrt,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % sqrt_add_le_add_sqrt
% 0.23/0.49 thf(fact_4803_take__bit__signed__take__bit,axiom,
% 0.23/0.49 ! [M: nat,N: nat,A: int] :
% 0.23/0.49 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 0.23/0.49 = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_signed_take_bit
% 0.23/0.49 thf(fact_4804_le__real__sqrt__sumsq,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] : ( ord_less_eq_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % le_real_sqrt_sumsq
% 0.23/0.49 thf(fact_4805_and__less__eq,axiom,
% 0.23/0.49 ! [L2: int,K: int] :
% 0.23/0.49 ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.23/0.49 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_less_eq
% 0.23/0.49 thf(fact_4806_AND__upper1_H_H,axiom,
% 0.23/0.49 ! [Y3: int,Z: int,Ya: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.49 => ( ( ord_less_int @ Y3 @ Z )
% 0.23/0.49 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % AND_upper1''
% 0.23/0.49 thf(fact_4807_AND__upper2_H_H,axiom,
% 0.23/0.49 ! [Y3: int,Z: int,X4: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.49 => ( ( ord_less_int @ Y3 @ Z )
% 0.23/0.49 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ Z ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % AND_upper2''
% 0.23/0.49 thf(fact_4808_take__bit__decr__eq,axiom,
% 0.23/0.49 ! [N: nat,K: int] :
% 0.23/0.49 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.23/0.49 != zero_zero_int )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 0.23/0.49 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_decr_eq
% 0.23/0.49 thf(fact_4809_real__of__int__div4,axiom,
% 0.23/0.49 ! [N: int,X4: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_of_int_div4
% 0.23/0.49 thf(fact_4810_real__of__int__div,axiom,
% 0.23/0.49 ! [D: int,N: int] :
% 0.23/0.49 ( ( dvd_dvd_int @ D @ N )
% 0.23/0.49 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 0.23/0.49 = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_of_int_div
% 0.23/0.49 thf(fact_4811_less__mask,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.23/0.49 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % less_mask
% 0.23/0.49 thf(fact_4812_take__bit__eq__mask__iff__exp__dvd,axiom,
% 0.23/0.49 ! [N: nat,K: int] :
% 0.23/0.49 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.23/0.49 = ( bit_se2000444600071755411sk_int @ N ) )
% 0.23/0.49 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_eq_mask_iff_exp_dvd
% 0.23/0.49 thf(fact_4813_even__and__iff,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] :
% 0.23/0.49 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B2 ) )
% 0.23/0.49 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.23/0.49 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % even_and_iff
% 0.23/0.49 thf(fact_4814_even__and__iff,axiom,
% 0.23/0.49 ! [A: int,B2: int] :
% 0.23/0.49 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
% 0.23/0.49 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.23/0.49 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % even_and_iff
% 0.23/0.49 thf(fact_4815_even__and__iff,axiom,
% 0.23/0.49 ! [A: nat,B2: nat] :
% 0.23/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
% 0.23/0.49 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.23/0.49 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % even_and_iff
% 0.23/0.49 thf(fact_4816_sqrt2__less__2,axiom,
% 0.23/0.49 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.23/0.49
% 0.23/0.49 % sqrt2_less_2
% 0.23/0.49 thf(fact_4817_of__int__nonneg,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.23/0.49 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_nonneg
% 0.23/0.49 thf(fact_4818_of__int__nonneg,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.23/0.49 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_nonneg
% 0.23/0.49 thf(fact_4819_of__int__nonneg,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.23/0.49 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_nonneg
% 0.23/0.49 thf(fact_4820_of__int__pos,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.23/0.49 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_pos
% 0.23/0.49 thf(fact_4821_of__int__pos,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.23/0.49 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_pos
% 0.23/0.49 thf(fact_4822_of__int__pos,axiom,
% 0.23/0.49 ! [Z: int] :
% 0.23/0.49 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.23/0.49 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_pos
% 0.23/0.49 thf(fact_4823_even__and__iff__int,axiom,
% 0.23/0.49 ! [K: int,L2: int] :
% 0.23/0.49 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 0.23/0.49 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.23/0.49 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % even_and_iff_int
% 0.23/0.49 thf(fact_4824_of__int__neg__numeral,axiom,
% 0.23/0.49 ! [K: num] :
% 0.23/0.49 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_neg_numeral
% 0.23/0.49 thf(fact_4825_of__int__neg__numeral,axiom,
% 0.23/0.49 ! [K: num] :
% 0.23/0.49 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_neg_numeral
% 0.23/0.49 thf(fact_4826_of__int__neg__numeral,axiom,
% 0.23/0.49 ! [K: num] :
% 0.23/0.49 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_neg_numeral
% 0.23/0.49 thf(fact_4827_of__int__neg__numeral,axiom,
% 0.23/0.49 ! [K: num] :
% 0.23/0.49 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_neg_numeral
% 0.23/0.49 thf(fact_4828_of__int__neg__numeral,axiom,
% 0.23/0.49 ! [K: num] :
% 0.23/0.49 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_neg_numeral
% 0.23/0.49 thf(fact_4829_int__le__real__less,axiom,
% 0.23/0.49 ( ord_less_eq_int
% 0.23/0.49 = ( ^ [N2: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % int_le_real_less
% 0.23/0.49 thf(fact_4830_int__less__real__le,axiom,
% 0.23/0.49 ( ord_less_int
% 0.23/0.49 = ( ^ [N2: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % int_less_real_le
% 0.23/0.49 thf(fact_4831_real__of__int__div__aux,axiom,
% 0.23/0.49 ! [X4: int,D: int] :
% 0.23/0.49 ( ( divide_divide_real @ ( ring_1_of_int_real @ X4 ) @ ( ring_1_of_int_real @ D ) )
% 0.23/0.49 = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X4 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X4 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_of_int_div_aux
% 0.23/0.49 thf(fact_4832_take__bit__Suc__bit0,axiom,
% 0.23/0.49 ! [N: nat,K: num] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.23/0.49 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc_bit0
% 0.23/0.49 thf(fact_4833_take__bit__Suc__bit0,axiom,
% 0.23/0.49 ! [N: nat,K: num] :
% 0.23/0.49 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.23/0.49 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc_bit0
% 0.23/0.49 thf(fact_4834_take__bit__eq__mod,axiom,
% 0.23/0.49 ( bit_se1745604003318907178nteger
% 0.23/0.49 = ( ^ [N2: nat,A4: code_integer] : ( modulo364778990260209775nteger @ A4 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_eq_mod
% 0.23/0.49 thf(fact_4835_take__bit__eq__mod,axiom,
% 0.23/0.49 ( bit_se2923211474154528505it_int
% 0.23/0.49 = ( ^ [N2: nat,A4: int] : ( modulo_modulo_int @ A4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_eq_mod
% 0.23/0.49 thf(fact_4836_take__bit__eq__mod,axiom,
% 0.23/0.49 ( bit_se2925701944663578781it_nat
% 0.23/0.49 = ( ^ [N2: nat,A4: nat] : ( modulo_modulo_nat @ A4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_eq_mod
% 0.23/0.49 thf(fact_4837_one__and__eq,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
% 0.23/0.49 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % one_and_eq
% 0.23/0.49 thf(fact_4838_one__and__eq,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
% 0.23/0.49 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % one_and_eq
% 0.23/0.49 thf(fact_4839_one__and__eq,axiom,
% 0.23/0.49 ! [A: nat] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
% 0.23/0.49 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % one_and_eq
% 0.23/0.49 thf(fact_4840_and__one__eq,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
% 0.23/0.49 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_one_eq
% 0.23/0.49 thf(fact_4841_and__one__eq,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
% 0.23/0.49 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_one_eq
% 0.23/0.49 thf(fact_4842_and__one__eq,axiom,
% 0.23/0.49 ! [A: nat] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
% 0.23/0.49 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_one_eq
% 0.23/0.49 thf(fact_4843_take__bit__nat__eq__self__iff,axiom,
% 0.23/0.49 ! [N: nat,M: nat] :
% 0.23/0.49 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.23/0.49 = M )
% 0.23/0.49 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_nat_eq_self_iff
% 0.23/0.49 thf(fact_4844_take__bit__nat__less__exp,axiom,
% 0.23/0.49 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_nat_less_exp
% 0.23/0.49 thf(fact_4845_take__bit__nat__eq__self,axiom,
% 0.23/0.49 ! [M: nat,N: nat] :
% 0.23/0.49 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.49 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.23/0.49 = M ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_nat_eq_self
% 0.23/0.49 thf(fact_4846_real__less__rsqrt,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 0.23/0.49 => ( ord_less_real @ X4 @ ( sqrt @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_less_rsqrt
% 0.23/0.49 thf(fact_4847_real__le__rsqrt,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 0.23/0.49 => ( ord_less_eq_real @ X4 @ ( sqrt @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_le_rsqrt
% 0.23/0.49 thf(fact_4848_sqrt__le__D,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ Y3 )
% 0.23/0.49 => ( ord_less_eq_real @ X4 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % sqrt_le_D
% 0.23/0.49 thf(fact_4849_take__bit__nat__def,axiom,
% 0.23/0.49 ( bit_se2925701944663578781it_nat
% 0.23/0.49 = ( ^ [N2: nat,M6: nat] : ( modulo_modulo_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_nat_def
% 0.23/0.49 thf(fact_4850_take__bit__int__less__exp,axiom,
% 0.23/0.49 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_int_less_exp
% 0.23/0.49 thf(fact_4851_take__bit__int__def,axiom,
% 0.23/0.49 ( bit_se2923211474154528505it_int
% 0.23/0.49 = ( ^ [N2: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_int_def
% 0.23/0.49 thf(fact_4852_real__of__int__div2,axiom,
% 0.23/0.49 ! [N: int,X4: 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 @ X4 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_of_int_div2
% 0.23/0.49 thf(fact_4853_real__of__int__div3,axiom,
% 0.23/0.49 ! [N: int,X4: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X4 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) ) @ one_one_real ) ).
% 0.23/0.49
% 0.23/0.49 % real_of_int_div3
% 0.23/0.49 thf(fact_4854_take__bit__eq__0__iff,axiom,
% 0.23/0.49 ! [N: nat,A: code_integer] :
% 0.23/0.49 ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.23/0.49 = zero_z3403309356797280102nteger )
% 0.23/0.49 = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_eq_0_iff
% 0.23/0.49 thf(fact_4855_take__bit__eq__0__iff,axiom,
% 0.23/0.49 ! [N: nat,A: int] :
% 0.23/0.49 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.23/0.49 = zero_zero_int )
% 0.23/0.49 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_eq_0_iff
% 0.23/0.49 thf(fact_4856_take__bit__eq__0__iff,axiom,
% 0.23/0.49 ! [N: nat,A: nat] :
% 0.23/0.49 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.23/0.49 = zero_zero_nat )
% 0.23/0.49 = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_eq_0_iff
% 0.23/0.49 thf(fact_4857_take__bit__numeral__bit0,axiom,
% 0.23/0.49 ! [L2: num,K: num] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.23/0.49 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_numeral_bit0
% 0.23/0.49 thf(fact_4858_take__bit__numeral__bit0,axiom,
% 0.23/0.49 ! [L2: num,K: num] :
% 0.23/0.49 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.23/0.49 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_numeral_bit0
% 0.23/0.49 thf(fact_4859_take__bit__nat__less__self__iff,axiom,
% 0.23/0.49 ! [N: nat,M: nat] :
% 0.23/0.49 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 0.23/0.49 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_nat_less_self_iff
% 0.23/0.49 thf(fact_4860_real__sqrt__unique,axiom,
% 0.23/0.49 ! [Y3: real,X4: real] :
% 0.23/0.49 ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.49 = X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ( sqrt @ X4 )
% 0.23/0.49 = Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_unique
% 0.23/0.49 thf(fact_4861_real__le__lsqrt,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ( ord_less_eq_real @ X4 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.49 => ( ord_less_eq_real @ ( sqrt @ X4 ) @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_le_lsqrt
% 0.23/0.49 thf(fact_4862_lemma__real__divide__sqrt__less,axiom,
% 0.23/0.49 ! [U: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ U )
% 0.23/0.49 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 0.23/0.49
% 0.23/0.49 % lemma_real_divide_sqrt_less
% 0.23/0.49 thf(fact_4863_Suc__mask__eq__exp,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.23/0.49 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % Suc_mask_eq_exp
% 0.23/0.49 thf(fact_4864_real__sqrt__sum__squares__eq__cancel2,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.49 = Y3 )
% 0.23/0.49 => ( X4 = zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_sum_squares_eq_cancel2
% 0.23/0.49 thf(fact_4865_real__sqrt__sum__squares__eq__cancel,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.49 = X4 )
% 0.23/0.49 => ( Y3 = zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_sum_squares_eq_cancel
% 0.23/0.49 thf(fact_4866_mask__nat__less__exp,axiom,
% 0.23/0.49 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % mask_nat_less_exp
% 0.23/0.49 thf(fact_4867_real__sqrt__sum__squares__triangle__ineq,axiom,
% 0.23/0.49 ! [A: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B2 @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_sum_squares_triangle_ineq
% 0.23/0.49 thf(fact_4868_real__sqrt__sum__squares__ge2,axiom,
% 0.23/0.49 ! [Y3: real,X4: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_sum_squares_ge2
% 0.23/0.49 thf(fact_4869_real__sqrt__sum__squares__ge1,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] : ( ord_less_eq_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_sum_squares_ge1
% 0.23/0.49 thf(fact_4870_take__bit__Suc__minus__bit0,axiom,
% 0.23/0.49 ! [N: nat,K: num] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.23/0.49 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc_minus_bit0
% 0.23/0.49 thf(fact_4871_even__of__int__iff,axiom,
% 0.23/0.49 ! [K: int] :
% 0.23/0.49 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 0.23/0.49 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % even_of_int_iff
% 0.23/0.49 thf(fact_4872_even__of__int__iff,axiom,
% 0.23/0.49 ! [K: int] :
% 0.23/0.49 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 0.23/0.49 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % even_of_int_iff
% 0.23/0.49 thf(fact_4873_take__bit__int__greater__eq__self__iff,axiom,
% 0.23/0.49 ! [K: int,N: nat] :
% 0.23/0.49 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.23/0.49 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_int_greater_eq_self_iff
% 0.23/0.49 thf(fact_4874_take__bit__int__less__self__iff,axiom,
% 0.23/0.49 ! [N: nat,K: int] :
% 0.23/0.49 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.23/0.49 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_int_less_self_iff
% 0.23/0.49 thf(fact_4875_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
% 0.23/0.49 = ( N = zero_zero_nat ) ) ).
% 0.23/0.49
% 0.23/0.49 % semiring_bit_operations_class.even_mask_iff
% 0.23/0.49 thf(fact_4876_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.23/0.49 = ( N = zero_zero_nat ) ) ).
% 0.23/0.49
% 0.23/0.49 % semiring_bit_operations_class.even_mask_iff
% 0.23/0.49 thf(fact_4877_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.23/0.49 = ( N = zero_zero_nat ) ) ).
% 0.23/0.49
% 0.23/0.49 % semiring_bit_operations_class.even_mask_iff
% 0.23/0.49 thf(fact_4878_real__less__lsqrt,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ( ord_less_real @ X4 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.49 => ( ord_less_real @ ( sqrt @ X4 ) @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_less_lsqrt
% 0.23/0.49 thf(fact_4879_sqrt__sum__squares__le__sum,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % sqrt_sum_squares_le_sum
% 0.23/0.49 thf(fact_4880_sqrt__even__pow2,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.49 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.49 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % sqrt_even_pow2
% 0.23/0.49 thf(fact_4881_take__bit__int__eq__self__iff,axiom,
% 0.23/0.49 ! [N: nat,K: int] :
% 0.23/0.49 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.23/0.49 = K )
% 0.23/0.49 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.49 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_int_eq_self_iff
% 0.23/0.49 thf(fact_4882_take__bit__int__eq__self,axiom,
% 0.23/0.49 ! [K: int,N: nat] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.49 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.23/0.49 = K ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_int_eq_self
% 0.23/0.49 thf(fact_4883_and__int__rec,axiom,
% 0.23/0.49 ( bit_se725231765392027082nd_int
% 0.23/0.49 = ( ^ [K3: int,L: int] :
% 0.23/0.49 ( plus_plus_int
% 0.23/0.49 @ ( zero_n2684676970156552555ol_int
% 0.23/0.49 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.23/0.49 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.23/0.49 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_int_rec
% 0.23/0.49 thf(fact_4884_mask__nat__def,axiom,
% 0.23/0.49 ( bit_se2002935070580805687sk_nat
% 0.23/0.49 = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % mask_nat_def
% 0.23/0.49 thf(fact_4885_take__bit__numeral__minus__bit0,axiom,
% 0.23/0.49 ! [L2: num,K: num] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.23/0.49 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_numeral_minus_bit0
% 0.23/0.49 thf(fact_4886_mask__half__int,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.49 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % mask_half_int
% 0.23/0.49 thf(fact_4887_take__bit__incr__eq,axiom,
% 0.23/0.49 ! [N: nat,K: int] :
% 0.23/0.49 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.23/0.49 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.23/0.49 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_incr_eq
% 0.23/0.49 thf(fact_4888_mask__int__def,axiom,
% 0.23/0.49 ( bit_se2000444600071755411sk_int
% 0.23/0.49 = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % mask_int_def
% 0.23/0.49 thf(fact_4889_take__bit__Suc__minus__1__eq,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.49 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc_minus_1_eq
% 0.23/0.49 thf(fact_4890_take__bit__Suc__minus__1__eq,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.49 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc_minus_1_eq
% 0.23/0.49 thf(fact_4891_take__bit__Suc__bit1,axiom,
% 0.23/0.49 ! [N: nat,K: num] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.23/0.49 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc_bit1
% 0.23/0.49 thf(fact_4892_take__bit__Suc__bit1,axiom,
% 0.23/0.49 ! [N: nat,K: num] :
% 0.23/0.49 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.23/0.49 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc_bit1
% 0.23/0.49 thf(fact_4893_take__bit__numeral__minus__1__eq,axiom,
% 0.23/0.49 ! [K: num] :
% 0.23/0.49 ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.49 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_numeral_minus_1_eq
% 0.23/0.49 thf(fact_4894_take__bit__numeral__minus__1__eq,axiom,
% 0.23/0.49 ! [K: num] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.49 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_numeral_minus_1_eq
% 0.23/0.49 thf(fact_4895_take__bit__Suc,axiom,
% 0.23/0.49 ! [N: nat,A: code_integer] :
% 0.23/0.49 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
% 0.23/0.49 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc
% 0.23/0.49 thf(fact_4896_take__bit__Suc,axiom,
% 0.23/0.49 ! [N: nat,A: int] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 0.23/0.49 = ( 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 ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc
% 0.23/0.49 thf(fact_4897_take__bit__Suc,axiom,
% 0.23/0.49 ! [N: nat,A: nat] :
% 0.23/0.49 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
% 0.23/0.49 = ( 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 ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc
% 0.23/0.49 thf(fact_4898_mask__eq__exp__minus__1,axiom,
% 0.23/0.49 ( bit_se2002935070580805687sk_nat
% 0.23/0.49 = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % mask_eq_exp_minus_1
% 0.23/0.49 thf(fact_4899_mask__eq__exp__minus__1,axiom,
% 0.23/0.49 ( bit_se2000444600071755411sk_int
% 0.23/0.49 = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % mask_eq_exp_minus_1
% 0.23/0.49 thf(fact_4900_real__sqrt__power__even,axiom,
% 0.23/0.49 ! [N: nat,X4: real] :
% 0.23/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.49 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( power_power_real @ ( sqrt @ X4 ) @ N )
% 0.23/0.49 = ( power_power_real @ X4 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_power_even
% 0.23/0.49 thf(fact_4901_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 0.23/0.49 ! [X4: real,Y3: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( 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 ) ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_sum_squares_mult_ge_zero
% 0.23/0.49 thf(fact_4902_arith__geo__mean__sqrt,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X4 @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % arith_geo_mean_sqrt
% 0.23/0.49 thf(fact_4903_take__bit__int__less__eq,axiom,
% 0.23/0.49 ! [N: nat,K: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.23/0.49 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.49 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_int_less_eq
% 0.23/0.49 thf(fact_4904_take__bit__int__greater__eq,axiom,
% 0.23/0.49 ! [K: int,N: nat] :
% 0.23/0.49 ( ( ord_less_int @ K @ zero_zero_int )
% 0.23/0.49 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_int_greater_eq
% 0.23/0.49 thf(fact_4905_signed__take__bit__eq__take__bit__shift,axiom,
% 0.23/0.49 ( bit_ri631733984087533419it_int
% 0.23/0.49 = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % signed_take_bit_eq_take_bit_shift
% 0.23/0.49 thf(fact_4906_stable__imp__take__bit__eq,axiom,
% 0.23/0.49 ! [A: code_integer,N: nat] :
% 0.23/0.49 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.49 = A )
% 0.23/0.49 => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.23/0.49 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.23/0.49 = zero_z3403309356797280102nteger ) )
% 0.23/0.49 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.23/0.49 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.23/0.49 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % stable_imp_take_bit_eq
% 0.23/0.49 thf(fact_4907_stable__imp__take__bit__eq,axiom,
% 0.23/0.49 ! [A: int,N: nat] :
% 0.23/0.49 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.49 = A )
% 0.23/0.49 => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.23/0.49 = zero_zero_int ) )
% 0.23/0.49 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.23/0.49 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % stable_imp_take_bit_eq
% 0.23/0.49 thf(fact_4908_stable__imp__take__bit__eq,axiom,
% 0.23/0.49 ! [A: nat,N: nat] :
% 0.23/0.49 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.49 = A )
% 0.23/0.49 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.23/0.49 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.23/0.49 = zero_zero_nat ) )
% 0.23/0.49 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.23/0.49 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.23/0.49 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % stable_imp_take_bit_eq
% 0.23/0.49 thf(fact_4909_take__bit__numeral__bit1,axiom,
% 0.23/0.49 ! [L2: num,K: num] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.23/0.49 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_numeral_bit1
% 0.23/0.49 thf(fact_4910_take__bit__numeral__bit1,axiom,
% 0.23/0.49 ! [L2: num,K: num] :
% 0.23/0.49 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.23/0.49 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_numeral_bit1
% 0.23/0.49 thf(fact_4911_take__bit__minus__small__eq,axiom,
% 0.23/0.49 ! [K: int,N: nat] :
% 0.23/0.49 ( ( ord_less_int @ zero_zero_int @ K )
% 0.23/0.49 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.49 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 0.23/0.49 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_minus_small_eq
% 0.23/0.49 thf(fact_4912_one__le__exp__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X4 ) )
% 0.23/0.49 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % one_le_exp_iff
% 0.23/0.49 thf(fact_4913_exp__le__one__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( exp_real @ X4 ) @ one_one_real )
% 0.23/0.49 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_le_one_iff
% 0.23/0.49 thf(fact_4914_exp__less__one__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ ( exp_real @ X4 ) @ one_one_real )
% 0.23/0.49 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_less_one_iff
% 0.23/0.49 thf(fact_4915_one__less__exp__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ one_one_real @ ( exp_real @ X4 ) )
% 0.23/0.49 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % one_less_exp_iff
% 0.23/0.49 thf(fact_4916_real__exp__bound__lemma,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.49 => ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_exp_bound_lemma
% 0.23/0.49 thf(fact_4917_exp__bound,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.49 => ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_bound
% 0.23/0.49 thf(fact_4918_arsinh__real__aux,axiom,
% 0.23/0.49 ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % arsinh_real_aux
% 0.23/0.49 thf(fact_4919_exp__less__cancel__iff,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
% 0.23/0.49 = ( ord_less_real @ X4 @ Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_less_cancel_iff
% 0.23/0.49 thf(fact_4920_exp__less__mono,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.49 => ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_less_mono
% 0.23/0.49 thf(fact_4921_exp__le__cancel__iff,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
% 0.23/0.49 = ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_le_cancel_iff
% 0.23/0.49 thf(fact_4922_exp__zero,axiom,
% 0.23/0.49 ( ( exp_complex @ zero_zero_complex )
% 0.23/0.49 = one_one_complex ) ).
% 0.23/0.49
% 0.23/0.49 % exp_zero
% 0.23/0.49 thf(fact_4923_exp__zero,axiom,
% 0.23/0.49 ( ( exp_real @ zero_zero_real )
% 0.23/0.49 = one_one_real ) ).
% 0.23/0.49
% 0.23/0.49 % exp_zero
% 0.23/0.49 thf(fact_4924_exp__eq__one__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ( exp_real @ X4 )
% 0.23/0.49 = one_one_real )
% 0.23/0.49 = ( X4 = zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_eq_one_iff
% 0.23/0.49 thf(fact_4925_and__nat__numerals_I3_J,axiom,
% 0.23/0.49 ! [X4: num] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 0.23/0.49 = zero_zero_nat ) ).
% 0.23/0.49
% 0.23/0.49 % and_nat_numerals(3)
% 0.23/0.49 thf(fact_4926_and__nat__numerals_I1_J,axiom,
% 0.23/0.49 ! [Y3: num] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.23/0.49 = zero_zero_nat ) ).
% 0.23/0.49
% 0.23/0.49 % and_nat_numerals(1)
% 0.23/0.49 thf(fact_4927_and__nat__numerals_I2_J,axiom,
% 0.23/0.49 ! [Y3: num] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.23/0.49 = one_one_nat ) ).
% 0.23/0.49
% 0.23/0.49 % and_nat_numerals(2)
% 0.23/0.49 thf(fact_4928_and__nat__numerals_I4_J,axiom,
% 0.23/0.49 ! [X4: num] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 0.23/0.49 = one_one_nat ) ).
% 0.23/0.49
% 0.23/0.49 % and_nat_numerals(4)
% 0.23/0.49 thf(fact_4929_Suc__0__and__eq,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.23/0.49 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % Suc_0_and_eq
% 0.23/0.49 thf(fact_4930_and__Suc__0__eq,axiom,
% 0.23/0.49 ! [N: nat] :
% 0.23/0.49 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.23/0.49 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_Suc_0_eq
% 0.23/0.49 thf(fact_4931_exp__times__arg__commute,axiom,
% 0.23/0.49 ! [A3: complex] :
% 0.23/0.49 ( ( times_times_complex @ ( exp_complex @ A3 ) @ A3 )
% 0.23/0.49 = ( times_times_complex @ A3 @ ( exp_complex @ A3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_times_arg_commute
% 0.23/0.49 thf(fact_4932_exp__times__arg__commute,axiom,
% 0.23/0.49 ! [A3: real] :
% 0.23/0.49 ( ( times_times_real @ ( exp_real @ A3 ) @ A3 )
% 0.23/0.49 = ( times_times_real @ A3 @ ( exp_real @ A3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_times_arg_commute
% 0.23/0.49 thf(fact_4933_exp__less__cancel,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
% 0.23/0.49 => ( ord_less_real @ X4 @ Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_less_cancel
% 0.23/0.49 thf(fact_4934_and__nat__unfold,axiom,
% 0.23/0.49 ( bit_se727722235901077358nd_nat
% 0.23/0.49 = ( ^ [M6: nat,N2: nat] :
% 0.23/0.49 ( if_nat
% 0.23/0.49 @ ( ( M6 = zero_zero_nat )
% 0.23/0.49 | ( N2 = zero_zero_nat ) )
% 0.23/0.49 @ zero_zero_nat
% 0.23/0.49 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_nat_unfold
% 0.23/0.49 thf(fact_4935_and__nat__rec,axiom,
% 0.23/0.49 ( bit_se727722235901077358nd_nat
% 0.23/0.49 = ( ^ [M6: nat,N2: nat] :
% 0.23/0.49 ( plus_plus_nat
% 0.23/0.49 @ ( zero_n2687167440665602831ol_nat
% 0.23/0.49 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 0.23/0.49 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.23/0.49 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % and_nat_rec
% 0.23/0.49 thf(fact_4936_mult__exp__exp,axiom,
% 0.23/0.49 ! [X4: complex,Y3: complex] :
% 0.23/0.49 ( ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y3 ) )
% 0.23/0.49 = ( exp_complex @ ( plus_plus_complex @ X4 @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % mult_exp_exp
% 0.23/0.49 thf(fact_4937_mult__exp__exp,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
% 0.23/0.49 = ( exp_real @ ( plus_plus_real @ X4 @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % mult_exp_exp
% 0.23/0.49 thf(fact_4938_exp__add__commuting,axiom,
% 0.23/0.49 ! [X4: complex,Y3: complex] :
% 0.23/0.49 ( ( ( times_times_complex @ X4 @ Y3 )
% 0.23/0.49 = ( times_times_complex @ Y3 @ X4 ) )
% 0.23/0.49 => ( ( exp_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
% 0.23/0.49 = ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_add_commuting
% 0.23/0.49 thf(fact_4939_exp__add__commuting,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ( times_times_real @ X4 @ Y3 )
% 0.23/0.49 = ( times_times_real @ Y3 @ X4 ) )
% 0.23/0.49 => ( ( exp_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.49 = ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_add_commuting
% 0.23/0.49 thf(fact_4940_exp__diff,axiom,
% 0.23/0.49 ! [X4: complex,Y3: complex] :
% 0.23/0.49 ( ( exp_complex @ ( minus_minus_complex @ X4 @ Y3 ) )
% 0.23/0.49 = ( divide1717551699836669952omplex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_diff
% 0.23/0.49 thf(fact_4941_exp__diff,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( exp_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 0.23/0.49 = ( divide_divide_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_diff
% 0.23/0.49 thf(fact_4942_not__exp__less__zero,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ~ ( ord_less_real @ ( exp_real @ X4 ) @ zero_zero_real ) ).
% 0.23/0.49
% 0.23/0.49 % not_exp_less_zero
% 0.23/0.49 thf(fact_4943_exp__gt__zero,axiom,
% 0.23/0.49 ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_gt_zero
% 0.23/0.49 thf(fact_4944_exp__total,axiom,
% 0.23/0.49 ! [Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ? [X3: real] :
% 0.23/0.49 ( ( exp_real @ X3 )
% 0.23/0.49 = Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_total
% 0.23/0.49 thf(fact_4945_exp__ge__zero,axiom,
% 0.23/0.49 ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_ge_zero
% 0.23/0.49 thf(fact_4946_not__exp__le__zero,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ~ ( ord_less_eq_real @ ( exp_real @ X4 ) @ zero_zero_real ) ).
% 0.23/0.49
% 0.23/0.49 % not_exp_le_zero
% 0.23/0.49 thf(fact_4947_exp__minus__inverse,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) )
% 0.23/0.49 = one_one_real ) ).
% 0.23/0.49
% 0.23/0.49 % exp_minus_inverse
% 0.23/0.49 thf(fact_4948_exp__minus__inverse,axiom,
% 0.23/0.49 ! [X4: complex] :
% 0.23/0.49 ( ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) )
% 0.23/0.49 = one_one_complex ) ).
% 0.23/0.49
% 0.23/0.49 % exp_minus_inverse
% 0.23/0.49 thf(fact_4949_exp__gt__one,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ord_less_real @ one_one_real @ ( exp_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_gt_one
% 0.23/0.49 thf(fact_4950_exp__ge__add__one__self,axiom,
% 0.23/0.49 ! [X4: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( exp_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_ge_add_one_self
% 0.23/0.49 thf(fact_4951_exp__ge__add__one__self__aux,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( exp_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_ge_add_one_self_aux
% 0.23/0.49 thf(fact_4952_lemma__exp__total,axiom,
% 0.23/0.49 ! [Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ one_one_real @ Y3 )
% 0.23/0.49 => ? [X3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.23/0.49 & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y3 @ one_one_real ) )
% 0.23/0.49 & ( ( exp_real @ X3 )
% 0.23/0.49 = Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % lemma_exp_total
% 0.23/0.49 thf(fact_4953_exp__le,axiom,
% 0.23/0.49 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_le
% 0.23/0.49 thf(fact_4954_exp__double,axiom,
% 0.23/0.49 ! [Z: complex] :
% 0.23/0.49 ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 0.23/0.49 = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_double
% 0.23/0.49 thf(fact_4955_exp__double,axiom,
% 0.23/0.49 ! [Z: real] :
% 0.23/0.49 ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 0.23/0.49 = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_double
% 0.23/0.49 thf(fact_4956_exp__half__le2,axiom,
% 0.23/0.49 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_half_le2
% 0.23/0.49 thf(fact_4957_arcosh__1,axiom,
% 0.23/0.49 ( ( arcosh_real @ one_one_real )
% 0.23/0.49 = zero_zero_real ) ).
% 0.23/0.49
% 0.23/0.49 % arcosh_1
% 0.23/0.49 thf(fact_4958_floor__exists1,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ? [X3: int] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X4 )
% 0.23/0.49 & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 0.23/0.49 & ! [Y5: int] :
% 0.23/0.49 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X4 )
% 0.23/0.49 & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.23/0.49 => ( Y5 = X3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % floor_exists1
% 0.23/0.49 thf(fact_4959_floor__exists1,axiom,
% 0.23/0.49 ! [X4: rat] :
% 0.23/0.49 ? [X3: int] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X4 )
% 0.23/0.49 & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 0.23/0.49 & ! [Y5: int] :
% 0.23/0.49 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X4 )
% 0.23/0.49 & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.23/0.49 => ( Y5 = X3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % floor_exists1
% 0.23/0.49 thf(fact_4960_floor__exists,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ? [Z2: int] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X4 )
% 0.23/0.49 & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % floor_exists
% 0.23/0.49 thf(fact_4961_floor__exists,axiom,
% 0.23/0.49 ! [X4: rat] :
% 0.23/0.49 ? [Z2: int] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X4 )
% 0.23/0.49 & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % floor_exists
% 0.23/0.49 thf(fact_4962_take__bit__numeral__minus__bit1,axiom,
% 0.23/0.49 ! [L2: num,K: num] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.23/0.49 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_numeral_minus_bit1
% 0.23/0.49 thf(fact_4963_take__bit__Suc__minus__bit1,axiom,
% 0.23/0.49 ! [N: nat,K: num] :
% 0.23/0.49 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.23/0.49 = ( 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 ) ) ).
% 0.23/0.49
% 0.23/0.49 % take_bit_Suc_minus_bit1
% 0.23/0.49 thf(fact_4964_tanh__real__altdef,axiom,
% 0.23/0.49 ( tanh_real
% 0.23/0.49 = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_real_altdef
% 0.23/0.49 thf(fact_4965_ln__one__minus__pos__lower__bound,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.49 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X4 ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_one_minus_pos_lower_bound
% 0.23/0.49 thf(fact_4966_tanh__real__less__iff,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) )
% 0.23/0.49 = ( ord_less_real @ X4 @ Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_real_less_iff
% 0.23/0.49 thf(fact_4967_tanh__real__le__iff,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) )
% 0.23/0.49 = ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_real_le_iff
% 0.23/0.49 thf(fact_4968_ln__one,axiom,
% 0.23/0.49 ( ( ln_ln_real @ one_one_real )
% 0.23/0.49 = zero_zero_real ) ).
% 0.23/0.49
% 0.23/0.49 % ln_one
% 0.23/0.49 thf(fact_4969_ln__inj__iff,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ( ( ln_ln_real @ X4 )
% 0.23/0.49 = ( ln_ln_real @ Y3 ) )
% 0.23/0.49 = ( X4 = Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_inj_iff
% 0.23/0.49 thf(fact_4970_ln__less__cancel__iff,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) )
% 0.23/0.49 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_less_cancel_iff
% 0.23/0.49 thf(fact_4971_tanh__real__neg__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ ( tanh_real @ X4 ) @ zero_zero_real )
% 0.23/0.49 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_real_neg_iff
% 0.23/0.49 thf(fact_4972_tanh__real__pos__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X4 ) )
% 0.23/0.49 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_real_pos_iff
% 0.23/0.49 thf(fact_4973_tanh__real__nonpos__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( tanh_real @ X4 ) @ zero_zero_real )
% 0.23/0.49 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_real_nonpos_iff
% 0.23/0.49 thf(fact_4974_tanh__real__nonneg__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X4 ) )
% 0.23/0.49 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_real_nonneg_iff
% 0.23/0.49 thf(fact_4975_pred__numeral__inc,axiom,
% 0.23/0.49 ! [K: num] :
% 0.23/0.49 ( ( pred_numeral @ ( inc @ K ) )
% 0.23/0.49 = ( numeral_numeral_nat @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % pred_numeral_inc
% 0.23/0.49 thf(fact_4976_ln__le__cancel__iff,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) )
% 0.23/0.49 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_le_cancel_iff
% 0.23/0.49 thf(fact_4977_ln__eq__zero__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ( ln_ln_real @ X4 )
% 0.23/0.49 = zero_zero_real )
% 0.23/0.49 = ( X4 = one_one_real ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_eq_zero_iff
% 0.23/0.49 thf(fact_4978_ln__gt__zero__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
% 0.23/0.49 = ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_gt_zero_iff
% 0.23/0.49 thf(fact_4979_ln__less__zero__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
% 0.23/0.49 = ( ord_less_real @ X4 @ one_one_real ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_less_zero_iff
% 0.23/0.49 thf(fact_4980_exp__ln,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( exp_real @ ( ln_ln_real @ X4 ) )
% 0.23/0.49 = X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_ln
% 0.23/0.49 thf(fact_4981_exp__ln__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ( exp_real @ ( ln_ln_real @ X4 ) )
% 0.23/0.49 = X4 )
% 0.23/0.49 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % exp_ln_iff
% 0.23/0.49 thf(fact_4982_ln__le__zero__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
% 0.23/0.49 = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_le_zero_iff
% 0.23/0.49 thf(fact_4983_ln__ge__zero__iff,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
% 0.23/0.49 = ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_ge_zero_iff
% 0.23/0.49 thf(fact_4984_add__neg__numeral__special_I5_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_neg_numeral_special(5)
% 0.23/0.49 thf(fact_4985_add__neg__numeral__special_I5_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_neg_numeral_special(5)
% 0.23/0.49 thf(fact_4986_add__neg__numeral__special_I5_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.23/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_neg_numeral_special(5)
% 0.23/0.49 thf(fact_4987_add__neg__numeral__special_I5_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_neg_numeral_special(5)
% 0.23/0.49 thf(fact_4988_add__neg__numeral__special_I5_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_neg_numeral_special(5)
% 0.23/0.49 thf(fact_4989_add__neg__numeral__special_I6_J,axiom,
% 0.23/0.49 ! [M: num] :
% 0.23/0.49 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_neg_numeral_special(6)
% 0.23/0.49 thf(fact_4990_add__neg__numeral__special_I6_J,axiom,
% 0.23/0.49 ! [M: num] :
% 0.23/0.49 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_neg_numeral_special(6)
% 0.23/0.49 thf(fact_4991_add__neg__numeral__special_I6_J,axiom,
% 0.23/0.49 ! [M: num] :
% 0.23/0.49 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_neg_numeral_special(6)
% 0.23/0.49 thf(fact_4992_add__neg__numeral__special_I6_J,axiom,
% 0.23/0.49 ! [M: num] :
% 0.23/0.49 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_neg_numeral_special(6)
% 0.23/0.49 thf(fact_4993_add__neg__numeral__special_I6_J,axiom,
% 0.23/0.49 ! [M: num] :
% 0.23/0.49 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_neg_numeral_special(6)
% 0.23/0.49 thf(fact_4994_diff__numeral__special_I5_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 0.23/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % diff_numeral_special(5)
% 0.23/0.49 thf(fact_4995_diff__numeral__special_I5_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % diff_numeral_special(5)
% 0.23/0.49 thf(fact_4996_diff__numeral__special_I5_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 0.23/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % diff_numeral_special(5)
% 0.23/0.49 thf(fact_4997_diff__numeral__special_I5_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 0.23/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % diff_numeral_special(5)
% 0.23/0.49 thf(fact_4998_diff__numeral__special_I5_J,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 0.23/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % diff_numeral_special(5)
% 0.23/0.49 thf(fact_4999_diff__numeral__special_I6_J,axiom,
% 0.23/0.49 ! [M: num] :
% 0.23/0.49 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.49 = ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % diff_numeral_special(6)
% 0.23/0.49 thf(fact_5000_diff__numeral__special_I6_J,axiom,
% 0.23/0.49 ! [M: num] :
% 0.23/0.49 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.49 = ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % diff_numeral_special(6)
% 0.23/0.49 thf(fact_5001_diff__numeral__special_I6_J,axiom,
% 0.23/0.49 ! [M: num] :
% 0.23/0.49 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.49 = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % diff_numeral_special(6)
% 0.23/0.49 thf(fact_5002_diff__numeral__special_I6_J,axiom,
% 0.23/0.49 ! [M: num] :
% 0.23/0.49 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.49 = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % diff_numeral_special(6)
% 0.23/0.49 thf(fact_5003_diff__numeral__special_I6_J,axiom,
% 0.23/0.49 ! [M: num] :
% 0.23/0.49 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.49 = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % diff_numeral_special(6)
% 0.23/0.49 thf(fact_5004_num__induct,axiom,
% 0.23/0.49 ! [P: num > $o,X4: num] :
% 0.23/0.49 ( ( P @ one )
% 0.23/0.49 => ( ! [X3: num] :
% 0.23/0.49 ( ( P @ X3 )
% 0.23/0.49 => ( P @ ( inc @ X3 ) ) )
% 0.23/0.49 => ( P @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % num_induct
% 0.23/0.49 thf(fact_5005_add__inc,axiom,
% 0.23/0.49 ! [X4: num,Y3: num] :
% 0.23/0.49 ( ( plus_plus_num @ X4 @ ( inc @ Y3 ) )
% 0.23/0.49 = ( inc @ ( plus_plus_num @ X4 @ Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_inc
% 0.23/0.49 thf(fact_5006_ln__less__self,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ord_less_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_less_self
% 0.23/0.49 thf(fact_5007_tanh__real__lt__1,axiom,
% 0.23/0.49 ! [X4: real] : ( ord_less_real @ ( tanh_real @ X4 ) @ one_one_real ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_real_lt_1
% 0.23/0.49 thf(fact_5008_inc_Osimps_I1_J,axiom,
% 0.23/0.49 ( ( inc @ one )
% 0.23/0.49 = ( bit0 @ one ) ) ).
% 0.23/0.49
% 0.23/0.49 % inc.simps(1)
% 0.23/0.49 thf(fact_5009_inc_Osimps_I2_J,axiom,
% 0.23/0.49 ! [X4: num] :
% 0.23/0.49 ( ( inc @ ( bit0 @ X4 ) )
% 0.23/0.49 = ( bit1 @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % inc.simps(2)
% 0.23/0.49 thf(fact_5010_inc_Osimps_I3_J,axiom,
% 0.23/0.49 ! [X4: num] :
% 0.23/0.49 ( ( inc @ ( bit1 @ X4 ) )
% 0.23/0.49 = ( bit0 @ ( inc @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % inc.simps(3)
% 0.23/0.49 thf(fact_5011_ln__bound,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_bound
% 0.23/0.49 thf(fact_5012_ln__gt__zero,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.49 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_gt_zero
% 0.23/0.49 thf(fact_5013_ln__less__zero,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.23/0.49 => ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_less_zero
% 0.23/0.49 thf(fact_5014_ln__gt__zero__imp__gt__one,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
% 0.23/0.49 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_gt_zero_imp_gt_one
% 0.23/0.49 thf(fact_5015_ln__ge__zero,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ one_one_real @ X4 )
% 0.23/0.49 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_ge_zero
% 0.23/0.49 thf(fact_5016_add__One,axiom,
% 0.23/0.49 ! [X4: num] :
% 0.23/0.49 ( ( plus_plus_num @ X4 @ one )
% 0.23/0.49 = ( inc @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % add_One
% 0.23/0.49 thf(fact_5017_mult__inc,axiom,
% 0.23/0.49 ! [X4: num,Y3: num] :
% 0.23/0.49 ( ( times_times_num @ X4 @ ( inc @ Y3 ) )
% 0.23/0.49 = ( plus_plus_num @ ( times_times_num @ X4 @ Y3 ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % mult_inc
% 0.23/0.49 thf(fact_5018_tanh__real__gt__neg1,axiom,
% 0.23/0.49 ! [X4: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_real_gt_neg1
% 0.23/0.49 thf(fact_5019_ln__ge__zero__imp__ge__one,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
% 0.23/0.49 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_ge_zero_imp_ge_one
% 0.23/0.49 thf(fact_5020_ln__add__one__self__le__self,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_add_one_self_le_self
% 0.23/0.49 thf(fact_5021_ln__mult,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ( ln_ln_real @ ( times_times_real @ X4 @ Y3 ) )
% 0.23/0.49 = ( plus_plus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_mult
% 0.23/0.49 thf(fact_5022_ln__eq__minus__one,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ( ln_ln_real @ X4 )
% 0.23/0.49 = ( minus_minus_real @ X4 @ one_one_real ) )
% 0.23/0.49 => ( X4 = one_one_real ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_eq_minus_one
% 0.23/0.49 thf(fact_5023_ln__div,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ( ln_ln_real @ ( divide_divide_real @ X4 @ Y3 ) )
% 0.23/0.49 = ( minus_minus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_div
% 0.23/0.49 thf(fact_5024_ln__ge__iff,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X4 ) )
% 0.23/0.49 = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_ge_iff
% 0.23/0.49 thf(fact_5025_ln__x__over__x__mono,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.49 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_x_over_x_mono
% 0.23/0.49 thf(fact_5026_numeral__inc,axiom,
% 0.23/0.49 ! [X4: num] :
% 0.23/0.49 ( ( numera6690914467698888265omplex @ ( inc @ X4 ) )
% 0.23/0.49 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X4 ) @ one_one_complex ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_inc
% 0.23/0.49 thf(fact_5027_numeral__inc,axiom,
% 0.23/0.49 ! [X4: num] :
% 0.23/0.49 ( ( numeral_numeral_real @ ( inc @ X4 ) )
% 0.23/0.49 = ( plus_plus_real @ ( numeral_numeral_real @ X4 ) @ one_one_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_inc
% 0.23/0.49 thf(fact_5028_numeral__inc,axiom,
% 0.23/0.49 ! [X4: num] :
% 0.23/0.49 ( ( numeral_numeral_rat @ ( inc @ X4 ) )
% 0.23/0.49 = ( plus_plus_rat @ ( numeral_numeral_rat @ X4 ) @ one_one_rat ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_inc
% 0.23/0.49 thf(fact_5029_numeral__inc,axiom,
% 0.23/0.49 ! [X4: num] :
% 0.23/0.49 ( ( numeral_numeral_nat @ ( inc @ X4 ) )
% 0.23/0.49 = ( plus_plus_nat @ ( numeral_numeral_nat @ X4 ) @ one_one_nat ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_inc
% 0.23/0.49 thf(fact_5030_numeral__inc,axiom,
% 0.23/0.49 ! [X4: num] :
% 0.23/0.49 ( ( numeral_numeral_int @ ( inc @ X4 ) )
% 0.23/0.49 = ( plus_plus_int @ ( numeral_numeral_int @ X4 ) @ one_one_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % numeral_inc
% 0.23/0.49 thf(fact_5031_ln__2__less__1,axiom,
% 0.23/0.49 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 0.23/0.49
% 0.23/0.49 % ln_2_less_1
% 0.23/0.49 thf(fact_5032_ln__le__minus__one,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_le_minus_one
% 0.23/0.49 thf(fact_5033_ln__diff__le,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.49 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X4 @ Y3 ) @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_diff_le
% 0.23/0.49 thf(fact_5034_ln__add__one__self__le__self2,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.49 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_add_one_self_le_self2
% 0.23/0.49 thf(fact_5035_tanh__ln__real,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( tanh_real @ ( ln_ln_real @ X4 ) )
% 0.23/0.49 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_ln_real
% 0.23/0.49 thf(fact_5036_arcosh__real__def,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ one_one_real @ X4 )
% 0.23/0.49 => ( ( arcosh_real @ X4 )
% 0.23/0.49 = ( ln_ln_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % arcosh_real_def
% 0.23/0.49 thf(fact_5037_ln__one__minus__pos__upper__bound,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.23/0.49 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X4 ) ) @ ( uminus_uminus_real @ X4 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_one_minus_pos_upper_bound
% 0.23/0.49 thf(fact_5038_ln__sqrt,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ln_ln_real @ ( sqrt @ X4 ) )
% 0.23/0.49 = ( divide_divide_real @ ( ln_ln_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_sqrt
% 0.23/0.49 thf(fact_5039_exists__least__lemma,axiom,
% 0.23/0.49 ! [P: nat > $o] :
% 0.23/0.49 ( ~ ( P @ zero_zero_nat )
% 0.23/0.49 => ( ? [X_12: nat] : ( P @ X_12 )
% 0.23/0.49 => ? [N4: nat] :
% 0.23/0.49 ( ~ ( P @ N4 )
% 0.23/0.49 & ( P @ ( suc @ N4 ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % exists_least_lemma
% 0.23/0.49 thf(fact_5040_ex__le__of__int,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ? [Z2: int] : ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % ex_le_of_int
% 0.23/0.49 thf(fact_5041_ex__le__of__int,axiom,
% 0.23/0.49 ! [X4: rat] :
% 0.23/0.49 ? [Z2: int] : ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % ex_le_of_int
% 0.23/0.49 thf(fact_5042_ex__of__int__less,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X4 ) ).
% 0.23/0.49
% 0.23/0.49 % ex_of_int_less
% 0.23/0.49 thf(fact_5043_ex__of__int__less,axiom,
% 0.23/0.49 ! [X4: rat] :
% 0.23/0.49 ? [Z2: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X4 ) ).
% 0.23/0.49
% 0.23/0.49 % ex_of_int_less
% 0.23/0.49 thf(fact_5044_ex__less__of__int,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ? [Z2: int] : ( ord_less_real @ X4 @ ( ring_1_of_int_real @ Z2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % ex_less_of_int
% 0.23/0.49 thf(fact_5045_ex__less__of__int,axiom,
% 0.23/0.49 ! [X4: rat] :
% 0.23/0.49 ? [Z2: int] : ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ Z2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % ex_less_of_int
% 0.23/0.49 thf(fact_5046_tanh__altdef,axiom,
% 0.23/0.49 ( tanh_real
% 0.23/0.49 = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_altdef
% 0.23/0.49 thf(fact_5047_tanh__altdef,axiom,
% 0.23/0.49 ( tanh_complex
% 0.23/0.49 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % tanh_altdef
% 0.23/0.49 thf(fact_5048_ln__one__plus__pos__lower__bound,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.49 => ( ord_less_eq_real @ ( minus_minus_real @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % ln_one_plus_pos_lower_bound
% 0.23/0.49 thf(fact_5049_artanh__def,axiom,
% 0.23/0.49 ( artanh_real
% 0.23/0.49 = ( ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X ) @ ( minus_minus_real @ one_one_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % artanh_def
% 0.23/0.49 thf(fact_5050_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.49 => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.23/0.49 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ln_one_plus_x_minus_x_bound_nonpos
% 0.23/0.49 thf(fact_5051_round__unique,axiom,
% 0.23/0.49 ! [X4: real,Y3: int] :
% 0.23/0.49 ( ( ord_less_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
% 0.23/0.49 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.23/0.49 => ( ( archim8280529875227126926d_real @ X4 )
% 0.23/0.49 = Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % round_unique
% 0.23/0.49 thf(fact_5052_round__unique,axiom,
% 0.23/0.49 ! [X4: rat,Y3: int] :
% 0.23/0.49 ( ( ord_less_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
% 0.23/0.49 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.49 => ( ( archim7778729529865785530nd_rat @ X4 )
% 0.23/0.49 = Y3 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % round_unique
% 0.23/0.49 thf(fact_5053_arsinh__real__def,axiom,
% 0.23/0.49 ( arsinh_real
% 0.23/0.49 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % arsinh_real_def
% 0.23/0.49 thf(fact_5054_abs__ln__one__plus__x__minus__x__bound,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.49 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ln_one_plus_x_minus_x_bound
% 0.23/0.49 thf(fact_5055_of__int__round__gt,axiom,
% 0.23/0.49 ! [X4: real] : ( ord_less_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_round_gt
% 0.23/0.49 thf(fact_5056_of__int__round__gt,axiom,
% 0.23/0.49 ! [X4: rat] : ( ord_less_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_round_gt
% 0.23/0.49 thf(fact_5057_of__int__round__ge,axiom,
% 0.23/0.49 ! [X4: real] : ( ord_less_eq_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_round_ge
% 0.23/0.49 thf(fact_5058_of__int__round__ge,axiom,
% 0.23/0.49 ! [X4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % of_int_round_ge
% 0.23/0.49 thf(fact_5059_abs__abs,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.23/0.49 = ( abs_abs_real @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_abs
% 0.23/0.49 thf(fact_5060_abs__abs,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.23/0.49 = ( abs_abs_int @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_abs
% 0.23/0.49 thf(fact_5061_abs__abs,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.23/0.49 = ( abs_abs_Code_integer @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_abs
% 0.23/0.49 thf(fact_5062_abs__abs,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.23/0.49 = ( abs_abs_rat @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_abs
% 0.23/0.49 thf(fact_5063_abs__idempotent,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.23/0.49 = ( abs_abs_real @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_idempotent
% 0.23/0.49 thf(fact_5064_abs__idempotent,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.23/0.49 = ( abs_abs_int @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_idempotent
% 0.23/0.49 thf(fact_5065_abs__idempotent,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.23/0.49 = ( abs_abs_Code_integer @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_idempotent
% 0.23/0.49 thf(fact_5066_abs__idempotent,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.23/0.49 = ( abs_abs_rat @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_idempotent
% 0.23/0.49 thf(fact_5067_abs__0,axiom,
% 0.23/0.49 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.23/0.49 = zero_z3403309356797280102nteger ) ).
% 0.23/0.49
% 0.23/0.49 % abs_0
% 0.23/0.49 thf(fact_5068_abs__0,axiom,
% 0.23/0.49 ( ( abs_abs_real @ zero_zero_real )
% 0.23/0.49 = zero_zero_real ) ).
% 0.23/0.49
% 0.23/0.49 % abs_0
% 0.23/0.49 thf(fact_5069_abs__0,axiom,
% 0.23/0.49 ( ( abs_abs_rat @ zero_zero_rat )
% 0.23/0.49 = zero_zero_rat ) ).
% 0.23/0.49
% 0.23/0.49 % abs_0
% 0.23/0.49 thf(fact_5070_abs__0,axiom,
% 0.23/0.49 ( ( abs_abs_int @ zero_zero_int )
% 0.23/0.49 = zero_zero_int ) ).
% 0.23/0.49
% 0.23/0.49 % abs_0
% 0.23/0.49 thf(fact_5071_abs__0__eq,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( zero_z3403309356797280102nteger
% 0.23/0.49 = ( abs_abs_Code_integer @ A ) )
% 0.23/0.49 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_0_eq
% 0.23/0.49 thf(fact_5072_abs__0__eq,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( zero_zero_real
% 0.23/0.49 = ( abs_abs_real @ A ) )
% 0.23/0.49 = ( A = zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_0_eq
% 0.23/0.49 thf(fact_5073_abs__0__eq,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( zero_zero_rat
% 0.23/0.49 = ( abs_abs_rat @ A ) )
% 0.23/0.49 = ( A = zero_zero_rat ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_0_eq
% 0.23/0.49 thf(fact_5074_abs__0__eq,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( zero_zero_int
% 0.23/0.49 = ( abs_abs_int @ A ) )
% 0.23/0.49 = ( A = zero_zero_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_0_eq
% 0.23/0.49 thf(fact_5075_abs__eq__0,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( ( abs_abs_Code_integer @ A )
% 0.23/0.49 = zero_z3403309356797280102nteger )
% 0.23/0.49 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_0
% 0.23/0.49 thf(fact_5076_abs__eq__0,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( ( abs_abs_real @ A )
% 0.23/0.49 = zero_zero_real )
% 0.23/0.49 = ( A = zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_0
% 0.23/0.49 thf(fact_5077_abs__eq__0,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( ( abs_abs_rat @ A )
% 0.23/0.49 = zero_zero_rat )
% 0.23/0.49 = ( A = zero_zero_rat ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_0
% 0.23/0.49 thf(fact_5078_abs__eq__0,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( ( abs_abs_int @ A )
% 0.23/0.49 = zero_zero_int )
% 0.23/0.49 = ( A = zero_zero_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_0
% 0.23/0.49 thf(fact_5079_abs__zero,axiom,
% 0.23/0.49 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.23/0.49 = zero_z3403309356797280102nteger ) ).
% 0.23/0.49
% 0.23/0.49 % abs_zero
% 0.23/0.49 thf(fact_5080_abs__zero,axiom,
% 0.23/0.49 ( ( abs_abs_real @ zero_zero_real )
% 0.23/0.49 = zero_zero_real ) ).
% 0.23/0.49
% 0.23/0.49 % abs_zero
% 0.23/0.49 thf(fact_5081_abs__zero,axiom,
% 0.23/0.49 ( ( abs_abs_rat @ zero_zero_rat )
% 0.23/0.49 = zero_zero_rat ) ).
% 0.23/0.49
% 0.23/0.49 % abs_zero
% 0.23/0.49 thf(fact_5082_abs__zero,axiom,
% 0.23/0.49 ( ( abs_abs_int @ zero_zero_int )
% 0.23/0.49 = zero_zero_int ) ).
% 0.23/0.49
% 0.23/0.49 % abs_zero
% 0.23/0.49 thf(fact_5083_abs__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 0.23/0.49 = ( numera6620942414471956472nteger @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_numeral
% 0.23/0.49 thf(fact_5084_abs__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 0.23/0.49 = ( numeral_numeral_real @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_numeral
% 0.23/0.49 thf(fact_5085_abs__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 0.23/0.49 = ( numeral_numeral_rat @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_numeral
% 0.23/0.49 thf(fact_5086_abs__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 0.23/0.49 = ( numeral_numeral_int @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_numeral
% 0.23/0.49 thf(fact_5087_abs__mult__self__eq,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 0.23/0.49 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult_self_eq
% 0.23/0.49 thf(fact_5088_abs__mult__self__eq,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 0.23/0.49 = ( times_times_real @ A @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult_self_eq
% 0.23/0.49 thf(fact_5089_abs__mult__self__eq,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 0.23/0.49 = ( times_times_rat @ A @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult_self_eq
% 0.23/0.49 thf(fact_5090_abs__mult__self__eq,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 0.23/0.49 = ( times_times_int @ A @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult_self_eq
% 0.23/0.49 thf(fact_5091_abs__1,axiom,
% 0.23/0.49 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.23/0.49 = one_one_Code_integer ) ).
% 0.23/0.49
% 0.23/0.49 % abs_1
% 0.23/0.49 thf(fact_5092_abs__1,axiom,
% 0.23/0.49 ( ( abs_abs_complex @ one_one_complex )
% 0.23/0.49 = one_one_complex ) ).
% 0.23/0.49
% 0.23/0.49 % abs_1
% 0.23/0.49 thf(fact_5093_abs__1,axiom,
% 0.23/0.49 ( ( abs_abs_real @ one_one_real )
% 0.23/0.49 = one_one_real ) ).
% 0.23/0.49
% 0.23/0.49 % abs_1
% 0.23/0.49 thf(fact_5094_abs__1,axiom,
% 0.23/0.49 ( ( abs_abs_rat @ one_one_rat )
% 0.23/0.49 = one_one_rat ) ).
% 0.23/0.49
% 0.23/0.49 % abs_1
% 0.23/0.49 thf(fact_5095_abs__1,axiom,
% 0.23/0.49 ( ( abs_abs_int @ one_one_int )
% 0.23/0.49 = one_one_int ) ).
% 0.23/0.49
% 0.23/0.49 % abs_1
% 0.23/0.49 thf(fact_5096_abs__add__abs,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) )
% 0.23/0.49 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_add_abs
% 0.23/0.49 thf(fact_5097_abs__add__abs,axiom,
% 0.23/0.49 ! [A: real,B2: real] :
% 0.23/0.49 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) )
% 0.23/0.49 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_add_abs
% 0.23/0.49 thf(fact_5098_abs__add__abs,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] :
% 0.23/0.49 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) )
% 0.23/0.49 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_add_abs
% 0.23/0.49 thf(fact_5099_abs__add__abs,axiom,
% 0.23/0.49 ! [A: int,B2: int] :
% 0.23/0.49 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) )
% 0.23/0.49 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_add_abs
% 0.23/0.49 thf(fact_5100_abs__divide,axiom,
% 0.23/0.49 ! [A: complex,B2: complex] :
% 0.23/0.49 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.23/0.49 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_divide
% 0.23/0.49 thf(fact_5101_abs__divide,axiom,
% 0.23/0.49 ! [A: real,B2: real] :
% 0.23/0.49 ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
% 0.23/0.49 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_divide
% 0.23/0.49 thf(fact_5102_abs__divide,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] :
% 0.23/0.49 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.23/0.49 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_divide
% 0.23/0.49 thf(fact_5103_abs__minus,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.23/0.49 = ( abs_abs_real @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus
% 0.23/0.49 thf(fact_5104_abs__minus,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.23/0.49 = ( abs_abs_int @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus
% 0.23/0.49 thf(fact_5105_abs__minus,axiom,
% 0.23/0.49 ! [A: complex] :
% 0.23/0.49 ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 0.23/0.49 = ( abs_abs_complex @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus
% 0.23/0.49 thf(fact_5106_abs__minus,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.49 = ( abs_abs_Code_integer @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus
% 0.23/0.49 thf(fact_5107_abs__minus,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.23/0.49 = ( abs_abs_rat @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus
% 0.23/0.49 thf(fact_5108_abs__minus__cancel,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.23/0.49 = ( abs_abs_real @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus_cancel
% 0.23/0.49 thf(fact_5109_abs__minus__cancel,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.23/0.49 = ( abs_abs_int @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus_cancel
% 0.23/0.49 thf(fact_5110_abs__minus__cancel,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.49 = ( abs_abs_Code_integer @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus_cancel
% 0.23/0.49 thf(fact_5111_abs__minus__cancel,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.23/0.49 = ( abs_abs_rat @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus_cancel
% 0.23/0.49 thf(fact_5112_abs__dvd__iff,axiom,
% 0.23/0.49 ! [M: real,K: real] :
% 0.23/0.49 ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 0.23/0.49 = ( dvd_dvd_real @ M @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_dvd_iff
% 0.23/0.49 thf(fact_5113_abs__dvd__iff,axiom,
% 0.23/0.49 ! [M: int,K: int] :
% 0.23/0.49 ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 0.23/0.49 = ( dvd_dvd_int @ M @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_dvd_iff
% 0.23/0.49 thf(fact_5114_abs__dvd__iff,axiom,
% 0.23/0.49 ! [M: code_integer,K: code_integer] :
% 0.23/0.49 ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 0.23/0.49 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_dvd_iff
% 0.23/0.49 thf(fact_5115_abs__dvd__iff,axiom,
% 0.23/0.49 ! [M: rat,K: rat] :
% 0.23/0.49 ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 0.23/0.49 = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_dvd_iff
% 0.23/0.49 thf(fact_5116_dvd__abs__iff,axiom,
% 0.23/0.49 ! [M: real,K: real] :
% 0.23/0.49 ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 0.23/0.49 = ( dvd_dvd_real @ M @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % dvd_abs_iff
% 0.23/0.49 thf(fact_5117_dvd__abs__iff,axiom,
% 0.23/0.49 ! [M: int,K: int] :
% 0.23/0.49 ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 0.23/0.49 = ( dvd_dvd_int @ M @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % dvd_abs_iff
% 0.23/0.49 thf(fact_5118_dvd__abs__iff,axiom,
% 0.23/0.49 ! [M: code_integer,K: code_integer] :
% 0.23/0.49 ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 0.23/0.49 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % dvd_abs_iff
% 0.23/0.49 thf(fact_5119_dvd__abs__iff,axiom,
% 0.23/0.49 ! [M: rat,K: rat] :
% 0.23/0.49 ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 0.23/0.49 = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % dvd_abs_iff
% 0.23/0.49 thf(fact_5120_abs__bool__eq,axiom,
% 0.23/0.49 ! [P: $o] :
% 0.23/0.49 ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.23/0.49 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_bool_eq
% 0.23/0.49 thf(fact_5121_abs__bool__eq,axiom,
% 0.23/0.49 ! [P: $o] :
% 0.23/0.49 ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.23/0.49 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_bool_eq
% 0.23/0.49 thf(fact_5122_abs__bool__eq,axiom,
% 0.23/0.49 ! [P: $o] :
% 0.23/0.49 ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.23/0.49 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_bool_eq
% 0.23/0.49 thf(fact_5123_abs__bool__eq,axiom,
% 0.23/0.49 ! [P: $o] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 0.23/0.49 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_bool_eq
% 0.23/0.49 thf(fact_5124_abs__le__zero__iff,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.23/0.49 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_zero_iff
% 0.23/0.49 thf(fact_5125_abs__le__zero__iff,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.23/0.49 = ( A = zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_zero_iff
% 0.23/0.49 thf(fact_5126_abs__le__zero__iff,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 0.23/0.49 = ( A = zero_zero_rat ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_zero_iff
% 0.23/0.49 thf(fact_5127_abs__le__zero__iff,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 0.23/0.49 = ( A = zero_zero_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_zero_iff
% 0.23/0.49 thf(fact_5128_abs__le__self__iff,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 0.23/0.49 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_self_iff
% 0.23/0.49 thf(fact_5129_abs__le__self__iff,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.23/0.49 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_self_iff
% 0.23/0.49 thf(fact_5130_abs__le__self__iff,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 0.23/0.49 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_self_iff
% 0.23/0.49 thf(fact_5131_abs__le__self__iff,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 0.23/0.49 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_self_iff
% 0.23/0.49 thf(fact_5132_abs__of__nonneg,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.23/0.49 => ( ( abs_abs_Code_integer @ A )
% 0.23/0.49 = A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_nonneg
% 0.23/0.49 thf(fact_5133_abs__of__nonneg,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.49 => ( ( abs_abs_real @ A )
% 0.23/0.49 = A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_nonneg
% 0.23/0.49 thf(fact_5134_abs__of__nonneg,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.49 => ( ( abs_abs_rat @ A )
% 0.23/0.49 = A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_nonneg
% 0.23/0.49 thf(fact_5135_abs__of__nonneg,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.23/0.49 => ( ( abs_abs_int @ A )
% 0.23/0.49 = A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_nonneg
% 0.23/0.49 thf(fact_5136_zero__less__abs__iff,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 0.23/0.49 = ( A != zero_z3403309356797280102nteger ) ) ).
% 0.23/0.49
% 0.23/0.49 % zero_less_abs_iff
% 0.23/0.49 thf(fact_5137_zero__less__abs__iff,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.23/0.49 = ( A != zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % zero_less_abs_iff
% 0.23/0.49 thf(fact_5138_zero__less__abs__iff,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 0.23/0.49 = ( A != zero_zero_rat ) ) ).
% 0.23/0.49
% 0.23/0.49 % zero_less_abs_iff
% 0.23/0.49 thf(fact_5139_zero__less__abs__iff,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 0.23/0.49 = ( A != zero_zero_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % zero_less_abs_iff
% 0.23/0.49 thf(fact_5140_abs__neg__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.49 = ( numeral_numeral_real @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_neg_numeral
% 0.23/0.49 thf(fact_5141_abs__neg__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.49 = ( numeral_numeral_int @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_neg_numeral
% 0.23/0.49 thf(fact_5142_abs__neg__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.49 = ( numera6620942414471956472nteger @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_neg_numeral
% 0.23/0.49 thf(fact_5143_abs__neg__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.49 = ( numeral_numeral_rat @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_neg_numeral
% 0.23/0.49 thf(fact_5144_abs__neg__one,axiom,
% 0.23/0.49 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.49 = one_one_real ) ).
% 0.23/0.49
% 0.23/0.49 % abs_neg_one
% 0.23/0.49 thf(fact_5145_abs__neg__one,axiom,
% 0.23/0.49 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.49 = one_one_int ) ).
% 0.23/0.49
% 0.23/0.49 % abs_neg_one
% 0.23/0.49 thf(fact_5146_abs__neg__one,axiom,
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.49 = one_one_Code_integer ) ).
% 0.23/0.49
% 0.23/0.49 % abs_neg_one
% 0.23/0.49 thf(fact_5147_abs__neg__one,axiom,
% 0.23/0.49 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.49 = one_one_rat ) ).
% 0.23/0.49
% 0.23/0.49 % abs_neg_one
% 0.23/0.49 thf(fact_5148_abs__power__minus,axiom,
% 0.23/0.49 ! [A: real,N: nat] :
% 0.23/0.49 ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.23/0.49 = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_power_minus
% 0.23/0.49 thf(fact_5149_abs__power__minus,axiom,
% 0.23/0.49 ! [A: int,N: nat] :
% 0.23/0.49 ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.23/0.49 = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_power_minus
% 0.23/0.49 thf(fact_5150_abs__power__minus,axiom,
% 0.23/0.49 ! [A: code_integer,N: nat] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.23/0.49 = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_power_minus
% 0.23/0.49 thf(fact_5151_abs__power__minus,axiom,
% 0.23/0.49 ! [A: rat,N: nat] :
% 0.23/0.49 ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.23/0.49 = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_power_minus
% 0.23/0.49 thf(fact_5152_real__sqrt__abs2,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( sqrt @ ( times_times_real @ X4 @ X4 ) )
% 0.23/0.49 = ( abs_abs_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_abs2
% 0.23/0.49 thf(fact_5153_real__sqrt__mult__self,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 0.23/0.49 = ( abs_abs_real @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_mult_self
% 0.23/0.49 thf(fact_5154_round__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 0.23/0.49 = ( numeral_numeral_int @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % round_numeral
% 0.23/0.49 thf(fact_5155_round__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 0.23/0.49 = ( numeral_numeral_int @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % round_numeral
% 0.23/0.49 thf(fact_5156_round__1,axiom,
% 0.23/0.49 ( ( archim8280529875227126926d_real @ one_one_real )
% 0.23/0.49 = one_one_int ) ).
% 0.23/0.49
% 0.23/0.49 % round_1
% 0.23/0.49 thf(fact_5157_round__1,axiom,
% 0.23/0.49 ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 0.23/0.49 = one_one_int ) ).
% 0.23/0.49
% 0.23/0.49 % round_1
% 0.23/0.49 thf(fact_5158_zero__le__divide__abs__iff,axiom,
% 0.23/0.49 ! [A: real,B2: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) )
% 0.23/0.49 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.49 | ( B2 = zero_zero_real ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % zero_le_divide_abs_iff
% 0.23/0.49 thf(fact_5159_zero__le__divide__abs__iff,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) )
% 0.23/0.49 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.49 | ( B2 = zero_zero_rat ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % zero_le_divide_abs_iff
% 0.23/0.49 thf(fact_5160_divide__le__0__abs__iff,axiom,
% 0.23/0.49 ! [A: real,B2: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) @ zero_zero_real )
% 0.23/0.49 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.23/0.49 | ( B2 = zero_zero_real ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % divide_le_0_abs_iff
% 0.23/0.49 thf(fact_5161_divide__le__0__abs__iff,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) @ zero_zero_rat )
% 0.23/0.49 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.23/0.49 | ( B2 = zero_zero_rat ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % divide_le_0_abs_iff
% 0.23/0.49 thf(fact_5162_abs__of__nonpos,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.23/0.49 => ( ( abs_abs_real @ A )
% 0.23/0.49 = ( uminus_uminus_real @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_nonpos
% 0.23/0.49 thf(fact_5163_abs__of__nonpos,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 0.23/0.49 => ( ( abs_abs_Code_integer @ A )
% 0.23/0.49 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_nonpos
% 0.23/0.49 thf(fact_5164_abs__of__nonpos,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.23/0.49 => ( ( abs_abs_rat @ A )
% 0.23/0.49 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_nonpos
% 0.23/0.49 thf(fact_5165_abs__of__nonpos,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.23/0.49 => ( ( abs_abs_int @ A )
% 0.23/0.49 = ( uminus_uminus_int @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_nonpos
% 0.23/0.49 thf(fact_5166_artanh__minus__real,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.49 => ( ( artanh_real @ ( uminus_uminus_real @ X4 ) )
% 0.23/0.49 = ( uminus_uminus_real @ ( artanh_real @ X4 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % artanh_minus_real
% 0.23/0.49 thf(fact_5167_zero__less__power__abs__iff,axiom,
% 0.23/0.49 ! [A: code_integer,N: nat] :
% 0.23/0.49 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 0.23/0.49 = ( ( A != zero_z3403309356797280102nteger )
% 0.23/0.49 | ( N = zero_zero_nat ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % zero_less_power_abs_iff
% 0.23/0.49 thf(fact_5168_zero__less__power__abs__iff,axiom,
% 0.23/0.49 ! [A: real,N: nat] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.23/0.49 = ( ( A != zero_zero_real )
% 0.23/0.49 | ( N = zero_zero_nat ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % zero_less_power_abs_iff
% 0.23/0.49 thf(fact_5169_zero__less__power__abs__iff,axiom,
% 0.23/0.49 ! [A: rat,N: nat] :
% 0.23/0.49 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 0.23/0.49 = ( ( A != zero_zero_rat )
% 0.23/0.49 | ( N = zero_zero_nat ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % zero_less_power_abs_iff
% 0.23/0.49 thf(fact_5170_zero__less__power__abs__iff,axiom,
% 0.23/0.49 ! [A: int,N: nat] :
% 0.23/0.49 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 0.23/0.49 = ( ( A != zero_zero_int )
% 0.23/0.49 | ( N = zero_zero_nat ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % zero_less_power_abs_iff
% 0.23/0.49 thf(fact_5171_abs__power2,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.49 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_power2
% 0.23/0.49 thf(fact_5172_abs__power2,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.49 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_power2
% 0.23/0.49 thf(fact_5173_abs__power2,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.49 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_power2
% 0.23/0.49 thf(fact_5174_abs__power2,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.49 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_power2
% 0.23/0.49 thf(fact_5175_power2__abs,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.49 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % power2_abs
% 0.23/0.49 thf(fact_5176_power2__abs,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.49 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % power2_abs
% 0.23/0.49 thf(fact_5177_power2__abs,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.49 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % power2_abs
% 0.23/0.49 thf(fact_5178_power2__abs,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.49 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % power2_abs
% 0.23/0.49 thf(fact_5179_round__neg__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % round_neg_numeral
% 0.23/0.49 thf(fact_5180_round__neg__numeral,axiom,
% 0.23/0.49 ! [N: num] :
% 0.23/0.49 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.23/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % round_neg_numeral
% 0.23/0.49 thf(fact_5181_power__even__abs__numeral,axiom,
% 0.23/0.49 ! [W: num,A: code_integer] :
% 0.23/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.23/0.49 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.23/0.49 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % power_even_abs_numeral
% 0.23/0.49 thf(fact_5182_power__even__abs__numeral,axiom,
% 0.23/0.49 ! [W: num,A: rat] :
% 0.23/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.23/0.49 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.23/0.49 = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % power_even_abs_numeral
% 0.23/0.49 thf(fact_5183_power__even__abs__numeral,axiom,
% 0.23/0.49 ! [W: num,A: real] :
% 0.23/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.23/0.49 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.23/0.49 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % power_even_abs_numeral
% 0.23/0.49 thf(fact_5184_power__even__abs__numeral,axiom,
% 0.23/0.49 ! [W: num,A: int] :
% 0.23/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.23/0.49 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.23/0.49 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % power_even_abs_numeral
% 0.23/0.49 thf(fact_5185_real__sqrt__abs,axiom,
% 0.23/0.49 ! [X4: real] :
% 0.23/0.49 ( ( sqrt @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.49 = ( abs_abs_real @ X4 ) ) ).
% 0.23/0.49
% 0.23/0.49 % real_sqrt_abs
% 0.23/0.49 thf(fact_5186_abs__ge__self,axiom,
% 0.23/0.49 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_self
% 0.23/0.49 thf(fact_5187_abs__ge__self,axiom,
% 0.23/0.49 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_self
% 0.23/0.49 thf(fact_5188_abs__ge__self,axiom,
% 0.23/0.49 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_self
% 0.23/0.49 thf(fact_5189_abs__ge__self,axiom,
% 0.23/0.49 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_self
% 0.23/0.49 thf(fact_5190_abs__le__D1,axiom,
% 0.23/0.49 ! [A: real,B2: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 0.23/0.49 => ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_D1
% 0.23/0.49 thf(fact_5191_abs__le__D1,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] :
% 0.23/0.49 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 0.23/0.49 => ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_D1
% 0.23/0.49 thf(fact_5192_abs__le__D1,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 0.23/0.49 => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_D1
% 0.23/0.49 thf(fact_5193_abs__le__D1,axiom,
% 0.23/0.49 ! [A: int,B2: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 0.23/0.49 => ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_D1
% 0.23/0.49 thf(fact_5194_abs__eq__0__iff,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( ( abs_abs_Code_integer @ A )
% 0.23/0.49 = zero_z3403309356797280102nteger )
% 0.23/0.49 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_0_iff
% 0.23/0.49 thf(fact_5195_abs__eq__0__iff,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( ( abs_abs_real @ A )
% 0.23/0.49 = zero_zero_real )
% 0.23/0.49 = ( A = zero_zero_real ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_0_iff
% 0.23/0.49 thf(fact_5196_abs__eq__0__iff,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( ( abs_abs_rat @ A )
% 0.23/0.49 = zero_zero_rat )
% 0.23/0.49 = ( A = zero_zero_rat ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_0_iff
% 0.23/0.49 thf(fact_5197_abs__eq__0__iff,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( ( abs_abs_int @ A )
% 0.23/0.49 = zero_zero_int )
% 0.23/0.49 = ( A = zero_zero_int ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_0_iff
% 0.23/0.49 thf(fact_5198_abs__mult,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 0.23/0.49 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult
% 0.23/0.49 thf(fact_5199_abs__mult,axiom,
% 0.23/0.49 ! [A: real,B2: real] :
% 0.23/0.49 ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
% 0.23/0.49 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult
% 0.23/0.49 thf(fact_5200_abs__mult,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] :
% 0.23/0.49 ( ( abs_abs_rat @ ( times_times_rat @ A @ B2 ) )
% 0.23/0.49 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult
% 0.23/0.49 thf(fact_5201_abs__mult,axiom,
% 0.23/0.49 ! [A: int,B2: int] :
% 0.23/0.49 ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
% 0.23/0.49 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult
% 0.23/0.49 thf(fact_5202_abs__one,axiom,
% 0.23/0.49 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.23/0.49 = one_one_Code_integer ) ).
% 0.23/0.49
% 0.23/0.49 % abs_one
% 0.23/0.49 thf(fact_5203_abs__one,axiom,
% 0.23/0.49 ( ( abs_abs_real @ one_one_real )
% 0.23/0.49 = one_one_real ) ).
% 0.23/0.49
% 0.23/0.49 % abs_one
% 0.23/0.49 thf(fact_5204_abs__one,axiom,
% 0.23/0.49 ( ( abs_abs_rat @ one_one_rat )
% 0.23/0.49 = one_one_rat ) ).
% 0.23/0.49
% 0.23/0.49 % abs_one
% 0.23/0.49 thf(fact_5205_abs__one,axiom,
% 0.23/0.49 ( ( abs_abs_int @ one_one_int )
% 0.23/0.49 = one_one_int ) ).
% 0.23/0.49
% 0.23/0.49 % abs_one
% 0.23/0.49 thf(fact_5206_abs__minus__commute,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 0.23/0.49 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus_commute
% 0.23/0.49 thf(fact_5207_abs__minus__commute,axiom,
% 0.23/0.49 ! [A: real,B2: real] :
% 0.23/0.49 ( ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) )
% 0.23/0.49 = ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus_commute
% 0.23/0.49 thf(fact_5208_abs__minus__commute,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] :
% 0.23/0.49 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) )
% 0.23/0.49 = ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus_commute
% 0.23/0.49 thf(fact_5209_abs__minus__commute,axiom,
% 0.23/0.49 ! [A: int,B2: int] :
% 0.23/0.49 ( ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) )
% 0.23/0.49 = ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_minus_commute
% 0.23/0.49 thf(fact_5210_abs__eq__iff,axiom,
% 0.23/0.49 ! [X4: real,Y3: real] :
% 0.23/0.49 ( ( ( abs_abs_real @ X4 )
% 0.23/0.49 = ( abs_abs_real @ Y3 ) )
% 0.23/0.49 = ( ( X4 = Y3 )
% 0.23/0.49 | ( X4
% 0.23/0.49 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_iff
% 0.23/0.49 thf(fact_5211_abs__eq__iff,axiom,
% 0.23/0.49 ! [X4: int,Y3: int] :
% 0.23/0.49 ( ( ( abs_abs_int @ X4 )
% 0.23/0.49 = ( abs_abs_int @ Y3 ) )
% 0.23/0.49 = ( ( X4 = Y3 )
% 0.23/0.49 | ( X4
% 0.23/0.49 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_iff
% 0.23/0.49 thf(fact_5212_abs__eq__iff,axiom,
% 0.23/0.49 ! [X4: code_integer,Y3: code_integer] :
% 0.23/0.49 ( ( ( abs_abs_Code_integer @ X4 )
% 0.23/0.49 = ( abs_abs_Code_integer @ Y3 ) )
% 0.23/0.49 = ( ( X4 = Y3 )
% 0.23/0.49 | ( X4
% 0.23/0.49 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_iff
% 0.23/0.49 thf(fact_5213_abs__eq__iff,axiom,
% 0.23/0.49 ! [X4: rat,Y3: rat] :
% 0.23/0.49 ( ( ( abs_abs_rat @ X4 )
% 0.23/0.49 = ( abs_abs_rat @ Y3 ) )
% 0.23/0.49 = ( ( X4 = Y3 )
% 0.23/0.49 | ( X4
% 0.23/0.49 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_eq_iff
% 0.23/0.49 thf(fact_5214_power__abs,axiom,
% 0.23/0.49 ! [A: code_integer,N: nat] :
% 0.23/0.49 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.23/0.49 = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % power_abs
% 0.23/0.49 thf(fact_5215_power__abs,axiom,
% 0.23/0.49 ! [A: rat,N: nat] :
% 0.23/0.49 ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
% 0.23/0.49 = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % power_abs
% 0.23/0.49 thf(fact_5216_power__abs,axiom,
% 0.23/0.49 ! [A: real,N: nat] :
% 0.23/0.49 ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 0.23/0.49 = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % power_abs
% 0.23/0.49 thf(fact_5217_power__abs,axiom,
% 0.23/0.49 ! [A: int,N: nat] :
% 0.23/0.49 ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 0.23/0.49 = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.23/0.49
% 0.23/0.49 % power_abs
% 0.23/0.49 thf(fact_5218_dvd__if__abs__eq,axiom,
% 0.23/0.49 ! [L2: real,K: real] :
% 0.23/0.49 ( ( ( abs_abs_real @ L2 )
% 0.23/0.49 = ( abs_abs_real @ K ) )
% 0.23/0.49 => ( dvd_dvd_real @ L2 @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % dvd_if_abs_eq
% 0.23/0.49 thf(fact_5219_dvd__if__abs__eq,axiom,
% 0.23/0.49 ! [L2: int,K: int] :
% 0.23/0.49 ( ( ( abs_abs_int @ L2 )
% 0.23/0.49 = ( abs_abs_int @ K ) )
% 0.23/0.49 => ( dvd_dvd_int @ L2 @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % dvd_if_abs_eq
% 0.23/0.49 thf(fact_5220_dvd__if__abs__eq,axiom,
% 0.23/0.49 ! [L2: code_integer,K: code_integer] :
% 0.23/0.49 ( ( ( abs_abs_Code_integer @ L2 )
% 0.23/0.49 = ( abs_abs_Code_integer @ K ) )
% 0.23/0.49 => ( dvd_dvd_Code_integer @ L2 @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % dvd_if_abs_eq
% 0.23/0.49 thf(fact_5221_dvd__if__abs__eq,axiom,
% 0.23/0.49 ! [L2: rat,K: rat] :
% 0.23/0.49 ( ( ( abs_abs_rat @ L2 )
% 0.23/0.49 = ( abs_abs_rat @ K ) )
% 0.23/0.49 => ( dvd_dvd_rat @ L2 @ K ) ) ).
% 0.23/0.49
% 0.23/0.49 % dvd_if_abs_eq
% 0.23/0.49 thf(fact_5222_round__diff__minimal,axiom,
% 0.23/0.49 ! [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 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % round_diff_minimal
% 0.23/0.49 thf(fact_5223_round__diff__minimal,axiom,
% 0.23/0.49 ! [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 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % round_diff_minimal
% 0.23/0.49 thf(fact_5224_abs__ge__zero,axiom,
% 0.23/0.49 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_zero
% 0.23/0.49 thf(fact_5225_abs__ge__zero,axiom,
% 0.23/0.49 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_zero
% 0.23/0.49 thf(fact_5226_abs__ge__zero,axiom,
% 0.23/0.49 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_zero
% 0.23/0.49 thf(fact_5227_abs__ge__zero,axiom,
% 0.23/0.49 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_zero
% 0.23/0.49 thf(fact_5228_abs__not__less__zero,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 0.23/0.49
% 0.23/0.49 % abs_not_less_zero
% 0.23/0.49 thf(fact_5229_abs__not__less__zero,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.23/0.49
% 0.23/0.49 % abs_not_less_zero
% 0.23/0.49 thf(fact_5230_abs__not__less__zero,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 0.23/0.49
% 0.23/0.49 % abs_not_less_zero
% 0.23/0.49 thf(fact_5231_abs__not__less__zero,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 0.23/0.49
% 0.23/0.49 % abs_not_less_zero
% 0.23/0.49 thf(fact_5232_abs__of__pos,axiom,
% 0.23/0.49 ! [A: code_integer] :
% 0.23/0.49 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.23/0.49 => ( ( abs_abs_Code_integer @ A )
% 0.23/0.49 = A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_pos
% 0.23/0.49 thf(fact_5233_abs__of__pos,axiom,
% 0.23/0.49 ! [A: real] :
% 0.23/0.49 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.49 => ( ( abs_abs_real @ A )
% 0.23/0.49 = A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_pos
% 0.23/0.49 thf(fact_5234_abs__of__pos,axiom,
% 0.23/0.49 ! [A: rat] :
% 0.23/0.49 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.49 => ( ( abs_abs_rat @ A )
% 0.23/0.49 = A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_pos
% 0.23/0.49 thf(fact_5235_abs__of__pos,axiom,
% 0.23/0.49 ! [A: int] :
% 0.23/0.49 ( ( ord_less_int @ zero_zero_int @ A )
% 0.23/0.49 => ( ( abs_abs_int @ A )
% 0.23/0.49 = A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_of_pos
% 0.23/0.49 thf(fact_5236_abs__triangle__ineq,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq
% 0.23/0.49 thf(fact_5237_abs__triangle__ineq,axiom,
% 0.23/0.49 ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq
% 0.23/0.49 thf(fact_5238_abs__triangle__ineq,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq
% 0.23/0.49 thf(fact_5239_abs__triangle__ineq,axiom,
% 0.23/0.49 ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq
% 0.23/0.49 thf(fact_5240_abs__mult__less,axiom,
% 0.23/0.49 ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
% 0.23/0.49 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 0.23/0.49 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B2 ) @ D )
% 0.23/0.49 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult_less
% 0.23/0.49 thf(fact_5241_abs__mult__less,axiom,
% 0.23/0.49 ! [A: real,C: real,B2: real,D: real] :
% 0.23/0.49 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 0.23/0.49 => ( ( ord_less_real @ ( abs_abs_real @ B2 ) @ D )
% 0.23/0.49 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult_less
% 0.23/0.49 thf(fact_5242_abs__mult__less,axiom,
% 0.23/0.49 ! [A: rat,C: rat,B2: rat,D: rat] :
% 0.23/0.49 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 0.23/0.49 => ( ( ord_less_rat @ ( abs_abs_rat @ B2 ) @ D )
% 0.23/0.49 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult_less
% 0.23/0.49 thf(fact_5243_abs__mult__less,axiom,
% 0.23/0.49 ! [A: int,C: int,B2: int,D: int] :
% 0.23/0.49 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 0.23/0.49 => ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D )
% 0.23/0.49 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_mult_less
% 0.23/0.49 thf(fact_5244_abs__triangle__ineq2,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq2
% 0.23/0.49 thf(fact_5245_abs__triangle__ineq2,axiom,
% 0.23/0.49 ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq2
% 0.23/0.49 thf(fact_5246_abs__triangle__ineq2,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq2
% 0.23/0.49 thf(fact_5247_abs__triangle__ineq2,axiom,
% 0.23/0.49 ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq2
% 0.23/0.49 thf(fact_5248_abs__triangle__ineq3,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq3
% 0.23/0.49 thf(fact_5249_abs__triangle__ineq3,axiom,
% 0.23/0.49 ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq3
% 0.23/0.49 thf(fact_5250_abs__triangle__ineq3,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq3
% 0.23/0.49 thf(fact_5251_abs__triangle__ineq3,axiom,
% 0.23/0.49 ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq3
% 0.23/0.49 thf(fact_5252_abs__triangle__ineq2__sym,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq2_sym
% 0.23/0.49 thf(fact_5253_abs__triangle__ineq2__sym,axiom,
% 0.23/0.49 ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq2_sym
% 0.23/0.49 thf(fact_5254_abs__triangle__ineq2__sym,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq2_sym
% 0.23/0.49 thf(fact_5255_abs__triangle__ineq2__sym,axiom,
% 0.23/0.49 ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_triangle_ineq2_sym
% 0.23/0.49 thf(fact_5256_nonzero__abs__divide,axiom,
% 0.23/0.49 ! [B2: real,A: real] :
% 0.23/0.49 ( ( B2 != zero_zero_real )
% 0.23/0.49 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
% 0.23/0.49 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % nonzero_abs_divide
% 0.23/0.49 thf(fact_5257_nonzero__abs__divide,axiom,
% 0.23/0.49 ! [B2: rat,A: rat] :
% 0.23/0.49 ( ( B2 != zero_zero_rat )
% 0.23/0.49 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.23/0.49 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % nonzero_abs_divide
% 0.23/0.49 thf(fact_5258_abs__ge__minus__self,axiom,
% 0.23/0.49 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_minus_self
% 0.23/0.49 thf(fact_5259_abs__ge__minus__self,axiom,
% 0.23/0.49 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_minus_self
% 0.23/0.49 thf(fact_5260_abs__ge__minus__self,axiom,
% 0.23/0.49 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_minus_self
% 0.23/0.49 thf(fact_5261_abs__ge__minus__self,axiom,
% 0.23/0.49 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_ge_minus_self
% 0.23/0.49 thf(fact_5262_abs__le__iff,axiom,
% 0.23/0.49 ! [A: real,B2: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 0.23/0.49 = ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.49 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_iff
% 0.23/0.49 thf(fact_5263_abs__le__iff,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] :
% 0.23/0.49 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 0.23/0.49 = ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 0.23/0.49 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_iff
% 0.23/0.49 thf(fact_5264_abs__le__iff,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 0.23/0.49 = ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.49 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_iff
% 0.23/0.49 thf(fact_5265_abs__le__iff,axiom,
% 0.23/0.49 ! [A: int,B2: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 0.23/0.49 = ( ( ord_less_eq_int @ A @ B2 )
% 0.23/0.49 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_iff
% 0.23/0.49 thf(fact_5266_abs__le__D2,axiom,
% 0.23/0.49 ! [A: real,B2: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 0.23/0.49 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_D2
% 0.23/0.49 thf(fact_5267_abs__le__D2,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] :
% 0.23/0.49 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 0.23/0.49 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_D2
% 0.23/0.49 thf(fact_5268_abs__le__D2,axiom,
% 0.23/0.49 ! [A: rat,B2: rat] :
% 0.23/0.49 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 0.23/0.49 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_D2
% 0.23/0.49 thf(fact_5269_abs__le__D2,axiom,
% 0.23/0.49 ! [A: int,B2: int] :
% 0.23/0.49 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 0.23/0.49 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_le_D2
% 0.23/0.49 thf(fact_5270_abs__leI,axiom,
% 0.23/0.49 ! [A: real,B2: real] :
% 0.23/0.49 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.49 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.23/0.49 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 ) ) ) ).
% 0.23/0.49
% 0.23/0.49 % abs_leI
% 0.23/0.49 thf(fact_5271_abs__leI,axiom,
% 0.23/0.49 ! [A: code_integer,B2: code_integer] :
% 0.23/0.50 ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 0.23/0.50 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.23/0.50 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_leI
% 0.23/0.50 thf(fact_5272_abs__leI,axiom,
% 0.23/0.50 ! [A: rat,B2: rat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.50 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.23/0.50 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_leI
% 0.23/0.50 thf(fact_5273_abs__leI,axiom,
% 0.23/0.50 ! [A: int,B2: int] :
% 0.23/0.50 ( ( ord_less_eq_int @ A @ B2 )
% 0.23/0.50 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.23/0.50 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_leI
% 0.23/0.50 thf(fact_5274_abs__less__iff,axiom,
% 0.23/0.50 ! [A: real,B2: real] :
% 0.23/0.50 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B2 )
% 0.23/0.50 = ( ( ord_less_real @ A @ B2 )
% 0.23/0.50 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_less_iff
% 0.23/0.50 thf(fact_5275_abs__less__iff,axiom,
% 0.23/0.50 ! [A: int,B2: int] :
% 0.23/0.50 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B2 )
% 0.23/0.50 = ( ( ord_less_int @ A @ B2 )
% 0.23/0.50 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_less_iff
% 0.23/0.50 thf(fact_5276_abs__less__iff,axiom,
% 0.23/0.50 ! [A: code_integer,B2: code_integer] :
% 0.23/0.50 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 0.23/0.50 = ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 0.23/0.50 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_less_iff
% 0.23/0.50 thf(fact_5277_abs__less__iff,axiom,
% 0.23/0.50 ! [A: rat,B2: rat] :
% 0.23/0.50 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B2 )
% 0.23/0.50 = ( ( ord_less_rat @ A @ B2 )
% 0.23/0.50 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_less_iff
% 0.23/0.50 thf(fact_5278_dense__eq0__I,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ! [E2: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.23/0.50 => ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ E2 ) )
% 0.23/0.50 => ( X4 = zero_zero_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % dense_eq0_I
% 0.23/0.50 thf(fact_5279_dense__eq0__I,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ! [E2: rat] :
% 0.23/0.50 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.23/0.50 => ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ E2 ) )
% 0.23/0.50 => ( X4 = zero_zero_rat ) ) ).
% 0.23/0.50
% 0.23/0.50 % dense_eq0_I
% 0.23/0.50 thf(fact_5280_abs__eq__mult,axiom,
% 0.23/0.50 ! [A: code_integer,B2: code_integer] :
% 0.23/0.50 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.23/0.50 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 0.23/0.50 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.23/0.50 | ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) )
% 0.23/0.50 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 0.23/0.50 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_eq_mult
% 0.23/0.50 thf(fact_5281_abs__eq__mult,axiom,
% 0.23/0.50 ! [A: real,B2: real] :
% 0.23/0.50 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.50 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.23/0.50 & ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.23/0.50 | ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
% 0.23/0.50 => ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
% 0.23/0.50 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_eq_mult
% 0.23/0.50 thf(fact_5282_abs__eq__mult,axiom,
% 0.23/0.50 ! [A: rat,B2: rat] :
% 0.23/0.50 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.50 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.23/0.50 & ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.23/0.50 | ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
% 0.23/0.50 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B2 ) )
% 0.23/0.50 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_eq_mult
% 0.23/0.50 thf(fact_5283_abs__eq__mult,axiom,
% 0.23/0.50 ! [A: int,B2: int] :
% 0.23/0.50 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.23/0.50 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.23/0.50 & ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.23/0.50 | ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
% 0.23/0.50 => ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
% 0.23/0.50 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_eq_mult
% 0.23/0.50 thf(fact_5284_abs__mult__pos,axiom,
% 0.23/0.50 ! [X4: code_integer,Y3: code_integer] :
% 0.23/0.50 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 0.23/0.50 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X4 )
% 0.23/0.50 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_mult_pos
% 0.23/0.50 thf(fact_5285_abs__mult__pos,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X4 )
% 0.23/0.50 = ( abs_abs_real @ ( times_times_real @ Y3 @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_mult_pos
% 0.23/0.50 thf(fact_5286_abs__mult__pos,axiom,
% 0.23/0.50 ! [X4: rat,Y3: rat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.23/0.50 => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X4 )
% 0.23/0.50 = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_mult_pos
% 0.23/0.50 thf(fact_5287_abs__mult__pos,axiom,
% 0.23/0.50 ! [X4: int,Y3: int] :
% 0.23/0.50 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.50 => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X4 )
% 0.23/0.50 = ( abs_abs_int @ ( times_times_int @ Y3 @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_mult_pos
% 0.23/0.50 thf(fact_5288_abs__minus__le__zero,axiom,
% 0.23/0.50 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 0.23/0.50
% 0.23/0.50 % abs_minus_le_zero
% 0.23/0.50 thf(fact_5289_abs__minus__le__zero,axiom,
% 0.23/0.50 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 0.23/0.50
% 0.23/0.50 % abs_minus_le_zero
% 0.23/0.50 thf(fact_5290_abs__minus__le__zero,axiom,
% 0.23/0.50 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 0.23/0.50
% 0.23/0.50 % abs_minus_le_zero
% 0.23/0.50 thf(fact_5291_abs__minus__le__zero,axiom,
% 0.23/0.50 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 0.23/0.50
% 0.23/0.50 % abs_minus_le_zero
% 0.23/0.50 thf(fact_5292_eq__abs__iff_H,axiom,
% 0.23/0.50 ! [A: real,B2: real] :
% 0.23/0.50 ( ( A
% 0.23/0.50 = ( abs_abs_real @ B2 ) )
% 0.23/0.50 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.50 & ( ( B2 = A )
% 0.23/0.50 | ( B2
% 0.23/0.50 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % eq_abs_iff'
% 0.23/0.50 thf(fact_5293_eq__abs__iff_H,axiom,
% 0.23/0.50 ! [A: code_integer,B2: code_integer] :
% 0.23/0.50 ( ( A
% 0.23/0.50 = ( abs_abs_Code_integer @ B2 ) )
% 0.23/0.50 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.23/0.50 & ( ( B2 = A )
% 0.23/0.50 | ( B2
% 0.23/0.50 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % eq_abs_iff'
% 0.23/0.50 thf(fact_5294_eq__abs__iff_H,axiom,
% 0.23/0.50 ! [A: rat,B2: rat] :
% 0.23/0.50 ( ( A
% 0.23/0.50 = ( abs_abs_rat @ B2 ) )
% 0.23/0.50 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.50 & ( ( B2 = A )
% 0.23/0.50 | ( B2
% 0.23/0.50 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % eq_abs_iff'
% 0.23/0.50 thf(fact_5295_eq__abs__iff_H,axiom,
% 0.23/0.50 ! [A: int,B2: int] :
% 0.23/0.50 ( ( A
% 0.23/0.50 = ( abs_abs_int @ B2 ) )
% 0.23/0.50 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.23/0.50 & ( ( B2 = A )
% 0.23/0.50 | ( B2
% 0.23/0.50 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % eq_abs_iff'
% 0.23/0.50 thf(fact_5296_abs__eq__iff_H,axiom,
% 0.23/0.50 ! [A: real,B2: real] :
% 0.23/0.50 ( ( ( abs_abs_real @ A )
% 0.23/0.50 = B2 )
% 0.23/0.50 = ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.23/0.50 & ( ( A = B2 )
% 0.23/0.50 | ( A
% 0.23/0.50 = ( uminus_uminus_real @ B2 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_eq_iff'
% 0.23/0.50 thf(fact_5297_abs__eq__iff_H,axiom,
% 0.23/0.50 ! [A: code_integer,B2: code_integer] :
% 0.23/0.50 ( ( ( abs_abs_Code_integer @ A )
% 0.23/0.50 = B2 )
% 0.23/0.50 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.23/0.50 & ( ( A = B2 )
% 0.23/0.50 | ( A
% 0.23/0.50 = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_eq_iff'
% 0.23/0.50 thf(fact_5298_abs__eq__iff_H,axiom,
% 0.23/0.50 ! [A: rat,B2: rat] :
% 0.23/0.50 ( ( ( abs_abs_rat @ A )
% 0.23/0.50 = B2 )
% 0.23/0.50 = ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.23/0.50 & ( ( A = B2 )
% 0.23/0.50 | ( A
% 0.23/0.50 = ( uminus_uminus_rat @ B2 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_eq_iff'
% 0.23/0.50 thf(fact_5299_abs__eq__iff_H,axiom,
% 0.23/0.50 ! [A: int,B2: int] :
% 0.23/0.50 ( ( ( abs_abs_int @ A )
% 0.23/0.50 = B2 )
% 0.23/0.50 = ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.23/0.50 & ( ( A = B2 )
% 0.23/0.50 | ( A
% 0.23/0.50 = ( uminus_uminus_int @ B2 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_eq_iff'
% 0.23/0.50 thf(fact_5300_abs__div__pos,axiom,
% 0.23/0.50 ! [Y3: real,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.50 => ( ( divide_divide_real @ ( abs_abs_real @ X4 ) @ Y3 )
% 0.23/0.50 = ( abs_abs_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_div_pos
% 0.23/0.50 thf(fact_5301_abs__div__pos,axiom,
% 0.23/0.50 ! [Y3: rat,X4: rat] :
% 0.23/0.50 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.23/0.50 => ( ( divide_divide_rat @ ( abs_abs_rat @ X4 ) @ Y3 )
% 0.23/0.50 = ( abs_abs_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_div_pos
% 0.23/0.50 thf(fact_5302_zero__le__power__abs,axiom,
% 0.23/0.50 ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_le_power_abs
% 0.23/0.50 thf(fact_5303_zero__le__power__abs,axiom,
% 0.23/0.50 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_le_power_abs
% 0.23/0.50 thf(fact_5304_zero__le__power__abs,axiom,
% 0.23/0.50 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_le_power_abs
% 0.23/0.50 thf(fact_5305_zero__le__power__abs,axiom,
% 0.23/0.50 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_le_power_abs
% 0.23/0.50 thf(fact_5306_abs__if,axiom,
% 0.23/0.50 ( abs_abs_real
% 0.23/0.50 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_if
% 0.23/0.50 thf(fact_5307_abs__if,axiom,
% 0.23/0.50 ( abs_abs_int
% 0.23/0.50 = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_if
% 0.23/0.50 thf(fact_5308_abs__if,axiom,
% 0.23/0.50 ( abs_abs_Code_integer
% 0.23/0.50 = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_if
% 0.23/0.50 thf(fact_5309_abs__if,axiom,
% 0.23/0.50 ( abs_abs_rat
% 0.23/0.50 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_if
% 0.23/0.50 thf(fact_5310_abs__if__raw,axiom,
% 0.23/0.50 ( abs_abs_real
% 0.23/0.50 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_if_raw
% 0.23/0.50 thf(fact_5311_abs__if__raw,axiom,
% 0.23/0.50 ( abs_abs_int
% 0.23/0.50 = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_if_raw
% 0.23/0.50 thf(fact_5312_abs__if__raw,axiom,
% 0.23/0.50 ( abs_abs_Code_integer
% 0.23/0.50 = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_if_raw
% 0.23/0.50 thf(fact_5313_abs__if__raw,axiom,
% 0.23/0.50 ( abs_abs_rat
% 0.23/0.50 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_if_raw
% 0.23/0.50 thf(fact_5314_abs__of__neg,axiom,
% 0.23/0.50 ! [A: real] :
% 0.23/0.50 ( ( ord_less_real @ A @ zero_zero_real )
% 0.23/0.50 => ( ( abs_abs_real @ A )
% 0.23/0.50 = ( uminus_uminus_real @ A ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_of_neg
% 0.23/0.50 thf(fact_5315_abs__of__neg,axiom,
% 0.23/0.50 ! [A: int] :
% 0.23/0.50 ( ( ord_less_int @ A @ zero_zero_int )
% 0.23/0.50 => ( ( abs_abs_int @ A )
% 0.23/0.50 = ( uminus_uminus_int @ A ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_of_neg
% 0.23/0.50 thf(fact_5316_abs__of__neg,axiom,
% 0.23/0.50 ! [A: code_integer] :
% 0.23/0.50 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.23/0.50 => ( ( abs_abs_Code_integer @ A )
% 0.23/0.50 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_of_neg
% 0.23/0.50 thf(fact_5317_abs__of__neg,axiom,
% 0.23/0.50 ! [A: rat] :
% 0.23/0.50 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.23/0.50 => ( ( abs_abs_rat @ A )
% 0.23/0.50 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_of_neg
% 0.23/0.50 thf(fact_5318_abs__triangle__ineq4,axiom,
% 0.23/0.50 ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_triangle_ineq4
% 0.23/0.50 thf(fact_5319_abs__triangle__ineq4,axiom,
% 0.23/0.50 ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_triangle_ineq4
% 0.23/0.50 thf(fact_5320_abs__triangle__ineq4,axiom,
% 0.23/0.50 ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_triangle_ineq4
% 0.23/0.50 thf(fact_5321_abs__triangle__ineq4,axiom,
% 0.23/0.50 ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_triangle_ineq4
% 0.23/0.50 thf(fact_5322_abs__diff__triangle__ineq,axiom,
% 0.23/0.50 ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ D ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_triangle_ineq
% 0.23/0.50 thf(fact_5323_abs__diff__triangle__ineq,axiom,
% 0.23/0.50 ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_triangle_ineq
% 0.23/0.50 thf(fact_5324_abs__diff__triangle__ineq,axiom,
% 0.23/0.50 ! [A: rat,B2: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_triangle_ineq
% 0.23/0.50 thf(fact_5325_abs__diff__triangle__ineq,axiom,
% 0.23/0.50 ! [A: int,B2: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_triangle_ineq
% 0.23/0.50 thf(fact_5326_abs__diff__le__iff,axiom,
% 0.23/0.50 ! [X4: code_integer,A: code_integer,R2: code_integer] :
% 0.23/0.50 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ A ) ) @ R2 )
% 0.23/0.50 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X4 )
% 0.23/0.50 & ( ord_le3102999989581377725nteger @ X4 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_le_iff
% 0.23/0.50 thf(fact_5327_abs__diff__le__iff,axiom,
% 0.23/0.50 ! [X4: real,A: real,R2: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ A ) ) @ R2 )
% 0.23/0.50 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X4 )
% 0.23/0.50 & ( ord_less_eq_real @ X4 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_le_iff
% 0.23/0.50 thf(fact_5328_abs__diff__le__iff,axiom,
% 0.23/0.50 ! [X4: rat,A: rat,R2: rat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ A ) ) @ R2 )
% 0.23/0.50 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X4 )
% 0.23/0.50 & ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_le_iff
% 0.23/0.50 thf(fact_5329_abs__diff__le__iff,axiom,
% 0.23/0.50 ! [X4: int,A: int,R2: int] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ A ) ) @ R2 )
% 0.23/0.50 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X4 )
% 0.23/0.50 & ( ord_less_eq_int @ X4 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_le_iff
% 0.23/0.50 thf(fact_5330_abs__diff__less__iff,axiom,
% 0.23/0.50 ! [X4: code_integer,A: code_integer,R2: code_integer] :
% 0.23/0.50 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ A ) ) @ R2 )
% 0.23/0.50 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X4 )
% 0.23/0.50 & ( ord_le6747313008572928689nteger @ X4 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_less_iff
% 0.23/0.50 thf(fact_5331_abs__diff__less__iff,axiom,
% 0.23/0.50 ! [X4: real,A: real,R2: real] :
% 0.23/0.50 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ A ) ) @ R2 )
% 0.23/0.50 = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X4 )
% 0.23/0.50 & ( ord_less_real @ X4 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_less_iff
% 0.23/0.50 thf(fact_5332_abs__diff__less__iff,axiom,
% 0.23/0.50 ! [X4: rat,A: rat,R2: rat] :
% 0.23/0.50 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ A ) ) @ R2 )
% 0.23/0.50 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X4 )
% 0.23/0.50 & ( ord_less_rat @ X4 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_less_iff
% 0.23/0.50 thf(fact_5333_abs__diff__less__iff,axiom,
% 0.23/0.50 ! [X4: int,A: int,R2: int] :
% 0.23/0.50 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ A ) ) @ R2 )
% 0.23/0.50 = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X4 )
% 0.23/0.50 & ( ord_less_int @ X4 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_diff_less_iff
% 0.23/0.50 thf(fact_5334_round__mono,axiom,
% 0.23/0.50 ! [X4: rat,Y3: rat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.50 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X4 ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % round_mono
% 0.23/0.50 thf(fact_5335_abs__real__def,axiom,
% 0.23/0.50 ( abs_abs_real
% 0.23/0.50 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_real_def
% 0.23/0.50 thf(fact_5336_lemma__interval__lt,axiom,
% 0.23/0.50 ! [A: real,X4: real,B2: real] :
% 0.23/0.50 ( ( ord_less_real @ A @ X4 )
% 0.23/0.50 => ( ( ord_less_real @ X4 @ B2 )
% 0.23/0.50 => ? [D3: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.23/0.50 & ! [Y5: real] :
% 0.23/0.50 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y5 ) ) @ D3 )
% 0.23/0.50 => ( ( ord_less_real @ A @ Y5 )
% 0.23/0.50 & ( ord_less_real @ Y5 @ B2 ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % lemma_interval_lt
% 0.23/0.50 thf(fact_5337_sin__bound__lemma,axiom,
% 0.23/0.50 ! [X4: real,Y3: real,U: real,V: real] :
% 0.23/0.50 ( ( X4 = Y3 )
% 0.23/0.50 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 0.23/0.50 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X4 @ U ) @ Y3 ) ) @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % sin_bound_lemma
% 0.23/0.50 thf(fact_5338_abs__add__one__gt__zero,axiom,
% 0.23/0.50 ! [X4: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_add_one_gt_zero
% 0.23/0.50 thf(fact_5339_abs__add__one__gt__zero,axiom,
% 0.23/0.50 ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_add_one_gt_zero
% 0.23/0.50 thf(fact_5340_abs__add__one__gt__zero,axiom,
% 0.23/0.50 ! [X4: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_add_one_gt_zero
% 0.23/0.50 thf(fact_5341_abs__add__one__gt__zero,axiom,
% 0.23/0.50 ! [X4: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_add_one_gt_zero
% 0.23/0.50 thf(fact_5342_of__int__leD,axiom,
% 0.23/0.50 ! [N: int,X4: code_integer] :
% 0.23/0.50 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X4 )
% 0.23/0.50 => ( ( N = zero_zero_int )
% 0.23/0.50 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_leD
% 0.23/0.50 thf(fact_5343_of__int__leD,axiom,
% 0.23/0.50 ! [N: int,X4: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X4 )
% 0.23/0.50 => ( ( N = zero_zero_int )
% 0.23/0.50 | ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_leD
% 0.23/0.50 thf(fact_5344_of__int__leD,axiom,
% 0.23/0.50 ! [N: int,X4: rat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X4 )
% 0.23/0.50 => ( ( N = zero_zero_int )
% 0.23/0.50 | ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_leD
% 0.23/0.50 thf(fact_5345_of__int__leD,axiom,
% 0.23/0.50 ! [N: int,X4: int] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X4 )
% 0.23/0.50 => ( ( N = zero_zero_int )
% 0.23/0.50 | ( ord_less_eq_int @ one_one_int @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_leD
% 0.23/0.50 thf(fact_5346_of__int__lessD,axiom,
% 0.23/0.50 ! [N: int,X4: code_integer] :
% 0.23/0.50 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X4 )
% 0.23/0.50 => ( ( N = zero_zero_int )
% 0.23/0.50 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_lessD
% 0.23/0.50 thf(fact_5347_of__int__lessD,axiom,
% 0.23/0.50 ! [N: int,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X4 )
% 0.23/0.50 => ( ( N = zero_zero_int )
% 0.23/0.50 | ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_lessD
% 0.23/0.50 thf(fact_5348_of__int__lessD,axiom,
% 0.23/0.50 ! [N: int,X4: rat] :
% 0.23/0.50 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X4 )
% 0.23/0.50 => ( ( N = zero_zero_int )
% 0.23/0.50 | ( ord_less_rat @ one_one_rat @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_lessD
% 0.23/0.50 thf(fact_5349_of__int__lessD,axiom,
% 0.23/0.50 ! [N: int,X4: int] :
% 0.23/0.50 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X4 )
% 0.23/0.50 => ( ( N = zero_zero_int )
% 0.23/0.50 | ( ord_less_int @ one_one_int @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_lessD
% 0.23/0.50 thf(fact_5350_lemma__interval,axiom,
% 0.23/0.50 ! [A: real,X4: real,B2: real] :
% 0.23/0.50 ( ( ord_less_real @ A @ X4 )
% 0.23/0.50 => ( ( ord_less_real @ X4 @ B2 )
% 0.23/0.50 => ? [D3: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.23/0.50 & ! [Y5: real] :
% 0.23/0.50 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y5 ) ) @ D3 )
% 0.23/0.50 => ( ( ord_less_eq_real @ A @ Y5 )
% 0.23/0.50 & ( ord_less_eq_real @ Y5 @ B2 ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % lemma_interval
% 0.23/0.50 thf(fact_5351_of__int__round__abs__le,axiom,
% 0.23/0.50 ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) @ X4 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_round_abs_le
% 0.23/0.50 thf(fact_5352_of__int__round__abs__le,axiom,
% 0.23/0.50 ! [X4: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) @ X4 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_round_abs_le
% 0.23/0.50 thf(fact_5353_round__unique_H,axiom,
% 0.23/0.50 ! [X4: real,N: int] :
% 0.23/0.50 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.50 => ( ( archim8280529875227126926d_real @ X4 )
% 0.23/0.50 = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % round_unique'
% 0.23/0.50 thf(fact_5354_round__unique_H,axiom,
% 0.23/0.50 ! [X4: rat,N: int] :
% 0.23/0.50 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.23/0.50 => ( ( archim7778729529865785530nd_rat @ X4 )
% 0.23/0.50 = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % round_unique'
% 0.23/0.50 thf(fact_5355_abs__le__square__iff,axiom,
% 0.23/0.50 ! [X4: code_integer,Y3: code_integer] :
% 0.23/0.50 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ ( abs_abs_Code_integer @ Y3 ) )
% 0.23/0.50 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_le_square_iff
% 0.23/0.50 thf(fact_5356_abs__le__square__iff,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ Y3 ) )
% 0.23/0.50 = ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_le_square_iff
% 0.23/0.50 thf(fact_5357_abs__le__square__iff,axiom,
% 0.23/0.50 ! [X4: rat,Y3: rat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ ( abs_abs_rat @ Y3 ) )
% 0.23/0.50 = ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_le_square_iff
% 0.23/0.50 thf(fact_5358_abs__le__square__iff,axiom,
% 0.23/0.50 ! [X4: int,Y3: int] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ ( abs_abs_int @ Y3 ) )
% 0.23/0.50 = ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_le_square_iff
% 0.23/0.50 thf(fact_5359_abs__square__eq__1,axiom,
% 0.23/0.50 ! [X4: code_integer] :
% 0.23/0.50 ( ( ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.50 = one_one_Code_integer )
% 0.23/0.50 = ( ( abs_abs_Code_integer @ X4 )
% 0.23/0.50 = one_one_Code_integer ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_eq_1
% 0.23/0.50 thf(fact_5360_abs__square__eq__1,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.50 = one_one_rat )
% 0.23/0.50 = ( ( abs_abs_rat @ X4 )
% 0.23/0.50 = one_one_rat ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_eq_1
% 0.23/0.50 thf(fact_5361_abs__square__eq__1,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.50 = one_one_real )
% 0.23/0.50 = ( ( abs_abs_real @ X4 )
% 0.23/0.50 = one_one_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_eq_1
% 0.23/0.50 thf(fact_5362_abs__square__eq__1,axiom,
% 0.23/0.50 ! [X4: int] :
% 0.23/0.50 ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.50 = one_one_int )
% 0.23/0.50 = ( ( abs_abs_int @ X4 )
% 0.23/0.50 = one_one_int ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_eq_1
% 0.23/0.50 thf(fact_5363_power__even__abs,axiom,
% 0.23/0.50 ! [N: nat,A: code_integer] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 0.23/0.50 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power_even_abs
% 0.23/0.50 thf(fact_5364_power__even__abs,axiom,
% 0.23/0.50 ! [N: nat,A: rat] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 0.23/0.50 = ( power_power_rat @ A @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power_even_abs
% 0.23/0.50 thf(fact_5365_power__even__abs,axiom,
% 0.23/0.50 ! [N: nat,A: real] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 0.23/0.50 = ( power_power_real @ A @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power_even_abs
% 0.23/0.50 thf(fact_5366_power__even__abs,axiom,
% 0.23/0.50 ! [N: nat,A: int] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 0.23/0.50 = ( power_power_int @ A @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power_even_abs
% 0.23/0.50 thf(fact_5367_power2__le__iff__abs__le,axiom,
% 0.23/0.50 ! [Y3: code_integer,X4: code_integer] :
% 0.23/0.50 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
% 0.23/0.50 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.50 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ Y3 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power2_le_iff_abs_le
% 0.23/0.50 thf(fact_5368_power2__le__iff__abs__le,axiom,
% 0.23/0.50 ! [Y3: real,X4: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.50 => ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.50 = ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ Y3 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power2_le_iff_abs_le
% 0.23/0.50 thf(fact_5369_power2__le__iff__abs__le,axiom,
% 0.23/0.50 ! [Y3: rat,X4: rat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.23/0.50 => ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.50 = ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ Y3 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power2_le_iff_abs_le
% 0.23/0.50 thf(fact_5370_power2__le__iff__abs__le,axiom,
% 0.23/0.50 ! [Y3: int,X4: int] :
% 0.23/0.50 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.50 => ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.50 = ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ Y3 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power2_le_iff_abs_le
% 0.23/0.50 thf(fact_5371_abs__square__le__1,axiom,
% 0.23/0.50 ! [X4: code_integer] :
% 0.23/0.50 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.23/0.50 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_le_1
% 0.23/0.50 thf(fact_5372_abs__square__le__1,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.23/0.50 = ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_le_1
% 0.23/0.50 thf(fact_5373_abs__square__le__1,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.23/0.50 = ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_le_1
% 0.23/0.50 thf(fact_5374_abs__square__le__1,axiom,
% 0.23/0.50 ! [X4: int] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.23/0.50 = ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ one_one_int ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_le_1
% 0.23/0.50 thf(fact_5375_abs__square__less__1,axiom,
% 0.23/0.50 ! [X4: code_integer] :
% 0.23/0.50 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.23/0.50 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_less_1
% 0.23/0.50 thf(fact_5376_abs__square__less__1,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.23/0.50 = ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_less_1
% 0.23/0.50 thf(fact_5377_abs__square__less__1,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.23/0.50 = ( ord_less_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_less_1
% 0.23/0.50 thf(fact_5378_abs__square__less__1,axiom,
% 0.23/0.50 ! [X4: int] :
% 0.23/0.50 ( ( ord_less_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.23/0.50 = ( ord_less_int @ ( abs_abs_int @ X4 ) @ one_one_int ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_square_less_1
% 0.23/0.50 thf(fact_5379_power__mono__even,axiom,
% 0.23/0.50 ! [N: nat,A: code_integer,B2: code_integer] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) )
% 0.23/0.50 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power_mono_even
% 0.23/0.50 thf(fact_5380_power__mono__even,axiom,
% 0.23/0.50 ! [N: nat,A: real,B2: real] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) )
% 0.23/0.50 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power_mono_even
% 0.23/0.50 thf(fact_5381_power__mono__even,axiom,
% 0.23/0.50 ! [N: nat,A: rat,B2: rat] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) )
% 0.23/0.50 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power_mono_even
% 0.23/0.50 thf(fact_5382_power__mono__even,axiom,
% 0.23/0.50 ! [N: nat,A: int,B2: int] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) )
% 0.23/0.50 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % power_mono_even
% 0.23/0.50 thf(fact_5383_sqrt__ge__absD,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( sqrt @ Y3 ) )
% 0.23/0.50 => ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).
% 0.23/0.50
% 0.23/0.50 % sqrt_ge_absD
% 0.23/0.50 thf(fact_5384_real__sqrt__ge__abs1,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_sqrt_ge_abs1
% 0.23/0.50 thf(fact_5385_real__sqrt__ge__abs2,axiom,
% 0.23/0.50 ! [Y3: real,X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_sqrt_ge_abs2
% 0.23/0.50 thf(fact_5386_sqrt__sum__squares__le__sum__abs,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ Y3 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % sqrt_sum_squares_le_sum_abs
% 0.23/0.50 thf(fact_5387_cos__x__y__le__one,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 0.23/0.50
% 0.23/0.50 % cos_x_y_le_one
% 0.23/0.50 thf(fact_5388_real__sqrt__sum__squares__less,axiom,
% 0.23/0.50 ! [X4: real,U: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.23/0.50 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.23/0.50 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_sqrt_sum_squares_less
% 0.23/0.50 thf(fact_5389_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.50 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_ln_one_plus_x_minus_x_bound_nonneg
% 0.23/0.50 thf(fact_5390_of__int__round__le,axiom,
% 0.23/0.50 ! [X4: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_round_le
% 0.23/0.50 thf(fact_5391_of__int__round__le,axiom,
% 0.23/0.50 ! [X4: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_round_le
% 0.23/0.50 thf(fact_5392_abs__sqrt__wlog,axiom,
% 0.23/0.50 ! [P: code_integer > code_integer > $o,X4: code_integer] :
% 0.23/0.50 ( ! [X3: code_integer] :
% 0.23/0.50 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 0.23/0.50 => ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.50 => ( P @ ( abs_abs_Code_integer @ X4 ) @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_sqrt_wlog
% 0.23/0.50 thf(fact_5393_abs__sqrt__wlog,axiom,
% 0.23/0.50 ! [P: real > real > $o,X4: real] :
% 0.23/0.50 ( ! [X3: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.23/0.50 => ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.50 => ( P @ ( abs_abs_real @ X4 ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_sqrt_wlog
% 0.23/0.50 thf(fact_5394_abs__sqrt__wlog,axiom,
% 0.23/0.50 ! [P: rat > rat > $o,X4: rat] :
% 0.23/0.50 ( ! [X3: rat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.23/0.50 => ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.50 => ( P @ ( abs_abs_rat @ X4 ) @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_sqrt_wlog
% 0.23/0.50 thf(fact_5395_abs__sqrt__wlog,axiom,
% 0.23/0.50 ! [P: int > int > $o,X4: int] :
% 0.23/0.50 ( ! [X3: int] :
% 0.23/0.50 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.23/0.50 => ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.50 => ( P @ ( abs_abs_int @ X4 ) @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_sqrt_wlog
% 0.23/0.50 thf(fact_5396_arctan__double,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.50 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X4 ) )
% 0.23/0.50 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % arctan_double
% 0.23/0.50 thf(fact_5397_arctan__half,axiom,
% 0.23/0.50 ( arctan
% 0.23/0.50 = ( ^ [X: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % arctan_half
% 0.23/0.50 thf(fact_5398_log__base__10__eq1,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.50 = ( 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 @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_base_10_eq1
% 0.23/0.50 thf(fact_5399_signed__take__bit__eq__take__bit__minus,axiom,
% 0.23/0.50 ( bit_ri631733984087533419it_int
% 0.23/0.50 = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % signed_take_bit_eq_take_bit_minus
% 0.23/0.50 thf(fact_5400_log__base__10__eq2,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.50 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_base_10_eq2
% 0.23/0.50 thf(fact_5401_exp__ge__one__minus__x__over__n__power__n,axiom,
% 0.23/0.50 ! [X4: real,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.50 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.50 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % exp_ge_one_minus_x_over_n_power_n
% 0.23/0.50 thf(fact_5402_of__nat__eq__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ( semiri8010041392384452111omplex @ M )
% 0.23/0.50 = ( semiri8010041392384452111omplex @ N ) )
% 0.23/0.50 = ( M = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_iff
% 0.23/0.50 thf(fact_5403_of__nat__eq__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ( semiri5074537144036343181t_real @ M )
% 0.23/0.50 = ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.50 = ( M = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_iff
% 0.23/0.50 thf(fact_5404_of__nat__eq__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ( semiri681578069525770553at_rat @ M )
% 0.23/0.50 = ( semiri681578069525770553at_rat @ N ) )
% 0.23/0.50 = ( M = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_iff
% 0.23/0.50 thf(fact_5405_of__nat__eq__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.23/0.50 = ( semiri1316708129612266289at_nat @ N ) )
% 0.23/0.50 = ( M = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_iff
% 0.23/0.50 thf(fact_5406_of__nat__eq__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ( semiri1314217659103216013at_int @ M )
% 0.23/0.50 = ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.50 = ( M = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_iff
% 0.23/0.50 thf(fact_5407_abs__of__nat,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.23/0.50 = ( semiri4939895301339042750nteger @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_of_nat
% 0.23/0.50 thf(fact_5408_abs__of__nat,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.50 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_of_nat
% 0.23/0.50 thf(fact_5409_abs__of__nat,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.23/0.50 = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_of_nat
% 0.23/0.50 thf(fact_5410_abs__of__nat,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.50 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_of_nat
% 0.23/0.50 thf(fact_5411_of__nat__eq__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( ( semiri8010041392384452111omplex @ M )
% 0.23/0.50 = zero_zero_complex )
% 0.23/0.50 = ( M = zero_zero_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_0_iff
% 0.23/0.50 thf(fact_5412_of__nat__eq__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( ( semiri5074537144036343181t_real @ M )
% 0.23/0.50 = zero_zero_real )
% 0.23/0.50 = ( M = zero_zero_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_0_iff
% 0.23/0.50 thf(fact_5413_of__nat__eq__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( ( semiri681578069525770553at_rat @ M )
% 0.23/0.50 = zero_zero_rat )
% 0.23/0.50 = ( M = zero_zero_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_0_iff
% 0.23/0.50 thf(fact_5414_of__nat__eq__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.23/0.50 = zero_zero_nat )
% 0.23/0.50 = ( M = zero_zero_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_0_iff
% 0.23/0.50 thf(fact_5415_of__nat__eq__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( ( semiri1314217659103216013at_int @ M )
% 0.23/0.50 = zero_zero_int )
% 0.23/0.50 = ( M = zero_zero_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_0_iff
% 0.23/0.50 thf(fact_5416_of__nat__0__eq__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( zero_zero_complex
% 0.23/0.50 = ( semiri8010041392384452111omplex @ N ) )
% 0.23/0.50 = ( zero_zero_nat = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_eq_iff
% 0.23/0.50 thf(fact_5417_of__nat__0__eq__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( zero_zero_real
% 0.23/0.50 = ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.50 = ( zero_zero_nat = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_eq_iff
% 0.23/0.50 thf(fact_5418_of__nat__0__eq__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( zero_zero_rat
% 0.23/0.50 = ( semiri681578069525770553at_rat @ N ) )
% 0.23/0.50 = ( zero_zero_nat = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_eq_iff
% 0.23/0.50 thf(fact_5419_of__nat__0__eq__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( zero_zero_nat
% 0.23/0.50 = ( semiri1316708129612266289at_nat @ N ) )
% 0.23/0.50 = ( zero_zero_nat = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_eq_iff
% 0.23/0.50 thf(fact_5420_of__nat__0__eq__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( zero_zero_int
% 0.23/0.50 = ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.50 = ( zero_zero_nat = N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_eq_iff
% 0.23/0.50 thf(fact_5421_of__nat__0,axiom,
% 0.23/0.50 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 0.23/0.50 = zero_zero_complex ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0
% 0.23/0.50 thf(fact_5422_of__nat__0,axiom,
% 0.23/0.50 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 0.23/0.50 = zero_zero_real ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0
% 0.23/0.50 thf(fact_5423_of__nat__0,axiom,
% 0.23/0.50 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 0.23/0.50 = zero_zero_rat ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0
% 0.23/0.50 thf(fact_5424_of__nat__0,axiom,
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 0.23/0.50 = zero_zero_nat ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0
% 0.23/0.50 thf(fact_5425_of__nat__0,axiom,
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.23/0.50 = zero_zero_int ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0
% 0.23/0.50 thf(fact_5426_of__nat__less__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.50 = ( ord_less_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_iff
% 0.23/0.50 thf(fact_5427_of__nat__less__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.23/0.50 = ( ord_less_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_iff
% 0.23/0.50 thf(fact_5428_of__nat__less__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.23/0.50 = ( ord_less_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_iff
% 0.23/0.50 thf(fact_5429_of__nat__less__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.50 = ( ord_less_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_iff
% 0.23/0.50 thf(fact_5430_of__nat__numeral,axiom,
% 0.23/0.50 ! [N: num] :
% 0.23/0.50 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 0.23/0.50 = ( numera6690914467698888265omplex @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_numeral
% 0.23/0.50 thf(fact_5431_of__nat__numeral,axiom,
% 0.23/0.50 ! [N: num] :
% 0.23/0.50 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 0.23/0.50 = ( numeral_numeral_real @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_numeral
% 0.23/0.50 thf(fact_5432_of__nat__numeral,axiom,
% 0.23/0.50 ! [N: num] :
% 0.23/0.50 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 0.23/0.50 = ( numeral_numeral_rat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_numeral
% 0.23/0.50 thf(fact_5433_of__nat__numeral,axiom,
% 0.23/0.50 ! [N: num] :
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 0.23/0.50 = ( numeral_numeral_nat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_numeral
% 0.23/0.50 thf(fact_5434_of__nat__numeral,axiom,
% 0.23/0.50 ! [N: num] :
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.23/0.50 = ( numeral_numeral_int @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_numeral
% 0.23/0.50 thf(fact_5435_of__nat__le__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.50 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_iff
% 0.23/0.50 thf(fact_5436_of__nat__le__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.23/0.50 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_iff
% 0.23/0.50 thf(fact_5437_of__nat__le__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.23/0.50 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_iff
% 0.23/0.50 thf(fact_5438_of__nat__le__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.50 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_iff
% 0.23/0.50 thf(fact_5439_of__nat__add,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
% 0.23/0.50 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_add
% 0.23/0.50 thf(fact_5440_of__nat__add,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 0.23/0.50 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_add
% 0.23/0.50 thf(fact_5441_of__nat__add,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 0.23/0.50 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_add
% 0.23/0.50 thf(fact_5442_of__nat__add,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 0.23/0.50 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_add
% 0.23/0.50 thf(fact_5443_of__nat__add,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 0.23/0.50 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_add
% 0.23/0.50 thf(fact_5444_of__nat__mult,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 0.23/0.50 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mult
% 0.23/0.50 thf(fact_5445_of__nat__mult,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 0.23/0.50 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mult
% 0.23/0.50 thf(fact_5446_of__nat__mult,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 0.23/0.50 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mult
% 0.23/0.50 thf(fact_5447_of__nat__mult,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 0.23/0.50 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mult
% 0.23/0.50 thf(fact_5448_of__nat__mult,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 0.23/0.50 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mult
% 0.23/0.50 thf(fact_5449_of__nat__1,axiom,
% 0.23/0.50 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 0.23/0.50 = one_one_complex ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_1
% 0.23/0.50 thf(fact_5450_of__nat__1,axiom,
% 0.23/0.50 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 0.23/0.50 = one_one_real ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_1
% 0.23/0.50 thf(fact_5451_of__nat__1,axiom,
% 0.23/0.50 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 0.23/0.50 = one_one_rat ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_1
% 0.23/0.50 thf(fact_5452_of__nat__1,axiom,
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 0.23/0.50 = one_one_nat ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_1
% 0.23/0.50 thf(fact_5453_of__nat__1,axiom,
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.23/0.50 = one_one_int ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_1
% 0.23/0.50 thf(fact_5454_of__nat__1__eq__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( one_one_complex
% 0.23/0.50 = ( semiri8010041392384452111omplex @ N ) )
% 0.23/0.50 = ( N = one_one_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_1_eq_iff
% 0.23/0.50 thf(fact_5455_of__nat__1__eq__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( one_one_real
% 0.23/0.50 = ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.50 = ( N = one_one_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_1_eq_iff
% 0.23/0.50 thf(fact_5456_of__nat__1__eq__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( one_one_rat
% 0.23/0.50 = ( semiri681578069525770553at_rat @ N ) )
% 0.23/0.50 = ( N = one_one_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_1_eq_iff
% 0.23/0.50 thf(fact_5457_of__nat__1__eq__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( one_one_nat
% 0.23/0.50 = ( semiri1316708129612266289at_nat @ N ) )
% 0.23/0.50 = ( N = one_one_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_1_eq_iff
% 0.23/0.50 thf(fact_5458_of__nat__1__eq__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( one_one_int
% 0.23/0.50 = ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.50 = ( N = one_one_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_1_eq_iff
% 0.23/0.50 thf(fact_5459_of__nat__eq__1__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ( semiri8010041392384452111omplex @ N )
% 0.23/0.50 = one_one_complex )
% 0.23/0.50 = ( N = one_one_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_1_iff
% 0.23/0.50 thf(fact_5460_of__nat__eq__1__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ( semiri5074537144036343181t_real @ N )
% 0.23/0.50 = one_one_real )
% 0.23/0.50 = ( N = one_one_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_1_iff
% 0.23/0.50 thf(fact_5461_of__nat__eq__1__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ( semiri681578069525770553at_rat @ N )
% 0.23/0.50 = one_one_rat )
% 0.23/0.50 = ( N = one_one_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_1_iff
% 0.23/0.50 thf(fact_5462_of__nat__eq__1__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ( semiri1316708129612266289at_nat @ N )
% 0.23/0.50 = one_one_nat )
% 0.23/0.50 = ( N = one_one_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_1_iff
% 0.23/0.50 thf(fact_5463_of__nat__eq__1__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ( semiri1314217659103216013at_int @ N )
% 0.23/0.50 = one_one_int )
% 0.23/0.50 = ( N = one_one_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_1_iff
% 0.23/0.50 thf(fact_5464_of__nat__power,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 0.23/0.50 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power
% 0.23/0.50 thf(fact_5465_of__nat__power,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 0.23/0.50 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power
% 0.23/0.50 thf(fact_5466_of__nat__power,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 0.23/0.50 = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power
% 0.23/0.50 thf(fact_5467_of__nat__power,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 0.23/0.50 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power
% 0.23/0.50 thf(fact_5468_of__nat__power,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 0.23/0.50 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power
% 0.23/0.50 thf(fact_5469_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W )
% 0.23/0.50 = ( semiri8010041392384452111omplex @ X4 ) )
% 0.23/0.50 = ( ( power_power_nat @ B2 @ W )
% 0.23/0.50 = X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5470_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W )
% 0.23/0.50 = ( semiri5074537144036343181t_real @ X4 ) )
% 0.23/0.50 = ( ( power_power_nat @ B2 @ W )
% 0.23/0.50 = X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5471_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W )
% 0.23/0.50 = ( semiri681578069525770553at_rat @ X4 ) )
% 0.23/0.50 = ( ( power_power_nat @ B2 @ W )
% 0.23/0.50 = X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5472_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W )
% 0.23/0.50 = ( semiri1316708129612266289at_nat @ X4 ) )
% 0.23/0.50 = ( ( power_power_nat @ B2 @ W )
% 0.23/0.50 = X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5473_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W )
% 0.23/0.50 = ( semiri1314217659103216013at_int @ X4 ) )
% 0.23/0.50 = ( ( power_power_nat @ B2 @ W )
% 0.23/0.50 = X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_eq_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5474_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ( semiri8010041392384452111omplex @ X4 )
% 0.23/0.50 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W ) )
% 0.23/0.50 = ( X4
% 0.23/0.50 = ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_eq_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5475_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ( semiri5074537144036343181t_real @ X4 )
% 0.23/0.50 = ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 0.23/0.50 = ( X4
% 0.23/0.50 = ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_eq_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5476_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ( semiri681578069525770553at_rat @ X4 )
% 0.23/0.50 = ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
% 0.23/0.50 = ( X4
% 0.23/0.50 = ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_eq_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5477_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ( semiri1316708129612266289at_nat @ X4 )
% 0.23/0.50 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 0.23/0.50 = ( X4
% 0.23/0.50 = ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_eq_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5478_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ( semiri1314217659103216013at_int @ X4 )
% 0.23/0.50 = ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 0.23/0.50 = ( X4
% 0.23/0.50 = ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_eq_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5479_zabs__less__one__iff,axiom,
% 0.23/0.50 ! [Z: int] :
% 0.23/0.50 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 0.23/0.50 = ( Z = zero_zero_int ) ) ).
% 0.23/0.50
% 0.23/0.50 % zabs_less_one_iff
% 0.23/0.50 thf(fact_5480_log__one,axiom,
% 0.23/0.50 ! [A: real] :
% 0.23/0.50 ( ( log @ A @ one_one_real )
% 0.23/0.50 = zero_zero_real ) ).
% 0.23/0.50
% 0.23/0.50 % log_one
% 0.23/0.50 thf(fact_5481_arctan__less__zero__iff,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_real @ ( arctan @ X4 ) @ zero_zero_real )
% 0.23/0.50 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % arctan_less_zero_iff
% 0.23/0.50 thf(fact_5482_zero__less__arctan__iff,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ ( arctan @ X4 ) )
% 0.23/0.50 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_less_arctan_iff
% 0.23/0.50 thf(fact_5483_zero__le__arctan__iff,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_le_arctan_iff
% 0.23/0.50 thf(fact_5484_arctan__le__zero__iff,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( arctan @ X4 ) @ zero_zero_real )
% 0.23/0.50 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % arctan_le_zero_iff
% 0.23/0.50 thf(fact_5485_of__nat__of__bool,axiom,
% 0.23/0.50 ! [P: $o] :
% 0.23/0.50 ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.23/0.50 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_of_bool
% 0.23/0.50 thf(fact_5486_of__nat__of__bool,axiom,
% 0.23/0.50 ! [P: $o] :
% 0.23/0.50 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.23/0.50 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_of_bool
% 0.23/0.50 thf(fact_5487_of__nat__of__bool,axiom,
% 0.23/0.50 ! [P: $o] :
% 0.23/0.50 ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.23/0.50 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_of_bool
% 0.23/0.50 thf(fact_5488_of__nat__of__bool,axiom,
% 0.23/0.50 ! [P: $o] :
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.23/0.50 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_of_bool
% 0.23/0.50 thf(fact_5489_of__nat__of__bool,axiom,
% 0.23/0.50 ! [P: $o] :
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.23/0.50 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_of_bool
% 0.23/0.50 thf(fact_5490_of__nat__of__bool,axiom,
% 0.23/0.50 ! [P: $o] :
% 0.23/0.50 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.23/0.50 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_of_bool
% 0.23/0.50 thf(fact_5491_of__nat__le__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 0.23/0.50 = ( M = zero_zero_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_0_iff
% 0.23/0.50 thf(fact_5492_of__nat__le__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 0.23/0.50 = ( M = zero_zero_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_0_iff
% 0.23/0.50 thf(fact_5493_of__nat__le__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 0.23/0.50 = ( M = zero_zero_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_0_iff
% 0.23/0.50 thf(fact_5494_of__nat__le__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 0.23/0.50 = ( M = zero_zero_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_0_iff
% 0.23/0.50 thf(fact_5495_of__nat__Suc,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 0.23/0.50 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_Suc
% 0.23/0.50 thf(fact_5496_of__nat__Suc,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 0.23/0.50 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_Suc
% 0.23/0.50 thf(fact_5497_of__nat__Suc,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 0.23/0.50 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_Suc
% 0.23/0.50 thf(fact_5498_of__nat__Suc,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 0.23/0.50 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_Suc
% 0.23/0.50 thf(fact_5499_of__nat__Suc,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 0.23/0.50 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_Suc
% 0.23/0.50 thf(fact_5500_bit__numeral__Bit0__Suc__iff,axiom,
% 0.23/0.50 ! [M: num,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 0.23/0.50 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_Bit0_Suc_iff
% 0.23/0.50 thf(fact_5501_bit__numeral__Bit0__Suc__iff,axiom,
% 0.23/0.50 ! [M: num,N: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 0.23/0.50 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_Bit0_Suc_iff
% 0.23/0.50 thf(fact_5502_bit__numeral__Bit1__Suc__iff,axiom,
% 0.23/0.50 ! [M: num,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 0.23/0.50 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_Bit1_Suc_iff
% 0.23/0.50 thf(fact_5503_bit__numeral__Bit1__Suc__iff,axiom,
% 0.23/0.50 ! [M: num,N: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 0.23/0.50 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_Bit1_Suc_iff
% 0.23/0.50 thf(fact_5504_log__eq__one,axiom,
% 0.23/0.50 ! [A: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.50 => ( ( A != one_one_real )
% 0.23/0.50 => ( ( log @ A @ A )
% 0.23/0.50 = one_one_real ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_eq_one
% 0.23/0.50 thf(fact_5505_log__less__cancel__iff,axiom,
% 0.23/0.50 ! [A: real,X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.50 => ( ( ord_less_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) )
% 0.23/0.50 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_less_cancel_iff
% 0.23/0.50 thf(fact_5506_log__less__one__cancel__iff,axiom,
% 0.23/0.50 ! [A: real,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_real @ ( log @ A @ X4 ) @ one_one_real )
% 0.23/0.50 = ( ord_less_real @ X4 @ A ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_less_one_cancel_iff
% 0.23/0.50 thf(fact_5507_one__less__log__cancel__iff,axiom,
% 0.23/0.50 ! [A: real,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_real @ one_one_real @ ( log @ A @ X4 ) )
% 0.23/0.50 = ( ord_less_real @ A @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % one_less_log_cancel_iff
% 0.23/0.50 thf(fact_5508_log__less__zero__cancel__iff,axiom,
% 0.23/0.50 ! [A: real,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_real @ ( log @ A @ X4 ) @ zero_zero_real )
% 0.23/0.50 = ( ord_less_real @ X4 @ one_one_real ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_less_zero_cancel_iff
% 0.23/0.50 thf(fact_5509_zero__less__log__cancel__iff,axiom,
% 0.23/0.50 ! [A: real,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X4 ) )
% 0.23/0.50 = ( ord_less_real @ one_one_real @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_less_log_cancel_iff
% 0.23/0.50 thf(fact_5510_signed__take__bit__nonnegative__iff,axiom,
% 0.23/0.50 ! [N: nat,K: int] :
% 0.23/0.50 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.23/0.50 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % signed_take_bit_nonnegative_iff
% 0.23/0.50 thf(fact_5511_signed__take__bit__negative__iff,axiom,
% 0.23/0.50 ! [N: nat,K: int] :
% 0.23/0.50 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 0.23/0.50 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % signed_take_bit_negative_iff
% 0.23/0.50 thf(fact_5512_of__nat__0__less__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.50 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_less_iff
% 0.23/0.50 thf(fact_5513_of__nat__0__less__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.23/0.50 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_less_iff
% 0.23/0.50 thf(fact_5514_of__nat__0__less__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 0.23/0.50 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_less_iff
% 0.23/0.50 thf(fact_5515_of__nat__0__less__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.50 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_less_iff
% 0.23/0.50 thf(fact_5516_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X4 ) )
% 0.23/0.50 = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5517_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5518_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X4 ) )
% 0.23/0.50 = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5519_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X4 ) )
% 0.23/0.50 = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5520_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 0.23/0.50 = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_less_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5521_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
% 0.23/0.50 = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_less_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5522_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 0.23/0.50 = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_less_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5523_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 0.23/0.50 = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_less_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5524_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [Y3: nat,X4: num,N: nat] :
% 0.23/0.50 ( ( ( semiri8010041392384452111omplex @ Y3 )
% 0.23/0.50 = ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N ) )
% 0.23/0.50 = ( Y3
% 0.23/0.50 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_of_nat_eq_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5525_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [Y3: nat,X4: num,N: nat] :
% 0.23/0.50 ( ( ( semiri5074537144036343181t_real @ Y3 )
% 0.23/0.50 = ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 0.23/0.50 = ( Y3
% 0.23/0.50 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_of_nat_eq_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5526_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [Y3: nat,X4: num,N: nat] :
% 0.23/0.50 ( ( ( semiri681578069525770553at_rat @ Y3 )
% 0.23/0.50 = ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 0.23/0.50 = ( Y3
% 0.23/0.50 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_of_nat_eq_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5527_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [Y3: nat,X4: num,N: nat] :
% 0.23/0.50 ( ( ( semiri1316708129612266289at_nat @ Y3 )
% 0.23/0.50 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
% 0.23/0.50 = ( Y3
% 0.23/0.50 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_of_nat_eq_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5528_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [Y3: nat,X4: num,N: nat] :
% 0.23/0.50 ( ( ( semiri1314217659103216013at_int @ Y3 )
% 0.23/0.50 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
% 0.23/0.50 = ( Y3
% 0.23/0.50 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_of_nat_eq_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5529_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: num,N: nat,Y3: nat] :
% 0.23/0.50 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N )
% 0.23/0.50 = ( semiri8010041392384452111omplex @ Y3 ) )
% 0.23/0.50 = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 0.23/0.50 = Y3 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_eq_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5530_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: num,N: nat,Y3: nat] :
% 0.23/0.50 ( ( ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N )
% 0.23/0.50 = ( semiri5074537144036343181t_real @ Y3 ) )
% 0.23/0.50 = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 0.23/0.50 = Y3 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_eq_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5531_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: num,N: nat,Y3: nat] :
% 0.23/0.50 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N )
% 0.23/0.50 = ( semiri681578069525770553at_rat @ Y3 ) )
% 0.23/0.50 = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 0.23/0.50 = Y3 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_eq_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5532_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: num,N: nat,Y3: nat] :
% 0.23/0.50 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 0.23/0.50 = ( semiri1316708129612266289at_nat @ Y3 ) )
% 0.23/0.50 = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 0.23/0.50 = Y3 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_eq_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5533_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: num,N: nat,Y3: nat] :
% 0.23/0.50 ( ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 0.23/0.50 = ( semiri1314217659103216013at_int @ Y3 ) )
% 0.23/0.50 = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 0.23/0.50 = Y3 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_eq_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5534_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5535_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5536_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5537_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.23/0.50 ! [B2: nat,W: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_of_nat_power_cancel_iff
% 0.23/0.50 thf(fact_5538_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 0.23/0.50 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_le_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5539_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
% 0.23/0.50 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_le_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5540_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 0.23/0.50 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_le_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5541_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,B2: nat,W: nat] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 0.23/0.50 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_power_le_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5542_numeral__less__real__of__nat__iff,axiom,
% 0.23/0.50 ! [W: num,N: nat] :
% 0.23/0.50 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.50 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_less_real_of_nat_iff
% 0.23/0.50 thf(fact_5543_real__of__nat__less__numeral__iff,axiom,
% 0.23/0.50 ! [N: nat,W: num] :
% 0.23/0.50 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 0.23/0.50 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_of_nat_less_numeral_iff
% 0.23/0.50 thf(fact_5544_numeral__le__real__of__nat__iff,axiom,
% 0.23/0.50 ! [N: num,M: nat] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 0.23/0.50 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_le_real_of_nat_iff
% 0.23/0.50 thf(fact_5545_zero__le__log__cancel__iff,axiom,
% 0.23/0.50 ! [A: real,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_le_log_cancel_iff
% 0.23/0.50 thf(fact_5546_log__le__zero__cancel__iff,axiom,
% 0.23/0.50 ! [A: real,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ zero_zero_real )
% 0.23/0.50 = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_le_zero_cancel_iff
% 0.23/0.50 thf(fact_5547_one__le__log__cancel__iff,axiom,
% 0.23/0.50 ! [A: real,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_real @ A @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % one_le_log_cancel_iff
% 0.23/0.50 thf(fact_5548_log__le__one__cancel__iff,axiom,
% 0.23/0.50 ! [A: real,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ one_one_real )
% 0.23/0.50 = ( ord_less_eq_real @ X4 @ A ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_le_one_cancel_iff
% 0.23/0.50 thf(fact_5549_log__le__cancel__iff,axiom,
% 0.23/0.50 ! [A: real,X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.50 => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) )
% 0.23/0.50 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_le_cancel_iff
% 0.23/0.50 thf(fact_5550_bit__numeral__simps_I2_J,axiom,
% 0.23/0.50 ! [W: num,N: num] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.23/0.50 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_simps(2)
% 0.23/0.50 thf(fact_5551_bit__numeral__simps_I2_J,axiom,
% 0.23/0.50 ! [W: num,N: num] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.23/0.50 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_simps(2)
% 0.23/0.50 thf(fact_5552_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 0.23/0.50 ! [W: num,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 0.23/0.50 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_minus_numeral_Bit0_Suc_iff
% 0.23/0.50 thf(fact_5553_bit__numeral__simps_I3_J,axiom,
% 0.23/0.50 ! [W: num,N: num] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.23/0.50 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_simps(3)
% 0.23/0.50 thf(fact_5554_bit__numeral__simps_I3_J,axiom,
% 0.23/0.50 ! [W: num,N: num] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.23/0.50 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_simps(3)
% 0.23/0.50 thf(fact_5555_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 0.23/0.50 ! [W: num,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 0.23/0.50 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_minus_numeral_Bit1_Suc_iff
% 0.23/0.50 thf(fact_5556_of__nat__zero__less__power__iff,axiom,
% 0.23/0.50 ! [X4: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X4 ) @ N ) )
% 0.23/0.50 = ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 0.23/0.50 | ( N = zero_zero_nat ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_zero_less_power_iff
% 0.23/0.50 thf(fact_5557_of__nat__zero__less__power__iff,axiom,
% 0.23/0.50 ! [X4: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X4 ) @ N ) )
% 0.23/0.50 = ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 0.23/0.50 | ( N = zero_zero_nat ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_zero_less_power_iff
% 0.23/0.50 thf(fact_5558_of__nat__zero__less__power__iff,axiom,
% 0.23/0.50 ! [X4: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ N ) )
% 0.23/0.50 = ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 0.23/0.50 | ( N = zero_zero_nat ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_zero_less_power_iff
% 0.23/0.50 thf(fact_5559_of__nat__zero__less__power__iff,axiom,
% 0.23/0.50 ! [X4: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X4 ) @ N ) )
% 0.23/0.50 = ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 0.23/0.50 | ( N = zero_zero_nat ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_zero_less_power_iff
% 0.23/0.50 thf(fact_5560_bit__0,axiom,
% 0.23/0.50 ! [A: code_integer] :
% 0.23/0.50 ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 0.23/0.50 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_0
% 0.23/0.50 thf(fact_5561_bit__0,axiom,
% 0.23/0.50 ! [A: int] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 0.23/0.50 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_0
% 0.23/0.50 thf(fact_5562_bit__0,axiom,
% 0.23/0.50 ! [A: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 0.23/0.50 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_0
% 0.23/0.50 thf(fact_5563_log__pow__cancel,axiom,
% 0.23/0.50 ! [A: real,B2: nat] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.50 => ( ( A != one_one_real )
% 0.23/0.50 => ( ( log @ A @ ( power_power_real @ A @ B2 ) )
% 0.23/0.50 = ( semiri5074537144036343181t_real @ B2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_pow_cancel
% 0.23/0.50 thf(fact_5564_bit__minus__numeral__int_I1_J,axiom,
% 0.23/0.50 ! [W: num,N: num] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.23/0.50 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_minus_numeral_int(1)
% 0.23/0.50 thf(fact_5565_bit__minus__numeral__int_I2_J,axiom,
% 0.23/0.50 ! [W: num,N: num] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.23/0.50 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_minus_numeral_int(2)
% 0.23/0.50 thf(fact_5566_even__of__nat,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.23/0.50 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % even_of_nat
% 0.23/0.50 thf(fact_5567_even__of__nat,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.23/0.50 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % even_of_nat
% 0.23/0.50 thf(fact_5568_even__of__nat,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.50 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % even_of_nat
% 0.23/0.50 thf(fact_5569_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [I: num,N: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X4 ) )
% 0.23/0.50 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_less_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5570_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [I: num,N: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_less_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5571_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [I: num,N: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X4 ) )
% 0.23/0.50 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_less_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5572_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [I: num,N: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X4 ) )
% 0.23/0.50 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_less_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5573_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,I: num,N: nat] :
% 0.23/0.50 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.23/0.50 = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5574_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,I: num,N: nat] :
% 0.23/0.50 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.23/0.50 = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5575_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,I: num,N: nat] :
% 0.23/0.50 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.23/0.50 = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5576_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,I: num,N: nat] :
% 0.23/0.50 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.23/0.50 = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5577_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [I: num,N: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_le_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5578_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [I: num,N: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_le_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5579_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [I: num,N: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_le_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5580_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.23/0.50 ! [I: num,N: nat,X4: nat] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_power_le_of_nat_cancel_iff
% 0.23/0.50 thf(fact_5581_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,I: num,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.23/0.50 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5582_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,I: num,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.23/0.50 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5583_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,I: num,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.23/0.50 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5584_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.23/0.50 ! [X4: nat,I: num,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.23/0.50 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_le_numeral_power_cancel_iff
% 0.23/0.50 thf(fact_5585_bit__mod__2__iff,axiom,
% 0.23/0.50 ! [A: code_integer,N: nat] :
% 0.23/0.50 ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
% 0.23/0.50 = ( ( N = zero_zero_nat )
% 0.23/0.50 & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_mod_2_iff
% 0.23/0.50 thf(fact_5586_bit__mod__2__iff,axiom,
% 0.23/0.50 ! [A: int,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
% 0.23/0.50 = ( ( N = zero_zero_nat )
% 0.23/0.50 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_mod_2_iff
% 0.23/0.50 thf(fact_5587_bit__mod__2__iff,axiom,
% 0.23/0.50 ! [A: nat,N: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.23/0.50 = ( ( N = zero_zero_nat )
% 0.23/0.50 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_mod_2_iff
% 0.23/0.50 thf(fact_5588_bit__numeral__iff,axiom,
% 0.23/0.50 ! [M: num,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N )
% 0.23/0.50 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_iff
% 0.23/0.50 thf(fact_5589_bit__numeral__iff,axiom,
% 0.23/0.50 ! [M: num,N: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N )
% 0.23/0.50 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_iff
% 0.23/0.50 thf(fact_5590_bit__of__nat__iff__bit,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
% 0.23/0.50 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_of_nat_iff_bit
% 0.23/0.50 thf(fact_5591_bit__of__nat__iff__bit,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
% 0.23/0.50 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_of_nat_iff_bit
% 0.23/0.50 thf(fact_5592_bit__disjunctive__add__iff,axiom,
% 0.23/0.50 ! [A: int,B2: int,N: nat] :
% 0.23/0.50 ( ! [N4: nat] :
% 0.23/0.50 ( ~ ( bit_se1146084159140164899it_int @ A @ N4 )
% 0.23/0.50 | ~ ( bit_se1146084159140164899it_int @ B2 @ N4 ) )
% 0.23/0.50 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B2 ) @ N )
% 0.23/0.50 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.23/0.50 | ( bit_se1146084159140164899it_int @ B2 @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_disjunctive_add_iff
% 0.23/0.50 thf(fact_5593_bit__disjunctive__add__iff,axiom,
% 0.23/0.50 ! [A: nat,B2: nat,N: nat] :
% 0.23/0.50 ( ! [N4: nat] :
% 0.23/0.50 ( ~ ( bit_se1148574629649215175it_nat @ A @ N4 )
% 0.23/0.50 | ~ ( bit_se1148574629649215175it_nat @ B2 @ N4 ) )
% 0.23/0.50 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
% 0.23/0.50 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.23/0.50 | ( bit_se1148574629649215175it_nat @ B2 @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_disjunctive_add_iff
% 0.23/0.50 thf(fact_5594_real__arch__simple,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ? [N4: nat] : ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_arch_simple
% 0.23/0.50 thf(fact_5595_real__arch__simple,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ? [N4: nat] : ( ord_less_eq_rat @ X4 @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_arch_simple
% 0.23/0.50 thf(fact_5596_reals__Archimedean2,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ? [N4: nat] : ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % reals_Archimedean2
% 0.23/0.50 thf(fact_5597_reals__Archimedean2,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ? [N4: nat] : ( ord_less_rat @ X4 @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % reals_Archimedean2
% 0.23/0.50 thf(fact_5598_mult__of__nat__commute,axiom,
% 0.23/0.50 ! [X4: nat,Y3: complex] :
% 0.23/0.50 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X4 ) @ Y3 )
% 0.23/0.50 = ( times_times_complex @ Y3 @ ( semiri8010041392384452111omplex @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % mult_of_nat_commute
% 0.23/0.50 thf(fact_5599_mult__of__nat__commute,axiom,
% 0.23/0.50 ! [X4: nat,Y3: real] :
% 0.23/0.50 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ Y3 )
% 0.23/0.50 = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % mult_of_nat_commute
% 0.23/0.50 thf(fact_5600_mult__of__nat__commute,axiom,
% 0.23/0.50 ! [X4: nat,Y3: rat] :
% 0.23/0.50 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X4 ) @ Y3 )
% 0.23/0.50 = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % mult_of_nat_commute
% 0.23/0.50 thf(fact_5601_mult__of__nat__commute,axiom,
% 0.23/0.50 ! [X4: nat,Y3: nat] :
% 0.23/0.50 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ Y3 )
% 0.23/0.50 = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % mult_of_nat_commute
% 0.23/0.50 thf(fact_5602_mult__of__nat__commute,axiom,
% 0.23/0.50 ! [X4: nat,Y3: int] :
% 0.23/0.50 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X4 ) @ Y3 )
% 0.23/0.50 = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % mult_of_nat_commute
% 0.23/0.50 thf(fact_5603_bit__and__iff,axiom,
% 0.23/0.50 ! [A: int,B2: int,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ A @ B2 ) @ N )
% 0.23/0.50 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.23/0.50 & ( bit_se1146084159140164899it_int @ B2 @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_and_iff
% 0.23/0.50 thf(fact_5604_bit__and__iff,axiom,
% 0.23/0.50 ! [A: nat,B2: nat,N: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ ( bit_se727722235901077358nd_nat @ A @ B2 ) @ N )
% 0.23/0.50 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.23/0.50 & ( bit_se1148574629649215175it_nat @ B2 @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_and_iff
% 0.23/0.50 thf(fact_5605_bit__and__int__iff,axiom,
% 0.23/0.50 ! [K: int,L2: int,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N )
% 0.23/0.50 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.23/0.50 & ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_and_int_iff
% 0.23/0.50 thf(fact_5606_arctan__less__iff,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) )
% 0.23/0.50 = ( ord_less_real @ X4 @ Y3 ) ) ).
% 0.23/0.50
% 0.23/0.50 % arctan_less_iff
% 0.23/0.50 thf(fact_5607_arctan__monotone,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.50 => ( ord_less_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % arctan_monotone
% 0.23/0.50 thf(fact_5608_arctan__le__iff,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) )
% 0.23/0.50 = ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 0.23/0.50
% 0.23/0.50 % arctan_le_iff
% 0.23/0.50 thf(fact_5609_arctan__monotone_H,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.50 => ( ord_less_eq_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % arctan_monotone'
% 0.23/0.50 thf(fact_5610_bit__unset__bit__iff,axiom,
% 0.23/0.50 ! [M: nat,A: int,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( bit_se4203085406695923979it_int @ M @ A ) @ N )
% 0.23/0.50 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.23/0.50 & ( M != N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_unset_bit_iff
% 0.23/0.50 thf(fact_5611_bit__unset__bit__iff,axiom,
% 0.23/0.50 ! [M: nat,A: nat,N: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ ( bit_se4205575877204974255it_nat @ M @ A ) @ N )
% 0.23/0.50 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.23/0.50 & ( M != N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_unset_bit_iff
% 0.23/0.50 thf(fact_5612_less__log__of__power,axiom,
% 0.23/0.50 ! [B2: real,N: nat,M: real] :
% 0.23/0.50 ( ( ord_less_real @ ( power_power_real @ B2 @ N ) @ M )
% 0.23/0.50 => ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.50 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % less_log_of_power
% 0.23/0.50 thf(fact_5613_log__of__power__eq,axiom,
% 0.23/0.50 ! [M: nat,B2: real,N: nat] :
% 0.23/0.50 ( ( ( semiri5074537144036343181t_real @ M )
% 0.23/0.50 = ( power_power_real @ B2 @ N ) )
% 0.23/0.50 => ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.50 => ( ( semiri5074537144036343181t_real @ N )
% 0.23/0.50 = ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_of_power_eq
% 0.23/0.50 thf(fact_5614_of__nat__less__of__int__iff,axiom,
% 0.23/0.50 ! [N: nat,X4: int] :
% 0.23/0.50 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X4 ) )
% 0.23/0.50 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_of_int_iff
% 0.23/0.50 thf(fact_5615_of__nat__less__of__int__iff,axiom,
% 0.23/0.50 ! [N: nat,X4: int] :
% 0.23/0.50 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_of_int_iff
% 0.23/0.50 thf(fact_5616_of__nat__less__of__int__iff,axiom,
% 0.23/0.50 ! [N: nat,X4: int] :
% 0.23/0.50 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X4 ) )
% 0.23/0.50 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_of_int_iff
% 0.23/0.50 thf(fact_5617_le__log__of__power,axiom,
% 0.23/0.50 ! [B2: real,N: nat,M: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( power_power_real @ B2 @ N ) @ M )
% 0.23/0.50 => ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.50 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % le_log_of_power
% 0.23/0.50 thf(fact_5618_log__base__pow,axiom,
% 0.23/0.50 ! [A: real,N: nat,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.50 => ( ( log @ ( power_power_real @ A @ N ) @ X4 )
% 0.23/0.50 = ( divide_divide_real @ ( log @ A @ X4 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_base_pow
% 0.23/0.50 thf(fact_5619_log__nat__power,axiom,
% 0.23/0.50 ! [X4: real,B2: real,N: nat] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( log @ B2 @ ( power_power_real @ X4 @ N ) )
% 0.23/0.50 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_nat_power
% 0.23/0.50 thf(fact_5620_not__bit__1__Suc,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % not_bit_1_Suc
% 0.23/0.50 thf(fact_5621_not__bit__1__Suc,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % not_bit_1_Suc
% 0.23/0.50 thf(fact_5622_bit__1__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
% 0.23/0.50 = ( N = zero_zero_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_1_iff
% 0.23/0.50 thf(fact_5623_bit__1__iff,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
% 0.23/0.50 = ( N = zero_zero_nat ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_1_iff
% 0.23/0.50 thf(fact_5624_bit__numeral__simps_I1_J,axiom,
% 0.23/0.50 ! [N: num] :
% 0.23/0.50 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_simps(1)
% 0.23/0.50 thf(fact_5625_bit__numeral__simps_I1_J,axiom,
% 0.23/0.50 ! [N: num] :
% 0.23/0.50 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_numeral_simps(1)
% 0.23/0.50 thf(fact_5626_of__nat__0__le__iff,axiom,
% 0.23/0.50 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_le_iff
% 0.23/0.50 thf(fact_5627_of__nat__0__le__iff,axiom,
% 0.23/0.50 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_le_iff
% 0.23/0.50 thf(fact_5628_of__nat__0__le__iff,axiom,
% 0.23/0.50 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_le_iff
% 0.23/0.50 thf(fact_5629_of__nat__0__le__iff,axiom,
% 0.23/0.50 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_0_le_iff
% 0.23/0.50 thf(fact_5630_of__nat__less__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_0_iff
% 0.23/0.50 thf(fact_5631_of__nat__less__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_0_iff
% 0.23/0.50 thf(fact_5632_of__nat__less__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_0_iff
% 0.23/0.50 thf(fact_5633_of__nat__less__0__iff,axiom,
% 0.23/0.50 ! [M: nat] :
% 0.23/0.50 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_0_iff
% 0.23/0.50 thf(fact_5634_bit__take__bit__iff,axiom,
% 0.23/0.50 ! [M: nat,A: int,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
% 0.23/0.50 = ( ( ord_less_nat @ N @ M )
% 0.23/0.50 & ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_take_bit_iff
% 0.23/0.50 thf(fact_5635_bit__take__bit__iff,axiom,
% 0.23/0.50 ! [M: nat,A: nat,N: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
% 0.23/0.50 = ( ( ord_less_nat @ N @ M )
% 0.23/0.50 & ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_take_bit_iff
% 0.23/0.50 thf(fact_5636_of__nat__neq__0,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 0.23/0.50 != zero_zero_complex ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_neq_0
% 0.23/0.50 thf(fact_5637_of__nat__neq__0,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 0.23/0.50 != zero_zero_real ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_neq_0
% 0.23/0.50 thf(fact_5638_of__nat__neq__0,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 0.23/0.50 != zero_zero_rat ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_neq_0
% 0.23/0.50 thf(fact_5639_of__nat__neq__0,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 0.23/0.50 != zero_zero_nat ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_neq_0
% 0.23/0.50 thf(fact_5640_of__nat__neq__0,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.23/0.50 != zero_zero_int ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_neq_0
% 0.23/0.50 thf(fact_5641_bit__of__bool__iff,axiom,
% 0.23/0.50 ! [B2: $o,N: nat] :
% 0.23/0.50 ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B2 ) @ N )
% 0.23/0.50 = ( B2
% 0.23/0.50 & ( N = zero_zero_nat ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_of_bool_iff
% 0.23/0.50 thf(fact_5642_bit__of__bool__iff,axiom,
% 0.23/0.50 ! [B2: $o,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ N )
% 0.23/0.50 = ( B2
% 0.23/0.50 & ( N = zero_zero_nat ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_of_bool_iff
% 0.23/0.50 thf(fact_5643_bit__of__bool__iff,axiom,
% 0.23/0.50 ! [B2: $o,N: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ N )
% 0.23/0.50 = ( B2
% 0.23/0.50 & ( N = zero_zero_nat ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_of_bool_iff
% 0.23/0.50 thf(fact_5644_div__mult2__eq_H,axiom,
% 0.23/0.50 ! [A: nat,M: nat,N: nat] :
% 0.23/0.50 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.23/0.50 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % div_mult2_eq'
% 0.23/0.50 thf(fact_5645_div__mult2__eq_H,axiom,
% 0.23/0.50 ! [A: int,M: nat,N: nat] :
% 0.23/0.50 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.23/0.50 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % div_mult2_eq'
% 0.23/0.50 thf(fact_5646_less__imp__of__nat__less,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_nat @ M @ N )
% 0.23/0.50 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % less_imp_of_nat_less
% 0.23/0.50 thf(fact_5647_less__imp__of__nat__less,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_nat @ M @ N )
% 0.23/0.50 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % less_imp_of_nat_less
% 0.23/0.50 thf(fact_5648_less__imp__of__nat__less,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_nat @ M @ N )
% 0.23/0.50 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % less_imp_of_nat_less
% 0.23/0.50 thf(fact_5649_less__imp__of__nat__less,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_nat @ M @ N )
% 0.23/0.50 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % less_imp_of_nat_less
% 0.23/0.50 thf(fact_5650_of__nat__less__imp__less,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.50 => ( ord_less_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_imp_less
% 0.23/0.50 thf(fact_5651_of__nat__less__imp__less,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.23/0.50 => ( ord_less_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_imp_less
% 0.23/0.50 thf(fact_5652_of__nat__less__imp__less,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.23/0.50 => ( ord_less_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_imp_less
% 0.23/0.50 thf(fact_5653_of__nat__less__imp__less,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.50 => ( ord_less_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_imp_less
% 0.23/0.50 thf(fact_5654_of__nat__mono,axiom,
% 0.23/0.50 ! [I: nat,J: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.23/0.50 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mono
% 0.23/0.50 thf(fact_5655_of__nat__mono,axiom,
% 0.23/0.50 ! [I: nat,J: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.23/0.50 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mono
% 0.23/0.50 thf(fact_5656_of__nat__mono,axiom,
% 0.23/0.50 ! [I: nat,J: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.23/0.50 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mono
% 0.23/0.50 thf(fact_5657_of__nat__mono,axiom,
% 0.23/0.50 ! [I: nat,J: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.23/0.50 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mono
% 0.23/0.50 thf(fact_5658_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 0.23/0.50 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.23/0.50 thf(fact_5659_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 0.23/0.50 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.23/0.50 thf(fact_5660_log__def,axiom,
% 0.23/0.50 ( log
% 0.23/0.50 = ( ^ [A4: real,X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ A4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_def
% 0.23/0.50 thf(fact_5661_of__nat__dvd__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.23/0.50 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_dvd_iff
% 0.23/0.50 thf(fact_5662_of__nat__dvd__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.23/0.50 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_dvd_iff
% 0.23/0.50 thf(fact_5663_of__nat__dvd__iff,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.50 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_dvd_iff
% 0.23/0.50 thf(fact_5664_signed__take__bit__eq__if__positive,axiom,
% 0.23/0.50 ! [A: int,N: nat] :
% 0.23/0.50 ( ~ ( bit_se1146084159140164899it_int @ A @ N )
% 0.23/0.50 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 0.23/0.50 = ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % signed_take_bit_eq_if_positive
% 0.23/0.50 thf(fact_5665_abs__zmult__eq__1,axiom,
% 0.23/0.50 ! [M: int,N: int] :
% 0.23/0.50 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 0.23/0.50 = one_one_int )
% 0.23/0.50 => ( ( abs_abs_int @ M )
% 0.23/0.50 = one_one_int ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_zmult_eq_1
% 0.23/0.50 thf(fact_5666_of__nat__mod,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 0.23/0.50 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mod
% 0.23/0.50 thf(fact_5667_of__nat__mod,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.23/0.50 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mod
% 0.23/0.50 thf(fact_5668_of__nat__mod,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 0.23/0.50 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mod
% 0.23/0.50 thf(fact_5669_abs__div,axiom,
% 0.23/0.50 ! [Y3: int,X4: int] :
% 0.23/0.50 ( ( dvd_dvd_int @ Y3 @ X4 )
% 0.23/0.50 => ( ( abs_abs_int @ ( divide_divide_int @ X4 @ Y3 ) )
% 0.23/0.50 = ( divide_divide_int @ ( abs_abs_int @ X4 ) @ ( abs_abs_int @ Y3 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_div
% 0.23/0.50 thf(fact_5670_log2__of__power__eq,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( M
% 0.23/0.50 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.50 => ( ( semiri5074537144036343181t_real @ N )
% 0.23/0.50 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log2_of_power_eq
% 0.23/0.50 thf(fact_5671_log__of__power__less,axiom,
% 0.23/0.50 ! [M: nat,B2: real,N: nat] :
% 0.23/0.50 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
% 0.23/0.50 => ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.50 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.50 => ( ord_less_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_of_power_less
% 0.23/0.50 thf(fact_5672_take__bit__of__nat,axiom,
% 0.23/0.50 ! [N: nat,M: nat] :
% 0.23/0.50 ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 0.23/0.50 = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % take_bit_of_nat
% 0.23/0.50 thf(fact_5673_take__bit__of__nat,axiom,
% 0.23/0.50 ! [N: nat,M: nat] :
% 0.23/0.50 ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 0.23/0.50 = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % take_bit_of_nat
% 0.23/0.50 thf(fact_5674_of__nat__and__eq,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 0.23/0.50 = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_and_eq
% 0.23/0.50 thf(fact_5675_of__nat__and__eq,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 0.23/0.50 = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_and_eq
% 0.23/0.50 thf(fact_5676_of__nat__mask__eq,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.23/0.50 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mask_eq
% 0.23/0.50 thf(fact_5677_of__nat__mask__eq,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.23/0.50 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_mask_eq
% 0.23/0.50 thf(fact_5678_log__of__power__le,axiom,
% 0.23/0.50 ! [M: nat,B2: real,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
% 0.23/0.50 => ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.50 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.50 => ( ord_less_eq_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_of_power_le
% 0.23/0.50 thf(fact_5679_ex__less__of__nat__mult,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ? [N4: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ex_less_of_nat_mult
% 0.23/0.50 thf(fact_5680_ex__less__of__nat__mult,axiom,
% 0.23/0.50 ! [X4: rat,Y3: rat] :
% 0.23/0.50 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 0.23/0.50 => ? [N4: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N4 ) @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ex_less_of_nat_mult
% 0.23/0.50 thf(fact_5681_of__nat__diff,axiom,
% 0.23/0.50 ! [N: nat,M: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.50 => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 0.23/0.50 = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_diff
% 0.23/0.50 thf(fact_5682_of__nat__diff,axiom,
% 0.23/0.50 ! [N: nat,M: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.50 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 0.23/0.50 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_diff
% 0.23/0.50 thf(fact_5683_of__nat__diff,axiom,
% 0.23/0.50 ! [N: nat,M: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.50 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 0.23/0.50 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_diff
% 0.23/0.50 thf(fact_5684_of__nat__diff,axiom,
% 0.23/0.50 ! [N: nat,M: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.50 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 0.23/0.50 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_diff
% 0.23/0.50 thf(fact_5685_of__nat__diff,axiom,
% 0.23/0.50 ! [N: nat,M: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.50 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 0.23/0.50 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_diff
% 0.23/0.50 thf(fact_5686_bit__not__int__iff_H,axiom,
% 0.23/0.50 ! [K: int,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 0.23/0.50 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_not_int_iff'
% 0.23/0.50 thf(fact_5687_exp__of__nat__mult,axiom,
% 0.23/0.50 ! [N: nat,X4: complex] :
% 0.23/0.50 ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X4 ) )
% 0.23/0.50 = ( power_power_complex @ ( exp_complex @ X4 ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % exp_of_nat_mult
% 0.23/0.50 thf(fact_5688_exp__of__nat__mult,axiom,
% 0.23/0.50 ! [N: nat,X4: real] :
% 0.23/0.50 ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) )
% 0.23/0.50 = ( power_power_real @ ( exp_real @ X4 ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % exp_of_nat_mult
% 0.23/0.50 thf(fact_5689_exp__of__nat2__mult,axiom,
% 0.23/0.50 ! [X4: complex,N: nat] :
% 0.23/0.50 ( ( exp_complex @ ( times_times_complex @ X4 @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.23/0.50 = ( power_power_complex @ ( exp_complex @ X4 ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % exp_of_nat2_mult
% 0.23/0.50 thf(fact_5690_exp__of__nat2__mult,axiom,
% 0.23/0.50 ! [X4: real,N: nat] :
% 0.23/0.50 ( ( exp_real @ ( times_times_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.23/0.50 = ( power_power_real @ ( exp_real @ X4 ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % exp_of_nat2_mult
% 0.23/0.50 thf(fact_5691_zabs__def,axiom,
% 0.23/0.50 ( abs_abs_int
% 0.23/0.50 = ( ^ [I2: int] : ( if_int @ ( ord_less_int @ I2 @ zero_zero_int ) @ ( uminus_uminus_int @ I2 ) @ I2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % zabs_def
% 0.23/0.50 thf(fact_5692_reals__Archimedean3,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ! [Y5: real] :
% 0.23/0.50 ? [N4: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % reals_Archimedean3
% 0.23/0.50 thf(fact_5693_log__ln,axiom,
% 0.23/0.50 ( ln_ln_real
% 0.23/0.50 = ( log @ ( exp_real @ one_one_real ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_ln
% 0.23/0.50 thf(fact_5694_real__of__nat__div4,axiom,
% 0.23/0.50 ! [N: nat,X4: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_of_nat_div4
% 0.23/0.50 thf(fact_5695_abs__mod__less,axiom,
% 0.23/0.50 ! [L2: int,K: int] :
% 0.23/0.50 ( ( L2 != zero_zero_int )
% 0.23/0.50 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % abs_mod_less
% 0.23/0.50 thf(fact_5696_real__of__nat__div,axiom,
% 0.23/0.50 ! [D: nat,N: nat] :
% 0.23/0.50 ( ( dvd_dvd_nat @ D @ N )
% 0.23/0.50 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 0.23/0.50 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_of_nat_div
% 0.23/0.50 thf(fact_5697_less__log2__of__power,axiom,
% 0.23/0.50 ! [N: nat,M: nat] :
% 0.23/0.50 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.23/0.50 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % less_log2_of_power
% 0.23/0.50 thf(fact_5698_le__log2__of__power,axiom,
% 0.23/0.50 ! [N: nat,M: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.23/0.50 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % le_log2_of_power
% 0.23/0.50 thf(fact_5699_flip__bit__eq__if,axiom,
% 0.23/0.50 ( bit_se2159334234014336723it_int
% 0.23/0.50 = ( ^ [N2: nat,A4: int] : ( if_nat_int_int @ ( bit_se1146084159140164899it_int @ A4 @ N2 ) @ bit_se4203085406695923979it_int @ bit_se7879613467334960850it_int @ N2 @ A4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % flip_bit_eq_if
% 0.23/0.50 thf(fact_5700_flip__bit__eq__if,axiom,
% 0.23/0.50 ( bit_se2161824704523386999it_nat
% 0.23/0.50 = ( ^ [N2: nat,A4: nat] : ( if_nat_nat_nat @ ( bit_se1148574629649215175it_nat @ A4 @ N2 ) @ bit_se4205575877204974255it_nat @ bit_se7882103937844011126it_nat @ N2 @ A4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % flip_bit_eq_if
% 0.23/0.50 thf(fact_5701_log2__of__power__less,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.50 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.50 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log2_of_power_less
% 0.23/0.50 thf(fact_5702_log__base__change,axiom,
% 0.23/0.50 ! [A: real,B2: real,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.50 => ( ( A != one_one_real )
% 0.23/0.50 => ( ( log @ B2 @ X4 )
% 0.23/0.50 = ( divide_divide_real @ ( log @ A @ X4 ) @ ( log @ A @ B2 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_base_change
% 0.23/0.50 thf(fact_5703_mod__mult2__eq_H,axiom,
% 0.23/0.50 ! [A: code_integer,M: nat,N: nat] :
% 0.23/0.50 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 0.23/0.50 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % mod_mult2_eq'
% 0.23/0.50 thf(fact_5704_mod__mult2__eq_H,axiom,
% 0.23/0.50 ! [A: nat,M: nat,N: nat] :
% 0.23/0.50 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.23/0.50 = ( 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 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % mod_mult2_eq'
% 0.23/0.50 thf(fact_5705_mod__mult2__eq_H,axiom,
% 0.23/0.50 ! [A: int,M: nat,N: nat] :
% 0.23/0.50 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.23/0.50 = ( 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 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % mod_mult2_eq'
% 0.23/0.50 thf(fact_5706_bit__imp__take__bit__positive,axiom,
% 0.23/0.50 ! [N: nat,M: nat,K: int] :
% 0.23/0.50 ( ( ord_less_nat @ N @ M )
% 0.23/0.50 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.23/0.50 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_imp_take_bit_positive
% 0.23/0.50 thf(fact_5707_field__char__0__class_Oof__nat__div,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 0.23/0.50 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % field_char_0_class.of_nat_div
% 0.23/0.50 thf(fact_5708_field__char__0__class_Oof__nat__div,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 0.23/0.50 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % field_char_0_class.of_nat_div
% 0.23/0.50 thf(fact_5709_field__char__0__class_Oof__nat__div,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 0.23/0.50 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % field_char_0_class.of_nat_div
% 0.23/0.50 thf(fact_5710_bit__concat__bit__iff,axiom,
% 0.23/0.50 ! [M: nat,K: int,L2: int,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N )
% 0.23/0.50 = ( ( ( ord_less_nat @ N @ M )
% 0.23/0.50 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 0.23/0.50 | ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.50 & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_concat_bit_iff
% 0.23/0.50 thf(fact_5711_nat__less__real__le,axiom,
% 0.23/0.50 ( ord_less_nat
% 0.23/0.50 = ( ^ [N2: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % nat_less_real_le
% 0.23/0.50 thf(fact_5712_nat__le__real__less,axiom,
% 0.23/0.50 ( ord_less_eq_nat
% 0.23/0.50 = ( ^ [N2: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % nat_le_real_less
% 0.23/0.50 thf(fact_5713_zdvd__mult__cancel1,axiom,
% 0.23/0.50 ! [M: int,N: int] :
% 0.23/0.50 ( ( M != zero_zero_int )
% 0.23/0.50 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 0.23/0.50 = ( ( abs_abs_int @ N )
% 0.23/0.50 = one_one_int ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % zdvd_mult_cancel1
% 0.23/0.50 thf(fact_5714_log2__of__power__le,axiom,
% 0.23/0.50 ! [M: nat,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.50 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.50 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log2_of_power_le
% 0.23/0.50 thf(fact_5715_real__of__nat__div__aux,axiom,
% 0.23/0.50 ! [X4: nat,D: nat] :
% 0.23/0.50 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( semiri5074537144036343181t_real @ D ) )
% 0.23/0.50 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X4 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X4 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_of_nat_div_aux
% 0.23/0.50 thf(fact_5716_signed__take__bit__eq__concat__bit,axiom,
% 0.23/0.50 ( bit_ri631733984087533419it_int
% 0.23/0.50 = ( ^ [N2: nat,K3: int] : ( bit_concat_bit @ N2 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % signed_take_bit_eq_concat_bit
% 0.23/0.50 thf(fact_5717_exp__eq__0__imp__not__bit,axiom,
% 0.23/0.50 ! [N: nat,A: int] :
% 0.23/0.50 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 = zero_zero_int )
% 0.23/0.50 => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % exp_eq_0_imp_not_bit
% 0.23/0.50 thf(fact_5718_exp__eq__0__imp__not__bit,axiom,
% 0.23/0.50 ! [N: nat,A: nat] :
% 0.23/0.50 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 = zero_zero_nat )
% 0.23/0.50 => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % exp_eq_0_imp_not_bit
% 0.23/0.50 thf(fact_5719_bit__Suc,axiom,
% 0.23/0.50 ! [A: int,N: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
% 0.23/0.50 = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_Suc
% 0.23/0.50 thf(fact_5720_bit__Suc,axiom,
% 0.23/0.50 ! [A: nat,N: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
% 0.23/0.50 = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_Suc
% 0.23/0.50 thf(fact_5721_stable__imp__bit__iff__odd,axiom,
% 0.23/0.50 ! [A: code_integer,N: nat] :
% 0.23/0.50 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.50 = A )
% 0.23/0.50 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.23/0.50 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % stable_imp_bit_iff_odd
% 0.23/0.50 thf(fact_5722_stable__imp__bit__iff__odd,axiom,
% 0.23/0.50 ! [A: int,N: nat] :
% 0.23/0.50 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.50 = A )
% 0.23/0.50 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.23/0.50 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % stable_imp_bit_iff_odd
% 0.23/0.50 thf(fact_5723_stable__imp__bit__iff__odd,axiom,
% 0.23/0.50 ! [A: nat,N: nat] :
% 0.23/0.50 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.50 = A )
% 0.23/0.50 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.23/0.50 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % stable_imp_bit_iff_odd
% 0.23/0.50 thf(fact_5724_bit__iff__idd__imp__stable,axiom,
% 0.23/0.50 ! [A: code_integer] :
% 0.23/0.50 ( ! [N4: nat] :
% 0.23/0.50 ( ( bit_se9216721137139052372nteger @ A @ N4 )
% 0.23/0.50 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.23/0.50 => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.50 = A ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_iff_idd_imp_stable
% 0.23/0.50 thf(fact_5725_bit__iff__idd__imp__stable,axiom,
% 0.23/0.50 ! [A: int] :
% 0.23/0.50 ( ! [N4: nat] :
% 0.23/0.50 ( ( bit_se1146084159140164899it_int @ A @ N4 )
% 0.23/0.50 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.23/0.50 => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.50 = A ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_iff_idd_imp_stable
% 0.23/0.50 thf(fact_5726_bit__iff__idd__imp__stable,axiom,
% 0.23/0.50 ! [A: nat] :
% 0.23/0.50 ( ! [N4: nat] :
% 0.23/0.50 ( ( bit_se1148574629649215175it_nat @ A @ N4 )
% 0.23/0.50 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.23/0.50 => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.50 = A ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_iff_idd_imp_stable
% 0.23/0.50 thf(fact_5727_log__mult,axiom,
% 0.23/0.50 ! [A: real,X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.50 => ( ( A != one_one_real )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.50 => ( ( log @ A @ ( times_times_real @ X4 @ Y3 ) )
% 0.23/0.50 = ( plus_plus_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_mult
% 0.23/0.50 thf(fact_5728_nat__approx__posE,axiom,
% 0.23/0.50 ! [E: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ E )
% 0.23/0.50 => ~ ! [N4: nat] :
% 0.23/0.50 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ E ) ) ).
% 0.23/0.50
% 0.23/0.50 % nat_approx_posE
% 0.23/0.50 thf(fact_5729_nat__approx__posE,axiom,
% 0.23/0.50 ! [E: rat] :
% 0.23/0.50 ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.23/0.50 => ~ ! [N4: nat] :
% 0.23/0.50 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ E ) ) ).
% 0.23/0.50
% 0.23/0.50 % nat_approx_posE
% 0.23/0.50 thf(fact_5730_of__nat__less__two__power,axiom,
% 0.23/0.50 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_two_power
% 0.23/0.50 thf(fact_5731_of__nat__less__two__power,axiom,
% 0.23/0.50 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_two_power
% 0.23/0.50 thf(fact_5732_of__nat__less__two__power,axiom,
% 0.23/0.50 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_nat_less_two_power
% 0.23/0.50 thf(fact_5733_int__bit__bound,axiom,
% 0.23/0.50 ! [K: int] :
% 0.23/0.50 ~ ! [N4: nat] :
% 0.23/0.50 ( ! [M5: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ N4 @ M5 )
% 0.23/0.50 => ( ( bit_se1146084159140164899it_int @ K @ M5 )
% 0.23/0.50 = ( bit_se1146084159140164899it_int @ K @ N4 ) ) )
% 0.23/0.50 => ~ ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.23/0.50 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N4 @ one_one_nat ) )
% 0.23/0.50 = ( ~ ( bit_se1146084159140164899it_int @ K @ N4 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % int_bit_bound
% 0.23/0.50 thf(fact_5734_log__divide,axiom,
% 0.23/0.50 ! [A: real,X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.50 => ( ( A != one_one_real )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.50 => ( ( log @ A @ ( divide_divide_real @ X4 @ Y3 ) )
% 0.23/0.50 = ( minus_minus_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_divide
% 0.23/0.50 thf(fact_5735_inverse__of__nat__le,axiom,
% 0.23/0.50 ! [N: nat,M: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.50 => ( ( N != zero_zero_nat )
% 0.23/0.50 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % inverse_of_nat_le
% 0.23/0.50 thf(fact_5736_inverse__of__nat__le,axiom,
% 0.23/0.50 ! [N: nat,M: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.50 => ( ( N != zero_zero_nat )
% 0.23/0.50 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % inverse_of_nat_le
% 0.23/0.50 thf(fact_5737_exp__divide__power__eq,axiom,
% 0.23/0.50 ! [N: nat,X4: complex] :
% 0.23/0.50 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.50 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X4 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 0.23/0.50 = ( exp_complex @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % exp_divide_power_eq
% 0.23/0.50 thf(fact_5738_exp__divide__power__eq,axiom,
% 0.23/0.50 ! [N: nat,X4: real] :
% 0.23/0.50 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.50 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 0.23/0.50 = ( exp_real @ X4 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % exp_divide_power_eq
% 0.23/0.50 thf(fact_5739_even__abs__add__iff,axiom,
% 0.23/0.50 ! [K: int,L2: int] :
% 0.23/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
% 0.23/0.50 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % even_abs_add_iff
% 0.23/0.50 thf(fact_5740_even__add__abs__iff,axiom,
% 0.23/0.50 ! [K: int,L2: int] :
% 0.23/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
% 0.23/0.50 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % even_add_abs_iff
% 0.23/0.50 thf(fact_5741_real__archimedian__rdiv__eq__0,axiom,
% 0.23/0.50 ! [X4: real,C: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.23/0.50 => ( ! [M4: nat] :
% 0.23/0.50 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 0.23/0.50 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X4 ) @ C ) )
% 0.23/0.50 => ( X4 = zero_zero_real ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_archimedian_rdiv_eq_0
% 0.23/0.50 thf(fact_5742_real__of__nat__div2,axiom,
% 0.23/0.50 ! [N: nat,X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % real_of_nat_div2
% 0.23/0.50 thf(fact_5743_real__of__nat__div3,axiom,
% 0.23/0.50 ! [N: nat,X4: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) ) @ one_one_real ) ).
% 0.23/0.50
% 0.23/0.50 % real_of_nat_div3
% 0.23/0.50 thf(fact_5744_ln__realpow,axiom,
% 0.23/0.50 ! [X4: real,N: nat] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( ln_ln_real @ ( power_power_real @ X4 @ N ) )
% 0.23/0.50 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ln_realpow
% 0.23/0.50 thf(fact_5745_bit__iff__odd,axiom,
% 0.23/0.50 ( bit_se9216721137139052372nteger
% 0.23/0.50 = ( ^ [A4: code_integer,N2: nat] :
% 0.23/0.50 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A4 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_iff_odd
% 0.23/0.50 thf(fact_5746_bit__iff__odd,axiom,
% 0.23/0.50 ( bit_se1146084159140164899it_int
% 0.23/0.50 = ( ^ [A4: int,N2: nat] :
% 0.23/0.50 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_iff_odd
% 0.23/0.50 thf(fact_5747_bit__iff__odd,axiom,
% 0.23/0.50 ( bit_se1148574629649215175it_nat
% 0.23/0.50 = ( ^ [A4: nat,N2: nat] :
% 0.23/0.50 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_iff_odd
% 0.23/0.50 thf(fact_5748_and__exp__eq__0__iff__not__bit,axiom,
% 0.23/0.50 ! [A: int,N: nat] :
% 0.23/0.50 ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.50 = zero_zero_int )
% 0.23/0.50 = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % and_exp_eq_0_iff_not_bit
% 0.23/0.50 thf(fact_5749_and__exp__eq__0__iff__not__bit,axiom,
% 0.23/0.50 ! [A: nat,N: nat] :
% 0.23/0.50 ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.50 = zero_zero_nat )
% 0.23/0.50 = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % and_exp_eq_0_iff_not_bit
% 0.23/0.50 thf(fact_5750_log__eq__div__ln__mult__log,axiom,
% 0.23/0.50 ! [A: real,B2: real,X4: real] :
% 0.23/0.50 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.50 => ( ( A != one_one_real )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.23/0.50 => ( ( B2 != one_one_real )
% 0.23/0.50 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ( log @ A @ X4 )
% 0.23/0.50 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( ln_ln_real @ A ) ) @ ( log @ B2 @ X4 ) ) ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % log_eq_div_ln_mult_log
% 0.23/0.50 thf(fact_5751_bit__int__def,axiom,
% 0.23/0.50 ( bit_se1146084159140164899it_int
% 0.23/0.50 = ( ^ [K3: int,N2: nat] :
% 0.23/0.50 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_int_def
% 0.23/0.50 thf(fact_5752_nat__intermed__int__val,axiom,
% 0.23/0.50 ! [M: nat,N: nat,F: nat > int,K: int] :
% 0.23/0.50 ( ! [I3: nat] :
% 0.23/0.50 ( ( ( ord_less_eq_nat @ M @ I3 )
% 0.23/0.50 & ( ord_less_nat @ I3 @ N ) )
% 0.23/0.50 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.23/0.50 => ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.50 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.23/0.50 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.23/0.50 => ? [I3: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ M @ I3 )
% 0.23/0.50 & ( ord_less_eq_nat @ I3 @ N )
% 0.23/0.50 & ( ( F @ I3 )
% 0.23/0.50 = K ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % nat_intermed_int_val
% 0.23/0.50 thf(fact_5753_decr__lemma,axiom,
% 0.23/0.50 ! [D: int,X4: int,Z: int] :
% 0.23/0.50 ( ( ord_less_int @ zero_zero_int @ D )
% 0.23/0.50 => ( ord_less_int @ ( minus_minus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 0.23/0.50
% 0.23/0.50 % decr_lemma
% 0.23/0.50 thf(fact_5754_incr__lemma,axiom,
% 0.23/0.50 ! [D: int,Z: int,X4: int] :
% 0.23/0.50 ( ( ord_less_int @ zero_zero_int @ D )
% 0.23/0.50 => ( ord_less_int @ Z @ ( plus_plus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % incr_lemma
% 0.23/0.50 thf(fact_5755_linear__plus__1__le__power,axiom,
% 0.23/0.50 ! [X4: real,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.50 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X4 @ one_one_real ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % linear_plus_1_le_power
% 0.23/0.50 thf(fact_5756_Bernoulli__inequality,axiom,
% 0.23/0.50 ! [X4: real,N: nat] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.50 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X4 ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % Bernoulli_inequality
% 0.23/0.50 thf(fact_5757_even__bit__succ__iff,axiom,
% 0.23/0.50 ! [A: code_integer,N: nat] :
% 0.23/0.50 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.23/0.50 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
% 0.23/0.50 = ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.23/0.50 | ( N = zero_zero_nat ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % even_bit_succ_iff
% 0.23/0.50 thf(fact_5758_even__bit__succ__iff,axiom,
% 0.23/0.50 ! [A: int,N: nat] :
% 0.23/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.23/0.50 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
% 0.23/0.50 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.23/0.50 | ( N = zero_zero_nat ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % even_bit_succ_iff
% 0.23/0.50 thf(fact_5759_even__bit__succ__iff,axiom,
% 0.23/0.50 ! [A: nat,N: nat] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.23/0.50 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
% 0.23/0.50 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.23/0.50 | ( N = zero_zero_nat ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % even_bit_succ_iff
% 0.23/0.50 thf(fact_5760_odd__bit__iff__bit__pred,axiom,
% 0.23/0.50 ! [A: code_integer,N: nat] :
% 0.23/0.50 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.23/0.50 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.23/0.50 = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
% 0.23/0.50 | ( N = zero_zero_nat ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % odd_bit_iff_bit_pred
% 0.23/0.50 thf(fact_5761_odd__bit__iff__bit__pred,axiom,
% 0.23/0.50 ! [A: int,N: nat] :
% 0.23/0.50 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.23/0.50 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.23/0.50 = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
% 0.23/0.50 | ( N = zero_zero_nat ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % odd_bit_iff_bit_pred
% 0.23/0.50 thf(fact_5762_odd__bit__iff__bit__pred,axiom,
% 0.23/0.50 ! [A: nat,N: nat] :
% 0.23/0.50 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.23/0.50 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.23/0.50 = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
% 0.23/0.50 | ( N = zero_zero_nat ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % odd_bit_iff_bit_pred
% 0.23/0.50 thf(fact_5763_nat__ivt__aux,axiom,
% 0.23/0.50 ! [N: nat,F: nat > int,K: int] :
% 0.23/0.50 ( ! [I3: nat] :
% 0.23/0.50 ( ( ord_less_nat @ I3 @ N )
% 0.23/0.50 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.23/0.50 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.23/0.50 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.23/0.50 => ? [I3: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ I3 @ N )
% 0.23/0.50 & ( ( F @ I3 )
% 0.23/0.50 = K ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % nat_ivt_aux
% 0.23/0.50 thf(fact_5764_bit__sum__mult__2__cases,axiom,
% 0.23/0.50 ! [A: code_integer,B2: code_integer,N: nat] :
% 0.23/0.50 ( ! [J2: nat] :
% 0.23/0.50 ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
% 0.23/0.50 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ N )
% 0.23/0.50 = ( ( ( N = zero_zero_nat )
% 0.23/0.50 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.23/0.50 & ( ( N != zero_zero_nat )
% 0.23/0.50 => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_sum_mult_2_cases
% 0.23/0.50 thf(fact_5765_bit__sum__mult__2__cases,axiom,
% 0.23/0.50 ! [A: int,B2: int,N: nat] :
% 0.23/0.50 ( ! [J2: nat] :
% 0.23/0.50 ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
% 0.23/0.50 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ N )
% 0.23/0.50 = ( ( ( N = zero_zero_nat )
% 0.23/0.50 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.23/0.50 & ( ( N != zero_zero_nat )
% 0.23/0.50 => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_sum_mult_2_cases
% 0.23/0.50 thf(fact_5766_bit__sum__mult__2__cases,axiom,
% 0.23/0.50 ! [A: nat,B2: nat,N: nat] :
% 0.23/0.50 ( ! [J2: nat] :
% 0.23/0.50 ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
% 0.23/0.50 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ N )
% 0.23/0.50 = ( ( ( N = zero_zero_nat )
% 0.23/0.50 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.23/0.50 & ( ( N != zero_zero_nat )
% 0.23/0.50 => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_sum_mult_2_cases
% 0.23/0.50 thf(fact_5767_bit__rec,axiom,
% 0.23/0.50 ( bit_se9216721137139052372nteger
% 0.23/0.50 = ( ^ [A4: code_integer,N2: nat] :
% 0.23/0.50 ( ( ( N2 = zero_zero_nat )
% 0.23/0.50 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) )
% 0.23/0.50 & ( ( N2 != zero_zero_nat )
% 0.23/0.50 => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_rec
% 0.23/0.50 thf(fact_5768_bit__rec,axiom,
% 0.23/0.50 ( bit_se1146084159140164899it_int
% 0.23/0.50 = ( ^ [A4: int,N2: nat] :
% 0.23/0.50 ( ( ( N2 = zero_zero_nat )
% 0.23/0.50 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) )
% 0.23/0.50 & ( ( N2 != zero_zero_nat )
% 0.23/0.50 => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_rec
% 0.23/0.50 thf(fact_5769_bit__rec,axiom,
% 0.23/0.50 ( bit_se1148574629649215175it_nat
% 0.23/0.50 = ( ^ [A4: nat,N2: nat] :
% 0.23/0.50 ( ( ( N2 = zero_zero_nat )
% 0.23/0.50 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) )
% 0.23/0.50 & ( ( N2 != zero_zero_nat )
% 0.23/0.50 => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % bit_rec
% 0.23/0.50 thf(fact_5770_nat0__intermed__int__val,axiom,
% 0.23/0.50 ! [N: nat,F: nat > int,K: int] :
% 0.23/0.50 ( ! [I3: nat] :
% 0.23/0.50 ( ( ord_less_nat @ I3 @ N )
% 0.23/0.50 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.23/0.50 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.23/0.50 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.23/0.50 => ? [I3: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ I3 @ N )
% 0.23/0.50 & ( ( F @ I3 )
% 0.23/0.50 = K ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % nat0_intermed_int_val
% 0.23/0.50 thf(fact_5771_set__bit__eq,axiom,
% 0.23/0.50 ( bit_se7879613467334960850it_int
% 0.23/0.50 = ( ^ [N2: nat,K3: int] :
% 0.23/0.50 ( plus_plus_int @ K3
% 0.23/0.50 @ ( times_times_int
% 0.23/0.50 @ ( zero_n2684676970156552555ol_int
% 0.23/0.50 @ ~ ( bit_se1146084159140164899it_int @ K3 @ N2 ) )
% 0.23/0.50 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % set_bit_eq
% 0.23/0.50 thf(fact_5772_unset__bit__eq,axiom,
% 0.23/0.50 ( bit_se4203085406695923979it_int
% 0.23/0.50 = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % unset_bit_eq
% 0.23/0.50 thf(fact_5773_arctan__add,axiom,
% 0.23/0.50 ! [X4: real,Y3: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.50 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.23/0.50 => ( ( plus_plus_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) )
% 0.23/0.50 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % arctan_add
% 0.23/0.50 thf(fact_5774_take__bit__Suc__from__most,axiom,
% 0.23/0.50 ! [N: nat,K: int] :
% 0.23/0.50 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 0.23/0.50 = ( 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 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % take_bit_Suc_from_most
% 0.23/0.50 thf(fact_5775_Bernoulli__inequality__even,axiom,
% 0.23/0.50 ! [N: nat,X4: real] :
% 0.23/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X4 ) @ N ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % Bernoulli_inequality_even
% 0.23/0.50 thf(fact_5776_exp__ge__one__plus__x__over__n__power__n,axiom,
% 0.23/0.50 ! [N: nat,X4: real] :
% 0.23/0.50 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X4 )
% 0.23/0.50 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.50 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X4 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % exp_ge_one_plus_x_over_n_power_n
% 0.23/0.50 thf(fact_5777_lemma__termdiff3,axiom,
% 0.23/0.50 ! [H2: real,Z: real,K5: real,N: nat] :
% 0.23/0.50 ( ( H2 != zero_zero_real )
% 0.23/0.50 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 0.23/0.50 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
% 0.23/0.50 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % lemma_termdiff3
% 0.23/0.50 thf(fact_5778_lemma__termdiff3,axiom,
% 0.23/0.50 ! [H2: complex,Z: complex,K5: real,N: nat] :
% 0.23/0.50 ( ( H2 != zero_zero_complex )
% 0.23/0.50 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 0.23/0.50 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
% 0.23/0.50 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % lemma_termdiff3
% 0.23/0.50 thf(fact_5779_ceiling__log2__div2,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.23/0.50 = ( 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 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_log2_div2
% 0.23/0.50 thf(fact_5780_pochhammer__double,axiom,
% 0.23/0.50 ! [Z: complex,N: nat] :
% 0.23/0.50 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.50 = ( 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 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % pochhammer_double
% 0.23/0.50 thf(fact_5781_pochhammer__double,axiom,
% 0.23/0.50 ! [Z: real,N: nat] :
% 0.23/0.50 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.50 = ( 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 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % pochhammer_double
% 0.23/0.50 thf(fact_5782_pochhammer__double,axiom,
% 0.23/0.50 ! [Z: rat,N: nat] :
% 0.23/0.50 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.50 = ( 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 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % pochhammer_double
% 0.23/0.50 thf(fact_5783_central__binomial__lower__bound,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.50 => ( 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 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % central_binomial_lower_bound
% 0.23/0.50 thf(fact_5784_floor__log2__div2,axiom,
% 0.23/0.50 ! [N: nat] :
% 0.23/0.50 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.50 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.23/0.50 = ( 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 ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_log2_div2
% 0.23/0.50 thf(fact_5785_machin,axiom,
% 0.23/0.50 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.23/0.50 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % machin
% 0.23/0.50 thf(fact_5786_of__int__floor__cancel,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.50 = X4 )
% 0.23/0.50 = ( ? [N2: int] :
% 0.23/0.50 ( X4
% 0.23/0.50 = ( ring_1_of_int_real @ N2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_floor_cancel
% 0.23/0.50 thf(fact_5787_of__int__floor__cancel,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.50 = X4 )
% 0.23/0.50 = ( ? [N2: int] :
% 0.23/0.50 ( X4
% 0.23/0.50 = ( ring_1_of_int_rat @ N2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_floor_cancel
% 0.23/0.50 thf(fact_5788_of__int__ceiling__cancel,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) )
% 0.23/0.50 = X4 )
% 0.23/0.50 = ( ? [N2: int] :
% 0.23/0.50 ( X4
% 0.23/0.50 = ( ring_1_of_int_rat @ N2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_ceiling_cancel
% 0.23/0.50 thf(fact_5789_of__int__ceiling__cancel,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) )
% 0.23/0.50 = X4 )
% 0.23/0.50 = ( ? [N2: int] :
% 0.23/0.50 ( X4
% 0.23/0.50 = ( ring_1_of_int_real @ N2 ) ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % of_int_ceiling_cancel
% 0.23/0.50 thf(fact_5790_int__eq__iff__numeral,axiom,
% 0.23/0.50 ! [M: nat,V: num] :
% 0.23/0.50 ( ( ( semiri1314217659103216013at_int @ M )
% 0.23/0.50 = ( numeral_numeral_int @ V ) )
% 0.23/0.50 = ( M
% 0.23/0.50 = ( numeral_numeral_nat @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % int_eq_iff_numeral
% 0.23/0.50 thf(fact_5791_floor__numeral,axiom,
% 0.23/0.50 ! [V: num] :
% 0.23/0.50 ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
% 0.23/0.50 = ( numeral_numeral_int @ V ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_numeral
% 0.23/0.50 thf(fact_5792_floor__numeral,axiom,
% 0.23/0.50 ! [V: num] :
% 0.23/0.50 ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
% 0.23/0.50 = ( numeral_numeral_int @ V ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_numeral
% 0.23/0.50 thf(fact_5793_floor__one,axiom,
% 0.23/0.50 ( ( archim6058952711729229775r_real @ one_one_real )
% 0.23/0.50 = one_one_int ) ).
% 0.23/0.50
% 0.23/0.50 % floor_one
% 0.23/0.50 thf(fact_5794_floor__one,axiom,
% 0.23/0.50 ( ( archim3151403230148437115or_rat @ one_one_rat )
% 0.23/0.50 = one_one_int ) ).
% 0.23/0.50
% 0.23/0.50 % floor_one
% 0.23/0.50 thf(fact_5795_ceiling__numeral,axiom,
% 0.23/0.50 ! [V: num] :
% 0.23/0.50 ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
% 0.23/0.50 = ( numeral_numeral_int @ V ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_numeral
% 0.23/0.50 thf(fact_5796_ceiling__numeral,axiom,
% 0.23/0.50 ! [V: num] :
% 0.23/0.50 ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
% 0.23/0.50 = ( numeral_numeral_int @ V ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_numeral
% 0.23/0.50 thf(fact_5797_pochhammer__0,axiom,
% 0.23/0.50 ! [A: complex] :
% 0.23/0.50 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 0.23/0.50 = one_one_complex ) ).
% 0.23/0.50
% 0.23/0.50 % pochhammer_0
% 0.23/0.50 thf(fact_5798_pochhammer__0,axiom,
% 0.23/0.50 ! [A: real] :
% 0.23/0.50 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 0.23/0.50 = one_one_real ) ).
% 0.23/0.50
% 0.23/0.50 % pochhammer_0
% 0.23/0.50 thf(fact_5799_pochhammer__0,axiom,
% 0.23/0.50 ! [A: rat] :
% 0.23/0.50 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 0.23/0.50 = one_one_rat ) ).
% 0.23/0.50
% 0.23/0.50 % pochhammer_0
% 0.23/0.50 thf(fact_5800_pochhammer__0,axiom,
% 0.23/0.50 ! [A: nat] :
% 0.23/0.50 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 0.23/0.50 = one_one_nat ) ).
% 0.23/0.50
% 0.23/0.50 % pochhammer_0
% 0.23/0.50 thf(fact_5801_pochhammer__0,axiom,
% 0.23/0.50 ! [A: int] :
% 0.23/0.50 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 0.23/0.50 = one_one_int ) ).
% 0.23/0.50
% 0.23/0.50 % pochhammer_0
% 0.23/0.50 thf(fact_5802_ceiling__one,axiom,
% 0.23/0.50 ( ( archim2889992004027027881ng_rat @ one_one_rat )
% 0.23/0.50 = one_one_int ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_one
% 0.23/0.50 thf(fact_5803_ceiling__one,axiom,
% 0.23/0.50 ( ( archim7802044766580827645g_real @ one_one_real )
% 0.23/0.50 = one_one_int ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_one
% 0.23/0.50 thf(fact_5804_negative__zless,axiom,
% 0.23/0.50 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.23/0.50
% 0.23/0.50 % negative_zless
% 0.23/0.50 thf(fact_5805_ceiling__add__of__int,axiom,
% 0.23/0.50 ! [X4: rat,Z: int] :
% 0.23/0.50 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) )
% 0.23/0.50 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_add_of_int
% 0.23/0.50 thf(fact_5806_ceiling__add__of__int,axiom,
% 0.23/0.50 ! [X4: real,Z: int] :
% 0.23/0.50 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) )
% 0.23/0.50 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ Z ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_add_of_int
% 0.23/0.50 thf(fact_5807_floor__diff__of__int,axiom,
% 0.23/0.50 ! [X4: real,Z: int] :
% 0.23/0.50 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X4 ) @ Z ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_diff_of_int
% 0.23/0.50 thf(fact_5808_floor__diff__of__int,axiom,
% 0.23/0.50 ! [X4: rat,Z: int] :
% 0.23/0.50 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_diff_of_int
% 0.23/0.50 thf(fact_5809_ceiling__diff__of__int,axiom,
% 0.23/0.50 ! [X4: rat,Z: int] :
% 0.23/0.50 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_diff_of_int
% 0.23/0.50 thf(fact_5810_ceiling__diff__of__int,axiom,
% 0.23/0.50 ! [X4: real,Z: int] :
% 0.23/0.50 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ Z ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_diff_of_int
% 0.23/0.50 thf(fact_5811_zero__le__floor,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_le_floor
% 0.23/0.50 thf(fact_5812_zero__le__floor,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_rat @ zero_zero_rat @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_le_floor
% 0.23/0.50 thf(fact_5813_floor__less__zero,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ zero_zero_int )
% 0.23/0.50 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_less_zero
% 0.23/0.50 thf(fact_5814_floor__less__zero,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ zero_zero_int )
% 0.23/0.50 = ( ord_less_rat @ X4 @ zero_zero_rat ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_less_zero
% 0.23/0.50 thf(fact_5815_numeral__le__floor,axiom,
% 0.23/0.50 ! [V: num,X4: real] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_le_floor
% 0.23/0.50 thf(fact_5816_numeral__le__floor,axiom,
% 0.23/0.50 ! [V: num,X4: rat] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_le_floor
% 0.23/0.50 thf(fact_5817_zero__less__floor,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_real @ one_one_real @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_less_floor
% 0.23/0.50 thf(fact_5818_zero__less__floor,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_less_floor
% 0.23/0.50 thf(fact_5819_floor__le__zero,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ zero_zero_int )
% 0.23/0.50 = ( ord_less_real @ X4 @ one_one_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_le_zero
% 0.23/0.50 thf(fact_5820_floor__le__zero,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ zero_zero_int )
% 0.23/0.50 = ( ord_less_rat @ X4 @ one_one_rat ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_le_zero
% 0.23/0.50 thf(fact_5821_ceiling__le__zero,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ zero_zero_int )
% 0.23/0.50 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_le_zero
% 0.23/0.50 thf(fact_5822_ceiling__le__zero,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ zero_zero_int )
% 0.23/0.50 = ( ord_less_eq_rat @ X4 @ zero_zero_rat ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_le_zero
% 0.23/0.50 thf(fact_5823_floor__less__numeral,axiom,
% 0.23/0.50 ! [X4: real,V: num] :
% 0.23/0.50 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) )
% 0.23/0.50 = ( ord_less_real @ X4 @ ( numeral_numeral_real @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_less_numeral
% 0.23/0.50 thf(fact_5824_floor__less__numeral,axiom,
% 0.23/0.50 ! [X4: rat,V: num] :
% 0.23/0.50 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
% 0.23/0.50 = ( ord_less_rat @ X4 @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_less_numeral
% 0.23/0.50 thf(fact_5825_zero__less__ceiling,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_rat @ zero_zero_rat @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_less_ceiling
% 0.23/0.50 thf(fact_5826_zero__less__ceiling,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X4 ) )
% 0.23/0.50 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % zero_less_ceiling
% 0.23/0.50 thf(fact_5827_one__le__floor,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_real @ one_one_real @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % one_le_floor
% 0.23/0.50 thf(fact_5828_one__le__floor,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % one_le_floor
% 0.23/0.50 thf(fact_5829_ceiling__le__numeral,axiom,
% 0.23/0.50 ! [X4: real,V: num] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) )
% 0.23/0.50 = ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_le_numeral
% 0.23/0.50 thf(fact_5830_ceiling__le__numeral,axiom,
% 0.23/0.50 ! [X4: rat,V: num] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
% 0.23/0.50 = ( ord_less_eq_rat @ X4 @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_le_numeral
% 0.23/0.50 thf(fact_5831_floor__less__one,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
% 0.23/0.50 = ( ord_less_real @ X4 @ one_one_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_less_one
% 0.23/0.50 thf(fact_5832_floor__less__one,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
% 0.23/0.50 = ( ord_less_rat @ X4 @ one_one_rat ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_less_one
% 0.23/0.50 thf(fact_5833_ceiling__less__one,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int )
% 0.23/0.50 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_less_one
% 0.23/0.50 thf(fact_5834_ceiling__less__one,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int )
% 0.23/0.50 = ( ord_less_eq_rat @ X4 @ zero_zero_rat ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_less_one
% 0.23/0.50 thf(fact_5835_one__le__ceiling,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_rat @ zero_zero_rat @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % one_le_ceiling
% 0.23/0.50 thf(fact_5836_one__le__ceiling,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X4 ) )
% 0.23/0.50 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % one_le_ceiling
% 0.23/0.50 thf(fact_5837_numeral__less__ceiling,axiom,
% 0.23/0.50 ! [V: num,X4: real] :
% 0.23/0.50 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X4 ) )
% 0.23/0.50 = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_less_ceiling
% 0.23/0.50 thf(fact_5838_numeral__less__ceiling,axiom,
% 0.23/0.50 ! [V: num,X4: rat] :
% 0.23/0.50 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % numeral_less_ceiling
% 0.23/0.50 thf(fact_5839_floor__neg__numeral,axiom,
% 0.23/0.50 ! [V: num] :
% 0.23/0.50 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.23/0.50 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_neg_numeral
% 0.23/0.50 thf(fact_5840_floor__neg__numeral,axiom,
% 0.23/0.50 ! [V: num] :
% 0.23/0.50 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.23/0.50 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_neg_numeral
% 0.23/0.50 thf(fact_5841_ceiling__le__one,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int )
% 0.23/0.50 = ( ord_less_eq_real @ X4 @ one_one_real ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_le_one
% 0.23/0.50 thf(fact_5842_ceiling__le__one,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int )
% 0.23/0.50 = ( ord_less_eq_rat @ X4 @ one_one_rat ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_le_one
% 0.23/0.50 thf(fact_5843_one__less__ceiling,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X4 ) )
% 0.23/0.50 = ( ord_less_rat @ one_one_rat @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % one_less_ceiling
% 0.23/0.50 thf(fact_5844_one__less__ceiling,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X4 ) )
% 0.23/0.50 = ( ord_less_real @ one_one_real @ X4 ) ) ).
% 0.23/0.50
% 0.23/0.50 % one_less_ceiling
% 0.23/0.50 thf(fact_5845_ceiling__add__numeral,axiom,
% 0.23/0.50 ! [X4: real,V: num] :
% 0.23/0.50 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
% 0.23/0.50 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_add_numeral
% 0.23/0.50 thf(fact_5846_ceiling__add__numeral,axiom,
% 0.23/0.50 ! [X4: rat,V: num] :
% 0.23/0.50 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
% 0.23/0.50 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_add_numeral
% 0.23/0.50 thf(fact_5847_floor__diff__numeral,axiom,
% 0.23/0.50 ! [X4: real,V: num] :
% 0.23/0.50 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_diff_numeral
% 0.23/0.50 thf(fact_5848_floor__diff__numeral,axiom,
% 0.23/0.50 ! [X4: rat,V: num] :
% 0.23/0.50 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_diff_numeral
% 0.23/0.50 thf(fact_5849_ceiling__neg__numeral,axiom,
% 0.23/0.50 ! [V: num] :
% 0.23/0.50 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.23/0.50 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_neg_numeral
% 0.23/0.50 thf(fact_5850_ceiling__neg__numeral,axiom,
% 0.23/0.50 ! [V: num] :
% 0.23/0.50 ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.23/0.50 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_neg_numeral
% 0.23/0.50 thf(fact_5851_ceiling__add__one,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) )
% 0.23/0.50 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_add_one
% 0.23/0.50 thf(fact_5852_ceiling__add__one,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ one_one_real ) )
% 0.23/0.50 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_add_one
% 0.23/0.50 thf(fact_5853_floor__diff__one,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X4 @ one_one_real ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_diff_one
% 0.23/0.50 thf(fact_5854_floor__diff__one,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X4 @ one_one_rat ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ).
% 0.23/0.50
% 0.23/0.50 % floor_diff_one
% 0.23/0.50 thf(fact_5855_ceiling__diff__numeral,axiom,
% 0.23/0.50 ! [X4: real,V: num] :
% 0.23/0.50 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_diff_numeral
% 0.23/0.50 thf(fact_5856_ceiling__diff__numeral,axiom,
% 0.23/0.50 ! [X4: rat,V: num] :
% 0.23/0.50 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_diff_numeral
% 0.23/0.50 thf(fact_5857_ceiling__diff__one,axiom,
% 0.23/0.50 ! [X4: rat] :
% 0.23/0.50 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X4 @ one_one_rat ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_diff_one
% 0.23/0.50 thf(fact_5858_ceiling__diff__one,axiom,
% 0.23/0.50 ! [X4: real] :
% 0.23/0.50 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X4 @ one_one_real ) )
% 0.23/0.50 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int ) ) ).
% 0.23/0.50
% 0.23/0.50 % ceiling_diff_one
% 0.23/0.50 thf(fact_5859_floor__numeral__power,axiom,
% 0.23/0.50 ! [X4: num,N: nat] :
% 0.23/0.51 ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 0.23/0.51 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_numeral_power
% 0.23/0.51 thf(fact_5860_floor__numeral__power,axiom,
% 0.23/0.51 ! [X4: num,N: nat] :
% 0.23/0.51 ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 0.23/0.51 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_numeral_power
% 0.23/0.51 thf(fact_5861_ceiling__numeral__power,axiom,
% 0.23/0.51 ! [X4: num,N: nat] :
% 0.23/0.51 ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 0.23/0.51 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_numeral_power
% 0.23/0.51 thf(fact_5862_ceiling__numeral__power,axiom,
% 0.23/0.51 ! [X4: num,N: nat] :
% 0.23/0.51 ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 0.23/0.51 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_numeral_power
% 0.23/0.51 thf(fact_5863_floor__divide__eq__div__numeral,axiom,
% 0.23/0.51 ! [A: num,B2: num] :
% 0.23/0.51 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
% 0.23/0.51 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_divide_eq_div_numeral
% 0.23/0.51 thf(fact_5864_ceiling__less__zero,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ zero_zero_int )
% 0.23/0.51 = ( ord_less_eq_real @ X4 @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_less_zero
% 0.23/0.51 thf(fact_5865_ceiling__less__zero,axiom,
% 0.23/0.51 ! [X4: rat] :
% 0.23/0.51 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ zero_zero_int )
% 0.23/0.51 = ( ord_less_eq_rat @ X4 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_less_zero
% 0.23/0.51 thf(fact_5866_zero__le__ceiling,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X4 ) )
% 0.23/0.51 = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % zero_le_ceiling
% 0.23/0.51 thf(fact_5867_zero__le__ceiling,axiom,
% 0.23/0.51 ! [X4: rat] :
% 0.23/0.51 ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % zero_le_ceiling
% 0.23/0.51 thf(fact_5868_ceiling__divide__eq__div__numeral,axiom,
% 0.23/0.51 ! [A: num,B2: num] :
% 0.23/0.51 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
% 0.23/0.51 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_divide_eq_div_numeral
% 0.23/0.51 thf(fact_5869_numeral__less__floor,axiom,
% 0.23/0.51 ! [V: num,X4: real] :
% 0.23/0.51 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % numeral_less_floor
% 0.23/0.51 thf(fact_5870_numeral__less__floor,axiom,
% 0.23/0.51 ! [V: num,X4: rat] :
% 0.23/0.51 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % numeral_less_floor
% 0.23/0.51 thf(fact_5871_floor__le__numeral,axiom,
% 0.23/0.51 ! [X4: real,V: num] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) )
% 0.23/0.51 = ( ord_less_real @ X4 @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_le_numeral
% 0.23/0.51 thf(fact_5872_floor__le__numeral,axiom,
% 0.23/0.51 ! [X4: rat,V: num] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
% 0.23/0.51 = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_le_numeral
% 0.23/0.51 thf(fact_5873_one__less__floor,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % one_less_floor
% 0.23/0.51 thf(fact_5874_one__less__floor,axiom,
% 0.23/0.51 ! [X4: rat] :
% 0.23/0.51 ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % one_less_floor
% 0.23/0.51 thf(fact_5875_floor__le__one,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
% 0.23/0.51 = ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_le_one
% 0.23/0.51 thf(fact_5876_floor__le__one,axiom,
% 0.23/0.51 ! [X4: rat] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
% 0.23/0.51 = ( ord_less_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_le_one
% 0.23/0.51 thf(fact_5877_ceiling__less__numeral,axiom,
% 0.23/0.51 ! [X4: real,V: num] :
% 0.23/0.51 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) )
% 0.23/0.51 = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_less_numeral
% 0.23/0.51 thf(fact_5878_ceiling__less__numeral,axiom,
% 0.23/0.51 ! [X4: rat,V: num] :
% 0.23/0.51 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
% 0.23/0.51 = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_less_numeral
% 0.23/0.51 thf(fact_5879_numeral__le__ceiling,axiom,
% 0.23/0.51 ! [V: num,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X4 ) )
% 0.23/0.51 = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % numeral_le_ceiling
% 0.23/0.51 thf(fact_5880_numeral__le__ceiling,axiom,
% 0.23/0.51 ! [V: num,X4: rat] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % numeral_le_ceiling
% 0.23/0.51 thf(fact_5881_neg__numeral__le__floor,axiom,
% 0.23/0.51 ! [V: num,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % neg_numeral_le_floor
% 0.23/0.51 thf(fact_5882_neg__numeral__le__floor,axiom,
% 0.23/0.51 ! [V: num,X4: rat] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % neg_numeral_le_floor
% 0.23/0.51 thf(fact_5883_floor__less__neg__numeral,axiom,
% 0.23/0.51 ! [X4: real,V: num] :
% 0.23/0.51 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.23/0.51 = ( ord_less_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_less_neg_numeral
% 0.23/0.51 thf(fact_5884_floor__less__neg__numeral,axiom,
% 0.23/0.51 ! [X4: rat,V: num] :
% 0.23/0.51 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.23/0.51 = ( ord_less_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_less_neg_numeral
% 0.23/0.51 thf(fact_5885_ceiling__le__neg__numeral,axiom,
% 0.23/0.51 ! [X4: real,V: num] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.23/0.51 = ( ord_less_eq_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_le_neg_numeral
% 0.23/0.51 thf(fact_5886_ceiling__le__neg__numeral,axiom,
% 0.23/0.51 ! [X4: rat,V: num] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.23/0.51 = ( ord_less_eq_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_le_neg_numeral
% 0.23/0.51 thf(fact_5887_neg__numeral__less__ceiling,axiom,
% 0.23/0.51 ! [V: num,X4: real] :
% 0.23/0.51 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X4 ) )
% 0.23/0.51 = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % neg_numeral_less_ceiling
% 0.23/0.51 thf(fact_5888_neg__numeral__less__ceiling,axiom,
% 0.23/0.51 ! [V: num,X4: rat] :
% 0.23/0.51 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % neg_numeral_less_ceiling
% 0.23/0.51 thf(fact_5889_floor__one__divide__eq__div__numeral,axiom,
% 0.23/0.51 ! [B2: num] :
% 0.23/0.51 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) )
% 0.23/0.51 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_one_divide_eq_div_numeral
% 0.23/0.51 thf(fact_5890_floor__minus__divide__eq__div__numeral,axiom,
% 0.23/0.51 ! [A: num,B2: num] :
% 0.23/0.51 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
% 0.23/0.51 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_minus_divide_eq_div_numeral
% 0.23/0.51 thf(fact_5891_ceiling__minus__divide__eq__div__numeral,axiom,
% 0.23/0.51 ! [A: num,B2: num] :
% 0.23/0.51 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
% 0.23/0.51 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_minus_divide_eq_div_numeral
% 0.23/0.51 thf(fact_5892_neg__numeral__less__floor,axiom,
% 0.23/0.51 ! [V: num,X4: real] :
% 0.23/0.51 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % neg_numeral_less_floor
% 0.23/0.51 thf(fact_5893_neg__numeral__less__floor,axiom,
% 0.23/0.51 ! [V: num,X4: rat] :
% 0.23/0.51 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % neg_numeral_less_floor
% 0.23/0.51 thf(fact_5894_floor__le__neg__numeral,axiom,
% 0.23/0.51 ! [X4: real,V: num] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.23/0.51 = ( ord_less_real @ X4 @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_le_neg_numeral
% 0.23/0.51 thf(fact_5895_floor__le__neg__numeral,axiom,
% 0.23/0.51 ! [X4: rat,V: num] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.23/0.51 = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_le_neg_numeral
% 0.23/0.51 thf(fact_5896_ceiling__less__neg__numeral,axiom,
% 0.23/0.51 ! [X4: real,V: num] :
% 0.23/0.51 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.23/0.51 = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_less_neg_numeral
% 0.23/0.51 thf(fact_5897_ceiling__less__neg__numeral,axiom,
% 0.23/0.51 ! [X4: rat,V: num] :
% 0.23/0.51 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.23/0.51 = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_less_neg_numeral
% 0.23/0.51 thf(fact_5898_neg__numeral__le__ceiling,axiom,
% 0.23/0.51 ! [V: num,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X4 ) )
% 0.23/0.51 = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % neg_numeral_le_ceiling
% 0.23/0.51 thf(fact_5899_neg__numeral__le__ceiling,axiom,
% 0.23/0.51 ! [V: num,X4: rat] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % neg_numeral_le_ceiling
% 0.23/0.51 thf(fact_5900_floor__minus__one__divide__eq__div__numeral,axiom,
% 0.23/0.51 ! [B2: num] :
% 0.23/0.51 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) ) )
% 0.23/0.51 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_minus_one_divide_eq_div_numeral
% 0.23/0.51 thf(fact_5901_int__if,axiom,
% 0.23/0.51 ! [P: $o,A: nat,B2: nat] :
% 0.23/0.51 ( ( P
% 0.23/0.51 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
% 0.23/0.51 = ( semiri1314217659103216013at_int @ A ) ) )
% 0.23/0.51 & ( ~ P
% 0.23/0.51 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
% 0.23/0.51 = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_if
% 0.23/0.51 thf(fact_5902_nat__int__comparison_I1_J,axiom,
% 0.23/0.51 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.23/0.51 = ( ^ [A4: nat,B3: nat] :
% 0.23/0.51 ( ( semiri1314217659103216013at_int @ A4 )
% 0.23/0.51 = ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % nat_int_comparison(1)
% 0.23/0.51 thf(fact_5903_int__diff__cases,axiom,
% 0.23/0.51 ! [Z: int] :
% 0.23/0.51 ~ ! [M4: nat,N4: nat] :
% 0.23/0.51 ( Z
% 0.23/0.51 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_diff_cases
% 0.23/0.51 thf(fact_5904_ceiling__altdef,axiom,
% 0.23/0.51 ( archim7802044766580827645g_real
% 0.23/0.51 = ( ^ [X: real] :
% 0.23/0.51 ( if_int
% 0.23/0.51 @ ( X
% 0.23/0.51 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
% 0.23/0.51 @ ( archim6058952711729229775r_real @ X )
% 0.23/0.51 @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_altdef
% 0.23/0.51 thf(fact_5905_ceiling__altdef,axiom,
% 0.23/0.51 ( archim2889992004027027881ng_rat
% 0.23/0.51 = ( ^ [X: rat] :
% 0.23/0.51 ( if_int
% 0.23/0.51 @ ( X
% 0.23/0.51 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
% 0.23/0.51 @ ( archim3151403230148437115or_rat @ X )
% 0.23/0.51 @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_altdef
% 0.23/0.51 thf(fact_5906_ceiling__diff__floor__le__1,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim6058952711729229775r_real @ X4 ) ) @ one_one_int ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_diff_floor_le_1
% 0.23/0.51 thf(fact_5907_ceiling__diff__floor__le__1,axiom,
% 0.23/0.51 ! [X4: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim3151403230148437115or_rat @ X4 ) ) @ one_one_int ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_diff_floor_le_1
% 0.23/0.51 thf(fact_5908_floor__mono,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.51 => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_mono
% 0.23/0.51 thf(fact_5909_floor__mono,axiom,
% 0.23/0.51 ! [X4: rat,Y3: rat] :
% 0.23/0.51 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.51 => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_mono
% 0.23/0.51 thf(fact_5910_of__int__floor__le,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) @ X4 ) ).
% 0.23/0.51
% 0.23/0.51 % of_int_floor_le
% 0.23/0.51 thf(fact_5911_of__int__floor__le,axiom,
% 0.23/0.51 ! [X4: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) @ X4 ) ).
% 0.23/0.51
% 0.23/0.51 % of_int_floor_le
% 0.23/0.51 thf(fact_5912_floor__less__cancel,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) )
% 0.23/0.51 => ( ord_less_real @ X4 @ Y3 ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_less_cancel
% 0.23/0.51 thf(fact_5913_floor__less__cancel,axiom,
% 0.23/0.51 ! [X4: rat,Y3: rat] :
% 0.23/0.51 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) )
% 0.23/0.51 => ( ord_less_rat @ X4 @ Y3 ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_less_cancel
% 0.23/0.51 thf(fact_5914_not__bit__Suc__0__Suc,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % not_bit_Suc_0_Suc
% 0.23/0.51 thf(fact_5915_bit__Suc__0__iff,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.23/0.51 = ( N = zero_zero_nat ) ) ).
% 0.23/0.51
% 0.23/0.51 % bit_Suc_0_iff
% 0.23/0.51 thf(fact_5916_ceiling__mono,axiom,
% 0.23/0.51 ! [Y3: real,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ Y3 @ X4 )
% 0.23/0.51 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y3 ) @ ( archim7802044766580827645g_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_mono
% 0.23/0.51 thf(fact_5917_ceiling__mono,axiom,
% 0.23/0.51 ! [Y3: rat,X4: rat] :
% 0.23/0.51 ( ( ord_less_eq_rat @ Y3 @ X4 )
% 0.23/0.51 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_mono
% 0.23/0.51 thf(fact_5918_le__of__int__ceiling,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_of_int_ceiling
% 0.23/0.51 thf(fact_5919_le__of__int__ceiling,axiom,
% 0.23/0.51 ! [X4: rat] : ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_of_int_ceiling
% 0.23/0.51 thf(fact_5920_ceiling__less__cancel,axiom,
% 0.23/0.51 ! [X4: rat,Y3: rat] :
% 0.23/0.51 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
% 0.23/0.51 => ( ord_less_rat @ X4 @ Y3 ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_less_cancel
% 0.23/0.51 thf(fact_5921_ceiling__less__cancel,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim7802044766580827645g_real @ Y3 ) )
% 0.23/0.51 => ( ord_less_real @ X4 @ Y3 ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_less_cancel
% 0.23/0.51 thf(fact_5922_int__ops_I1_J,axiom,
% 0.23/0.51 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.23/0.51 = zero_zero_int ) ).
% 0.23/0.51
% 0.23/0.51 % int_ops(1)
% 0.23/0.51 thf(fact_5923_complex__mod__minus__le__complex__mod,axiom,
% 0.23/0.51 ! [X4: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % complex_mod_minus_le_complex_mod
% 0.23/0.51 thf(fact_5924_int__ops_I3_J,axiom,
% 0.23/0.51 ! [N: num] :
% 0.23/0.51 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.23/0.51 = ( numeral_numeral_int @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_ops(3)
% 0.23/0.51 thf(fact_5925_pochhammer__pos,axiom,
% 0.23/0.51 ! [X4: real,N: nat] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X4 @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_pos
% 0.23/0.51 thf(fact_5926_pochhammer__pos,axiom,
% 0.23/0.51 ! [X4: rat,N: nat] :
% 0.23/0.51 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 0.23/0.51 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X4 @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_pos
% 0.23/0.51 thf(fact_5927_pochhammer__pos,axiom,
% 0.23/0.51 ! [X4: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 0.23/0.51 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X4 @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_pos
% 0.23/0.51 thf(fact_5928_pochhammer__pos,axiom,
% 0.23/0.51 ! [X4: int,N: nat] :
% 0.23/0.51 ( ( ord_less_int @ zero_zero_int @ X4 )
% 0.23/0.51 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X4 @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_pos
% 0.23/0.51 thf(fact_5929_nat__int__comparison_I2_J,axiom,
% 0.23/0.51 ( ord_less_nat
% 0.23/0.51 = ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % nat_int_comparison(2)
% 0.23/0.51 thf(fact_5930_int__of__nat__induct,axiom,
% 0.23/0.51 ! [P: int > $o,Z: int] :
% 0.23/0.51 ( ! [N4: nat] : ( P @ ( semiri1314217659103216013at_int @ N4 ) )
% 0.23/0.51 => ( ! [N4: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
% 0.23/0.51 => ( P @ Z ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_of_nat_induct
% 0.23/0.51 thf(fact_5931_int__cases,axiom,
% 0.23/0.51 ! [Z: int] :
% 0.23/0.51 ( ! [N4: nat] :
% 0.23/0.51 ( Z
% 0.23/0.51 != ( semiri1314217659103216013at_int @ N4 ) )
% 0.23/0.51 => ~ ! [N4: nat] :
% 0.23/0.51 ( Z
% 0.23/0.51 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_cases
% 0.23/0.51 thf(fact_5932_pi__not__less__zero,axiom,
% 0.23/0.51 ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % pi_not_less_zero
% 0.23/0.51 thf(fact_5933_pi__gt__zero,axiom,
% 0.23/0.51 ord_less_real @ zero_zero_real @ pi ).
% 0.23/0.51
% 0.23/0.51 % pi_gt_zero
% 0.23/0.51 thf(fact_5934_pi__ge__zero,axiom,
% 0.23/0.51 ord_less_eq_real @ zero_zero_real @ pi ).
% 0.23/0.51
% 0.23/0.51 % pi_ge_zero
% 0.23/0.51 thf(fact_5935_nat__int__comparison_I3_J,axiom,
% 0.23/0.51 ( ord_less_eq_nat
% 0.23/0.51 = ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % nat_int_comparison(3)
% 0.23/0.51 thf(fact_5936_zle__int,axiom,
% 0.23/0.51 ! [M: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.51 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % zle_int
% 0.23/0.51 thf(fact_5937_complex__mod__triangle__ineq2,axiom,
% 0.23/0.51 ! [B2: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B2 @ A ) ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 0.23/0.51
% 0.23/0.51 % complex_mod_triangle_ineq2
% 0.23/0.51 thf(fact_5938_pochhammer__eq__0__mono,axiom,
% 0.23/0.51 ! [A: real,N: nat,M: nat] :
% 0.23/0.51 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.23/0.51 = zero_zero_real )
% 0.23/0.51 => ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.51 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.23/0.51 = zero_zero_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_eq_0_mono
% 0.23/0.51 thf(fact_5939_pochhammer__eq__0__mono,axiom,
% 0.23/0.51 ! [A: rat,N: nat,M: nat] :
% 0.23/0.51 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.23/0.51 = zero_zero_rat )
% 0.23/0.51 => ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.51 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.23/0.51 = zero_zero_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_eq_0_mono
% 0.23/0.51 thf(fact_5940_pochhammer__neq__0__mono,axiom,
% 0.23/0.51 ! [A: real,M: nat,N: nat] :
% 0.23/0.51 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.23/0.51 != zero_zero_real )
% 0.23/0.51 => ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.51 => ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.23/0.51 != zero_zero_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_neq_0_mono
% 0.23/0.51 thf(fact_5941_pochhammer__neq__0__mono,axiom,
% 0.23/0.51 ! [A: rat,M: nat,N: nat] :
% 0.23/0.51 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.23/0.51 != zero_zero_rat )
% 0.23/0.51 => ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.51 => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.23/0.51 != zero_zero_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_neq_0_mono
% 0.23/0.51 thf(fact_5942_int__ops_I5_J,axiom,
% 0.23/0.51 ! [A: nat,B2: nat] :
% 0.23/0.51 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B2 ) )
% 0.23/0.51 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_ops(5)
% 0.23/0.51 thf(fact_5943_int__plus,axiom,
% 0.23/0.51 ! [N: nat,M: nat] :
% 0.23/0.51 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 0.23/0.51 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_plus
% 0.23/0.51 thf(fact_5944_zadd__int__left,axiom,
% 0.23/0.51 ! [M: nat,N: nat,Z: int] :
% 0.23/0.51 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 0.23/0.51 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 0.23/0.51
% 0.23/0.51 % zadd_int_left
% 0.23/0.51 thf(fact_5945_int__ops_I2_J,axiom,
% 0.23/0.51 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.23/0.51 = one_one_int ) ).
% 0.23/0.51
% 0.23/0.51 % int_ops(2)
% 0.23/0.51 thf(fact_5946_int__ops_I7_J,axiom,
% 0.23/0.51 ! [A: nat,B2: nat] :
% 0.23/0.51 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B2 ) )
% 0.23/0.51 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_ops(7)
% 0.23/0.51 thf(fact_5947_zle__iff__zadd,axiom,
% 0.23/0.51 ( ord_less_eq_int
% 0.23/0.51 = ( ^ [W2: int,Z5: int] :
% 0.23/0.51 ? [N2: nat] :
% 0.23/0.51 ( Z5
% 0.23/0.51 = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % zle_iff_zadd
% 0.23/0.51 thf(fact_5948_not__int__zless__negative,axiom,
% 0.23/0.51 ! [N: nat,M: nat] :
% 0.23/0.51 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % not_int_zless_negative
% 0.23/0.51 thf(fact_5949_zdiv__int,axiom,
% 0.23/0.51 ! [A: nat,B2: nat] :
% 0.23/0.51 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B2 ) )
% 0.23/0.51 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % zdiv_int
% 0.23/0.51 thf(fact_5950_zmod__int,axiom,
% 0.23/0.51 ! [A: nat,B2: nat] :
% 0.23/0.51 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.23/0.51 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % zmod_int
% 0.23/0.51 thf(fact_5951_not__bit__Suc__0__numeral,axiom,
% 0.23/0.51 ! [N: num] :
% 0.23/0.51 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % not_bit_Suc_0_numeral
% 0.23/0.51 thf(fact_5952_floor__divide__of__nat__eq,axiom,
% 0.23/0.51 ! [M: nat,N: nat] :
% 0.23/0.51 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.23/0.51 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_divide_of_nat_eq
% 0.23/0.51 thf(fact_5953_floor__divide__of__nat__eq,axiom,
% 0.23/0.51 ! [M: nat,N: nat] :
% 0.23/0.51 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
% 0.23/0.51 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_divide_of_nat_eq
% 0.23/0.51 thf(fact_5954_le__floor__iff,axiom,
% 0.23/0.51 ! [Z: int,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_floor_iff
% 0.23/0.51 thf(fact_5955_le__floor__iff,axiom,
% 0.23/0.51 ! [Z: int,X4: rat] :
% 0.23/0.51 ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_floor_iff
% 0.23/0.51 thf(fact_5956_floor__less__iff,axiom,
% 0.23/0.51 ! [X4: real,Z: int] :
% 0.23/0.51 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
% 0.23/0.51 = ( ord_less_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_less_iff
% 0.23/0.51 thf(fact_5957_floor__less__iff,axiom,
% 0.23/0.51 ! [X4: rat,Z: int] :
% 0.23/0.51 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
% 0.23/0.51 = ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_less_iff
% 0.23/0.51 thf(fact_5958_le__floor__add,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_floor_add
% 0.23/0.51 thf(fact_5959_le__floor__add,axiom,
% 0.23/0.51 ! [X4: rat,Y3: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_floor_add
% 0.23/0.51 thf(fact_5960_int__add__floor,axiom,
% 0.23/0.51 ! [Z: int,X4: real] :
% 0.23/0.51 ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.51 = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_add_floor
% 0.23/0.51 thf(fact_5961_int__add__floor,axiom,
% 0.23/0.51 ! [Z: int,X4: rat] :
% 0.23/0.51 ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.51 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_add_floor
% 0.23/0.51 thf(fact_5962_floor__add__int,axiom,
% 0.23/0.51 ! [X4: real,Z: int] :
% 0.23/0.51 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
% 0.23/0.51 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_add_int
% 0.23/0.51 thf(fact_5963_floor__add__int,axiom,
% 0.23/0.51 ! [X4: rat,Z: int] :
% 0.23/0.51 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
% 0.23/0.51 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_add_int
% 0.23/0.51 thf(fact_5964_floor__divide__of__int__eq,axiom,
% 0.23/0.51 ! [K: int,L2: int] :
% 0.23/0.51 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L2 ) ) )
% 0.23/0.51 = ( divide_divide_int @ K @ L2 ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_divide_of_int_eq
% 0.23/0.51 thf(fact_5965_floor__divide__of__int__eq,axiom,
% 0.23/0.51 ! [K: int,L2: int] :
% 0.23/0.51 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L2 ) ) )
% 0.23/0.51 = ( divide_divide_int @ K @ L2 ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_divide_of_int_eq
% 0.23/0.51 thf(fact_5966_floor__power,axiom,
% 0.23/0.51 ! [X4: real,N: nat] :
% 0.23/0.51 ( ( X4
% 0.23/0.51 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) )
% 0.23/0.51 => ( ( archim6058952711729229775r_real @ ( power_power_real @ X4 @ N ) )
% 0.23/0.51 = ( power_power_int @ ( archim6058952711729229775r_real @ X4 ) @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_power
% 0.23/0.51 thf(fact_5967_floor__power,axiom,
% 0.23/0.51 ! [X4: rat,N: nat] :
% 0.23/0.51 ( ( X4
% 0.23/0.51 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) )
% 0.23/0.51 => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X4 @ N ) )
% 0.23/0.51 = ( power_power_int @ ( archim3151403230148437115or_rat @ X4 ) @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_power
% 0.23/0.51 thf(fact_5968_ceiling__le,axiom,
% 0.23/0.51 ! [X4: real,A: int] :
% 0.23/0.51 ( ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ A ) )
% 0.23/0.51 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ A ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_le
% 0.23/0.51 thf(fact_5969_ceiling__le,axiom,
% 0.23/0.51 ! [X4: rat,A: int] :
% 0.23/0.51 ( ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ A ) )
% 0.23/0.51 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ A ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_le
% 0.23/0.51 thf(fact_5970_ceiling__le__iff,axiom,
% 0.23/0.51 ! [X4: real,Z: int] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ Z )
% 0.23/0.51 = ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_le_iff
% 0.23/0.51 thf(fact_5971_ceiling__le__iff,axiom,
% 0.23/0.51 ! [X4: rat,Z: int] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z )
% 0.23/0.51 = ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_le_iff
% 0.23/0.51 thf(fact_5972_less__ceiling__iff,axiom,
% 0.23/0.51 ! [Z: int,X4: rat] :
% 0.23/0.51 ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % less_ceiling_iff
% 0.23/0.51 thf(fact_5973_less__ceiling__iff,axiom,
% 0.23/0.51 ! [Z: int,X4: real] :
% 0.23/0.51 ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X4 ) )
% 0.23/0.51 = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % less_ceiling_iff
% 0.23/0.51 thf(fact_5974_ceiling__add__le,axiom,
% 0.23/0.51 ! [X4: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_add_le
% 0.23/0.51 thf(fact_5975_ceiling__add__le,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim7802044766580827645g_real @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_add_le
% 0.23/0.51 thf(fact_5976_pochhammer__nonneg,axiom,
% 0.23/0.51 ! [X4: real,N: nat] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X4 @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_nonneg
% 0.23/0.51 thf(fact_5977_pochhammer__nonneg,axiom,
% 0.23/0.51 ! [X4: rat,N: nat] :
% 0.23/0.51 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 0.23/0.51 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X4 @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_nonneg
% 0.23/0.51 thf(fact_5978_pochhammer__nonneg,axiom,
% 0.23/0.51 ! [X4: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 0.23/0.51 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X4 @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_nonneg
% 0.23/0.51 thf(fact_5979_pochhammer__nonneg,axiom,
% 0.23/0.51 ! [X4: int,N: nat] :
% 0.23/0.51 ( ( ord_less_int @ zero_zero_int @ X4 )
% 0.23/0.51 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X4 @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_nonneg
% 0.23/0.51 thf(fact_5980_int__cases4,axiom,
% 0.23/0.51 ! [M: int] :
% 0.23/0.51 ( ! [N4: nat] :
% 0.23/0.51 ( M
% 0.23/0.51 != ( semiri1314217659103216013at_int @ N4 ) )
% 0.23/0.51 => ~ ! [N4: nat] :
% 0.23/0.51 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.23/0.51 => ( M
% 0.23/0.51 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_cases4
% 0.23/0.51 thf(fact_5981_pochhammer__0__left,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( ( N = zero_zero_nat )
% 0.23/0.51 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.23/0.51 = one_one_complex ) )
% 0.23/0.51 & ( ( N != zero_zero_nat )
% 0.23/0.51 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.23/0.51 = zero_zero_complex ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_0_left
% 0.23/0.51 thf(fact_5982_pochhammer__0__left,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( ( N = zero_zero_nat )
% 0.23/0.51 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.23/0.51 = one_one_real ) )
% 0.23/0.51 & ( ( N != zero_zero_nat )
% 0.23/0.51 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.23/0.51 = zero_zero_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_0_left
% 0.23/0.51 thf(fact_5983_pochhammer__0__left,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( ( N = zero_zero_nat )
% 0.23/0.51 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.23/0.51 = one_one_rat ) )
% 0.23/0.51 & ( ( N != zero_zero_nat )
% 0.23/0.51 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.23/0.51 = zero_zero_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_0_left
% 0.23/0.51 thf(fact_5984_pochhammer__0__left,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( ( N = zero_zero_nat )
% 0.23/0.51 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.23/0.51 = one_one_nat ) )
% 0.23/0.51 & ( ( N != zero_zero_nat )
% 0.23/0.51 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.23/0.51 = zero_zero_nat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_0_left
% 0.23/0.51 thf(fact_5985_pochhammer__0__left,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( ( N = zero_zero_nat )
% 0.23/0.51 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.23/0.51 = one_one_int ) )
% 0.23/0.51 & ( ( N != zero_zero_nat )
% 0.23/0.51 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.23/0.51 = zero_zero_int ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_0_left
% 0.23/0.51 thf(fact_5986_int__Suc,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.23/0.51 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_Suc
% 0.23/0.51 thf(fact_5987_int__ops_I4_J,axiom,
% 0.23/0.51 ! [A: nat] :
% 0.23/0.51 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 0.23/0.51 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_ops(4)
% 0.23/0.51 thf(fact_5988_zless__iff__Suc__zadd,axiom,
% 0.23/0.51 ( ord_less_int
% 0.23/0.51 = ( ^ [W2: int,Z5: int] :
% 0.23/0.51 ? [N2: nat] :
% 0.23/0.51 ( Z5
% 0.23/0.51 = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % zless_iff_Suc_zadd
% 0.23/0.51 thf(fact_5989_norm__exp,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X4 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_exp
% 0.23/0.51 thf(fact_5990_norm__exp,axiom,
% 0.23/0.51 ! [X4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X4 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_exp
% 0.23/0.51 thf(fact_5991_one__add__floor,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
% 0.23/0.51 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % one_add_floor
% 0.23/0.51 thf(fact_5992_one__add__floor,axiom,
% 0.23/0.51 ! [X4: rat] :
% 0.23/0.51 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
% 0.23/0.51 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % one_add_floor
% 0.23/0.51 thf(fact_5993_of__int__ceiling__le__add__one,axiom,
% 0.23/0.51 ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % of_int_ceiling_le_add_one
% 0.23/0.51 thf(fact_5994_of__int__ceiling__le__add__one,axiom,
% 0.23/0.51 ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 0.23/0.51
% 0.23/0.51 % of_int_ceiling_le_add_one
% 0.23/0.51 thf(fact_5995_of__int__ceiling__diff__one__le,axiom,
% 0.23/0.51 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 0.23/0.51
% 0.23/0.51 % of_int_ceiling_diff_one_le
% 0.23/0.51 thf(fact_5996_of__int__ceiling__diff__one__le,axiom,
% 0.23/0.51 ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 0.23/0.51
% 0.23/0.51 % of_int_ceiling_diff_one_le
% 0.23/0.51 thf(fact_5997_floor__eq,axiom,
% 0.23/0.51 ! [N: int,X4: real] :
% 0.23/0.51 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.23/0.51 => ( ( archim6058952711729229775r_real @ X4 )
% 0.23/0.51 = N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_eq
% 0.23/0.51 thf(fact_5998_real__of__int__floor__add__one__gt,axiom,
% 0.23/0.51 ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % real_of_int_floor_add_one_gt
% 0.23/0.51 thf(fact_5999_real__of__int__floor__add__one__ge,axiom,
% 0.23/0.51 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % real_of_int_floor_add_one_ge
% 0.23/0.51 thf(fact_6000_real__of__int__floor__gt__diff__one,axiom,
% 0.23/0.51 ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % real_of_int_floor_gt_diff_one
% 0.23/0.51 thf(fact_6001_real__of__int__floor__ge__diff__one,axiom,
% 0.23/0.51 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % real_of_int_floor_ge_diff_one
% 0.23/0.51 thf(fact_6002_ceiling__divide__eq__div,axiom,
% 0.23/0.51 ! [A: int,B2: int] :
% 0.23/0.51 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B2 ) ) )
% 0.23/0.51 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_divide_eq_div
% 0.23/0.51 thf(fact_6003_ceiling__divide__eq__div,axiom,
% 0.23/0.51 ! [A: int,B2: int] :
% 0.23/0.51 ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B2 ) ) )
% 0.23/0.51 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_divide_eq_div
% 0.23/0.51 thf(fact_6004_zero__less__imp__eq__int,axiom,
% 0.23/0.51 ! [K: int] :
% 0.23/0.51 ( ( ord_less_int @ zero_zero_int @ K )
% 0.23/0.51 => ? [N4: nat] :
% 0.23/0.51 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.23/0.51 & ( K
% 0.23/0.51 = ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % zero_less_imp_eq_int
% 0.23/0.51 thf(fact_6005_pos__int__cases,axiom,
% 0.23/0.51 ! [K: int] :
% 0.23/0.51 ( ( ord_less_int @ zero_zero_int @ K )
% 0.23/0.51 => ~ ! [N4: nat] :
% 0.23/0.51 ( ( K
% 0.23/0.51 = ( semiri1314217659103216013at_int @ N4 ) )
% 0.23/0.51 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pos_int_cases
% 0.23/0.51 thf(fact_6006_int__cases3,axiom,
% 0.23/0.51 ! [K: int] :
% 0.23/0.51 ( ( K != zero_zero_int )
% 0.23/0.51 => ( ! [N4: nat] :
% 0.23/0.51 ( ( K
% 0.23/0.51 = ( semiri1314217659103216013at_int @ N4 ) )
% 0.23/0.51 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
% 0.23/0.51 => ~ ! [N4: nat] :
% 0.23/0.51 ( ( K
% 0.23/0.51 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 0.23/0.51 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_cases3
% 0.23/0.51 thf(fact_6007_pi__less__4,axiom,
% 0.23/0.51 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pi_less_4
% 0.23/0.51 thf(fact_6008_pi__ge__two,axiom,
% 0.23/0.51 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 0.23/0.51
% 0.23/0.51 % pi_ge_two
% 0.23/0.51 thf(fact_6009_zmult__zless__mono2__lemma,axiom,
% 0.23/0.51 ! [I: int,J: int,K: nat] :
% 0.23/0.51 ( ( ord_less_int @ I @ J )
% 0.23/0.51 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.23/0.51 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % zmult_zless_mono2_lemma
% 0.23/0.51 thf(fact_6010_not__zle__0__negative,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % not_zle_0_negative
% 0.23/0.51 thf(fact_6011_negative__zless__0,axiom,
% 0.23/0.51 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 0.23/0.51
% 0.23/0.51 % negative_zless_0
% 0.23/0.51 thf(fact_6012_negD,axiom,
% 0.23/0.51 ! [X4: int] :
% 0.23/0.51 ( ( ord_less_int @ X4 @ zero_zero_int )
% 0.23/0.51 => ? [N4: nat] :
% 0.23/0.51 ( X4
% 0.23/0.51 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % negD
% 0.23/0.51 thf(fact_6013_pi__half__neq__two,axiom,
% 0.23/0.51 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.23/0.51 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pi_half_neq_two
% 0.23/0.51 thf(fact_6014_pochhammer__rec,axiom,
% 0.23/0.51 ! [A: complex,N: nat] :
% 0.23/0.51 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_rec
% 0.23/0.51 thf(fact_6015_pochhammer__rec,axiom,
% 0.23/0.51 ! [A: real,N: nat] :
% 0.23/0.51 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_rec
% 0.23/0.51 thf(fact_6016_pochhammer__rec,axiom,
% 0.23/0.51 ! [A: rat,N: nat] :
% 0.23/0.51 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_rec
% 0.23/0.51 thf(fact_6017_pochhammer__rec,axiom,
% 0.23/0.51 ! [A: nat,N: nat] :
% 0.23/0.51 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_rec
% 0.23/0.51 thf(fact_6018_pochhammer__rec,axiom,
% 0.23/0.51 ! [A: int,N: nat] :
% 0.23/0.51 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_rec
% 0.23/0.51 thf(fact_6019_pochhammer__Suc,axiom,
% 0.23/0.51 ! [A: complex,N: nat] :
% 0.23/0.51 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_Suc
% 0.23/0.51 thf(fact_6020_pochhammer__Suc,axiom,
% 0.23/0.51 ! [A: real,N: nat] :
% 0.23/0.51 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_Suc
% 0.23/0.51 thf(fact_6021_pochhammer__Suc,axiom,
% 0.23/0.51 ! [A: rat,N: nat] :
% 0.23/0.51 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_Suc
% 0.23/0.51 thf(fact_6022_pochhammer__Suc,axiom,
% 0.23/0.51 ! [A: nat,N: nat] :
% 0.23/0.51 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_Suc
% 0.23/0.51 thf(fact_6023_pochhammer__Suc,axiom,
% 0.23/0.51 ! [A: int,N: nat] :
% 0.23/0.51 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_Suc
% 0.23/0.51 thf(fact_6024_pochhammer__rec_H,axiom,
% 0.23/0.51 ! [Z: complex,N: nat] :
% 0.23/0.51 ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_rec'
% 0.23/0.51 thf(fact_6025_pochhammer__rec_H,axiom,
% 0.23/0.51 ! [Z: real,N: nat] :
% 0.23/0.51 ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_rec'
% 0.23/0.51 thf(fact_6026_pochhammer__rec_H,axiom,
% 0.23/0.51 ! [Z: rat,N: nat] :
% 0.23/0.51 ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_rec'
% 0.23/0.51 thf(fact_6027_pochhammer__rec_H,axiom,
% 0.23/0.51 ! [Z: nat,N: nat] :
% 0.23/0.51 ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_rec'
% 0.23/0.51 thf(fact_6028_pochhammer__rec_H,axiom,
% 0.23/0.51 ! [Z: int,N: nat] :
% 0.23/0.51 ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 0.23/0.51 = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_rec'
% 0.23/0.51 thf(fact_6029_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ord_less_nat @ N @ K )
% 0.23/0.51 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.23/0.51 = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_lemma
% 0.23/0.51 thf(fact_6030_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ord_less_nat @ N @ K )
% 0.23/0.51 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.23/0.51 = zero_zero_complex ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_lemma
% 0.23/0.51 thf(fact_6031_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ord_less_nat @ N @ K )
% 0.23/0.51 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.23/0.51 = zero_zero_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_lemma
% 0.23/0.51 thf(fact_6032_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ord_less_nat @ N @ K )
% 0.23/0.51 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.23/0.51 = zero_zero_rat ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_lemma
% 0.23/0.51 thf(fact_6033_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ord_less_nat @ N @ K )
% 0.23/0.51 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.23/0.51 = zero_zero_int ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_lemma
% 0.23/0.51 thf(fact_6034_pochhammer__of__nat__eq__0__iff,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.23/0.51 = zero_z3403309356797280102nteger )
% 0.23/0.51 = ( ord_less_nat @ N @ K ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_iff
% 0.23/0.51 thf(fact_6035_pochhammer__of__nat__eq__0__iff,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.23/0.51 = zero_zero_complex )
% 0.23/0.51 = ( ord_less_nat @ N @ K ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_iff
% 0.23/0.51 thf(fact_6036_pochhammer__of__nat__eq__0__iff,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.23/0.51 = zero_zero_real )
% 0.23/0.51 = ( ord_less_nat @ N @ K ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_iff
% 0.23/0.51 thf(fact_6037_pochhammer__of__nat__eq__0__iff,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.23/0.51 = zero_zero_rat )
% 0.23/0.51 = ( ord_less_nat @ N @ K ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_iff
% 0.23/0.51 thf(fact_6038_pochhammer__of__nat__eq__0__iff,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.23/0.51 = zero_zero_int )
% 0.23/0.51 = ( ord_less_nat @ N @ K ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_iff
% 0.23/0.51 thf(fact_6039_pochhammer__eq__0__iff,axiom,
% 0.23/0.51 ! [A: complex,N: nat] :
% 0.23/0.51 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.23/0.51 = zero_zero_complex )
% 0.23/0.51 = ( ? [K3: nat] :
% 0.23/0.51 ( ( ord_less_nat @ K3 @ N )
% 0.23/0.51 & ( A
% 0.23/0.51 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_eq_0_iff
% 0.23/0.51 thf(fact_6040_pochhammer__eq__0__iff,axiom,
% 0.23/0.51 ! [A: real,N: nat] :
% 0.23/0.51 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.23/0.51 = zero_zero_real )
% 0.23/0.51 = ( ? [K3: nat] :
% 0.23/0.51 ( ( ord_less_nat @ K3 @ N )
% 0.23/0.51 & ( A
% 0.23/0.51 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_eq_0_iff
% 0.23/0.51 thf(fact_6041_pochhammer__eq__0__iff,axiom,
% 0.23/0.51 ! [A: rat,N: nat] :
% 0.23/0.51 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.23/0.51 = zero_zero_rat )
% 0.23/0.51 = ( ? [K3: nat] :
% 0.23/0.51 ( ( ord_less_nat @ K3 @ N )
% 0.23/0.51 & ( A
% 0.23/0.51 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_eq_0_iff
% 0.23/0.51 thf(fact_6042_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.51 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.23/0.51 != zero_z3403309356797280102nteger ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_lemma'
% 0.23/0.51 thf(fact_6043_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.51 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.23/0.51 != zero_zero_complex ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_lemma'
% 0.23/0.51 thf(fact_6044_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.51 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.23/0.51 != zero_zero_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_lemma'
% 0.23/0.51 thf(fact_6045_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.51 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.23/0.51 != zero_zero_rat ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_lemma'
% 0.23/0.51 thf(fact_6046_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.51 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.23/0.51 != zero_zero_int ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_of_nat_eq_0_lemma'
% 0.23/0.51 thf(fact_6047_int__ops_I6_J,axiom,
% 0.23/0.51 ! [A: nat,B2: nat] :
% 0.23/0.51 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
% 0.23/0.51 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
% 0.23/0.51 = zero_zero_int ) )
% 0.23/0.51 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
% 0.23/0.51 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
% 0.23/0.51 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % int_ops(6)
% 0.23/0.51 thf(fact_6048_pochhammer__product_H,axiom,
% 0.23/0.51 ! [Z: complex,N: nat,M: nat] :
% 0.23/0.51 ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.23/0.51 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_product'
% 0.23/0.51 thf(fact_6049_pochhammer__product_H,axiom,
% 0.23/0.51 ! [Z: real,N: nat,M: nat] :
% 0.23/0.51 ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.23/0.51 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_product'
% 0.23/0.51 thf(fact_6050_pochhammer__product_H,axiom,
% 0.23/0.51 ! [Z: rat,N: nat,M: nat] :
% 0.23/0.51 ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.23/0.51 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_product'
% 0.23/0.51 thf(fact_6051_pochhammer__product_H,axiom,
% 0.23/0.51 ! [Z: nat,N: nat,M: nat] :
% 0.23/0.51 ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.23/0.51 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_product'
% 0.23/0.51 thf(fact_6052_pochhammer__product_H,axiom,
% 0.23/0.51 ! [Z: int,N: nat,M: nat] :
% 0.23/0.51 ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.23/0.51 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_product'
% 0.23/0.51 thf(fact_6053_floor__split,axiom,
% 0.23/0.51 ! [P: int > $o,T: real] :
% 0.23/0.51 ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 0.23/0.51 = ( ! [I2: int] :
% 0.23/0.51 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I2 ) @ T )
% 0.23/0.51 & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) ) )
% 0.23/0.51 => ( P @ I2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_split
% 0.23/0.51 thf(fact_6054_floor__split,axiom,
% 0.23/0.51 ! [P: int > $o,T: rat] :
% 0.23/0.51 ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 0.23/0.51 = ( ! [I2: int] :
% 0.23/0.51 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I2 ) @ T )
% 0.23/0.51 & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) ) )
% 0.23/0.51 => ( P @ I2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_split
% 0.23/0.51 thf(fact_6055_floor__eq__iff,axiom,
% 0.23/0.51 ! [X4: real,A: int] :
% 0.23/0.51 ( ( ( archim6058952711729229775r_real @ X4 )
% 0.23/0.51 = A )
% 0.23/0.51 = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X4 )
% 0.23/0.51 & ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_eq_iff
% 0.23/0.51 thf(fact_6056_floor__eq__iff,axiom,
% 0.23/0.51 ! [X4: rat,A: int] :
% 0.23/0.51 ( ( ( archim3151403230148437115or_rat @ X4 )
% 0.23/0.51 = A )
% 0.23/0.51 = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X4 )
% 0.23/0.51 & ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_eq_iff
% 0.23/0.51 thf(fact_6057_floor__unique,axiom,
% 0.23/0.51 ! [Z: int,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
% 0.23/0.51 => ( ( archim6058952711729229775r_real @ X4 )
% 0.23/0.51 = Z ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_unique
% 0.23/0.51 thf(fact_6058_floor__unique,axiom,
% 0.23/0.51 ! [Z: int,X4: rat] :
% 0.23/0.51 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X4 )
% 0.23/0.51 => ( ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
% 0.23/0.51 => ( ( archim3151403230148437115or_rat @ X4 )
% 0.23/0.51 = Z ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_unique
% 0.23/0.51 thf(fact_6059_le__mult__floor,axiom,
% 0.23/0.51 ! [A: real,B2: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.23/0.51 => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_mult_floor
% 0.23/0.51 thf(fact_6060_le__mult__floor,axiom,
% 0.23/0.51 ! [A: rat,B2: rat] :
% 0.23/0.51 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.51 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.23/0.51 => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_mult_floor
% 0.23/0.51 thf(fact_6061_less__floor__iff,axiom,
% 0.23/0.51 ! [Z: int,X4: real] :
% 0.23/0.51 ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % less_floor_iff
% 0.23/0.51 thf(fact_6062_less__floor__iff,axiom,
% 0.23/0.51 ! [Z: int,X4: rat] :
% 0.23/0.51 ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % less_floor_iff
% 0.23/0.51 thf(fact_6063_floor__le__iff,axiom,
% 0.23/0.51 ! [X4: real,Z: int] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
% 0.23/0.51 = ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_le_iff
% 0.23/0.51 thf(fact_6064_floor__le__iff,axiom,
% 0.23/0.51 ! [X4: rat,Z: int] :
% 0.23/0.51 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
% 0.23/0.51 = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_le_iff
% 0.23/0.51 thf(fact_6065_binomial__mono,axiom,
% 0.23/0.51 ! [K: nat,K6: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ K6 )
% 0.23/0.51 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.23/0.51 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_mono
% 0.23/0.51 thf(fact_6066_binomial__maximum_H,axiom,
% 0.23/0.51 ! [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 ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_maximum'
% 0.23/0.51 thf(fact_6067_binomial__maximum,axiom,
% 0.23/0.51 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_maximum
% 0.23/0.51 thf(fact_6068_binomial__antimono,axiom,
% 0.23/0.51 ! [K: nat,K6: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ K6 )
% 0.23/0.51 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 0.23/0.51 => ( ( ord_less_eq_nat @ K6 @ N )
% 0.23/0.51 => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_antimono
% 0.23/0.51 thf(fact_6069_floor__correct,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) @ X4 )
% 0.23/0.51 & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_correct
% 0.23/0.51 thf(fact_6070_floor__correct,axiom,
% 0.23/0.51 ! [X4: rat] :
% 0.23/0.51 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) @ X4 )
% 0.23/0.51 & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_correct
% 0.23/0.51 thf(fact_6071_ceiling__split,axiom,
% 0.23/0.51 ! [P: int > $o,T: real] :
% 0.23/0.51 ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 0.23/0.51 = ( ! [I2: int] :
% 0.23/0.51 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
% 0.23/0.51 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
% 0.23/0.51 => ( P @ I2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_split
% 0.23/0.51 thf(fact_6072_ceiling__split,axiom,
% 0.23/0.51 ! [P: int > $o,T: rat] :
% 0.23/0.51 ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 0.23/0.51 = ( ! [I2: int] :
% 0.23/0.51 ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
% 0.23/0.51 & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
% 0.23/0.51 => ( P @ I2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_split
% 0.23/0.51 thf(fact_6073_ceiling__eq__iff,axiom,
% 0.23/0.51 ! [X4: real,A: int] :
% 0.23/0.51 ( ( ( archim7802044766580827645g_real @ X4 )
% 0.23/0.51 = A )
% 0.23/0.51 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X4 )
% 0.23/0.51 & ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_eq_iff
% 0.23/0.51 thf(fact_6074_ceiling__eq__iff,axiom,
% 0.23/0.51 ! [X4: rat,A: int] :
% 0.23/0.51 ( ( ( archim2889992004027027881ng_rat @ X4 )
% 0.23/0.51 = A )
% 0.23/0.51 = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X4 )
% 0.23/0.51 & ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_eq_iff
% 0.23/0.51 thf(fact_6075_ceiling__unique,axiom,
% 0.23/0.51 ! [Z: int,X4: real] :
% 0.23/0.51 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z ) )
% 0.23/0.51 => ( ( archim7802044766580827645g_real @ X4 )
% 0.23/0.51 = Z ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_unique
% 0.23/0.51 thf(fact_6076_ceiling__unique,axiom,
% 0.23/0.51 ! [Z: int,X4: rat] :
% 0.23/0.51 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z ) )
% 0.23/0.51 => ( ( archim2889992004027027881ng_rat @ X4 )
% 0.23/0.51 = Z ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_unique
% 0.23/0.51 thf(fact_6077_ceiling__correct,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) @ one_one_real ) @ X4 )
% 0.23/0.51 & ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_correct
% 0.23/0.51 thf(fact_6078_ceiling__correct,axiom,
% 0.23/0.51 ! [X4: rat] :
% 0.23/0.51 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) @ one_one_rat ) @ X4 )
% 0.23/0.51 & ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_correct
% 0.23/0.51 thf(fact_6079_mult__ceiling__le,axiom,
% 0.23/0.51 ! [A: real,B2: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.23/0.51 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % mult_ceiling_le
% 0.23/0.51 thf(fact_6080_mult__ceiling__le,axiom,
% 0.23/0.51 ! [A: rat,B2: rat] :
% 0.23/0.51 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.51 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.23/0.51 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % mult_ceiling_le
% 0.23/0.51 thf(fact_6081_floor__log__nat__eq__if,axiom,
% 0.23/0.51 ! [B2: nat,N: nat,K: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 0.23/0.51 => ( ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.23/0.51 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.23/0.51 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.23/0.51 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_log_nat_eq_if
% 0.23/0.51 thf(fact_6082_ceiling__less__iff,axiom,
% 0.23/0.51 ! [X4: real,Z: int] :
% 0.23/0.51 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ Z )
% 0.23/0.51 = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_less_iff
% 0.23/0.51 thf(fact_6083_ceiling__less__iff,axiom,
% 0.23/0.51 ! [X4: rat,Z: int] :
% 0.23/0.51 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z )
% 0.23/0.51 = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_less_iff
% 0.23/0.51 thf(fact_6084_le__ceiling__iff,axiom,
% 0.23/0.51 ! [Z: int,X4: rat] :
% 0.23/0.51 ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X4 ) )
% 0.23/0.51 = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_ceiling_iff
% 0.23/0.51 thf(fact_6085_le__ceiling__iff,axiom,
% 0.23/0.51 ! [Z: int,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X4 ) )
% 0.23/0.51 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_ceiling_iff
% 0.23/0.51 thf(fact_6086_floor__eq2,axiom,
% 0.23/0.51 ! [N: int,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.23/0.51 => ( ( archim6058952711729229775r_real @ X4 )
% 0.23/0.51 = N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_eq2
% 0.23/0.51 thf(fact_6087_floor__divide__real__eq__div,axiom,
% 0.23/0.51 ! [B2: int,A: real] :
% 0.23/0.51 ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.23/0.51 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B2 ) ) )
% 0.23/0.51 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_divide_real_eq_div
% 0.23/0.51 thf(fact_6088_neg__int__cases,axiom,
% 0.23/0.51 ! [K: int] :
% 0.23/0.51 ( ( ord_less_int @ K @ zero_zero_int )
% 0.23/0.51 => ~ ! [N4: nat] :
% 0.23/0.51 ( ( K
% 0.23/0.51 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 0.23/0.51 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % neg_int_cases
% 0.23/0.51 thf(fact_6089_pi__half__neq__zero,axiom,
% 0.23/0.51 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.23/0.51 != zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % pi_half_neq_zero
% 0.23/0.51 thf(fact_6090_pi__half__less__two,axiom,
% 0.23/0.51 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.23/0.51
% 0.23/0.51 % pi_half_less_two
% 0.23/0.51 thf(fact_6091_pi__half__le__two,axiom,
% 0.23/0.51 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.23/0.51
% 0.23/0.51 % pi_half_le_two
% 0.23/0.51 thf(fact_6092_zdiff__int__split,axiom,
% 0.23/0.51 ! [P: int > $o,X4: nat,Y3: nat] :
% 0.23/0.51 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X4 @ Y3 ) ) )
% 0.23/0.51 = ( ( ( ord_less_eq_nat @ Y3 @ X4 )
% 0.23/0.51 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
% 0.23/0.51 & ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.51 => ( P @ zero_zero_int ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % zdiff_int_split
% 0.23/0.51 thf(fact_6093_floor__log__nat__eq__powr__iff,axiom,
% 0.23/0.51 ! [B2: nat,K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.23/0.51 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.23/0.51 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.23/0.51 = ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.51 = ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 0.23/0.51 & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_log_nat_eq_powr_iff
% 0.23/0.51 thf(fact_6094_pochhammer__product,axiom,
% 0.23/0.51 ! [M: nat,N: nat,Z: complex] :
% 0.23/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.51 => ( ( comm_s2602460028002588243omplex @ Z @ N )
% 0.23/0.51 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_product
% 0.23/0.51 thf(fact_6095_pochhammer__product,axiom,
% 0.23/0.51 ! [M: nat,N: nat,Z: real] :
% 0.23/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.51 => ( ( comm_s7457072308508201937r_real @ Z @ N )
% 0.23/0.51 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_product
% 0.23/0.51 thf(fact_6096_pochhammer__product,axiom,
% 0.23/0.51 ! [M: nat,N: nat,Z: rat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.51 => ( ( comm_s4028243227959126397er_rat @ Z @ N )
% 0.23/0.51 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_product
% 0.23/0.51 thf(fact_6097_pochhammer__product,axiom,
% 0.23/0.51 ! [M: nat,N: nat,Z: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.51 => ( ( comm_s4663373288045622133er_nat @ Z @ N )
% 0.23/0.51 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_product
% 0.23/0.51 thf(fact_6098_pochhammer__product,axiom,
% 0.23/0.51 ! [M: nat,N: nat,Z: int] :
% 0.23/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.51 => ( ( comm_s4660882817536571857er_int @ Z @ N )
% 0.23/0.51 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_product
% 0.23/0.51 thf(fact_6099_floor__divide__lower,axiom,
% 0.23/0.51 ! [Q2: real,P2: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.23/0.51 => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ Q2 ) @ P2 ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_divide_lower
% 0.23/0.51 thf(fact_6100_floor__divide__lower,axiom,
% 0.23/0.51 ! [Q2: rat,P2: rat] :
% 0.23/0.51 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.23/0.51 => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ Q2 ) @ P2 ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_divide_lower
% 0.23/0.51 thf(fact_6101_binomial__less__binomial__Suc,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_less_binomial_Suc
% 0.23/0.51 thf(fact_6102_binomial__strict__mono,axiom,
% 0.23/0.51 ! [K: nat,K6: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_nat @ K @ K6 )
% 0.23/0.51 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.23/0.51 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_strict_mono
% 0.23/0.51 thf(fact_6103_binomial__strict__antimono,axiom,
% 0.23/0.51 ! [K: nat,K6: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_nat @ K @ K6 )
% 0.23/0.51 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 0.23/0.51 => ( ( ord_less_eq_nat @ K6 @ N )
% 0.23/0.51 => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_strict_antimono
% 0.23/0.51 thf(fact_6104_central__binomial__odd,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.51 => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.51 = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % central_binomial_odd
% 0.23/0.51 thf(fact_6105_ceiling__divide__upper,axiom,
% 0.23/0.51 ! [Q2: real,P2: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.23/0.51 => ( ord_less_eq_real @ P2 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_divide_upper
% 0.23/0.51 thf(fact_6106_ceiling__divide__upper,axiom,
% 0.23/0.51 ! [Q2: rat,P2: rat] :
% 0.23/0.51 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.23/0.51 => ( ord_less_eq_rat @ P2 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_divide_upper
% 0.23/0.51 thf(fact_6107_bit__nat__def,axiom,
% 0.23/0.51 ( bit_se1148574629649215175it_nat
% 0.23/0.51 = ( ^ [M6: nat,N2: nat] :
% 0.23/0.51 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % bit_nat_def
% 0.23/0.51 thf(fact_6108_ceiling__log__nat__eq__if,axiom,
% 0.23/0.51 ! [B2: nat,N: nat,K: nat] :
% 0.23/0.51 ( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 0.23/0.51 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.23/0.51 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.23/0.51 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.23/0.51 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_log_nat_eq_if
% 0.23/0.51 thf(fact_6109_pi__half__gt__zero,axiom,
% 0.23/0.51 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pi_half_gt_zero
% 0.23/0.51 thf(fact_6110_pi__half__ge__zero,axiom,
% 0.23/0.51 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pi_half_ge_zero
% 0.23/0.51 thf(fact_6111_pochhammer__absorb__comp,axiom,
% 0.23/0.51 ! [R2: code_integer,K: nat] :
% 0.23/0.51 ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
% 0.23/0.51 = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_absorb_comp
% 0.23/0.51 thf(fact_6112_pochhammer__absorb__comp,axiom,
% 0.23/0.51 ! [R2: complex,K: nat] :
% 0.23/0.51 ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
% 0.23/0.51 = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_absorb_comp
% 0.23/0.51 thf(fact_6113_pochhammer__absorb__comp,axiom,
% 0.23/0.51 ! [R2: real,K: nat] :
% 0.23/0.51 ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
% 0.23/0.51 = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_absorb_comp
% 0.23/0.51 thf(fact_6114_pochhammer__absorb__comp,axiom,
% 0.23/0.51 ! [R2: rat,K: nat] :
% 0.23/0.51 ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
% 0.23/0.51 = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_absorb_comp
% 0.23/0.51 thf(fact_6115_pochhammer__absorb__comp,axiom,
% 0.23/0.51 ! [R2: int,K: nat] :
% 0.23/0.51 ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
% 0.23/0.51 = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_absorb_comp
% 0.23/0.51 thf(fact_6116_m2pi__less__pi,axiom,
% 0.23/0.51 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 0.23/0.51
% 0.23/0.51 % m2pi_less_pi
% 0.23/0.51 thf(fact_6117_ceiling__log__nat__eq__powr__iff,axiom,
% 0.23/0.51 ! [B2: nat,K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.23/0.51 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.23/0.51 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.23/0.51 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 0.23/0.51 = ( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 0.23/0.51 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_log_nat_eq_powr_iff
% 0.23/0.51 thf(fact_6118_floor__divide__upper,axiom,
% 0.23/0.51 ! [Q2: real,P2: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.23/0.51 => ( ord_less_real @ P2 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_divide_upper
% 0.23/0.51 thf(fact_6119_floor__divide__upper,axiom,
% 0.23/0.51 ! [Q2: rat,P2: rat] :
% 0.23/0.51 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.23/0.51 => ( ord_less_rat @ P2 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % floor_divide_upper
% 0.23/0.51 thf(fact_6120_arctan__ubound,axiom,
% 0.23/0.51 ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % arctan_ubound
% 0.23/0.51 thf(fact_6121_arctan__one,axiom,
% 0.23/0.51 ( ( arctan @ one_one_real )
% 0.23/0.51 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % arctan_one
% 0.23/0.51 thf(fact_6122_round__def,axiom,
% 0.23/0.51 ( archim8280529875227126926d_real
% 0.23/0.51 = ( ^ [X: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % round_def
% 0.23/0.51 thf(fact_6123_round__def,axiom,
% 0.23/0.51 ( archim7778729529865785530nd_rat
% 0.23/0.51 = ( ^ [X: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % round_def
% 0.23/0.51 thf(fact_6124_ceiling__divide__lower,axiom,
% 0.23/0.51 ! [Q2: real,P2: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.23/0.51 => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P2 ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_divide_lower
% 0.23/0.51 thf(fact_6125_ceiling__divide__lower,axiom,
% 0.23/0.51 ! [Q2: rat,P2: rat] :
% 0.23/0.51 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.23/0.51 => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P2 ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_divide_lower
% 0.23/0.51 thf(fact_6126_ceiling__eq,axiom,
% 0.23/0.51 ! [N: int,X4: real] :
% 0.23/0.51 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.23/0.51 => ( ( archim7802044766580827645g_real @ X4 )
% 0.23/0.51 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_eq
% 0.23/0.51 thf(fact_6127_ceiling__eq,axiom,
% 0.23/0.51 ! [N: int,X4: rat] :
% 0.23/0.51 ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 0.23/0.51 => ( ( archim2889992004027027881ng_rat @ X4 )
% 0.23/0.51 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_eq
% 0.23/0.51 thf(fact_6128_exp__bound__half,axiom,
% 0.23/0.51 ! [Z: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % exp_bound_half
% 0.23/0.51 thf(fact_6129_exp__bound__half,axiom,
% 0.23/0.51 ! [Z: complex] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % exp_bound_half
% 0.23/0.51 thf(fact_6130_minus__pi__half__less__zero,axiom,
% 0.23/0.51 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 0.23/0.51
% 0.23/0.51 % minus_pi_half_less_zero
% 0.23/0.51 thf(fact_6131_pochhammer__minus,axiom,
% 0.23/0.51 ! [B2: code_integer,K: nat] :
% 0.23/0.51 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K )
% 0.23/0.51 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_minus
% 0.23/0.51 thf(fact_6132_pochhammer__minus,axiom,
% 0.23/0.51 ! [B2: complex,K: nat] :
% 0.23/0.51 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K )
% 0.23/0.51 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_minus
% 0.23/0.51 thf(fact_6133_pochhammer__minus,axiom,
% 0.23/0.51 ! [B2: real,K: nat] :
% 0.23/0.51 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K )
% 0.23/0.51 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_minus
% 0.23/0.51 thf(fact_6134_pochhammer__minus,axiom,
% 0.23/0.51 ! [B2: rat,K: nat] :
% 0.23/0.51 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K )
% 0.23/0.51 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_minus
% 0.23/0.51 thf(fact_6135_pochhammer__minus,axiom,
% 0.23/0.51 ! [B2: int,K: nat] :
% 0.23/0.51 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K )
% 0.23/0.51 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_minus
% 0.23/0.51 thf(fact_6136_pochhammer__minus_H,axiom,
% 0.23/0.51 ! [B2: code_integer,K: nat] :
% 0.23/0.51 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 0.23/0.51 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_minus'
% 0.23/0.51 thf(fact_6137_pochhammer__minus_H,axiom,
% 0.23/0.51 ! [B2: complex,K: nat] :
% 0.23/0.51 ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 0.23/0.51 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_minus'
% 0.23/0.51 thf(fact_6138_pochhammer__minus_H,axiom,
% 0.23/0.51 ! [B2: real,K: nat] :
% 0.23/0.51 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 0.23/0.51 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_minus'
% 0.23/0.51 thf(fact_6139_pochhammer__minus_H,axiom,
% 0.23/0.51 ! [B2: rat,K: nat] :
% 0.23/0.51 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 0.23/0.51 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_minus'
% 0.23/0.51 thf(fact_6140_pochhammer__minus_H,axiom,
% 0.23/0.51 ! [B2: int,K: nat] :
% 0.23/0.51 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 0.23/0.51 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % pochhammer_minus'
% 0.23/0.51 thf(fact_6141_arctan__lbound,axiom,
% 0.23/0.51 ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).
% 0.23/0.51
% 0.23/0.51 % arctan_lbound
% 0.23/0.51 thf(fact_6142_arctan__bounded,axiom,
% 0.23/0.51 ! [Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 0.23/0.51 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % arctan_bounded
% 0.23/0.51 thf(fact_6143_exp__bound__lemma,axiom,
% 0.23/0.51 ! [Z: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( 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 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % exp_bound_lemma
% 0.23/0.51 thf(fact_6144_exp__bound__lemma,axiom,
% 0.23/0.51 ! [Z: complex] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( 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 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % exp_bound_lemma
% 0.23/0.51 thf(fact_6145_machin__Euler,axiom,
% 0.23/0.51 ( ( 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 ) ) ) ) ) ) ) ) ) ) )
% 0.23/0.51 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % machin_Euler
% 0.23/0.51 thf(fact_6146_norm__divide__numeral,axiom,
% 0.23/0.51 ! [A: real,W: num] :
% 0.23/0.51 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.51 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_divide_numeral
% 0.23/0.51 thf(fact_6147_norm__divide__numeral,axiom,
% 0.23/0.51 ! [A: complex,W: num] :
% 0.23/0.51 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.23/0.51 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_divide_numeral
% 0.23/0.51 thf(fact_6148_norm__mult__numeral1,axiom,
% 0.23/0.51 ! [W: num,A: real] :
% 0.23/0.51 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.23/0.51 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_mult_numeral1
% 0.23/0.51 thf(fact_6149_norm__mult__numeral1,axiom,
% 0.23/0.51 ! [W: num,A: complex] :
% 0.23/0.51 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.23/0.51 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_mult_numeral1
% 0.23/0.51 thf(fact_6150_norm__mult__numeral2,axiom,
% 0.23/0.51 ! [A: real,W: num] :
% 0.23/0.51 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.51 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_mult_numeral2
% 0.23/0.51 thf(fact_6151_norm__mult__numeral2,axiom,
% 0.23/0.51 ! [A: complex,W: num] :
% 0.23/0.51 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.23/0.51 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_mult_numeral2
% 0.23/0.51 thf(fact_6152_zero__less__binomial__iff,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 0.23/0.51 = ( ord_less_eq_nat @ K @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % zero_less_binomial_iff
% 0.23/0.51 thf(fact_6153_norm__neg__numeral,axiom,
% 0.23/0.51 ! [W: num] :
% 0.23/0.51 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.51 = ( numeral_numeral_real @ W ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_neg_numeral
% 0.23/0.51 thf(fact_6154_norm__neg__numeral,axiom,
% 0.23/0.51 ! [W: num] :
% 0.23/0.51 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.23/0.51 = ( numeral_numeral_real @ W ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_neg_numeral
% 0.23/0.51 thf(fact_6155_norm__le__zero__iff,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X4 ) @ zero_zero_real )
% 0.23/0.51 = ( X4 = zero_zero_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_le_zero_iff
% 0.23/0.51 thf(fact_6156_norm__le__zero__iff,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ zero_zero_real )
% 0.23/0.51 = ( X4 = zero_zero_complex ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_le_zero_iff
% 0.23/0.51 thf(fact_6157_binomial__Suc__n,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( binomial @ ( suc @ N ) @ N )
% 0.23/0.51 = ( suc @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_Suc_n
% 0.23/0.51 thf(fact_6158_binomial__n__n,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( binomial @ N @ N )
% 0.23/0.51 = one_one_nat ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_n_n
% 0.23/0.51 thf(fact_6159_norm__one,axiom,
% 0.23/0.51 ( ( real_V7735802525324610683m_real @ one_one_real )
% 0.23/0.51 = one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % norm_one
% 0.23/0.51 thf(fact_6160_norm__one,axiom,
% 0.23/0.51 ( ( real_V1022390504157884413omplex @ one_one_complex )
% 0.23/0.51 = one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % norm_one
% 0.23/0.51 thf(fact_6161_norm__numeral,axiom,
% 0.23/0.51 ! [W: num] :
% 0.23/0.51 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 0.23/0.51 = ( numeral_numeral_real @ W ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_numeral
% 0.23/0.51 thf(fact_6162_norm__numeral,axiom,
% 0.23/0.51 ! [W: num] :
% 0.23/0.51 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.23/0.51 = ( numeral_numeral_real @ W ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_numeral
% 0.23/0.51 thf(fact_6163_binomial__0__Suc,axiom,
% 0.23/0.51 ! [K: nat] :
% 0.23/0.51 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 0.23/0.51 = zero_zero_nat ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_0_Suc
% 0.23/0.51 thf(fact_6164_binomial__1,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 0.23/0.51 = N ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_1
% 0.23/0.51 thf(fact_6165_binomial__eq__0__iff,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ( binomial @ N @ K )
% 0.23/0.51 = zero_zero_nat )
% 0.23/0.51 = ( ord_less_nat @ N @ K ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_eq_0_iff
% 0.23/0.51 thf(fact_6166_binomial__Suc__Suc,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.23/0.51 = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_Suc_Suc
% 0.23/0.51 thf(fact_6167_binomial__n__0,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( binomial @ N @ zero_zero_nat )
% 0.23/0.51 = one_one_nat ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_n_0
% 0.23/0.51 thf(fact_6168_zero__less__norm__iff,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X4 ) )
% 0.23/0.51 = ( X4 != zero_zero_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % zero_less_norm_iff
% 0.23/0.51 thf(fact_6169_zero__less__norm__iff,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X4 ) )
% 0.23/0.51 = ( X4 != zero_zero_complex ) ) ).
% 0.23/0.51
% 0.23/0.51 % zero_less_norm_iff
% 0.23/0.51 thf(fact_6170_norm__minus__commute,axiom,
% 0.23/0.51 ! [A: real,B2: real] :
% 0.23/0.51 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) )
% 0.23/0.51 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_minus_commute
% 0.23/0.51 thf(fact_6171_norm__minus__commute,axiom,
% 0.23/0.51 ! [A: complex,B2: complex] :
% 0.23/0.51 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) )
% 0.23/0.51 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ A ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_minus_commute
% 0.23/0.51 thf(fact_6172_choose__one,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( binomial @ N @ one_one_nat )
% 0.23/0.51 = N ) ).
% 0.23/0.51
% 0.23/0.51 % choose_one
% 0.23/0.51 thf(fact_6173_norm__not__less__zero,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % norm_not_less_zero
% 0.23/0.51 thf(fact_6174_norm__ge__zero,axiom,
% 0.23/0.51 ! [X4: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_ge_zero
% 0.23/0.51 thf(fact_6175_norm__mult,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X4 @ Y3 ) )
% 0.23/0.51 = ( times_times_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_mult
% 0.23/0.51 thf(fact_6176_norm__mult,axiom,
% 0.23/0.51 ! [X4: complex,Y3: complex] :
% 0.23/0.51 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X4 @ Y3 ) )
% 0.23/0.51 = ( times_times_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_mult
% 0.23/0.51 thf(fact_6177_norm__divide,axiom,
% 0.23/0.51 ! [A: real,B2: real] :
% 0.23/0.51 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
% 0.23/0.51 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_divide
% 0.23/0.51 thf(fact_6178_norm__divide,axiom,
% 0.23/0.51 ! [A: complex,B2: complex] :
% 0.23/0.51 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.23/0.51 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_divide
% 0.23/0.51 thf(fact_6179_norm__power,axiom,
% 0.23/0.51 ! [X4: real,N: nat] :
% 0.23/0.51 ( ( real_V7735802525324610683m_real @ ( power_power_real @ X4 @ N ) )
% 0.23/0.51 = ( power_power_real @ ( real_V7735802525324610683m_real @ X4 ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_power
% 0.23/0.51 thf(fact_6180_norm__power,axiom,
% 0.23/0.51 ! [X4: complex,N: nat] :
% 0.23/0.51 ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X4 @ N ) )
% 0.23/0.51 = ( power_power_real @ ( real_V1022390504157884413omplex @ X4 ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_power
% 0.23/0.51 thf(fact_6181_binomial__eq__0,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ord_less_nat @ N @ K )
% 0.23/0.51 => ( ( binomial @ N @ K )
% 0.23/0.51 = zero_zero_nat ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_eq_0
% 0.23/0.51 thf(fact_6182_Suc__times__binomial,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 0.23/0.51 = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % Suc_times_binomial
% 0.23/0.51 thf(fact_6183_Suc__times__binomial__eq,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 0.23/0.51 = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % Suc_times_binomial_eq
% 0.23/0.51 thf(fact_6184_binomial__symmetric,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.51 => ( ( binomial @ N @ K )
% 0.23/0.51 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_symmetric
% 0.23/0.51 thf(fact_6185_choose__mult__lemma,axiom,
% 0.23/0.51 ! [M: nat,R2: nat,K: nat] :
% 0.23/0.51 ( ( 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 ) )
% 0.23/0.51 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % choose_mult_lemma
% 0.23/0.51 thf(fact_6186_binomial__le__pow,axiom,
% 0.23/0.51 ! [R2: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ R2 @ N )
% 0.23/0.51 => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_le_pow
% 0.23/0.51 thf(fact_6187_norm__uminus__minus,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) )
% 0.23/0.51 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_uminus_minus
% 0.23/0.51 thf(fact_6188_norm__uminus__minus,axiom,
% 0.23/0.51 ! [X4: complex,Y3: complex] :
% 0.23/0.51 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ Y3 ) )
% 0.23/0.51 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_uminus_minus
% 0.23/0.51 thf(fact_6189_nonzero__norm__divide,axiom,
% 0.23/0.51 ! [B2: real,A: real] :
% 0.23/0.51 ( ( B2 != zero_zero_real )
% 0.23/0.51 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
% 0.23/0.51 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % nonzero_norm_divide
% 0.23/0.51 thf(fact_6190_nonzero__norm__divide,axiom,
% 0.23/0.51 ! [B2: complex,A: complex] :
% 0.23/0.51 ( ( B2 != zero_zero_complex )
% 0.23/0.51 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.23/0.51 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % nonzero_norm_divide
% 0.23/0.51 thf(fact_6191_power__eq__imp__eq__norm,axiom,
% 0.23/0.51 ! [W: real,N: nat,Z: real] :
% 0.23/0.51 ( ( ( power_power_real @ W @ N )
% 0.23/0.51 = ( power_power_real @ Z @ N ) )
% 0.23/0.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.51 => ( ( real_V7735802525324610683m_real @ W )
% 0.23/0.51 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % power_eq_imp_eq_norm
% 0.23/0.51 thf(fact_6192_power__eq__imp__eq__norm,axiom,
% 0.23/0.51 ! [W: complex,N: nat,Z: complex] :
% 0.23/0.51 ( ( ( power_power_complex @ W @ N )
% 0.23/0.51 = ( power_power_complex @ Z @ N ) )
% 0.23/0.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.51 => ( ( real_V1022390504157884413omplex @ W )
% 0.23/0.51 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % power_eq_imp_eq_norm
% 0.23/0.51 thf(fact_6193_norm__mult__less,axiom,
% 0.23/0.51 ! [X4: real,R2: real,Y3: real,S: real] :
% 0.23/0.51 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ R2 )
% 0.23/0.51 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 0.23/0.51 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X4 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_mult_less
% 0.23/0.51 thf(fact_6194_norm__mult__less,axiom,
% 0.23/0.51 ! [X4: complex,R2: real,Y3: complex,S: real] :
% 0.23/0.51 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ R2 )
% 0.23/0.51 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 0.23/0.51 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X4 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_mult_less
% 0.23/0.51 thf(fact_6195_norm__mult__ineq,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X4 @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_mult_ineq
% 0.23/0.51 thf(fact_6196_norm__mult__ineq,axiom,
% 0.23/0.51 ! [X4: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X4 @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_mult_ineq
% 0.23/0.51 thf(fact_6197_norm__triangle__lt,axiom,
% 0.23/0.51 ! [X4: real,Y3: real,E: real] :
% 0.23/0.51 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.23/0.51 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ E ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_lt
% 0.23/0.51 thf(fact_6198_norm__triangle__lt,axiom,
% 0.23/0.51 ! [X4: complex,Y3: complex,E: real] :
% 0.23/0.51 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.23/0.51 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ E ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_lt
% 0.23/0.51 thf(fact_6199_norm__add__less,axiom,
% 0.23/0.51 ! [X4: real,R2: real,Y3: real,S: real] :
% 0.23/0.51 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ R2 )
% 0.23/0.51 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 0.23/0.51 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_add_less
% 0.23/0.51 thf(fact_6200_norm__add__less,axiom,
% 0.23/0.51 ! [X4: complex,R2: real,Y3: complex,S: real] :
% 0.23/0.51 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ R2 )
% 0.23/0.51 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 0.23/0.51 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_add_less
% 0.23/0.51 thf(fact_6201_norm__power__ineq,axiom,
% 0.23/0.51 ! [X4: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X4 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X4 ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_power_ineq
% 0.23/0.51 thf(fact_6202_norm__power__ineq,axiom,
% 0.23/0.51 ! [X4: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X4 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X4 ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_power_ineq
% 0.23/0.51 thf(fact_6203_norm__triangle__mono,axiom,
% 0.23/0.51 ! [A: real,R2: real,B2: real,S: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ S )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_mono
% 0.23/0.51 thf(fact_6204_norm__triangle__mono,axiom,
% 0.23/0.51 ! [A: complex,R2: real,B2: complex,S: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ S )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_mono
% 0.23/0.51 thf(fact_6205_norm__triangle__ineq,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_ineq
% 0.23/0.51 thf(fact_6206_norm__triangle__ineq,axiom,
% 0.23/0.51 ! [X4: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_ineq
% 0.23/0.51 thf(fact_6207_norm__triangle__le,axiom,
% 0.23/0.51 ! [X4: real,Y3: real,E: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ E ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_le
% 0.23/0.51 thf(fact_6208_norm__triangle__le,axiom,
% 0.23/0.51 ! [X4: complex,Y3: complex,E: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ E ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_le
% 0.23/0.51 thf(fact_6209_norm__add__leD,axiom,
% 0.23/0.51 ! [A: real,B2: real,C: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ C )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_add_leD
% 0.23/0.51 thf(fact_6210_norm__add__leD,axiom,
% 0.23/0.51 ! [A: complex,B2: complex,C: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ C )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_add_leD
% 0.23/0.51 thf(fact_6211_norm__diff__triangle__less,axiom,
% 0.23/0.51 ! [X4: real,Y3: real,E1: real,Z: real,E22: real] :
% 0.23/0.51 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ E1 )
% 0.23/0.51 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 0.23/0.51 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_diff_triangle_less
% 0.23/0.51 thf(fact_6212_norm__diff__triangle__less,axiom,
% 0.23/0.51 ! [X4: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 0.23/0.51 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) @ E1 )
% 0.23/0.51 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 0.23/0.51 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_diff_triangle_less
% 0.23/0.51 thf(fact_6213_norm__triangle__sub,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_sub
% 0.23/0.51 thf(fact_6214_norm__triangle__sub,axiom,
% 0.23/0.51 ! [X4: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_sub
% 0.23/0.51 thf(fact_6215_norm__triangle__ineq4,axiom,
% 0.23/0.51 ! [A: real,B2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_ineq4
% 0.23/0.51 thf(fact_6216_norm__triangle__ineq4,axiom,
% 0.23/0.51 ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_ineq4
% 0.23/0.51 thf(fact_6217_norm__diff__triangle__le,axiom,
% 0.23/0.51 ! [X4: real,Y3: real,E1: real,Z: real,E22: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ E1 )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_diff_triangle_le
% 0.23/0.51 thf(fact_6218_norm__diff__triangle__le,axiom,
% 0.23/0.51 ! [X4: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) @ E1 )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_diff_triangle_le
% 0.23/0.51 thf(fact_6219_norm__triangle__le__diff,axiom,
% 0.23/0.51 ! [X4: real,Y3: real,E: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ E ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_le_diff
% 0.23/0.51 thf(fact_6220_norm__triangle__le__diff,axiom,
% 0.23/0.51 ! [X4: complex,Y3: complex,E: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.23/0.51 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) @ E ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_le_diff
% 0.23/0.51 thf(fact_6221_norm__diff__ineq,axiom,
% 0.23/0.51 ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_diff_ineq
% 0.23/0.51 thf(fact_6222_norm__diff__ineq,axiom,
% 0.23/0.51 ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_diff_ineq
% 0.23/0.51 thf(fact_6223_norm__triangle__ineq2,axiom,
% 0.23/0.51 ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_ineq2
% 0.23/0.51 thf(fact_6224_norm__triangle__ineq2,axiom,
% 0.23/0.51 ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_ineq2
% 0.23/0.51 thf(fact_6225_zero__less__binomial,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.51 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % zero_less_binomial
% 0.23/0.51 thf(fact_6226_Suc__times__binomial__add,axiom,
% 0.23/0.51 ! [A: nat,B2: nat] :
% 0.23/0.51 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ ( suc @ A ) ) )
% 0.23/0.51 = ( times_times_nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ A ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % Suc_times_binomial_add
% 0.23/0.51 thf(fact_6227_binomial__Suc__Suc__eq__times,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.23/0.51 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_Suc_Suc_eq_times
% 0.23/0.51 thf(fact_6228_choose__mult,axiom,
% 0.23/0.51 ! [K: nat,M: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ M )
% 0.23/0.51 => ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.51 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 0.23/0.51 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % choose_mult
% 0.23/0.51 thf(fact_6229_binomial__absorb__comp,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 0.23/0.51 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_absorb_comp
% 0.23/0.51 thf(fact_6230_power__eq__1__iff,axiom,
% 0.23/0.51 ! [W: real,N: nat] :
% 0.23/0.51 ( ( ( power_power_real @ W @ N )
% 0.23/0.51 = one_one_real )
% 0.23/0.51 => ( ( ( real_V7735802525324610683m_real @ W )
% 0.23/0.51 = one_one_real )
% 0.23/0.51 | ( N = zero_zero_nat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % power_eq_1_iff
% 0.23/0.51 thf(fact_6231_power__eq__1__iff,axiom,
% 0.23/0.51 ! [W: complex,N: nat] :
% 0.23/0.51 ( ( ( power_power_complex @ W @ N )
% 0.23/0.51 = one_one_complex )
% 0.23/0.51 => ( ( ( real_V1022390504157884413omplex @ W )
% 0.23/0.51 = one_one_real )
% 0.23/0.51 | ( N = zero_zero_nat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % power_eq_1_iff
% 0.23/0.51 thf(fact_6232_norm__diff__triangle__ineq,axiom,
% 0.23/0.51 ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_diff_triangle_ineq
% 0.23/0.51 thf(fact_6233_norm__diff__triangle__ineq,axiom,
% 0.23/0.51 ! [A: complex,B2: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ D ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_diff_triangle_ineq
% 0.23/0.51 thf(fact_6234_norm__triangle__ineq3,axiom,
% 0.23/0.51 ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_ineq3
% 0.23/0.51 thf(fact_6235_norm__triangle__ineq3,axiom,
% 0.23/0.51 ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_triangle_ineq3
% 0.23/0.51 thf(fact_6236_binomial__absorption,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 0.23/0.51 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_absorption
% 0.23/0.51 thf(fact_6237_binomial__ge__n__over__k__pow__k,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.51 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_ge_n_over_k_pow_k
% 0.23/0.51 thf(fact_6238_binomial__ge__n__over__k__pow__k,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.51 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_ge_n_over_k_pow_k
% 0.23/0.51 thf(fact_6239_binomial__le__pow2,axiom,
% 0.23/0.51 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_le_pow2
% 0.23/0.51 thf(fact_6240_choose__reduce__nat,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.51 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.23/0.51 => ( ( binomial @ N @ K )
% 0.23/0.51 = ( 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 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % choose_reduce_nat
% 0.23/0.51 thf(fact_6241_times__binomial__minus1__eq,axiom,
% 0.23/0.51 ! [K: nat,N: nat] :
% 0.23/0.51 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.23/0.51 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 0.23/0.51 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % times_binomial_minus1_eq
% 0.23/0.51 thf(fact_6242_square__norm__one,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.51 = one_one_real )
% 0.23/0.51 => ( ( real_V7735802525324610683m_real @ X4 )
% 0.23/0.51 = one_one_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % square_norm_one
% 0.23/0.51 thf(fact_6243_square__norm__one,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.51 = one_one_complex )
% 0.23/0.51 => ( ( real_V1022390504157884413omplex @ X4 )
% 0.23/0.51 = one_one_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % square_norm_one
% 0.23/0.51 thf(fact_6244_norm__power__diff,axiom,
% 0.23/0.51 ! [Z: real,W: real,M: nat] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 0.23/0.51 => ( 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 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_power_diff
% 0.23/0.51 thf(fact_6245_norm__power__diff,axiom,
% 0.23/0.51 ! [Z: complex,W: complex,M: nat] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 0.23/0.51 => ( 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 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % norm_power_diff
% 0.23/0.51 thf(fact_6246_binomial__addition__formula,axiom,
% 0.23/0.51 ! [N: nat,K: nat] :
% 0.23/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.51 => ( ( binomial @ N @ ( suc @ K ) )
% 0.23/0.51 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % binomial_addition_formula
% 0.23/0.51 thf(fact_6247_choose__two,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.51 = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % choose_two
% 0.23/0.51 thf(fact_6248_sin__cos__npi,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( 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 ) ) ) )
% 0.23/0.51 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_cos_npi
% 0.23/0.51 thf(fact_6249_cos__pi__eq__zero,axiom,
% 0.23/0.51 ! [M: nat] :
% 0.23/0.51 ( ( 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 ) ) ) )
% 0.23/0.51 = zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % cos_pi_eq_zero
% 0.23/0.51 thf(fact_6250_round__altdef,axiom,
% 0.23/0.51 ( archim8280529875227126926d_real
% 0.23/0.51 = ( ^ [X: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X ) ) @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % round_altdef
% 0.23/0.51 thf(fact_6251_round__altdef,axiom,
% 0.23/0.51 ( archim7778729529865785530nd_rat
% 0.23/0.51 = ( ^ [X: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X ) ) @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % round_altdef
% 0.23/0.51 thf(fact_6252_ceiling__log__eq__powr__iff,axiom,
% 0.23/0.51 ! [X4: real,B2: real,K: nat] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.51 => ( ( ( archim7802044766580827645g_real @ ( log @ B2 @ X4 ) )
% 0.23/0.51 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 0.23/0.51 = ( ( ord_less_real @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ X4 )
% 0.23/0.51 & ( ord_less_eq_real @ X4 @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ceiling_log_eq_powr_iff
% 0.23/0.51 thf(fact_6253_cos__zero__iff,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ( cos_real @ X4 )
% 0.23/0.51 = zero_zero_real )
% 0.23/0.51 = ( ? [N2: nat] :
% 0.23/0.51 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.23/0.51 & ( X4
% 0.23/0.51 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.51 | ? [N2: nat] :
% 0.23/0.51 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.23/0.51 & ( X4
% 0.23/0.51 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_zero_iff
% 0.23/0.51 thf(fact_6254_cot__less__zero,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 0.23/0.51 => ( ord_less_real @ ( cot_real @ X4 ) @ zero_zero_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cot_less_zero
% 0.23/0.51 thf(fact_6255_powr__one__eq__one,axiom,
% 0.23/0.51 ! [A: real] :
% 0.23/0.51 ( ( powr_real @ one_one_real @ A )
% 0.23/0.51 = one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % powr_one_eq_one
% 0.23/0.51 thf(fact_6256_cos__zero,axiom,
% 0.23/0.51 ( ( cos_complex @ zero_zero_complex )
% 0.23/0.51 = one_one_complex ) ).
% 0.23/0.51
% 0.23/0.51 % cos_zero
% 0.23/0.51 thf(fact_6257_cos__zero,axiom,
% 0.23/0.51 ( ( cos_real @ zero_zero_real )
% 0.23/0.51 = one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % cos_zero
% 0.23/0.51 thf(fact_6258_powr__zero__eq__one,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ( X4 = zero_zero_real )
% 0.23/0.51 => ( ( powr_real @ X4 @ zero_zero_real )
% 0.23/0.51 = zero_zero_real ) )
% 0.23/0.51 & ( ( X4 != zero_zero_real )
% 0.23/0.51 => ( ( powr_real @ X4 @ zero_zero_real )
% 0.23/0.51 = one_one_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_zero_eq_one
% 0.23/0.51 thf(fact_6259_powr__gt__zero,axiom,
% 0.23/0.51 ! [X4: real,A: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X4 @ A ) )
% 0.23/0.51 = ( X4 != zero_zero_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_gt_zero
% 0.23/0.51 thf(fact_6260_powr__nonneg__iff,axiom,
% 0.23/0.51 ! [A: real,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( powr_real @ A @ X4 ) @ zero_zero_real )
% 0.23/0.51 = ( A = zero_zero_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_nonneg_iff
% 0.23/0.51 thf(fact_6261_powr__less__cancel__iff,axiom,
% 0.23/0.51 ! [X4: real,A: real,B2: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) )
% 0.23/0.51 = ( ord_less_real @ A @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_less_cancel_iff
% 0.23/0.51 thf(fact_6262_sin__pi__minus,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( sin_real @ ( minus_minus_real @ pi @ X4 ) )
% 0.23/0.51 = ( sin_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_pi_minus
% 0.23/0.51 thf(fact_6263_powr__eq__one__iff,axiom,
% 0.23/0.51 ! [A: real,X4: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ A )
% 0.23/0.51 => ( ( ( powr_real @ A @ X4 )
% 0.23/0.51 = one_one_real )
% 0.23/0.51 = ( X4 = zero_zero_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_eq_one_iff
% 0.23/0.51 thf(fact_6264_powr__one,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( powr_real @ X4 @ one_one_real )
% 0.23/0.51 = X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_one
% 0.23/0.51 thf(fact_6265_powr__one__gt__zero__iff,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ( powr_real @ X4 @ one_one_real )
% 0.23/0.51 = X4 )
% 0.23/0.51 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_one_gt_zero_iff
% 0.23/0.51 thf(fact_6266_powr__le__cancel__iff,axiom,
% 0.23/0.51 ! [X4: real,A: real,B2: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) )
% 0.23/0.51 = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_le_cancel_iff
% 0.23/0.51 thf(fact_6267_numeral__powr__numeral__real,axiom,
% 0.23/0.51 ! [M: num,N: num] :
% 0.23/0.51 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.23/0.51 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % numeral_powr_numeral_real
% 0.23/0.51 thf(fact_6268_cos__pi,axiom,
% 0.23/0.51 ( ( cos_real @ pi )
% 0.23/0.51 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_pi
% 0.23/0.51 thf(fact_6269_cos__periodic__pi2,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( cos_real @ ( plus_plus_real @ pi @ X4 ) )
% 0.23/0.51 = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_periodic_pi2
% 0.23/0.51 thf(fact_6270_cos__periodic__pi,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( cos_real @ ( plus_plus_real @ X4 @ pi ) )
% 0.23/0.51 = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_periodic_pi
% 0.23/0.51 thf(fact_6271_sin__periodic__pi2,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( sin_real @ ( plus_plus_real @ pi @ X4 ) )
% 0.23/0.51 = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_periodic_pi2
% 0.23/0.51 thf(fact_6272_sin__periodic__pi,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( sin_real @ ( plus_plus_real @ X4 @ pi ) )
% 0.23/0.51 = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_periodic_pi
% 0.23/0.51 thf(fact_6273_cos__pi__minus,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( cos_real @ ( minus_minus_real @ pi @ X4 ) )
% 0.23/0.51 = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_pi_minus
% 0.23/0.51 thf(fact_6274_cos__minus__pi,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( cos_real @ ( minus_minus_real @ X4 @ pi ) )
% 0.23/0.51 = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_minus_pi
% 0.23/0.51 thf(fact_6275_sin__minus__pi,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( sin_real @ ( minus_minus_real @ X4 @ pi ) )
% 0.23/0.51 = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_minus_pi
% 0.23/0.51 thf(fact_6276_sin__cos__squared__add3,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ X4 ) ) @ ( times_times_complex @ ( sin_complex @ X4 ) @ ( sin_complex @ X4 ) ) )
% 0.23/0.51 = one_one_complex ) ).
% 0.23/0.51
% 0.23/0.51 % sin_cos_squared_add3
% 0.23/0.51 thf(fact_6277_sin__cos__squared__add3,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ X4 ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ X4 ) ) )
% 0.23/0.51 = one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_cos_squared_add3
% 0.23/0.51 thf(fact_6278_log__powr__cancel,axiom,
% 0.23/0.51 ! [A: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ( A != one_one_real )
% 0.23/0.51 => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
% 0.23/0.51 = Y3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % log_powr_cancel
% 0.23/0.51 thf(fact_6279_powr__log__cancel,axiom,
% 0.23/0.51 ! [A: real,X4: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ( A != one_one_real )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( powr_real @ A @ ( log @ A @ X4 ) )
% 0.23/0.51 = X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_log_cancel
% 0.23/0.51 thf(fact_6280_sin__npi2,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.23/0.51 = zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_npi2
% 0.23/0.51 thf(fact_6281_sin__npi,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.23/0.51 = zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_npi
% 0.23/0.51 thf(fact_6282_sin__npi__int,axiom,
% 0.23/0.51 ! [N: int] :
% 0.23/0.51 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.23/0.51 = zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_npi_int
% 0.23/0.51 thf(fact_6283_cot__npi,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.23/0.51 = zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % cot_npi
% 0.23/0.51 thf(fact_6284_powr__numeral,axiom,
% 0.23/0.51 ! [X4: real,N: num] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( powr_real @ X4 @ ( numeral_numeral_real @ N ) )
% 0.23/0.51 = ( power_power_real @ X4 @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_numeral
% 0.23/0.51 thf(fact_6285_cos__pi__half,axiom,
% 0.23/0.51 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 = zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % cos_pi_half
% 0.23/0.51 thf(fact_6286_sin__two__pi,axiom,
% 0.23/0.51 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.23/0.51 = zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_two_pi
% 0.23/0.51 thf(fact_6287_sin__pi__half,axiom,
% 0.23/0.51 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 = one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_pi_half
% 0.23/0.51 thf(fact_6288_cos__two__pi,axiom,
% 0.23/0.51 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.23/0.51 = one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % cos_two_pi
% 0.23/0.51 thf(fact_6289_cos__periodic,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( cos_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.23/0.51 = ( cos_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_periodic
% 0.23/0.51 thf(fact_6290_sin__periodic,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( sin_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.23/0.51 = ( sin_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_periodic
% 0.23/0.51 thf(fact_6291_cos__2pi__minus,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X4 ) )
% 0.23/0.51 = ( cos_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_2pi_minus
% 0.23/0.51 thf(fact_6292_cos__npi,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.23/0.51 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_npi
% 0.23/0.51 thf(fact_6293_cos__npi2,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.23/0.51 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_npi2
% 0.23/0.51 thf(fact_6294_cot__periodic,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( cot_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.23/0.51 = ( cot_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % cot_periodic
% 0.23/0.51 thf(fact_6295_sin__cos__squared__add2,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.51 = one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_cos_squared_add2
% 0.23/0.51 thf(fact_6296_sin__cos__squared__add2,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.51 = one_one_complex ) ).
% 0.23/0.51
% 0.23/0.51 % sin_cos_squared_add2
% 0.23/0.51 thf(fact_6297_sin__cos__squared__add,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.51 = one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_cos_squared_add
% 0.23/0.51 thf(fact_6298_sin__cos__squared__add,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.51 = one_one_complex ) ).
% 0.23/0.51
% 0.23/0.51 % sin_cos_squared_add
% 0.23/0.51 thf(fact_6299_sin__2npi,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.23/0.51 = zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_2npi
% 0.23/0.51 thf(fact_6300_cos__2npi,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.23/0.51 = one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % cos_2npi
% 0.23/0.51 thf(fact_6301_sin__2pi__minus,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X4 ) )
% 0.23/0.51 = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_2pi_minus
% 0.23/0.51 thf(fact_6302_sin__int__2pin,axiom,
% 0.23/0.51 ! [N: int] :
% 0.23/0.51 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.23/0.51 = zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_int_2pin
% 0.23/0.51 thf(fact_6303_cos__int__2pin,axiom,
% 0.23/0.51 ! [N: int] :
% 0.23/0.51 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.23/0.51 = one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % cos_int_2pin
% 0.23/0.51 thf(fact_6304_cos__3over2__pi,axiom,
% 0.23/0.51 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.23/0.51 = zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % cos_3over2_pi
% 0.23/0.51 thf(fact_6305_square__powr__half,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( powr_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 = ( abs_abs_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % square_powr_half
% 0.23/0.51 thf(fact_6306_sin__3over2__pi,axiom,
% 0.23/0.51 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.23/0.51 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_3over2_pi
% 0.23/0.51 thf(fact_6307_cos__npi__int,axiom,
% 0.23/0.51 ! [N: int] :
% 0.23/0.51 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.23/0.51 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.23/0.51 = one_one_real ) )
% 0.23/0.51 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.23/0.51 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.23/0.51 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_npi_int
% 0.23/0.51 thf(fact_6308_sin__add,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( sin_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.51 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_add
% 0.23/0.51 thf(fact_6309_sin__diff,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( sin_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 0.23/0.51 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_diff
% 0.23/0.51 thf(fact_6310_cot__def,axiom,
% 0.23/0.51 ( cot_complex
% 0.23/0.51 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X ) @ ( sin_complex @ X ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cot_def
% 0.23/0.51 thf(fact_6311_cot__def,axiom,
% 0.23/0.51 ( cot_real
% 0.23/0.51 = ( ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cot_def
% 0.23/0.51 thf(fact_6312_polar__Ex,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ? [R3: real,A5: real] :
% 0.23/0.51 ( ( X4
% 0.23/0.51 = ( times_times_real @ R3 @ ( cos_real @ A5 ) ) )
% 0.23/0.51 & ( Y3
% 0.23/0.51 = ( times_times_real @ R3 @ ( sin_real @ A5 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % polar_Ex
% 0.23/0.51 thf(fact_6313_cos__one__sin__zero,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( ( cos_complex @ X4 )
% 0.23/0.51 = one_one_complex )
% 0.23/0.51 => ( ( sin_complex @ X4 )
% 0.23/0.51 = zero_zero_complex ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_one_sin_zero
% 0.23/0.51 thf(fact_6314_cos__one__sin__zero,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ( cos_real @ X4 )
% 0.23/0.51 = one_one_real )
% 0.23/0.51 => ( ( sin_real @ X4 )
% 0.23/0.51 = zero_zero_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_one_sin_zero
% 0.23/0.51 thf(fact_6315_powr__powr,axiom,
% 0.23/0.51 ! [X4: real,A: real,B2: real] :
% 0.23/0.51 ( ( powr_real @ ( powr_real @ X4 @ A ) @ B2 )
% 0.23/0.51 = ( powr_real @ X4 @ ( times_times_real @ A @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_powr
% 0.23/0.51 thf(fact_6316_cos__add,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( cos_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.51 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_add
% 0.23/0.51 thf(fact_6317_cos__diff,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( cos_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 0.23/0.51 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_diff
% 0.23/0.51 thf(fact_6318_sin__zero__norm__cos__one,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ( sin_real @ X4 )
% 0.23/0.51 = zero_zero_real )
% 0.23/0.51 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X4 ) )
% 0.23/0.51 = one_one_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_zero_norm_cos_one
% 0.23/0.51 thf(fact_6319_sin__zero__norm__cos__one,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( ( sin_complex @ X4 )
% 0.23/0.51 = zero_zero_complex )
% 0.23/0.51 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X4 ) )
% 0.23/0.51 = one_one_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_zero_norm_cos_one
% 0.23/0.51 thf(fact_6320_sin__zero__abs__cos__one,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ( sin_real @ X4 )
% 0.23/0.51 = zero_zero_real )
% 0.23/0.51 => ( ( abs_abs_real @ ( cos_real @ X4 ) )
% 0.23/0.51 = one_one_real ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_zero_abs_cos_one
% 0.23/0.51 thf(fact_6321_sin__double,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.51 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X4 ) ) @ ( cos_complex @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_double
% 0.23/0.51 thf(fact_6322_sin__double,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.51 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X4 ) ) @ ( cos_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_double
% 0.23/0.51 thf(fact_6323_sincos__principal__value,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ? [Y4: real] :
% 0.23/0.51 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 0.23/0.51 & ( ord_less_eq_real @ Y4 @ pi )
% 0.23/0.51 & ( ( sin_real @ Y4 )
% 0.23/0.51 = ( sin_real @ X4 ) )
% 0.23/0.51 & ( ( cos_real @ Y4 )
% 0.23/0.51 = ( cos_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sincos_principal_value
% 0.23/0.51 thf(fact_6324_powr__less__mono2__neg,axiom,
% 0.23/0.51 ! [A: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ A @ zero_zero_real )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.51 => ( ord_less_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X4 @ A ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_less_mono2_neg
% 0.23/0.51 thf(fact_6325_powr__non__neg,axiom,
% 0.23/0.51 ! [A: real,X4: real] :
% 0.23/0.51 ~ ( ord_less_real @ ( powr_real @ A @ X4 ) @ zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % powr_non_neg
% 0.23/0.51 thf(fact_6326_powr__mono2,axiom,
% 0.23/0.51 ! [A: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.51 => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_mono2
% 0.23/0.51 thf(fact_6327_powr__ge__pzero,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X4 @ Y3 ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_ge_pzero
% 0.23/0.51 thf(fact_6328_powr__less__mono,axiom,
% 0.23/0.51 ! [A: real,B2: real,X4: real] :
% 0.23/0.51 ( ( ord_less_real @ A @ B2 )
% 0.23/0.51 => ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.51 => ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_less_mono
% 0.23/0.51 thf(fact_6329_powr__less__cancel,axiom,
% 0.23/0.51 ! [X4: real,A: real,B2: real] :
% 0.23/0.51 ( ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) )
% 0.23/0.51 => ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.51 => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_less_cancel
% 0.23/0.51 thf(fact_6330_powr__mono,axiom,
% 0.23/0.51 ! [A: real,B2: real,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.51 => ( ( ord_less_eq_real @ one_one_real @ X4 )
% 0.23/0.51 => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_mono
% 0.23/0.51 thf(fact_6331_sin__x__le__x,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ord_less_eq_real @ ( sin_real @ X4 ) @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_x_le_x
% 0.23/0.51 thf(fact_6332_sin__le__one,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_real @ ( sin_real @ X4 ) @ one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_le_one
% 0.23/0.51 thf(fact_6333_cos__le__one,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_real @ ( cos_real @ X4 ) @ one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % cos_le_one
% 0.23/0.51 thf(fact_6334_abs__sin__x__le__abs__x,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X4 ) ) @ ( abs_abs_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % abs_sin_x_le_abs_x
% 0.23/0.51 thf(fact_6335_sin__cos__le1,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % sin_cos_le1
% 0.23/0.51 thf(fact_6336_frac__ge__0,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % frac_ge_0
% 0.23/0.51 thf(fact_6337_frac__ge__0,axiom,
% 0.23/0.51 ! [X4: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % frac_ge_0
% 0.23/0.51 thf(fact_6338_sin__squared__eq,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.51 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_squared_eq
% 0.23/0.51 thf(fact_6339_sin__squared__eq,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.51 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_squared_eq
% 0.23/0.51 thf(fact_6340_cos__squared__eq,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.51 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_squared_eq
% 0.23/0.51 thf(fact_6341_cos__squared__eq,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.51 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_squared_eq
% 0.23/0.51 thf(fact_6342_frac__lt__1,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X4 ) @ one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % frac_lt_1
% 0.23/0.51 thf(fact_6343_frac__lt__1,axiom,
% 0.23/0.51 ! [X4: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X4 ) @ one_one_rat ) ).
% 0.23/0.51
% 0.23/0.51 % frac_lt_1
% 0.23/0.51 thf(fact_6344_frac__1__eq,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ one_one_real ) )
% 0.23/0.51 = ( archim2898591450579166408c_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % frac_1_eq
% 0.23/0.51 thf(fact_6345_frac__1__eq,axiom,
% 0.23/0.51 ! [X4: rat] :
% 0.23/0.51 ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) )
% 0.23/0.51 = ( archimedean_frac_rat @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % frac_1_eq
% 0.23/0.51 thf(fact_6346_powr__mono2_H,axiom,
% 0.23/0.51 ! [A: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.51 => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X4 @ A ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_mono2'
% 0.23/0.51 thf(fact_6347_powr__less__mono2,axiom,
% 0.23/0.51 ! [A: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.51 => ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_less_mono2
% 0.23/0.51 thf(fact_6348_gr__one__powr,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.51 => ( ord_less_real @ one_one_real @ ( powr_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % gr_one_powr
% 0.23/0.51 thf(fact_6349_powr__inj,axiom,
% 0.23/0.51 ! [A: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ( A != one_one_real )
% 0.23/0.51 => ( ( ( powr_real @ A @ X4 )
% 0.23/0.51 = ( powr_real @ A @ Y3 ) )
% 0.23/0.51 = ( X4 = Y3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_inj
% 0.23/0.51 thf(fact_6350_powr__le1,axiom,
% 0.23/0.51 ! [A: real,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.51 => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ one_one_real ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_le1
% 0.23/0.51 thf(fact_6351_powr__mono__both,axiom,
% 0.23/0.51 ! [A: real,B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.51 => ( ( ord_less_eq_real @ one_one_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.51 => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ B2 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_mono_both
% 0.23/0.51 thf(fact_6352_ge__one__powr__ge__zero,axiom,
% 0.23/0.51 ! [X4: real,A: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ one_one_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X4 @ A ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ge_one_powr_ge_zero
% 0.23/0.51 thf(fact_6353_powr__divide,axiom,
% 0.23/0.51 ! [X4: real,Y3: real,A: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.51 => ( ( powr_real @ ( divide_divide_real @ X4 @ Y3 ) @ A )
% 0.23/0.51 = ( divide_divide_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_divide
% 0.23/0.51 thf(fact_6354_powr__mult,axiom,
% 0.23/0.51 ! [X4: real,Y3: real,A: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.51 => ( ( powr_real @ ( times_times_real @ X4 @ Y3 ) @ A )
% 0.23/0.51 = ( times_times_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_mult
% 0.23/0.51 thf(fact_6355_divide__powr__uminus,axiom,
% 0.23/0.51 ! [A: real,B2: real,C: real] :
% 0.23/0.51 ( ( divide_divide_real @ A @ ( powr_real @ B2 @ C ) )
% 0.23/0.51 = ( times_times_real @ A @ ( powr_real @ B2 @ ( uminus_uminus_real @ C ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % divide_powr_uminus
% 0.23/0.51 thf(fact_6356_sin__gt__zero,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ pi )
% 0.23/0.51 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_gt_zero
% 0.23/0.51 thf(fact_6357_sin__x__ge__neg__x,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ ( sin_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_x_ge_neg_x
% 0.23/0.51 thf(fact_6358_sin__ge__zero,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ pi )
% 0.23/0.51 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_ge_zero
% 0.23/0.51 thf(fact_6359_log__base__powr,axiom,
% 0.23/0.51 ! [A: real,B2: real,X4: real] :
% 0.23/0.51 ( ( A != zero_zero_real )
% 0.23/0.51 => ( ( log @ ( powr_real @ A @ B2 ) @ X4 )
% 0.23/0.51 = ( divide_divide_real @ ( log @ A @ X4 ) @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % log_base_powr
% 0.23/0.51 thf(fact_6360_ln__powr,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( X4 != zero_zero_real )
% 0.23/0.51 => ( ( ln_ln_real @ ( powr_real @ X4 @ Y3 ) )
% 0.23/0.51 = ( times_times_real @ Y3 @ ( ln_ln_real @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ln_powr
% 0.23/0.51 thf(fact_6361_log__powr,axiom,
% 0.23/0.51 ! [X4: real,B2: real,Y3: real] :
% 0.23/0.51 ( ( X4 != zero_zero_real )
% 0.23/0.51 => ( ( log @ B2 @ ( powr_real @ X4 @ Y3 ) )
% 0.23/0.51 = ( times_times_real @ Y3 @ ( log @ B2 @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % log_powr
% 0.23/0.51 thf(fact_6362_sin__ge__minus__one,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_ge_minus_one
% 0.23/0.51 thf(fact_6363_cos__monotone__0__pi__le,axiom,
% 0.23/0.51 ! [Y3: real,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.51 => ( ( ord_less_eq_real @ Y3 @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ pi )
% 0.23/0.51 => ( ord_less_eq_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_monotone_0_pi_le
% 0.23/0.51 thf(fact_6364_cos__mono__le__eq,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ pi )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.51 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) )
% 0.23/0.51 = ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_mono_le_eq
% 0.23/0.51 thf(fact_6365_cos__inj__pi,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ pi )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.51 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.23/0.51 => ( ( ( cos_real @ X4 )
% 0.23/0.51 = ( cos_real @ Y3 ) )
% 0.23/0.51 => ( X4 = Y3 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_inj_pi
% 0.23/0.51 thf(fact_6366_cos__ge__minus__one,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X4 ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_ge_minus_one
% 0.23/0.51 thf(fact_6367_abs__sin__le__one,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X4 ) ) @ one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % abs_sin_le_one
% 0.23/0.51 thf(fact_6368_abs__cos__le__one,axiom,
% 0.23/0.51 ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X4 ) ) @ one_one_real ) ).
% 0.23/0.51
% 0.23/0.51 % abs_cos_le_one
% 0.23/0.51 thf(fact_6369_cos__diff__cos,axiom,
% 0.23/0.51 ! [W: complex,Z: complex] :
% 0.23/0.51 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_diff_cos
% 0.23/0.51 thf(fact_6370_cos__diff__cos,axiom,
% 0.23/0.51 ! [W: real,Z: real] :
% 0.23/0.51 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_diff_cos
% 0.23/0.51 thf(fact_6371_sin__diff__sin,axiom,
% 0.23/0.51 ! [W: complex,Z: complex] :
% 0.23/0.51 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_diff_sin
% 0.23/0.51 thf(fact_6372_sin__diff__sin,axiom,
% 0.23/0.51 ! [W: real,Z: real] :
% 0.23/0.51 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_diff_sin
% 0.23/0.51 thf(fact_6373_sin__plus__sin,axiom,
% 0.23/0.51 ! [W: complex,Z: complex] :
% 0.23/0.51 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_plus_sin
% 0.23/0.51 thf(fact_6374_sin__plus__sin,axiom,
% 0.23/0.51 ! [W: real,Z: real] :
% 0.23/0.51 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_plus_sin
% 0.23/0.51 thf(fact_6375_cos__times__sin,axiom,
% 0.23/0.51 ! [W: complex,Z: complex] :
% 0.23/0.51 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 0.23/0.51 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_times_sin
% 0.23/0.51 thf(fact_6376_cos__times__sin,axiom,
% 0.23/0.51 ! [W: real,Z: real] :
% 0.23/0.51 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_times_sin
% 0.23/0.51 thf(fact_6377_sin__times__cos,axiom,
% 0.23/0.51 ! [W: complex,Z: complex] :
% 0.23/0.51 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 0.23/0.51 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_times_cos
% 0.23/0.51 thf(fact_6378_sin__times__cos,axiom,
% 0.23/0.51 ! [W: real,Z: real] :
% 0.23/0.51 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_times_cos
% 0.23/0.51 thf(fact_6379_sin__times__sin,axiom,
% 0.23/0.51 ! [W: complex,Z: complex] :
% 0.23/0.51 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.23/0.51 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_times_sin
% 0.23/0.51 thf(fact_6380_sin__times__sin,axiom,
% 0.23/0.51 ! [W: real,Z: real] :
% 0.23/0.51 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_times_sin
% 0.23/0.51 thf(fact_6381_cos__double,axiom,
% 0.23/0.51 ! [X4: complex] :
% 0.23/0.51 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.51 = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_double
% 0.23/0.51 thf(fact_6382_cos__double,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.51 = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_double
% 0.23/0.51 thf(fact_6383_powr__add,axiom,
% 0.23/0.51 ! [X4: real,A: real,B2: real] :
% 0.23/0.51 ( ( powr_real @ X4 @ ( plus_plus_real @ A @ B2 ) )
% 0.23/0.51 = ( times_times_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_add
% 0.23/0.51 thf(fact_6384_powr__diff,axiom,
% 0.23/0.51 ! [W: real,Z1: real,Z22: real] :
% 0.23/0.51 ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
% 0.23/0.51 = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_diff
% 0.23/0.51 thf(fact_6385_cos__double__sin,axiom,
% 0.23/0.51 ! [W: complex] :
% 0.23/0.51 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.23/0.51 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_double_sin
% 0.23/0.51 thf(fact_6386_cos__double__sin,axiom,
% 0.23/0.51 ! [W: real] :
% 0.23/0.51 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.23/0.51 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_double_sin
% 0.23/0.51 thf(fact_6387_powr__realpow,axiom,
% 0.23/0.51 ! [X4: real,N: nat] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( powr_real @ X4 @ ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.51 = ( power_power_real @ X4 @ N ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_realpow
% 0.23/0.51 thf(fact_6388_cos__two__neq__zero,axiom,
% 0.23/0.51 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.23/0.51 != zero_zero_real ) ).
% 0.23/0.51
% 0.23/0.51 % cos_two_neq_zero
% 0.23/0.51 thf(fact_6389_less__log__iff,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ Y3 @ ( log @ B2 @ X4 ) )
% 0.23/0.51 = ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % less_log_iff
% 0.23/0.51 thf(fact_6390_log__less__iff,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ ( log @ B2 @ X4 ) @ Y3 )
% 0.23/0.51 = ( ord_less_real @ X4 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % log_less_iff
% 0.23/0.51 thf(fact_6391_less__powr__iff,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ ( powr_real @ B2 @ Y3 ) )
% 0.23/0.51 = ( ord_less_real @ ( log @ B2 @ X4 ) @ Y3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % less_powr_iff
% 0.23/0.51 thf(fact_6392_powr__less__iff,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X4 )
% 0.23/0.51 = ( ord_less_real @ Y3 @ ( log @ B2 @ X4 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_less_iff
% 0.23/0.51 thf(fact_6393_cos__monotone__0__pi,axiom,
% 0.23/0.51 ! [Y3: real,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.51 => ( ( ord_less_real @ Y3 @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ pi )
% 0.23/0.51 => ( ord_less_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_monotone_0_pi
% 0.23/0.51 thf(fact_6394_cos__mono__less__eq,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ pi )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.51 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.23/0.51 => ( ( ord_less_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) )
% 0.23/0.51 = ( ord_less_real @ Y3 @ X4 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_mono_less_eq
% 0.23/0.51 thf(fact_6395_sin__eq__0__pi,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ pi )
% 0.23/0.51 => ( ( ( sin_real @ X4 )
% 0.23/0.51 = zero_zero_real )
% 0.23/0.51 => ( X4 = zero_zero_real ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_eq_0_pi
% 0.23/0.51 thf(fact_6396_sin__zero__pi__iff,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ pi )
% 0.23/0.51 => ( ( ( sin_real @ X4 )
% 0.23/0.51 = zero_zero_real )
% 0.23/0.51 = ( X4 = zero_zero_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_zero_pi_iff
% 0.23/0.51 thf(fact_6397_cos__monotone__minus__pi__0_H,axiom,
% 0.23/0.51 ! [Y3: real,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 0.23/0.51 => ( ( ord_less_eq_real @ Y3 @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.23/0.51 => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X4 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_monotone_minus_pi_0'
% 0.23/0.51 thf(fact_6398_sin__zero__iff__int2,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ( sin_real @ X4 )
% 0.23/0.51 = zero_zero_real )
% 0.23/0.51 = ( ? [I2: int] :
% 0.23/0.51 ( X4
% 0.23/0.51 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_zero_iff_int2
% 0.23/0.51 thf(fact_6399_frac__def,axiom,
% 0.23/0.51 ( archim2898591450579166408c_real
% 0.23/0.51 = ( ^ [X: real] : ( minus_minus_real @ X @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % frac_def
% 0.23/0.51 thf(fact_6400_frac__def,axiom,
% 0.23/0.51 ( archimedean_frac_rat
% 0.23/0.51 = ( ^ [X: rat] : ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % frac_def
% 0.23/0.51 thf(fact_6401_sincos__total__pi,axiom,
% 0.23/0.51 ! [Y3: real,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.51 => ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.51 = one_one_real )
% 0.23/0.51 => ? [T2: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ T2 )
% 0.23/0.51 & ( ord_less_eq_real @ T2 @ pi )
% 0.23/0.51 & ( X4
% 0.23/0.51 = ( cos_real @ T2 ) )
% 0.23/0.51 & ( Y3
% 0.23/0.51 = ( sin_real @ T2 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sincos_total_pi
% 0.23/0.51 thf(fact_6402_sin__cos__sqrt,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X4 ) )
% 0.23/0.51 => ( ( sin_real @ X4 )
% 0.23/0.51 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_cos_sqrt
% 0.23/0.51 thf(fact_6403_sin__expansion__lemma,axiom,
% 0.23/0.51 ! [X4: real,M: nat] :
% 0.23/0.51 ( ( sin_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.23/0.51 = ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_expansion_lemma
% 0.23/0.51 thf(fact_6404_powr__minus__divide,axiom,
% 0.23/0.51 ! [X4: real,A: real] :
% 0.23/0.51 ( ( powr_real @ X4 @ ( uminus_uminus_real @ A ) )
% 0.23/0.51 = ( divide_divide_real @ one_one_real @ ( powr_real @ X4 @ A ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_minus_divide
% 0.23/0.51 thf(fact_6405_cos__expansion__lemma,axiom,
% 0.23/0.51 ! [X4: real,M: nat] :
% 0.23/0.51 ( ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.23/0.51 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_expansion_lemma
% 0.23/0.51 thf(fact_6406_powr__neg__one,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( powr_real @ X4 @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.51 = ( divide_divide_real @ one_one_real @ X4 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_neg_one
% 0.23/0.51 thf(fact_6407_sin__gt__zero__02,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.23/0.51 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_gt_zero_02
% 0.23/0.51 thf(fact_6408_powr__mult__base,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( times_times_real @ X4 @ ( powr_real @ X4 @ Y3 ) )
% 0.23/0.51 = ( powr_real @ X4 @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_mult_base
% 0.23/0.51 thf(fact_6409_powr__le__iff,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X4 )
% 0.23/0.51 = ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X4 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_le_iff
% 0.23/0.51 thf(fact_6410_le__powr__iff,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ ( powr_real @ B2 @ Y3 ) )
% 0.23/0.51 = ( ord_less_eq_real @ ( log @ B2 @ X4 ) @ Y3 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_powr_iff
% 0.23/0.51 thf(fact_6411_log__le__iff,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( log @ B2 @ X4 ) @ Y3 )
% 0.23/0.51 = ( ord_less_eq_real @ X4 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % log_le_iff
% 0.23/0.51 thf(fact_6412_le__log__iff,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X4 ) )
% 0.23/0.51 = ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % le_log_iff
% 0.23/0.51 thf(fact_6413_cos__two__less__zero,axiom,
% 0.23/0.51 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.23/0.51
% 0.23/0.51 % cos_two_less_zero
% 0.23/0.51 thf(fact_6414_cos__two__le__zero,axiom,
% 0.23/0.51 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.23/0.51
% 0.23/0.51 % cos_two_le_zero
% 0.23/0.51 thf(fact_6415_cos__is__zero,axiom,
% 0.23/0.51 ? [X3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.23/0.51 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.23/0.51 & ( ( cos_real @ X3 )
% 0.23/0.51 = zero_zero_real )
% 0.23/0.51 & ! [Y5: real] :
% 0.23/0.51 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.23/0.51 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.23/0.51 & ( ( cos_real @ Y5 )
% 0.23/0.51 = zero_zero_real ) )
% 0.23/0.51 => ( Y5 = X3 ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_is_zero
% 0.23/0.51 thf(fact_6416_cos__monotone__minus__pi__0,axiom,
% 0.23/0.51 ! [Y3: real,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 0.23/0.51 => ( ( ord_less_real @ Y3 @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.23/0.51 => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X4 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_monotone_minus_pi_0
% 0.23/0.51 thf(fact_6417_cos__total,axiom,
% 0.23/0.51 ! [Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.51 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.51 => ? [X3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.23/0.51 & ( ord_less_eq_real @ X3 @ pi )
% 0.23/0.51 & ( ( cos_real @ X3 )
% 0.23/0.51 = Y3 )
% 0.23/0.51 & ! [Y5: real] :
% 0.23/0.51 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.23/0.51 & ( ord_less_eq_real @ Y5 @ pi )
% 0.23/0.51 & ( ( cos_real @ Y5 )
% 0.23/0.51 = Y3 ) )
% 0.23/0.51 => ( Y5 = X3 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_total
% 0.23/0.51 thf(fact_6418_frac__eq,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ( archim2898591450579166408c_real @ X4 )
% 0.23/0.51 = X4 )
% 0.23/0.51 = ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 & ( ord_less_real @ X4 @ one_one_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % frac_eq
% 0.23/0.51 thf(fact_6419_frac__eq,axiom,
% 0.23/0.51 ! [X4: rat] :
% 0.23/0.51 ( ( ( archimedean_frac_rat @ X4 )
% 0.23/0.51 = X4 )
% 0.23/0.51 = ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.23/0.51 & ( ord_less_rat @ X4 @ one_one_rat ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % frac_eq
% 0.23/0.51 thf(fact_6420_frac__add,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 0.23/0.51 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.51 = ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) ) )
% 0.23/0.51 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 0.23/0.51 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.51 = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % frac_add
% 0.23/0.51 thf(fact_6421_frac__add,axiom,
% 0.23/0.51 ! [X4: rat,Y3: rat] :
% 0.23/0.51 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 0.23/0.51 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
% 0.23/0.51 = ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) ) )
% 0.23/0.51 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 0.23/0.51 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
% 0.23/0.51 = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % frac_add
% 0.23/0.51 thf(fact_6422_sincos__total__pi__half,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.51 => ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.51 = one_one_real )
% 0.23/0.51 => ? [T2: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ T2 )
% 0.23/0.51 & ( ord_less_eq_real @ T2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 & ( X4
% 0.23/0.51 = ( cos_real @ T2 ) )
% 0.23/0.51 & ( Y3
% 0.23/0.51 = ( sin_real @ T2 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sincos_total_pi_half
% 0.23/0.51 thf(fact_6423_sincos__total__2pi__le,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.51 = one_one_real )
% 0.23/0.51 => ? [T2: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ T2 )
% 0.23/0.51 & ( ord_less_eq_real @ T2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.23/0.51 & ( X4
% 0.23/0.51 = ( cos_real @ T2 ) )
% 0.23/0.51 & ( Y3
% 0.23/0.51 = ( sin_real @ T2 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sincos_total_2pi_le
% 0.23/0.51 thf(fact_6424_sincos__total__2pi,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.51 = one_one_real )
% 0.23/0.51 => ~ ! [T2: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ zero_zero_real @ T2 )
% 0.23/0.51 => ( ( ord_less_real @ T2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.23/0.51 => ( ( X4
% 0.23/0.51 = ( cos_real @ T2 ) )
% 0.23/0.51 => ( Y3
% 0.23/0.51 != ( sin_real @ T2 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sincos_total_2pi
% 0.23/0.51 thf(fact_6425_ln__powr__bound,axiom,
% 0.23/0.51 ! [X4: real,A: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ one_one_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( divide_divide_real @ ( powr_real @ X4 @ A ) @ A ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ln_powr_bound
% 0.23/0.51 thf(fact_6426_ln__powr__bound2,axiom,
% 0.23/0.51 ! [X4: real,A: real] :
% 0.23/0.51 ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.51 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X4 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % ln_powr_bound2
% 0.23/0.51 thf(fact_6427_log__add__eq__powr,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.23/0.51 => ( ( B2 != one_one_real )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( plus_plus_real @ ( log @ B2 @ X4 ) @ Y3 )
% 0.23/0.51 = ( log @ B2 @ ( times_times_real @ X4 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % log_add_eq_powr
% 0.23/0.51 thf(fact_6428_add__log__eq__powr,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.23/0.51 => ( ( B2 != one_one_real )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( plus_plus_real @ Y3 @ ( log @ B2 @ X4 ) )
% 0.23/0.51 = ( log @ B2 @ ( times_times_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % add_log_eq_powr
% 0.23/0.51 thf(fact_6429_minus__log__eq__powr,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.23/0.51 => ( ( B2 != one_one_real )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( minus_minus_real @ Y3 @ ( log @ B2 @ X4 ) )
% 0.23/0.51 = ( log @ B2 @ ( divide_divide_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % minus_log_eq_powr
% 0.23/0.51 thf(fact_6430_sin__pi__divide__n__ge__0,axiom,
% 0.23/0.51 ! [N: nat] :
% 0.23/0.51 ( ( N != zero_zero_nat )
% 0.23/0.51 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_pi_divide_n_ge_0
% 0.23/0.51 thf(fact_6431_sin__45,axiom,
% 0.23/0.51 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.23/0.51 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_45
% 0.23/0.51 thf(fact_6432_cos__45,axiom,
% 0.23/0.51 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.23/0.51 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_45
% 0.23/0.51 thf(fact_6433_powr__def,axiom,
% 0.23/0.51 ( powr_real
% 0.23/0.51 = ( ^ [X: real,A4: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A4 @ ( ln_ln_real @ X ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % powr_def
% 0.23/0.51 thf(fact_6434_cos__plus__cos,axiom,
% 0.23/0.51 ! [W: complex,Z: complex] :
% 0.23/0.51 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_plus_cos
% 0.23/0.51 thf(fact_6435_cos__plus__cos,axiom,
% 0.23/0.51 ! [W: real,Z: real] :
% 0.23/0.51 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_plus_cos
% 0.23/0.51 thf(fact_6436_cos__times__cos,axiom,
% 0.23/0.51 ! [W: complex,Z: complex] :
% 0.23/0.51 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.23/0.51 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_times_cos
% 0.23/0.51 thf(fact_6437_cos__times__cos,axiom,
% 0.23/0.51 ! [W: real,Z: real] :
% 0.23/0.51 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.23/0.51 = ( 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 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_times_cos
% 0.23/0.51 thf(fact_6438_sin__gt__zero2,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_gt_zero2
% 0.23/0.51 thf(fact_6439_sin__lt__zero,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ pi @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.23/0.51 => ( ord_less_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_lt_zero
% 0.23/0.51 thf(fact_6440_cos__double__less__one,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.23/0.51 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_real ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_double_less_one
% 0.23/0.51 thf(fact_6441_sin__30,axiom,
% 0.23/0.51 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.23/0.51 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_30
% 0.23/0.51 thf(fact_6442_cos__gt__zero,axiom,
% 0.23/0.51 ! [X4: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_gt_zero
% 0.23/0.51 thf(fact_6443_sin__monotone__2pi__le,axiom,
% 0.23/0.51 ! [Y3: real,X4: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.23/0.51 => ( ( ord_less_eq_real @ Y3 @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X4 ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_monotone_2pi_le
% 0.23/0.51 thf(fact_6444_sin__mono__le__eq,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.23/0.51 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) )
% 0.23/0.51 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_mono_le_eq
% 0.23/0.51 thf(fact_6445_sin__inj__pi,axiom,
% 0.23/0.51 ! [X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.51 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.23/0.51 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.51 => ( ( ( sin_real @ X4 )
% 0.23/0.51 = ( sin_real @ Y3 ) )
% 0.23/0.51 => ( X4 = Y3 ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_inj_pi
% 0.23/0.51 thf(fact_6446_log__minus__eq__powr,axiom,
% 0.23/0.51 ! [B2: real,X4: real,Y3: real] :
% 0.23/0.51 ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.23/0.51 => ( ( B2 != one_one_real )
% 0.23/0.51 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.51 => ( ( minus_minus_real @ ( log @ B2 @ X4 ) @ Y3 )
% 0.23/0.51 = ( log @ B2 @ ( times_times_real @ X4 @ ( powr_real @ B2 @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % log_minus_eq_powr
% 0.23/0.51 thf(fact_6447_cos__60,axiom,
% 0.23/0.51 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.23/0.51 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % cos_60
% 0.23/0.51 thf(fact_6448_sin__60,axiom,
% 0.23/0.51 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.23/0.51 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.51
% 0.23/0.51 % sin_60
% 0.23/0.51 thf(fact_6449_cos__30,axiom,
% 0.23/0.51 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.23/0.52 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_30
% 0.23/0.52 thf(fact_6450_cos__one__2pi__int,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ( cos_real @ X4 )
% 0.23/0.52 = one_one_real )
% 0.23/0.52 = ( ? [X: int] :
% 0.23/0.52 ( X4
% 0.23/0.52 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_one_2pi_int
% 0.23/0.52 thf(fact_6451_cos__double__cos,axiom,
% 0.23/0.52 ! [W: complex] :
% 0.23/0.52 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.23/0.52 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_double_cos
% 0.23/0.52 thf(fact_6452_cos__double__cos,axiom,
% 0.23/0.52 ! [W: real] :
% 0.23/0.52 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.23/0.52 = ( 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 ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_double_cos
% 0.23/0.52 thf(fact_6453_cos__treble__cos,axiom,
% 0.23/0.52 ! [X4: complex] :
% 0.23/0.52 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X4 ) )
% 0.23/0.52 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_treble_cos
% 0.23/0.52 thf(fact_6454_cos__treble__cos,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X4 ) )
% 0.23/0.52 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_treble_cos
% 0.23/0.52 thf(fact_6455_powr__half__sqrt,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( powr_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 = ( sqrt @ X4 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % powr_half_sqrt
% 0.23/0.52 thf(fact_6456_powr__neg__numeral,axiom,
% 0.23/0.52 ! [X4: real,N: num] :
% 0.23/0.52 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( powr_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.23/0.52 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % powr_neg_numeral
% 0.23/0.52 thf(fact_6457_sin__le__zero,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ pi @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.23/0.52 => ( ord_less_eq_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_le_zero
% 0.23/0.52 thf(fact_6458_sin__less__zero,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 0.23/0.52 => ( ord_less_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_less_zero
% 0.23/0.52 thf(fact_6459_sin__monotone__2pi,axiom,
% 0.23/0.52 ! [Y3: real,X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_real @ Y3 @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X4 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_monotone_2pi
% 0.23/0.52 thf(fact_6460_sin__mono__less__eq,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( ord_less_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) )
% 0.23/0.52 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_mono_less_eq
% 0.23/0.52 thf(fact_6461_sin__total,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ? [X3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.23/0.52 & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 & ( ( sin_real @ X3 )
% 0.23/0.52 = Y3 )
% 0.23/0.52 & ! [Y5: real] :
% 0.23/0.52 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.23/0.52 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 & ( ( sin_real @ Y5 )
% 0.23/0.52 = Y3 ) )
% 0.23/0.52 => ( Y5 = X3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_total
% 0.23/0.52 thf(fact_6462_cos__gt__zero__pi,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_gt_zero_pi
% 0.23/0.52 thf(fact_6463_cos__ge__zero,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_ge_zero
% 0.23/0.52 thf(fact_6464_cos__one__2pi,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ( cos_real @ X4 )
% 0.23/0.52 = one_one_real )
% 0.23/0.52 = ( ? [X: nat] :
% 0.23/0.52 ( X4
% 0.23/0.52 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.23/0.52 | ? [X: nat] :
% 0.23/0.52 ( X4
% 0.23/0.52 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_one_2pi
% 0.23/0.52 thf(fact_6465_floor__add,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 0.23/0.52 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) )
% 0.23/0.52 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 0.23/0.52 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % floor_add
% 0.23/0.52 thf(fact_6466_floor__add,axiom,
% 0.23/0.52 ! [X4: rat,Y3: rat] :
% 0.23/0.52 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 0.23/0.52 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
% 0.23/0.52 = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) )
% 0.23/0.52 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 0.23/0.52 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
% 0.23/0.52 = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % floor_add
% 0.23/0.52 thf(fact_6467_sin__pi__divide__n__gt__0,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.52 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_pi_divide_n_gt_0
% 0.23/0.52 thf(fact_6468_sin__arctan,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( sin_real @ ( arctan @ X4 ) )
% 0.23/0.52 = ( divide_divide_real @ X4 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_arctan
% 0.23/0.52 thf(fact_6469_floor__log__eq__powr__iff,axiom,
% 0.23/0.52 ! [X4: real,B2: real,K: int] :
% 0.23/0.52 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.52 => ( ( ( archim6058952711729229775r_real @ ( log @ B2 @ X4 ) )
% 0.23/0.52 = K )
% 0.23/0.52 = ( ( ord_less_eq_real @ ( powr_real @ B2 @ ( ring_1_of_int_real @ K ) ) @ X4 )
% 0.23/0.52 & ( ord_less_real @ X4 @ ( powr_real @ B2 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % floor_log_eq_powr_iff
% 0.23/0.52 thf(fact_6470_cos__arctan,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( cos_real @ ( arctan @ X4 ) )
% 0.23/0.52 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_arctan
% 0.23/0.52 thf(fact_6471_cot__gt__zero,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cot_gt_zero
% 0.23/0.52 thf(fact_6472_sin__zero__iff__int,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ( sin_real @ X4 )
% 0.23/0.52 = zero_zero_real )
% 0.23/0.52 = ( ? [I2: int] :
% 0.23/0.52 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
% 0.23/0.52 & ( X4
% 0.23/0.52 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_zero_iff_int
% 0.23/0.52 thf(fact_6473_cos__zero__iff__int,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ( cos_real @ X4 )
% 0.23/0.52 = zero_zero_real )
% 0.23/0.52 = ( ? [I2: int] :
% 0.23/0.52 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
% 0.23/0.52 & ( X4
% 0.23/0.52 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_zero_iff_int
% 0.23/0.52 thf(fact_6474_sin__zero__lemma,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( ( sin_real @ X4 )
% 0.23/0.52 = zero_zero_real )
% 0.23/0.52 => ? [N4: nat] :
% 0.23/0.52 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 0.23/0.52 & ( X4
% 0.23/0.52 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_zero_lemma
% 0.23/0.52 thf(fact_6475_sin__zero__iff,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ( sin_real @ X4 )
% 0.23/0.52 = zero_zero_real )
% 0.23/0.52 = ( ? [N2: nat] :
% 0.23/0.52 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.23/0.52 & ( X4
% 0.23/0.52 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.52 | ? [N2: nat] :
% 0.23/0.52 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.23/0.52 & ( X4
% 0.23/0.52 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_zero_iff
% 0.23/0.52 thf(fact_6476_cos__zero__lemma,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( ( cos_real @ X4 )
% 0.23/0.52 = zero_zero_real )
% 0.23/0.52 => ? [N4: nat] :
% 0.23/0.52 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 0.23/0.52 & ( X4
% 0.23/0.52 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_zero_lemma
% 0.23/0.52 thf(fact_6477_tan__double,axiom,
% 0.23/0.52 ! [X4: complex] :
% 0.23/0.52 ( ( ( cos_complex @ X4 )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X4 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_double
% 0.23/0.52 thf(fact_6478_tan__double,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ( cos_real @ X4 )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.52 = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X4 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_double
% 0.23/0.52 thf(fact_6479_sin__tan,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( sin_real @ X4 )
% 0.23/0.52 = ( divide_divide_real @ ( tan_real @ X4 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_tan
% 0.23/0.52 thf(fact_6480_cos__tan,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( cos_real @ X4 )
% 0.23/0.52 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_tan
% 0.23/0.52 thf(fact_6481_arcosh__def,axiom,
% 0.23/0.52 ( arcosh_real
% 0.23/0.52 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcosh_def
% 0.23/0.52 thf(fact_6482_complex__unimodular__polar,axiom,
% 0.23/0.52 ! [Z: complex] :
% 0.23/0.52 ( ( ( real_V1022390504157884413omplex @ Z )
% 0.23/0.52 = one_one_real )
% 0.23/0.52 => ~ ! [T2: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ T2 )
% 0.23/0.52 => ( ( ord_less_real @ T2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.23/0.52 => ( Z
% 0.23/0.52 != ( complex2 @ ( cos_real @ T2 ) @ ( sin_real @ T2 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % complex_unimodular_polar
% 0.23/0.52 thf(fact_6483_cos__arcsin,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.52 => ( ( cos_real @ ( arcsin @ X4 ) )
% 0.23/0.52 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_arcsin
% 0.23/0.52 thf(fact_6484_of__real__1,axiom,
% 0.23/0.52 ( ( real_V1803761363581548252l_real @ one_one_real )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_1
% 0.23/0.52 thf(fact_6485_of__real__1,axiom,
% 0.23/0.52 ( ( real_V4546457046886955230omplex @ one_one_real )
% 0.23/0.52 = one_one_complex ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_1
% 0.23/0.52 thf(fact_6486_of__real__eq__1__iff,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ( real_V1803761363581548252l_real @ X4 )
% 0.23/0.52 = one_one_real )
% 0.23/0.52 = ( X4 = one_one_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_eq_1_iff
% 0.23/0.52 thf(fact_6487_of__real__eq__1__iff,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ( real_V4546457046886955230omplex @ X4 )
% 0.23/0.52 = one_one_complex )
% 0.23/0.52 = ( X4 = one_one_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_eq_1_iff
% 0.23/0.52 thf(fact_6488_of__real__mult,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( real_V1803761363581548252l_real @ ( times_times_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( times_times_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_mult
% 0.23/0.52 thf(fact_6489_of__real__mult,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( real_V4546457046886955230omplex @ ( times_times_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( times_times_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_mult
% 0.23/0.52 thf(fact_6490_of__real__numeral,axiom,
% 0.23/0.52 ! [W: num] :
% 0.23/0.52 ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
% 0.23/0.52 = ( numeral_numeral_real @ W ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_numeral
% 0.23/0.52 thf(fact_6491_of__real__numeral,axiom,
% 0.23/0.52 ! [W: num] :
% 0.23/0.52 ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
% 0.23/0.52 = ( numera6690914467698888265omplex @ W ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_numeral
% 0.23/0.52 thf(fact_6492_of__real__divide,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_divide
% 0.23/0.52 thf(fact_6493_of__real__divide,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_divide
% 0.23/0.52 thf(fact_6494_of__real__power,axiom,
% 0.23/0.52 ! [X4: real,N: nat] :
% 0.23/0.52 ( ( real_V1803761363581548252l_real @ ( power_power_real @ X4 @ N ) )
% 0.23/0.52 = ( power_power_real @ ( real_V1803761363581548252l_real @ X4 ) @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_power
% 0.23/0.52 thf(fact_6495_of__real__power,axiom,
% 0.23/0.52 ! [X4: real,N: nat] :
% 0.23/0.52 ( ( real_V4546457046886955230omplex @ ( power_power_real @ X4 @ N ) )
% 0.23/0.52 = ( power_power_complex @ ( real_V4546457046886955230omplex @ X4 ) @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_power
% 0.23/0.52 thf(fact_6496_of__real__add,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_add
% 0.23/0.52 thf(fact_6497_of__real__add,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_add
% 0.23/0.52 thf(fact_6498_of__real__diff,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_diff
% 0.23/0.52 thf(fact_6499_of__real__diff,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_diff
% 0.23/0.52 thf(fact_6500_tan__periodic__pi,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( tan_real @ ( plus_plus_real @ X4 @ pi ) )
% 0.23/0.52 = ( tan_real @ X4 ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_periodic_pi
% 0.23/0.52 thf(fact_6501_of__real__neg__numeral,axiom,
% 0.23/0.52 ! [W: num] :
% 0.23/0.52 ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.52 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_neg_numeral
% 0.23/0.52 thf(fact_6502_of__real__neg__numeral,axiom,
% 0.23/0.52 ! [W: num] :
% 0.23/0.52 ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_neg_numeral
% 0.23/0.52 thf(fact_6503_cos__of__real__pi,axiom,
% 0.23/0.52 ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
% 0.23/0.52 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_of_real_pi
% 0.23/0.52 thf(fact_6504_cos__of__real__pi,axiom,
% 0.23/0.52 ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_of_real_pi
% 0.23/0.52 thf(fact_6505_tan__npi,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.23/0.52 = zero_zero_real ) ).
% 0.23/0.52
% 0.23/0.52 % tan_npi
% 0.23/0.52 thf(fact_6506_tan__periodic__n,axiom,
% 0.23/0.52 ! [X4: real,N: num] :
% 0.23/0.52 ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 0.23/0.52 = ( tan_real @ X4 ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_periodic_n
% 0.23/0.52 thf(fact_6507_sin__arcsin,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.23/0.52 = Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_arcsin
% 0.23/0.52 thf(fact_6508_tan__periodic__nat,axiom,
% 0.23/0.52 ! [X4: real,N: nat] :
% 0.23/0.52 ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 0.23/0.52 = ( tan_real @ X4 ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_periodic_nat
% 0.23/0.52 thf(fact_6509_tan__periodic__int,axiom,
% 0.23/0.52 ! [X4: real,I: int] :
% 0.23/0.52 ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 0.23/0.52 = ( tan_real @ X4 ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_periodic_int
% 0.23/0.52 thf(fact_6510_norm__cos__sin,axiom,
% 0.23/0.52 ! [T: real] :
% 0.23/0.52 ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % norm_cos_sin
% 0.23/0.52 thf(fact_6511_norm__of__real__add1,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ one_one_real ) )
% 0.23/0.52 = ( abs_abs_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % norm_of_real_add1
% 0.23/0.52 thf(fact_6512_norm__of__real__add1,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ one_one_complex ) )
% 0.23/0.52 = ( abs_abs_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % norm_of_real_add1
% 0.23/0.52 thf(fact_6513_norm__of__real__addn,axiom,
% 0.23/0.52 ! [X4: real,B2: num] :
% 0.23/0.52 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( numeral_numeral_real @ B2 ) ) )
% 0.23/0.52 = ( abs_abs_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ B2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % norm_of_real_addn
% 0.23/0.52 thf(fact_6514_norm__of__real__addn,axiom,
% 0.23/0.52 ! [X4: real,B2: num] :
% 0.23/0.52 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( numera6690914467698888265omplex @ B2 ) ) )
% 0.23/0.52 = ( abs_abs_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ B2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % norm_of_real_addn
% 0.23/0.52 thf(fact_6515_arcsin__1,axiom,
% 0.23/0.52 ( ( arcsin @ one_one_real )
% 0.23/0.52 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_1
% 0.23/0.52 thf(fact_6516_tan__periodic,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.23/0.52 = ( tan_real @ X4 ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_periodic
% 0.23/0.52 thf(fact_6517_cos__of__real__pi__half,axiom,
% 0.23/0.52 ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 = zero_zero_real ) ).
% 0.23/0.52
% 0.23/0.52 % cos_of_real_pi_half
% 0.23/0.52 thf(fact_6518_cos__of__real__pi__half,axiom,
% 0.23/0.52 ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.23/0.52 = zero_zero_complex ) ).
% 0.23/0.52
% 0.23/0.52 % cos_of_real_pi_half
% 0.23/0.52 thf(fact_6519_sin__of__real__pi__half,axiom,
% 0.23/0.52 ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % sin_of_real_pi_half
% 0.23/0.52 thf(fact_6520_sin__of__real__pi__half,axiom,
% 0.23/0.52 ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.23/0.52 = one_one_complex ) ).
% 0.23/0.52
% 0.23/0.52 % sin_of_real_pi_half
% 0.23/0.52 thf(fact_6521_arcsin__minus__1,axiom,
% 0.23/0.52 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.52 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_minus_1
% 0.23/0.52 thf(fact_6522_complex__of__real__mult__Complex,axiom,
% 0.23/0.52 ! [R2: real,X4: real,Y3: real] :
% 0.23/0.52 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X4 @ Y3 ) )
% 0.23/0.52 = ( complex2 @ ( times_times_real @ R2 @ X4 ) @ ( times_times_real @ R2 @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % complex_of_real_mult_Complex
% 0.23/0.52 thf(fact_6523_Complex__mult__complex__of__real,axiom,
% 0.23/0.52 ! [X4: real,Y3: real,R2: real] :
% 0.23/0.52 ( ( times_times_complex @ ( complex2 @ X4 @ Y3 ) @ ( real_V4546457046886955230omplex @ R2 ) )
% 0.23/0.52 = ( complex2 @ ( times_times_real @ X4 @ R2 ) @ ( times_times_real @ Y3 @ R2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Complex_mult_complex_of_real
% 0.23/0.52 thf(fact_6524_complex__of__real__add__Complex,axiom,
% 0.23/0.52 ! [R2: real,X4: real,Y3: real] :
% 0.23/0.52 ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X4 @ Y3 ) )
% 0.23/0.52 = ( complex2 @ ( plus_plus_real @ R2 @ X4 ) @ Y3 ) ) ).
% 0.23/0.52
% 0.23/0.52 % complex_of_real_add_Complex
% 0.23/0.52 thf(fact_6525_Complex__add__complex__of__real,axiom,
% 0.23/0.52 ! [X4: real,Y3: real,R2: real] :
% 0.23/0.52 ( ( plus_plus_complex @ ( complex2 @ X4 @ Y3 ) @ ( real_V4546457046886955230omplex @ R2 ) )
% 0.23/0.52 = ( complex2 @ ( plus_plus_real @ X4 @ R2 ) @ Y3 ) ) ).
% 0.23/0.52
% 0.23/0.52 % Complex_add_complex_of_real
% 0.23/0.52 thf(fact_6526_complex__diff,axiom,
% 0.23/0.52 ! [A: real,B2: real,C: real,D: real] :
% 0.23/0.52 ( ( minus_minus_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
% 0.23/0.52 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % complex_diff
% 0.23/0.52 thf(fact_6527_Complex__eq__1,axiom,
% 0.23/0.52 ! [A: real,B2: real] :
% 0.23/0.52 ( ( ( complex2 @ A @ B2 )
% 0.23/0.52 = one_one_complex )
% 0.23/0.52 = ( ( A = one_one_real )
% 0.23/0.52 & ( B2 = zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Complex_eq_1
% 0.23/0.52 thf(fact_6528_one__complex_Ocode,axiom,
% 0.23/0.52 ( one_one_complex
% 0.23/0.52 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % one_complex.code
% 0.23/0.52 thf(fact_6529_Complex__eq__numeral,axiom,
% 0.23/0.52 ! [A: real,B2: real,W: num] :
% 0.23/0.52 ( ( ( complex2 @ A @ B2 )
% 0.23/0.52 = ( numera6690914467698888265omplex @ W ) )
% 0.23/0.52 = ( ( A
% 0.23/0.52 = ( numeral_numeral_real @ W ) )
% 0.23/0.52 & ( B2 = zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Complex_eq_numeral
% 0.23/0.52 thf(fact_6530_complex__add,axiom,
% 0.23/0.52 ! [A: real,B2: real,C: real,D: real] :
% 0.23/0.52 ( ( plus_plus_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
% 0.23/0.52 = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % complex_add
% 0.23/0.52 thf(fact_6531_nonzero__of__real__divide,axiom,
% 0.23/0.52 ! [Y3: real,X4: real] :
% 0.23/0.52 ( ( Y3 != zero_zero_real )
% 0.23/0.52 => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nonzero_of_real_divide
% 0.23/0.52 thf(fact_6532_nonzero__of__real__divide,axiom,
% 0.23/0.52 ! [Y3: real,X4: real] :
% 0.23/0.52 ( ( Y3 != zero_zero_real )
% 0.23/0.52 => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nonzero_of_real_divide
% 0.23/0.52 thf(fact_6533_Complex__eq__neg__1,axiom,
% 0.23/0.52 ! [A: real,B2: real] :
% 0.23/0.52 ( ( ( complex2 @ A @ B2 )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.52 = ( ( A
% 0.23/0.52 = ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.52 & ( B2 = zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Complex_eq_neg_1
% 0.23/0.52 thf(fact_6534_Complex__eq__neg__numeral,axiom,
% 0.23/0.52 ! [A: real,B2: real,W: num] :
% 0.23/0.52 ( ( ( complex2 @ A @ B2 )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.23/0.52 = ( ( A
% 0.23/0.52 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.52 & ( B2 = zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Complex_eq_neg_numeral
% 0.23/0.52 thf(fact_6535_complex__mult,axiom,
% 0.23/0.52 ! [A: real,B2: real,C: real,D: real] :
% 0.23/0.52 ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
% 0.23/0.52 = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % complex_mult
% 0.23/0.52 thf(fact_6536_arcsin__le__arcsin,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ord_less_eq_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_le_arcsin
% 0.23/0.52 thf(fact_6537_arcsin__minus,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.52 => ( ( arcsin @ ( uminus_uminus_real @ X4 ) )
% 0.23/0.52 = ( uminus_uminus_real @ ( arcsin @ X4 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_minus
% 0.23/0.52 thf(fact_6538_arcsin__le__mono,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) )
% 0.23/0.52 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_le_mono
% 0.23/0.52 thf(fact_6539_arcsin__eq__iff,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.23/0.52 => ( ( ( arcsin @ X4 )
% 0.23/0.52 = ( arcsin @ Y3 ) )
% 0.23/0.52 = ( X4 = Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_eq_iff
% 0.23/0.52 thf(fact_6540_norm__less__p1,axiom,
% 0.23/0.52 ! [X4: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X4 ) ) @ one_one_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % norm_less_p1
% 0.23/0.52 thf(fact_6541_norm__less__p1,axiom,
% 0.23/0.52 ! [X4: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X4 ) ) @ one_one_complex ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % norm_less_p1
% 0.23/0.52 thf(fact_6542_tan__def,axiom,
% 0.23/0.52 ( tan_complex
% 0.23/0.52 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X ) @ ( cos_complex @ X ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_def
% 0.23/0.52 thf(fact_6543_tan__def,axiom,
% 0.23/0.52 ( tan_real
% 0.23/0.52 = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ X ) @ ( cos_real @ X ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_def
% 0.23/0.52 thf(fact_6544_arcsin__less__arcsin,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ord_less_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_less_arcsin
% 0.23/0.52 thf(fact_6545_arcsin__less__mono,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.23/0.52 => ( ( ord_less_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) )
% 0.23/0.52 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_less_mono
% 0.23/0.52 thf(fact_6546_norm__of__real__diff,axiom,
% 0.23/0.52 ! [B2: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B2 ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % norm_of_real_diff
% 0.23/0.52 thf(fact_6547_norm__of__real__diff,axiom,
% 0.23/0.52 ! [B2: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B2 ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % norm_of_real_diff
% 0.23/0.52 thf(fact_6548_tan__45,axiom,
% 0.23/0.52 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % tan_45
% 0.23/0.52 thf(fact_6549_tan__60,axiom,
% 0.23/0.52 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.23/0.52 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_60
% 0.23/0.52 thf(fact_6550_cos__int__times__real,axiom,
% 0.23/0.52 ! [M: int,X4: real] :
% 0.23/0.52 ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X4 ) ) )
% 0.23/0.52 = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_int_times_real
% 0.23/0.52 thf(fact_6551_cos__int__times__real,axiom,
% 0.23/0.52 ! [M: int,X4: real] :
% 0.23/0.52 ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X4 ) ) )
% 0.23/0.52 = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_int_times_real
% 0.23/0.52 thf(fact_6552_sin__int__times__real,axiom,
% 0.23/0.52 ! [M: int,X4: real] :
% 0.23/0.52 ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X4 ) ) )
% 0.23/0.52 = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_int_times_real
% 0.23/0.52 thf(fact_6553_sin__int__times__real,axiom,
% 0.23/0.52 ! [M: int,X4: real] :
% 0.23/0.52 ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X4 ) ) )
% 0.23/0.52 = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_int_times_real
% 0.23/0.52 thf(fact_6554_cos__arcsin__nonzero,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.23/0.52 => ( ( cos_real @ ( arcsin @ X4 ) )
% 0.23/0.52 != zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_arcsin_nonzero
% 0.23/0.52 thf(fact_6555_lemma__tan__total,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.23/0.52 => ? [X3: real] :
% 0.23/0.52 ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.23/0.52 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 & ( ord_less_real @ Y3 @ ( tan_real @ X3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % lemma_tan_total
% 0.23/0.52 thf(fact_6556_tan__gt__zero,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_gt_zero
% 0.23/0.52 thf(fact_6557_tan__total,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ? [X3: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.23/0.52 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 & ( ( tan_real @ X3 )
% 0.23/0.52 = Y3 )
% 0.23/0.52 & ! [Y5: real] :
% 0.23/0.52 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.23/0.52 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 & ( ( tan_real @ Y5 )
% 0.23/0.52 = Y3 ) )
% 0.23/0.52 => ( Y5 = X3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_total
% 0.23/0.52 thf(fact_6558_tan__monotone,axiom,
% 0.23/0.52 ! [Y3: real,X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_real @ Y3 @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X4 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_monotone
% 0.23/0.52 thf(fact_6559_tan__monotone_H,axiom,
% 0.23/0.52 ! [Y3: real,X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( ord_less_real @ Y3 @ X4 )
% 0.23/0.52 = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X4 ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_monotone'
% 0.23/0.52 thf(fact_6560_tan__mono__lt__eq,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( ord_less_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) )
% 0.23/0.52 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_mono_lt_eq
% 0.23/0.52 thf(fact_6561_lemma__tan__total1,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ? [X3: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.23/0.52 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 & ( ( tan_real @ X3 )
% 0.23/0.52 = Y3 ) ) ).
% 0.23/0.52
% 0.23/0.52 % lemma_tan_total1
% 0.23/0.52 thf(fact_6562_tan__minus__45,axiom,
% 0.23/0.52 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.52 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_minus_45
% 0.23/0.52 thf(fact_6563_tan__inverse,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y3 ) )
% 0.23/0.52 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_inverse
% 0.23/0.52 thf(fact_6564_complex__norm,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( real_V1022390504157884413omplex @ ( complex2 @ X4 @ Y3 ) )
% 0.23/0.52 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % complex_norm
% 0.23/0.52 thf(fact_6565_add__tan__eq,axiom,
% 0.23/0.52 ! [X4: complex,Y3: complex] :
% 0.23/0.52 ( ( ( cos_complex @ X4 )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( ( cos_complex @ Y3 )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( plus_plus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % add_tan_eq
% 0.23/0.52 thf(fact_6566_add__tan__eq,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ( cos_real @ X4 )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( ( cos_real @ Y3 )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( plus_plus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) )
% 0.23/0.52 = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % add_tan_eq
% 0.23/0.52 thf(fact_6567_tan__cot_H,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) )
% 0.23/0.52 = ( cot_real @ X4 ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_cot'
% 0.23/0.52 thf(fact_6568_sin__cos__eq,axiom,
% 0.23/0.52 ( sin_real
% 0.23/0.52 = ( ^ [X: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_cos_eq
% 0.23/0.52 thf(fact_6569_sin__cos__eq,axiom,
% 0.23/0.52 ( sin_complex
% 0.23/0.52 = ( ^ [X: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_cos_eq
% 0.23/0.52 thf(fact_6570_cos__sin__eq,axiom,
% 0.23/0.52 ( cos_real
% 0.23/0.52 = ( ^ [X: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_sin_eq
% 0.23/0.52 thf(fact_6571_cos__sin__eq,axiom,
% 0.23/0.52 ( cos_complex
% 0.23/0.52 = ( ^ [X: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_sin_eq
% 0.23/0.52 thf(fact_6572_tan__pos__pi2__le,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_pos_pi2_le
% 0.23/0.52 thf(fact_6573_tan__total__pos,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.52 => ? [X3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.23/0.52 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 & ( ( tan_real @ X3 )
% 0.23/0.52 = Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_total_pos
% 0.23/0.52 thf(fact_6574_tan__less__zero,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 0.23/0.52 => ( ord_less_real @ ( tan_real @ X4 ) @ zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_less_zero
% 0.23/0.52 thf(fact_6575_tan__mono__le,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.52 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ord_less_eq_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_mono_le
% 0.23/0.52 thf(fact_6576_tan__mono__le__eq,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) )
% 0.23/0.52 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_mono_le_eq
% 0.23/0.52 thf(fact_6577_tan__bound__pi2,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.23/0.52 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X4 ) ) @ one_one_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_bound_pi2
% 0.23/0.52 thf(fact_6578_tan__30,axiom,
% 0.23/0.52 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.23/0.52 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_30
% 0.23/0.52 thf(fact_6579_arctan,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 0.23/0.52 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 & ( ( tan_real @ ( arctan @ Y3 ) )
% 0.23/0.52 = Y3 ) ) ).
% 0.23/0.52
% 0.23/0.52 % arctan
% 0.23/0.52 thf(fact_6580_arctan__tan,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( arctan @ ( tan_real @ X4 ) )
% 0.23/0.52 = X4 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arctan_tan
% 0.23/0.52 thf(fact_6581_arctan__unique,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( ( tan_real @ X4 )
% 0.23/0.52 = Y3 )
% 0.23/0.52 => ( ( arctan @ Y3 )
% 0.23/0.52 = X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arctan_unique
% 0.23/0.52 thf(fact_6582_lemma__tan__add1,axiom,
% 0.23/0.52 ! [X4: complex,Y3: complex] :
% 0.23/0.52 ( ( ( cos_complex @ X4 )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( ( cos_complex @ Y3 )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % lemma_tan_add1
% 0.23/0.52 thf(fact_6583_lemma__tan__add1,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ( cos_real @ X4 )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( ( cos_real @ Y3 )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) )
% 0.23/0.52 = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % lemma_tan_add1
% 0.23/0.52 thf(fact_6584_tan__diff,axiom,
% 0.23/0.52 ! [X4: complex,Y3: complex] :
% 0.23/0.52 ( ( ( cos_complex @ X4 )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( ( cos_complex @ Y3 )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( ( cos_complex @ ( minus_minus_complex @ X4 @ Y3 ) )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( tan_complex @ ( minus_minus_complex @ X4 @ Y3 ) )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_diff
% 0.23/0.52 thf(fact_6585_tan__diff,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ( cos_real @ X4 )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( ( cos_real @ Y3 )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( ( cos_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( tan_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_diff
% 0.23/0.52 thf(fact_6586_tan__add,axiom,
% 0.23/0.52 ! [X4: complex,Y3: complex] :
% 0.23/0.52 ( ( ( cos_complex @ X4 )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( ( cos_complex @ Y3 )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( ( cos_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
% 0.23/0.52 != zero_zero_complex )
% 0.23/0.52 => ( ( tan_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_add
% 0.23/0.52 thf(fact_6587_tan__add,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ( cos_real @ X4 )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( ( cos_real @ Y3 )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( ( cos_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( tan_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_add
% 0.23/0.52 thf(fact_6588_minus__sin__cos__eq,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( uminus_uminus_real @ ( sin_real @ X4 ) )
% 0.23/0.52 = ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % minus_sin_cos_eq
% 0.23/0.52 thf(fact_6589_minus__sin__cos__eq,axiom,
% 0.23/0.52 ! [X4: complex] :
% 0.23/0.52 ( ( uminus1482373934393186551omplex @ ( sin_complex @ X4 ) )
% 0.23/0.52 = ( cos_complex @ ( plus_plus_complex @ X4 @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % minus_sin_cos_eq
% 0.23/0.52 thf(fact_6590_tan__total__pi4,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.52 => ? [Z2: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
% 0.23/0.52 & ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.23/0.52 & ( ( tan_real @ Z2 )
% 0.23/0.52 = X4 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_total_pi4
% 0.23/0.52 thf(fact_6591_arcsin__lt__bounded,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.23/0.52 & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_lt_bounded
% 0.23/0.52 thf(fact_6592_arcsin__lbound,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_lbound
% 0.23/0.52 thf(fact_6593_arcsin__ubound,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_ubound
% 0.23/0.52 thf(fact_6594_arcsin__bounded,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.23/0.52 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_bounded
% 0.23/0.52 thf(fact_6595_arcsin__sin,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( arcsin @ ( sin_real @ X4 ) )
% 0.23/0.52 = X4 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_sin
% 0.23/0.52 thf(fact_6596_tan__half,axiom,
% 0.23/0.52 ( tan_complex
% 0.23/0.52 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ one_one_complex ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_half
% 0.23/0.52 thf(fact_6597_tan__half,axiom,
% 0.23/0.52 ( tan_real
% 0.23/0.52 = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % tan_half
% 0.23/0.52 thf(fact_6598_le__arcsin__iff,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X4 ) )
% 0.23/0.52 = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X4 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % le_arcsin_iff
% 0.23/0.52 thf(fact_6599_arcsin__le__iff,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( arcsin @ X4 ) @ Y3 )
% 0.23/0.52 = ( ord_less_eq_real @ X4 @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_le_iff
% 0.23/0.52 thf(fact_6600_arcsin__pi,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.23/0.52 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
% 0.23/0.52 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.23/0.52 = Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin_pi
% 0.23/0.52 thf(fact_6601_arcsin,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.23/0.52 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.23/0.52 = Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arcsin
% 0.23/0.52 thf(fact_6602_arsinh__def,axiom,
% 0.23/0.52 ( arsinh_real
% 0.23/0.52 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arsinh_def
% 0.23/0.52 thf(fact_6603_sin__arccos__abs,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.23/0.52 => ( ( sin_real @ ( arccos @ Y3 ) )
% 0.23/0.52 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_arccos_abs
% 0.23/0.52 thf(fact_6604_sin__arccos,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.52 => ( ( sin_real @ ( arccos @ X4 ) )
% 0.23/0.52 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_arccos
% 0.23/0.52 thf(fact_6605_arccos__cos__eq__abs__2pi,axiom,
% 0.23/0.52 ! [Theta: real] :
% 0.23/0.52 ~ ! [K2: int] :
% 0.23/0.52 ( ( arccos @ ( cos_real @ Theta ) )
% 0.23/0.52 != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_cos_eq_abs_2pi
% 0.23/0.52 thf(fact_6606_fact__double,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.52 = ( 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 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_double
% 0.23/0.52 thf(fact_6607_fact__double,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.52 = ( 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 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_double
% 0.23/0.52 thf(fact_6608_fact__double,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.52 = ( 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 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_double
% 0.23/0.52 thf(fact_6609_modulo__int__unfold,axiom,
% 0.23/0.52 ! [L2: int,K: int,N: nat,M: nat] :
% 0.23/0.52 ( ( ( ( ( sgn_sgn_int @ L2 )
% 0.23/0.52 = zero_zero_int )
% 0.23/0.52 | ( ( sgn_sgn_int @ K )
% 0.23/0.52 = zero_zero_int )
% 0.23/0.52 | ( N = zero_zero_nat ) )
% 0.23/0.52 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.23/0.52 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 0.23/0.52 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 0.23/0.52 = zero_zero_int )
% 0.23/0.52 | ( ( sgn_sgn_int @ K )
% 0.23/0.52 = zero_zero_int )
% 0.23/0.52 | ( N = zero_zero_nat ) )
% 0.23/0.52 => ( ( ( ( sgn_sgn_int @ K )
% 0.23/0.52 = ( sgn_sgn_int @ L2 ) )
% 0.23/0.52 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.23/0.52 = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 0.23/0.52 & ( ( ( sgn_sgn_int @ K )
% 0.23/0.52 != ( sgn_sgn_int @ L2 ) )
% 0.23/0.52 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.23/0.52 = ( times_times_int @ ( sgn_sgn_int @ L2 )
% 0.23/0.52 @ ( minus_minus_int
% 0.23/0.52 @ ( semiri1314217659103216013at_int
% 0.23/0.52 @ ( times_times_nat @ N
% 0.23/0.52 @ ( zero_n2687167440665602831ol_nat
% 0.23/0.52 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 0.23/0.52 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % modulo_int_unfold
% 0.23/0.52 thf(fact_6610_sgn__sgn,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
% 0.23/0.52 = ( sgn_sgn_int @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_sgn
% 0.23/0.52 thf(fact_6611_sgn__sgn,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
% 0.23/0.52 = ( sgn_sgn_real @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_sgn
% 0.23/0.52 thf(fact_6612_sgn__sgn,axiom,
% 0.23/0.52 ! [A: complex] :
% 0.23/0.52 ( ( sgn_sgn_complex @ ( sgn_sgn_complex @ A ) )
% 0.23/0.52 = ( sgn_sgn_complex @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_sgn
% 0.23/0.52 thf(fact_6613_sgn__sgn,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.23/0.52 = ( sgn_sgn_Code_integer @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_sgn
% 0.23/0.52 thf(fact_6614_sgn__sgn,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
% 0.23/0.52 = ( sgn_sgn_rat @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_sgn
% 0.23/0.52 thf(fact_6615_sgn__0,axiom,
% 0.23/0.52 ( ( sgn_sgn_complex @ zero_zero_complex )
% 0.23/0.52 = zero_zero_complex ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_0
% 0.23/0.52 thf(fact_6616_sgn__0,axiom,
% 0.23/0.52 ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 0.23/0.52 = zero_z3403309356797280102nteger ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_0
% 0.23/0.52 thf(fact_6617_sgn__0,axiom,
% 0.23/0.52 ( ( sgn_sgn_real @ zero_zero_real )
% 0.23/0.52 = zero_zero_real ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_0
% 0.23/0.52 thf(fact_6618_sgn__0,axiom,
% 0.23/0.52 ( ( sgn_sgn_rat @ zero_zero_rat )
% 0.23/0.52 = zero_zero_rat ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_0
% 0.23/0.52 thf(fact_6619_sgn__0,axiom,
% 0.23/0.52 ( ( sgn_sgn_int @ zero_zero_int )
% 0.23/0.52 = zero_zero_int ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_0
% 0.23/0.52 thf(fact_6620_sgn__1,axiom,
% 0.23/0.52 ( ( sgn_sgn_int @ one_one_int )
% 0.23/0.52 = one_one_int ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1
% 0.23/0.52 thf(fact_6621_sgn__1,axiom,
% 0.23/0.52 ( ( sgn_sgn_real @ one_one_real )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1
% 0.23/0.52 thf(fact_6622_sgn__1,axiom,
% 0.23/0.52 ( ( sgn_sgn_complex @ one_one_complex )
% 0.23/0.52 = one_one_complex ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1
% 0.23/0.52 thf(fact_6623_sgn__1,axiom,
% 0.23/0.52 ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
% 0.23/0.52 = one_one_Code_integer ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1
% 0.23/0.52 thf(fact_6624_sgn__1,axiom,
% 0.23/0.52 ( ( sgn_sgn_rat @ one_one_rat )
% 0.23/0.52 = one_one_rat ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1
% 0.23/0.52 thf(fact_6625_sgn__one,axiom,
% 0.23/0.52 ( ( sgn_sgn_real @ one_one_real )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_one
% 0.23/0.52 thf(fact_6626_sgn__one,axiom,
% 0.23/0.52 ( ( sgn_sgn_complex @ one_one_complex )
% 0.23/0.52 = one_one_complex ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_one
% 0.23/0.52 thf(fact_6627_sgn__divide,axiom,
% 0.23/0.52 ! [A: complex,B2: complex] :
% 0.23/0.52 ( ( sgn_sgn_complex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_divide
% 0.23/0.52 thf(fact_6628_sgn__divide,axiom,
% 0.23/0.52 ! [A: real,B2: real] :
% 0.23/0.52 ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B2 ) )
% 0.23/0.52 = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_divide
% 0.23/0.52 thf(fact_6629_sgn__divide,axiom,
% 0.23/0.52 ! [A: rat,B2: rat] :
% 0.23/0.52 ( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.23/0.52 = ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_divide
% 0.23/0.52 thf(fact_6630_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
% 0.23/0.52 = ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % idom_abs_sgn_class.sgn_minus
% 0.23/0.52 thf(fact_6631_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
% 0.23/0.52 = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % idom_abs_sgn_class.sgn_minus
% 0.23/0.52 thf(fact_6632_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.23/0.52 ! [A: complex] :
% 0.23/0.52 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ A ) )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % idom_abs_sgn_class.sgn_minus
% 0.23/0.52 thf(fact_6633_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.52 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % idom_abs_sgn_class.sgn_minus
% 0.23/0.52 thf(fact_6634_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
% 0.23/0.52 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % idom_abs_sgn_class.sgn_minus
% 0.23/0.52 thf(fact_6635_power__sgn,axiom,
% 0.23/0.52 ! [A: code_integer,N: nat] :
% 0.23/0.52 ( ( sgn_sgn_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.23/0.52 = ( power_8256067586552552935nteger @ ( sgn_sgn_Code_integer @ A ) @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % power_sgn
% 0.23/0.52 thf(fact_6636_power__sgn,axiom,
% 0.23/0.52 ! [A: rat,N: nat] :
% 0.23/0.52 ( ( sgn_sgn_rat @ ( power_power_rat @ A @ N ) )
% 0.23/0.52 = ( power_power_rat @ ( sgn_sgn_rat @ A ) @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % power_sgn
% 0.23/0.52 thf(fact_6637_power__sgn,axiom,
% 0.23/0.52 ! [A: real,N: nat] :
% 0.23/0.52 ( ( sgn_sgn_real @ ( power_power_real @ A @ N ) )
% 0.23/0.52 = ( power_power_real @ ( sgn_sgn_real @ A ) @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % power_sgn
% 0.23/0.52 thf(fact_6638_power__sgn,axiom,
% 0.23/0.52 ! [A: int,N: nat] :
% 0.23/0.52 ( ( sgn_sgn_int @ ( power_power_int @ A @ N ) )
% 0.23/0.52 = ( power_power_int @ ( sgn_sgn_int @ A ) @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % power_sgn
% 0.23/0.52 thf(fact_6639_sgn__greater,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
% 0.23/0.52 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_greater
% 0.23/0.52 thf(fact_6640_sgn__greater,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 0.23/0.52 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_greater
% 0.23/0.52 thf(fact_6641_sgn__greater,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
% 0.23/0.52 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_greater
% 0.23/0.52 thf(fact_6642_sgn__greater,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
% 0.23/0.52 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_greater
% 0.23/0.52 thf(fact_6643_sgn__less,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.23/0.52 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_less
% 0.23/0.52 thf(fact_6644_sgn__less,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 0.23/0.52 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_less
% 0.23/0.52 thf(fact_6645_sgn__less,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
% 0.23/0.52 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_less
% 0.23/0.52 thf(fact_6646_sgn__less,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
% 0.23/0.52 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_less
% 0.23/0.52 thf(fact_6647_divide__sgn,axiom,
% 0.23/0.52 ! [A: real,B2: real] :
% 0.23/0.52 ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B2 ) )
% 0.23/0.52 = ( times_times_real @ A @ ( sgn_sgn_real @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % divide_sgn
% 0.23/0.52 thf(fact_6648_divide__sgn,axiom,
% 0.23/0.52 ! [A: rat,B2: rat] :
% 0.23/0.52 ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B2 ) )
% 0.23/0.52 = ( times_times_rat @ A @ ( sgn_sgn_rat @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % divide_sgn
% 0.23/0.52 thf(fact_6649_fact__0,axiom,
% 0.23/0.52 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 0.23/0.52 = one_one_complex ) ).
% 0.23/0.52
% 0.23/0.52 % fact_0
% 0.23/0.52 thf(fact_6650_fact__0,axiom,
% 0.23/0.52 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 0.23/0.52 = one_one_rat ) ).
% 0.23/0.52
% 0.23/0.52 % fact_0
% 0.23/0.52 thf(fact_6651_fact__0,axiom,
% 0.23/0.52 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 0.23/0.52 = one_one_int ) ).
% 0.23/0.52
% 0.23/0.52 % fact_0
% 0.23/0.52 thf(fact_6652_fact__0,axiom,
% 0.23/0.52 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 0.23/0.52 = one_one_nat ) ).
% 0.23/0.52
% 0.23/0.52 % fact_0
% 0.23/0.52 thf(fact_6653_fact__0,axiom,
% 0.23/0.52 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % fact_0
% 0.23/0.52 thf(fact_6654_fact__1,axiom,
% 0.23/0.52 ( ( semiri5044797733671781792omplex @ one_one_nat )
% 0.23/0.52 = one_one_complex ) ).
% 0.23/0.52
% 0.23/0.52 % fact_1
% 0.23/0.52 thf(fact_6655_fact__1,axiom,
% 0.23/0.52 ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 0.23/0.52 = one_one_rat ) ).
% 0.23/0.52
% 0.23/0.52 % fact_1
% 0.23/0.52 thf(fact_6656_fact__1,axiom,
% 0.23/0.52 ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 0.23/0.52 = one_one_int ) ).
% 0.23/0.52
% 0.23/0.52 % fact_1
% 0.23/0.52 thf(fact_6657_fact__1,axiom,
% 0.23/0.52 ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 0.23/0.52 = one_one_nat ) ).
% 0.23/0.52
% 0.23/0.52 % fact_1
% 0.23/0.52 thf(fact_6658_fact__1,axiom,
% 0.23/0.52 ( ( semiri2265585572941072030t_real @ one_one_nat )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % fact_1
% 0.23/0.52 thf(fact_6659_arccos__1,axiom,
% 0.23/0.52 ( ( arccos @ one_one_real )
% 0.23/0.52 = zero_zero_real ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_1
% 0.23/0.52 thf(fact_6660_sgn__pos,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.23/0.52 => ( ( sgn_sgn_Code_integer @ A )
% 0.23/0.52 = one_one_Code_integer ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_pos
% 0.23/0.52 thf(fact_6661_sgn__pos,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.52 => ( ( sgn_sgn_real @ A )
% 0.23/0.52 = one_one_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_pos
% 0.23/0.52 thf(fact_6662_sgn__pos,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.52 => ( ( sgn_sgn_rat @ A )
% 0.23/0.52 = one_one_rat ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_pos
% 0.23/0.52 thf(fact_6663_sgn__pos,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( ord_less_int @ zero_zero_int @ A )
% 0.23/0.52 => ( ( sgn_sgn_int @ A )
% 0.23/0.52 = one_one_int ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_pos
% 0.23/0.52 thf(fact_6664_abs__sgn__eq__1,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( A != zero_z3403309356797280102nteger )
% 0.23/0.52 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.23/0.52 = one_one_Code_integer ) ) ).
% 0.23/0.52
% 0.23/0.52 % abs_sgn_eq_1
% 0.23/0.52 thf(fact_6665_abs__sgn__eq__1,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( A != zero_zero_real )
% 0.23/0.52 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.23/0.52 = one_one_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % abs_sgn_eq_1
% 0.23/0.52 thf(fact_6666_abs__sgn__eq__1,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( A != zero_zero_rat )
% 0.23/0.52 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.23/0.52 = one_one_rat ) ) ).
% 0.23/0.52
% 0.23/0.52 % abs_sgn_eq_1
% 0.23/0.52 thf(fact_6667_abs__sgn__eq__1,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( A != zero_zero_int )
% 0.23/0.52 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.23/0.52 = one_one_int ) ) ).
% 0.23/0.52
% 0.23/0.52 % abs_sgn_eq_1
% 0.23/0.52 thf(fact_6668_fact__Suc__0,axiom,
% 0.23/0.52 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 0.23/0.52 = one_one_complex ) ).
% 0.23/0.52
% 0.23/0.52 % fact_Suc_0
% 0.23/0.52 thf(fact_6669_fact__Suc__0,axiom,
% 0.23/0.52 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 0.23/0.52 = one_one_rat ) ).
% 0.23/0.52
% 0.23/0.52 % fact_Suc_0
% 0.23/0.52 thf(fact_6670_fact__Suc__0,axiom,
% 0.23/0.52 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 0.23/0.52 = one_one_int ) ).
% 0.23/0.52
% 0.23/0.52 % fact_Suc_0
% 0.23/0.52 thf(fact_6671_fact__Suc__0,axiom,
% 0.23/0.52 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 0.23/0.52 = one_one_nat ) ).
% 0.23/0.52
% 0.23/0.52 % fact_Suc_0
% 0.23/0.52 thf(fact_6672_fact__Suc__0,axiom,
% 0.23/0.52 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % fact_Suc_0
% 0.23/0.52 thf(fact_6673_fact__Suc,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
% 0.23/0.52 = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_Suc
% 0.23/0.52 thf(fact_6674_fact__Suc,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 0.23/0.52 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_Suc
% 0.23/0.52 thf(fact_6675_fact__Suc,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 0.23/0.52 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_Suc
% 0.23/0.52 thf(fact_6676_fact__Suc,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 0.23/0.52 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_Suc
% 0.23/0.52 thf(fact_6677_fact__Suc,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 0.23/0.52 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_Suc
% 0.23/0.52 thf(fact_6678_sgn__mult__self__eq,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
% 0.23/0.52 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult_self_eq
% 0.23/0.52 thf(fact_6679_sgn__mult__self__eq,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 0.23/0.52 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult_self_eq
% 0.23/0.52 thf(fact_6680_sgn__mult__self__eq,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
% 0.23/0.52 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult_self_eq
% 0.23/0.52 thf(fact_6681_sgn__mult__self__eq,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 0.23/0.52 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult_self_eq
% 0.23/0.52 thf(fact_6682_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.23/0.52 ! [A: complex] :
% 0.23/0.52 ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
% 0.23/0.52 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % idom_abs_sgn_class.abs_sgn
% 0.23/0.52 thf(fact_6683_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
% 0.23/0.52 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % idom_abs_sgn_class.abs_sgn
% 0.23/0.52 thf(fact_6684_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
% 0.23/0.52 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % idom_abs_sgn_class.abs_sgn
% 0.23/0.52 thf(fact_6685_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
% 0.23/0.52 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % idom_abs_sgn_class.abs_sgn
% 0.23/0.52 thf(fact_6686_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.23/0.52 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % idom_abs_sgn_class.abs_sgn
% 0.23/0.52 thf(fact_6687_sgn__abs,axiom,
% 0.23/0.52 ! [A: complex] :
% 0.23/0.52 ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
% 0.23/0.52 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_abs
% 0.23/0.52 thf(fact_6688_sgn__abs,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.23/0.52 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_abs
% 0.23/0.52 thf(fact_6689_sgn__abs,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.23/0.52 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_abs
% 0.23/0.52 thf(fact_6690_sgn__abs,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.23/0.52 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_abs
% 0.23/0.52 thf(fact_6691_sgn__abs,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.23/0.52 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_abs
% 0.23/0.52 thf(fact_6692_arccos__minus__1,axiom,
% 0.23/0.52 ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.52 = pi ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_minus_1
% 0.23/0.52 thf(fact_6693_sgn__mult__dvd__iff,axiom,
% 0.23/0.52 ! [R2: int,L2: int,K: int] :
% 0.23/0.52 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L2 ) @ K )
% 0.23/0.52 = ( ( dvd_dvd_int @ L2 @ K )
% 0.23/0.52 & ( ( R2 = zero_zero_int )
% 0.23/0.52 => ( K = zero_zero_int ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult_dvd_iff
% 0.23/0.52 thf(fact_6694_mult__sgn__dvd__iff,axiom,
% 0.23/0.52 ! [L2: int,R2: int,K: int] :
% 0.23/0.52 ( ( dvd_dvd_int @ ( times_times_int @ L2 @ ( sgn_sgn_int @ R2 ) ) @ K )
% 0.23/0.52 = ( ( dvd_dvd_int @ L2 @ K )
% 0.23/0.52 & ( ( R2 = zero_zero_int )
% 0.23/0.52 => ( K = zero_zero_int ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % mult_sgn_dvd_iff
% 0.23/0.52 thf(fact_6695_dvd__sgn__mult__iff,axiom,
% 0.23/0.52 ! [L2: int,R2: int,K: int] :
% 0.23/0.52 ( ( dvd_dvd_int @ L2 @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
% 0.23/0.52 = ( ( dvd_dvd_int @ L2 @ K )
% 0.23/0.52 | ( R2 = zero_zero_int ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % dvd_sgn_mult_iff
% 0.23/0.52 thf(fact_6696_dvd__mult__sgn__iff,axiom,
% 0.23/0.52 ! [L2: int,K: int,R2: int] :
% 0.23/0.52 ( ( dvd_dvd_int @ L2 @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
% 0.23/0.52 = ( ( dvd_dvd_int @ L2 @ K )
% 0.23/0.52 | ( R2 = zero_zero_int ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % dvd_mult_sgn_iff
% 0.23/0.52 thf(fact_6697_sgn__neg,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.23/0.52 => ( ( sgn_sgn_real @ A )
% 0.23/0.52 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_neg
% 0.23/0.52 thf(fact_6698_sgn__neg,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( ord_less_int @ A @ zero_zero_int )
% 0.23/0.52 => ( ( sgn_sgn_int @ A )
% 0.23/0.52 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_neg
% 0.23/0.52 thf(fact_6699_sgn__neg,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.23/0.52 => ( ( sgn_sgn_Code_integer @ A )
% 0.23/0.52 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_neg
% 0.23/0.52 thf(fact_6700_sgn__neg,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.23/0.52 => ( ( sgn_sgn_rat @ A )
% 0.23/0.52 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_neg
% 0.23/0.52 thf(fact_6701_fact__2,axiom,
% 0.23/0.52 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.52 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_2
% 0.23/0.52 thf(fact_6702_fact__2,axiom,
% 0.23/0.52 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.52 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_2
% 0.23/0.52 thf(fact_6703_fact__2,axiom,
% 0.23/0.52 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.52 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_2
% 0.23/0.52 thf(fact_6704_fact__2,axiom,
% 0.23/0.52 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.52 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_2
% 0.23/0.52 thf(fact_6705_fact__2,axiom,
% 0.23/0.52 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.52 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_2
% 0.23/0.52 thf(fact_6706_sgn__of__nat,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.23/0.52 = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_of_nat
% 0.23/0.52 thf(fact_6707_sgn__of__nat,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.23/0.52 = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_of_nat
% 0.23/0.52 thf(fact_6708_sgn__of__nat,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.52 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_of_nat
% 0.23/0.52 thf(fact_6709_sgn__of__nat,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.23/0.52 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_of_nat
% 0.23/0.52 thf(fact_6710_cos__arccos,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ( cos_real @ ( arccos @ Y3 ) )
% 0.23/0.52 = Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_arccos
% 0.23/0.52 thf(fact_6711_arccos__0,axiom,
% 0.23/0.52 ( ( arccos @ zero_zero_real )
% 0.23/0.52 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_0
% 0.23/0.52 thf(fact_6712_complex__exp__exists,axiom,
% 0.23/0.52 ! [Z: complex] :
% 0.23/0.52 ? [A5: complex,R3: real] :
% 0.23/0.52 ( Z
% 0.23/0.52 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( exp_complex @ A5 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % complex_exp_exists
% 0.23/0.52 thf(fact_6713_fact__mono__nat,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.52 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_mono_nat
% 0.23/0.52 thf(fact_6714_fact__ge__self,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_ge_self
% 0.23/0.52 thf(fact_6715_sgn__eq__0__iff,axiom,
% 0.23/0.52 ! [A: complex] :
% 0.23/0.52 ( ( ( sgn_sgn_complex @ A )
% 0.23/0.52 = zero_zero_complex )
% 0.23/0.52 = ( A = zero_zero_complex ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_eq_0_iff
% 0.23/0.52 thf(fact_6716_sgn__eq__0__iff,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( ( sgn_sgn_Code_integer @ A )
% 0.23/0.52 = zero_z3403309356797280102nteger )
% 0.23/0.52 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_eq_0_iff
% 0.23/0.52 thf(fact_6717_sgn__eq__0__iff,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( ( sgn_sgn_real @ A )
% 0.23/0.52 = zero_zero_real )
% 0.23/0.52 = ( A = zero_zero_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_eq_0_iff
% 0.23/0.52 thf(fact_6718_sgn__eq__0__iff,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( ( sgn_sgn_rat @ A )
% 0.23/0.52 = zero_zero_rat )
% 0.23/0.52 = ( A = zero_zero_rat ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_eq_0_iff
% 0.23/0.52 thf(fact_6719_sgn__eq__0__iff,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( ( sgn_sgn_int @ A )
% 0.23/0.52 = zero_zero_int )
% 0.23/0.52 = ( A = zero_zero_int ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_eq_0_iff
% 0.23/0.52 thf(fact_6720_sgn__0__0,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( ( sgn_sgn_Code_integer @ A )
% 0.23/0.52 = zero_z3403309356797280102nteger )
% 0.23/0.52 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_0_0
% 0.23/0.52 thf(fact_6721_sgn__0__0,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( ( sgn_sgn_real @ A )
% 0.23/0.52 = zero_zero_real )
% 0.23/0.52 = ( A = zero_zero_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_0_0
% 0.23/0.52 thf(fact_6722_sgn__0__0,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( ( sgn_sgn_rat @ A )
% 0.23/0.52 = zero_zero_rat )
% 0.23/0.52 = ( A = zero_zero_rat ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_0_0
% 0.23/0.52 thf(fact_6723_sgn__0__0,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( ( sgn_sgn_int @ A )
% 0.23/0.52 = zero_zero_int )
% 0.23/0.52 = ( A = zero_zero_int ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_0_0
% 0.23/0.52 thf(fact_6724_Real__Vector__Spaces_Osgn__mult,axiom,
% 0.23/0.52 ! [X4: complex,Y3: complex] :
% 0.23/0.52 ( ( sgn_sgn_complex @ ( times_times_complex @ X4 @ Y3 ) )
% 0.23/0.52 = ( times_times_complex @ ( sgn_sgn_complex @ X4 ) @ ( sgn_sgn_complex @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Real_Vector_Spaces.sgn_mult
% 0.23/0.52 thf(fact_6725_Real__Vector__Spaces_Osgn__mult,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( sgn_sgn_real @ ( times_times_real @ X4 @ Y3 ) )
% 0.23/0.52 = ( times_times_real @ ( sgn_sgn_real @ X4 ) @ ( sgn_sgn_real @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Real_Vector_Spaces.sgn_mult
% 0.23/0.52 thf(fact_6726_sgn__mult,axiom,
% 0.23/0.52 ! [A: complex,B2: complex] :
% 0.23/0.52 ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B2 ) )
% 0.23/0.52 = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult
% 0.23/0.52 thf(fact_6727_sgn__mult,axiom,
% 0.23/0.52 ! [A: code_integer,B2: code_integer] :
% 0.23/0.52 ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 0.23/0.52 = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult
% 0.23/0.52 thf(fact_6728_sgn__mult,axiom,
% 0.23/0.52 ! [A: real,B2: real] :
% 0.23/0.52 ( ( sgn_sgn_real @ ( times_times_real @ A @ B2 ) )
% 0.23/0.52 = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult
% 0.23/0.52 thf(fact_6729_sgn__mult,axiom,
% 0.23/0.52 ! [A: rat,B2: rat] :
% 0.23/0.52 ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B2 ) )
% 0.23/0.52 = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult
% 0.23/0.52 thf(fact_6730_sgn__mult,axiom,
% 0.23/0.52 ! [A: int,B2: int] :
% 0.23/0.52 ( ( sgn_sgn_int @ ( times_times_int @ A @ B2 ) )
% 0.23/0.52 = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult
% 0.23/0.52 thf(fact_6731_same__sgn__sgn__add,axiom,
% 0.23/0.52 ! [B2: code_integer,A: code_integer] :
% 0.23/0.52 ( ( ( sgn_sgn_Code_integer @ B2 )
% 0.23/0.52 = ( sgn_sgn_Code_integer @ A ) )
% 0.23/0.52 => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.23/0.52 = ( sgn_sgn_Code_integer @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % same_sgn_sgn_add
% 0.23/0.52 thf(fact_6732_same__sgn__sgn__add,axiom,
% 0.23/0.52 ! [B2: real,A: real] :
% 0.23/0.52 ( ( ( sgn_sgn_real @ B2 )
% 0.23/0.52 = ( sgn_sgn_real @ A ) )
% 0.23/0.52 => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B2 ) )
% 0.23/0.52 = ( sgn_sgn_real @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % same_sgn_sgn_add
% 0.23/0.52 thf(fact_6733_same__sgn__sgn__add,axiom,
% 0.23/0.52 ! [B2: rat,A: rat] :
% 0.23/0.52 ( ( ( sgn_sgn_rat @ B2 )
% 0.23/0.52 = ( sgn_sgn_rat @ A ) )
% 0.23/0.52 => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B2 ) )
% 0.23/0.52 = ( sgn_sgn_rat @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % same_sgn_sgn_add
% 0.23/0.52 thf(fact_6734_same__sgn__sgn__add,axiom,
% 0.23/0.52 ! [B2: int,A: int] :
% 0.23/0.52 ( ( ( sgn_sgn_int @ B2 )
% 0.23/0.52 = ( sgn_sgn_int @ A ) )
% 0.23/0.52 => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B2 ) )
% 0.23/0.52 = ( sgn_sgn_int @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % same_sgn_sgn_add
% 0.23/0.52 thf(fact_6735_fact__less__mono__nat,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.52 => ( ( ord_less_nat @ M @ N )
% 0.23/0.52 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_less_mono_nat
% 0.23/0.52 thf(fact_6736_sgn__not__eq__imp,axiom,
% 0.23/0.52 ! [B2: real,A: real] :
% 0.23/0.52 ( ( ( sgn_sgn_real @ B2 )
% 0.23/0.52 != ( sgn_sgn_real @ A ) )
% 0.23/0.52 => ( ( ( sgn_sgn_real @ A )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( ( sgn_sgn_real @ B2 )
% 0.23/0.52 != zero_zero_real )
% 0.23/0.52 => ( ( sgn_sgn_real @ A )
% 0.23/0.52 = ( uminus_uminus_real @ ( sgn_sgn_real @ B2 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_not_eq_imp
% 0.23/0.52 thf(fact_6737_sgn__not__eq__imp,axiom,
% 0.23/0.52 ! [B2: int,A: int] :
% 0.23/0.52 ( ( ( sgn_sgn_int @ B2 )
% 0.23/0.52 != ( sgn_sgn_int @ A ) )
% 0.23/0.52 => ( ( ( sgn_sgn_int @ A )
% 0.23/0.52 != zero_zero_int )
% 0.23/0.52 => ( ( ( sgn_sgn_int @ B2 )
% 0.23/0.52 != zero_zero_int )
% 0.23/0.52 => ( ( sgn_sgn_int @ A )
% 0.23/0.52 = ( uminus_uminus_int @ ( sgn_sgn_int @ B2 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_not_eq_imp
% 0.23/0.52 thf(fact_6738_sgn__not__eq__imp,axiom,
% 0.23/0.52 ! [B2: code_integer,A: code_integer] :
% 0.23/0.52 ( ( ( sgn_sgn_Code_integer @ B2 )
% 0.23/0.52 != ( sgn_sgn_Code_integer @ A ) )
% 0.23/0.52 => ( ( ( sgn_sgn_Code_integer @ A )
% 0.23/0.52 != zero_z3403309356797280102nteger )
% 0.23/0.52 => ( ( ( sgn_sgn_Code_integer @ B2 )
% 0.23/0.52 != zero_z3403309356797280102nteger )
% 0.23/0.52 => ( ( sgn_sgn_Code_integer @ A )
% 0.23/0.52 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B2 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_not_eq_imp
% 0.23/0.52 thf(fact_6739_sgn__not__eq__imp,axiom,
% 0.23/0.52 ! [B2: rat,A: rat] :
% 0.23/0.52 ( ( ( sgn_sgn_rat @ B2 )
% 0.23/0.52 != ( sgn_sgn_rat @ A ) )
% 0.23/0.52 => ( ( ( sgn_sgn_rat @ A )
% 0.23/0.52 != zero_zero_rat )
% 0.23/0.52 => ( ( ( sgn_sgn_rat @ B2 )
% 0.23/0.52 != zero_zero_rat )
% 0.23/0.52 => ( ( sgn_sgn_rat @ A )
% 0.23/0.52 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B2 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_not_eq_imp
% 0.23/0.52 thf(fact_6740_sgn__minus__1,axiom,
% 0.23/0.52 ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.52 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_minus_1
% 0.23/0.52 thf(fact_6741_sgn__minus__1,axiom,
% 0.23/0.52 ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.52 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_minus_1
% 0.23/0.52 thf(fact_6742_sgn__minus__1,axiom,
% 0.23/0.52 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_minus_1
% 0.23/0.52 thf(fact_6743_sgn__minus__1,axiom,
% 0.23/0.52 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.52 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_minus_1
% 0.23/0.52 thf(fact_6744_sgn__minus__1,axiom,
% 0.23/0.52 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.52 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_minus_1
% 0.23/0.52 thf(fact_6745_fact__ge__zero,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_ge_zero
% 0.23/0.52 thf(fact_6746_fact__ge__zero,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_ge_zero
% 0.23/0.52 thf(fact_6747_fact__ge__zero,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_ge_zero
% 0.23/0.52 thf(fact_6748_fact__ge__zero,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_ge_zero
% 0.23/0.52 thf(fact_6749_linordered__idom__class_Oabs__sgn,axiom,
% 0.23/0.52 ( abs_abs_Code_integer
% 0.23/0.52 = ( ^ [K3: code_integer] : ( times_3573771949741848930nteger @ K3 @ ( sgn_sgn_Code_integer @ K3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % linordered_idom_class.abs_sgn
% 0.23/0.52 thf(fact_6750_linordered__idom__class_Oabs__sgn,axiom,
% 0.23/0.52 ( abs_abs_real
% 0.23/0.52 = ( ^ [K3: real] : ( times_times_real @ K3 @ ( sgn_sgn_real @ K3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % linordered_idom_class.abs_sgn
% 0.23/0.52 thf(fact_6751_linordered__idom__class_Oabs__sgn,axiom,
% 0.23/0.52 ( abs_abs_rat
% 0.23/0.52 = ( ^ [K3: rat] : ( times_times_rat @ K3 @ ( sgn_sgn_rat @ K3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % linordered_idom_class.abs_sgn
% 0.23/0.52 thf(fact_6752_linordered__idom__class_Oabs__sgn,axiom,
% 0.23/0.52 ( abs_abs_int
% 0.23/0.52 = ( ^ [K3: int] : ( times_times_int @ K3 @ ( sgn_sgn_int @ K3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % linordered_idom_class.abs_sgn
% 0.23/0.52 thf(fact_6753_abs__mult__sgn,axiom,
% 0.23/0.52 ! [A: complex] :
% 0.23/0.52 ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % abs_mult_sgn
% 0.23/0.52 thf(fact_6754_abs__mult__sgn,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % abs_mult_sgn
% 0.23/0.52 thf(fact_6755_abs__mult__sgn,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % abs_mult_sgn
% 0.23/0.52 thf(fact_6756_abs__mult__sgn,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % abs_mult_sgn
% 0.23/0.52 thf(fact_6757_abs__mult__sgn,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % abs_mult_sgn
% 0.23/0.52 thf(fact_6758_sgn__mult__abs,axiom,
% 0.23/0.52 ! [A: complex] :
% 0.23/0.52 ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult_abs
% 0.23/0.52 thf(fact_6759_sgn__mult__abs,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult_abs
% 0.23/0.52 thf(fact_6760_sgn__mult__abs,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult_abs
% 0.23/0.52 thf(fact_6761_sgn__mult__abs,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult_abs
% 0.23/0.52 thf(fact_6762_sgn__mult__abs,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mult_abs
% 0.23/0.52 thf(fact_6763_mult__sgn__abs,axiom,
% 0.23/0.52 ! [X4: code_integer] :
% 0.23/0.52 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X4 ) @ ( abs_abs_Code_integer @ X4 ) )
% 0.23/0.52 = X4 ) ).
% 0.23/0.52
% 0.23/0.52 % mult_sgn_abs
% 0.23/0.52 thf(fact_6764_mult__sgn__abs,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( times_times_real @ ( sgn_sgn_real @ X4 ) @ ( abs_abs_real @ X4 ) )
% 0.23/0.52 = X4 ) ).
% 0.23/0.52
% 0.23/0.52 % mult_sgn_abs
% 0.23/0.52 thf(fact_6765_mult__sgn__abs,axiom,
% 0.23/0.52 ! [X4: rat] :
% 0.23/0.52 ( ( times_times_rat @ ( sgn_sgn_rat @ X4 ) @ ( abs_abs_rat @ X4 ) )
% 0.23/0.52 = X4 ) ).
% 0.23/0.52
% 0.23/0.52 % mult_sgn_abs
% 0.23/0.52 thf(fact_6766_mult__sgn__abs,axiom,
% 0.23/0.52 ! [X4: int] :
% 0.23/0.52 ( ( times_times_int @ ( sgn_sgn_int @ X4 ) @ ( abs_abs_int @ X4 ) )
% 0.23/0.52 = X4 ) ).
% 0.23/0.52
% 0.23/0.52 % mult_sgn_abs
% 0.23/0.52 thf(fact_6767_fact__not__neg,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 0.23/0.52
% 0.23/0.52 % fact_not_neg
% 0.23/0.52 thf(fact_6768_fact__not__neg,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 0.23/0.52
% 0.23/0.52 % fact_not_neg
% 0.23/0.52 thf(fact_6769_fact__not__neg,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 0.23/0.52
% 0.23/0.52 % fact_not_neg
% 0.23/0.52 thf(fact_6770_fact__not__neg,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 0.23/0.52
% 0.23/0.52 % fact_not_neg
% 0.23/0.52 thf(fact_6771_fact__gt__zero,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_gt_zero
% 0.23/0.52 thf(fact_6772_fact__gt__zero,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_gt_zero
% 0.23/0.52 thf(fact_6773_fact__gt__zero,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_gt_zero
% 0.23/0.52 thf(fact_6774_fact__gt__zero,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_gt_zero
% 0.23/0.52 thf(fact_6775_int__sgnE,axiom,
% 0.23/0.52 ! [K: int] :
% 0.23/0.52 ~ ! [N4: nat,L3: int] :
% 0.23/0.52 ( K
% 0.23/0.52 != ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % int_sgnE
% 0.23/0.52 thf(fact_6776_same__sgn__abs__add,axiom,
% 0.23/0.52 ! [B2: code_integer,A: code_integer] :
% 0.23/0.52 ( ( ( sgn_sgn_Code_integer @ B2 )
% 0.23/0.52 = ( sgn_sgn_Code_integer @ A ) )
% 0.23/0.52 => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.23/0.52 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % same_sgn_abs_add
% 0.23/0.52 thf(fact_6777_same__sgn__abs__add,axiom,
% 0.23/0.52 ! [B2: real,A: real] :
% 0.23/0.52 ( ( ( sgn_sgn_real @ B2 )
% 0.23/0.52 = ( sgn_sgn_real @ A ) )
% 0.23/0.52 => ( ( abs_abs_real @ ( plus_plus_real @ A @ B2 ) )
% 0.23/0.52 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % same_sgn_abs_add
% 0.23/0.52 thf(fact_6778_same__sgn__abs__add,axiom,
% 0.23/0.52 ! [B2: rat,A: rat] :
% 0.23/0.52 ( ( ( sgn_sgn_rat @ B2 )
% 0.23/0.52 = ( sgn_sgn_rat @ A ) )
% 0.23/0.52 => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B2 ) )
% 0.23/0.52 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % same_sgn_abs_add
% 0.23/0.52 thf(fact_6779_same__sgn__abs__add,axiom,
% 0.23/0.52 ! [B2: int,A: int] :
% 0.23/0.52 ( ( ( sgn_sgn_int @ B2 )
% 0.23/0.52 = ( sgn_sgn_int @ A ) )
% 0.23/0.52 => ( ( abs_abs_int @ ( plus_plus_int @ A @ B2 ) )
% 0.23/0.52 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % same_sgn_abs_add
% 0.23/0.52 thf(fact_6780_fact__ge__1,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_ge_1
% 0.23/0.52 thf(fact_6781_fact__ge__1,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_ge_1
% 0.23/0.52 thf(fact_6782_fact__ge__1,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_ge_1
% 0.23/0.52 thf(fact_6783_fact__ge__1,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_ge_1
% 0.23/0.52 thf(fact_6784_fact__mono,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.52 => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_mono
% 0.23/0.52 thf(fact_6785_fact__mono,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.52 => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_mono
% 0.23/0.52 thf(fact_6786_fact__mono,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.52 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_mono
% 0.23/0.52 thf(fact_6787_fact__mono,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.52 => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_mono
% 0.23/0.52 thf(fact_6788_div__eq__sgn__abs,axiom,
% 0.23/0.52 ! [K: int,L2: int] :
% 0.23/0.52 ( ( ( sgn_sgn_int @ K )
% 0.23/0.52 = ( sgn_sgn_int @ L2 ) )
% 0.23/0.52 => ( ( divide_divide_int @ K @ L2 )
% 0.23/0.52 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % div_eq_sgn_abs
% 0.23/0.52 thf(fact_6789_fact__dvd,axiom,
% 0.23/0.52 ! [N: nat,M: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.52 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_dvd
% 0.23/0.52 thf(fact_6790_fact__dvd,axiom,
% 0.23/0.52 ! [N: nat,M: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.52 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_dvd
% 0.23/0.52 thf(fact_6791_fact__dvd,axiom,
% 0.23/0.52 ! [N: nat,M: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.52 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_dvd
% 0.23/0.52 thf(fact_6792_fact__dvd,axiom,
% 0.23/0.52 ! [N: nat,M: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.52 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_dvd
% 0.23/0.52 thf(fact_6793_pochhammer__fact,axiom,
% 0.23/0.52 ( semiri5044797733671781792omplex
% 0.23/0.52 = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 0.23/0.52
% 0.23/0.52 % pochhammer_fact
% 0.23/0.52 thf(fact_6794_pochhammer__fact,axiom,
% 0.23/0.52 ( semiri773545260158071498ct_rat
% 0.23/0.52 = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 0.23/0.52
% 0.23/0.52 % pochhammer_fact
% 0.23/0.52 thf(fact_6795_pochhammer__fact,axiom,
% 0.23/0.52 ( semiri1406184849735516958ct_int
% 0.23/0.52 = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 0.23/0.52
% 0.23/0.52 % pochhammer_fact
% 0.23/0.52 thf(fact_6796_pochhammer__fact,axiom,
% 0.23/0.52 ( semiri1408675320244567234ct_nat
% 0.23/0.52 = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 0.23/0.52
% 0.23/0.52 % pochhammer_fact
% 0.23/0.52 thf(fact_6797_pochhammer__fact,axiom,
% 0.23/0.52 ( semiri2265585572941072030t_real
% 0.23/0.52 = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % pochhammer_fact
% 0.23/0.52 thf(fact_6798_fact__ge__Suc__0__nat,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_ge_Suc_0_nat
% 0.23/0.52 thf(fact_6799_sgn__1__pos,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( ( sgn_sgn_Code_integer @ A )
% 0.23/0.52 = one_one_Code_integer )
% 0.23/0.52 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1_pos
% 0.23/0.52 thf(fact_6800_sgn__1__pos,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( ( sgn_sgn_real @ A )
% 0.23/0.52 = one_one_real )
% 0.23/0.52 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1_pos
% 0.23/0.52 thf(fact_6801_sgn__1__pos,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( ( sgn_sgn_rat @ A )
% 0.23/0.52 = one_one_rat )
% 0.23/0.52 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1_pos
% 0.23/0.52 thf(fact_6802_sgn__1__pos,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( ( sgn_sgn_int @ A )
% 0.23/0.52 = one_one_int )
% 0.23/0.52 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1_pos
% 0.23/0.52 thf(fact_6803_abs__sgn__eq,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( ( A = zero_z3403309356797280102nteger )
% 0.23/0.52 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.23/0.52 = zero_z3403309356797280102nteger ) )
% 0.23/0.52 & ( ( A != zero_z3403309356797280102nteger )
% 0.23/0.52 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.23/0.52 = one_one_Code_integer ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % abs_sgn_eq
% 0.23/0.52 thf(fact_6804_abs__sgn__eq,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( ( A = zero_zero_real )
% 0.23/0.52 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.23/0.52 = zero_zero_real ) )
% 0.23/0.52 & ( ( A != zero_zero_real )
% 0.23/0.52 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.23/0.52 = one_one_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % abs_sgn_eq
% 0.23/0.52 thf(fact_6805_abs__sgn__eq,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( ( A = zero_zero_rat )
% 0.23/0.52 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.23/0.52 = zero_zero_rat ) )
% 0.23/0.52 & ( ( A != zero_zero_rat )
% 0.23/0.52 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.23/0.52 = one_one_rat ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % abs_sgn_eq
% 0.23/0.52 thf(fact_6806_abs__sgn__eq,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( ( A = zero_zero_int )
% 0.23/0.52 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.23/0.52 = zero_zero_int ) )
% 0.23/0.52 & ( ( A != zero_zero_int )
% 0.23/0.52 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.23/0.52 = one_one_int ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % abs_sgn_eq
% 0.23/0.52 thf(fact_6807_dvd__fact,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.23/0.52 => ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.52 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % dvd_fact
% 0.23/0.52 thf(fact_6808_fact__less__mono,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.52 => ( ( ord_less_nat @ M @ N )
% 0.23/0.52 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_less_mono
% 0.23/0.52 thf(fact_6809_fact__less__mono,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.52 => ( ( ord_less_nat @ M @ N )
% 0.23/0.52 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_less_mono
% 0.23/0.52 thf(fact_6810_fact__less__mono,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.52 => ( ( ord_less_nat @ M @ N )
% 0.23/0.52 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_less_mono
% 0.23/0.52 thf(fact_6811_fact__less__mono,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.52 => ( ( ord_less_nat @ M @ N )
% 0.23/0.52 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_less_mono
% 0.23/0.52 thf(fact_6812_sgn__mod,axiom,
% 0.23/0.52 ! [L2: int,K: int] :
% 0.23/0.52 ( ( L2 != zero_zero_int )
% 0.23/0.52 => ( ~ ( dvd_dvd_int @ L2 @ K )
% 0.23/0.52 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
% 0.23/0.52 = ( sgn_sgn_int @ L2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_mod
% 0.23/0.52 thf(fact_6813_fact__mod,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.52 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 0.23/0.52 = zero_zero_int ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_mod
% 0.23/0.52 thf(fact_6814_fact__mod,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.52 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 0.23/0.52 = zero_z3403309356797280102nteger ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_mod
% 0.23/0.52 thf(fact_6815_fact__mod,axiom,
% 0.23/0.52 ! [M: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.52 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 0.23/0.52 = zero_zero_nat ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_mod
% 0.23/0.52 thf(fact_6816_fact__fact__dvd__fact,axiom,
% 0.23/0.52 ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_fact_dvd_fact
% 0.23/0.52 thf(fact_6817_fact__fact__dvd__fact,axiom,
% 0.23/0.52 ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_fact_dvd_fact
% 0.23/0.52 thf(fact_6818_fact__fact__dvd__fact,axiom,
% 0.23/0.52 ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_fact_dvd_fact
% 0.23/0.52 thf(fact_6819_fact__fact__dvd__fact,axiom,
% 0.23/0.52 ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_fact_dvd_fact
% 0.23/0.52 thf(fact_6820_fact__fact__dvd__fact,axiom,
% 0.23/0.52 ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_fact_dvd_fact
% 0.23/0.52 thf(fact_6821_fact__le__power,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_le_power
% 0.23/0.52 thf(fact_6822_fact__le__power,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_le_power
% 0.23/0.52 thf(fact_6823_fact__le__power,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_le_power
% 0.23/0.52 thf(fact_6824_fact__le__power,axiom,
% 0.23/0.52 ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_le_power
% 0.23/0.52 thf(fact_6825_arccos__le__arccos,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X4 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_le_arccos
% 0.23/0.52 thf(fact_6826_arccos__eq__iff,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.52 & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
% 0.23/0.52 => ( ( ( arccos @ X4 )
% 0.23/0.52 = ( arccos @ Y3 ) )
% 0.23/0.52 = ( X4 = Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_eq_iff
% 0.23/0.52 thf(fact_6827_arccos__le__mono,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( arccos @ X4 ) @ ( arccos @ Y3 ) )
% 0.23/0.52 = ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_le_mono
% 0.23/0.52 thf(fact_6828_fact__diff__Suc,axiom,
% 0.23/0.52 ! [N: nat,M: nat] :
% 0.23/0.52 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.23/0.52 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 0.23/0.52 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_diff_Suc
% 0.23/0.52 thf(fact_6829_sgn__if,axiom,
% 0.23/0.52 ( sgn_sgn_real
% 0.23/0.52 = ( ^ [X: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_if
% 0.23/0.52 thf(fact_6830_sgn__if,axiom,
% 0.23/0.52 ( sgn_sgn_int
% 0.23/0.52 = ( ^ [X: int] : ( if_int @ ( X = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_if
% 0.23/0.52 thf(fact_6831_sgn__if,axiom,
% 0.23/0.52 ( sgn_sgn_Code_integer
% 0.23/0.52 = ( ^ [X: code_integer] : ( if_Code_integer @ ( X = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_if
% 0.23/0.52 thf(fact_6832_sgn__if,axiom,
% 0.23/0.52 ( sgn_sgn_rat
% 0.23/0.52 = ( ^ [X: rat] : ( if_rat @ ( X = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_if
% 0.23/0.52 thf(fact_6833_sgn__1__neg,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( ( sgn_sgn_real @ A )
% 0.23/0.52 = ( uminus_uminus_real @ one_one_real ) )
% 0.23/0.52 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1_neg
% 0.23/0.52 thf(fact_6834_sgn__1__neg,axiom,
% 0.23/0.52 ! [A: int] :
% 0.23/0.52 ( ( ( sgn_sgn_int @ A )
% 0.23/0.52 = ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.52 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1_neg
% 0.23/0.52 thf(fact_6835_sgn__1__neg,axiom,
% 0.23/0.52 ! [A: code_integer] :
% 0.23/0.52 ( ( ( sgn_sgn_Code_integer @ A )
% 0.23/0.52 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.52 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1_neg
% 0.23/0.52 thf(fact_6836_sgn__1__neg,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( ( sgn_sgn_rat @ A )
% 0.23/0.52 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.23/0.52 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_1_neg
% 0.23/0.52 thf(fact_6837_zsgn__def,axiom,
% 0.23/0.52 ( sgn_sgn_int
% 0.23/0.52 = ( ^ [I2: int] : ( if_int @ ( I2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % zsgn_def
% 0.23/0.52 thf(fact_6838_fact__div__fact__le__pow,axiom,
% 0.23/0.52 ! [R2: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ R2 @ N )
% 0.23/0.52 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_div_fact_le_pow
% 0.23/0.52 thf(fact_6839_binomial__fact__lemma,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 0.23/0.52 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % binomial_fact_lemma
% 0.23/0.52 thf(fact_6840_norm__sgn,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ( X4 = zero_zero_real )
% 0.23/0.52 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X4 ) )
% 0.23/0.52 = zero_zero_real ) )
% 0.23/0.52 & ( ( X4 != zero_zero_real )
% 0.23/0.52 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X4 ) )
% 0.23/0.52 = one_one_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % norm_sgn
% 0.23/0.52 thf(fact_6841_norm__sgn,axiom,
% 0.23/0.52 ! [X4: complex] :
% 0.23/0.52 ( ( ( X4 = zero_zero_complex )
% 0.23/0.52 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X4 ) )
% 0.23/0.52 = zero_zero_real ) )
% 0.23/0.52 & ( ( X4 != zero_zero_complex )
% 0.23/0.52 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X4 ) )
% 0.23/0.52 = one_one_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % norm_sgn
% 0.23/0.52 thf(fact_6842_div__sgn__abs__cancel,axiom,
% 0.23/0.52 ! [V: int,K: int,L2: int] :
% 0.23/0.52 ( ( V != zero_zero_int )
% 0.23/0.52 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L2 ) ) )
% 0.23/0.52 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % div_sgn_abs_cancel
% 0.23/0.52 thf(fact_6843_div__dvd__sgn__abs,axiom,
% 0.23/0.52 ! [L2: int,K: int] :
% 0.23/0.52 ( ( dvd_dvd_int @ L2 @ K )
% 0.23/0.52 => ( ( divide_divide_int @ K @ L2 )
% 0.23/0.52 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % div_dvd_sgn_abs
% 0.23/0.52 thf(fact_6844_choose__dvd,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % choose_dvd
% 0.23/0.52 thf(fact_6845_choose__dvd,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % choose_dvd
% 0.23/0.52 thf(fact_6846_choose__dvd,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % choose_dvd
% 0.23/0.52 thf(fact_6847_choose__dvd,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % choose_dvd
% 0.23/0.52 thf(fact_6848_choose__dvd,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % choose_dvd
% 0.23/0.52 thf(fact_6849_fact__numeral,axiom,
% 0.23/0.52 ! [K: num] :
% 0.23/0.52 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 0.23/0.52 = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_numeral
% 0.23/0.52 thf(fact_6850_fact__numeral,axiom,
% 0.23/0.52 ! [K: num] :
% 0.23/0.52 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 0.23/0.52 = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_numeral
% 0.23/0.52 thf(fact_6851_fact__numeral,axiom,
% 0.23/0.52 ! [K: num] :
% 0.23/0.52 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 0.23/0.52 = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_numeral
% 0.23/0.52 thf(fact_6852_fact__numeral,axiom,
% 0.23/0.52 ! [K: num] :
% 0.23/0.52 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 0.23/0.52 = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_numeral
% 0.23/0.52 thf(fact_6853_fact__numeral,axiom,
% 0.23/0.52 ! [K: num] :
% 0.23/0.52 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 0.23/0.52 = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_numeral
% 0.23/0.52 thf(fact_6854_arccos__lbound,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_lbound
% 0.23/0.52 thf(fact_6855_arccos__less__arccos,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X4 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_less_arccos
% 0.23/0.52 thf(fact_6856_arccos__less__mono,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.23/0.52 => ( ( ord_less_real @ ( arccos @ X4 ) @ ( arccos @ Y3 ) )
% 0.23/0.52 = ( ord_less_real @ Y3 @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_less_mono
% 0.23/0.52 thf(fact_6857_arccos__ubound,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_ubound
% 0.23/0.52 thf(fact_6858_arccos__cos,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ pi )
% 0.23/0.52 => ( ( arccos @ ( cos_real @ X4 ) )
% 0.23/0.52 = X4 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_cos
% 0.23/0.52 thf(fact_6859_cos__arccos__abs,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.23/0.52 => ( ( cos_real @ ( arccos @ Y3 ) )
% 0.23/0.52 = Y3 ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_arccos_abs
% 0.23/0.52 thf(fact_6860_arccos__cos__eq__abs,axiom,
% 0.23/0.52 ! [Theta: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 0.23/0.52 => ( ( arccos @ ( cos_real @ Theta ) )
% 0.23/0.52 = ( abs_abs_real @ Theta ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_cos_eq_abs
% 0.23/0.52 thf(fact_6861_binomial__altdef__nat,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( ( binomial @ N @ K )
% 0.23/0.52 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % binomial_altdef_nat
% 0.23/0.52 thf(fact_6862_square__fact__le__2__fact,axiom,
% 0.23/0.52 ! [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 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % square_fact_le_2_fact
% 0.23/0.52 thf(fact_6863_arccos__lt__bounded,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.23/0.52 & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_lt_bounded
% 0.23/0.52 thf(fact_6864_arccos__bounded,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.23/0.52 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_bounded
% 0.23/0.52 thf(fact_6865_sin__arccos__nonzero,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.23/0.52 => ( ( sin_real @ ( arccos @ X4 ) )
% 0.23/0.52 != zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_arccos_nonzero
% 0.23/0.52 thf(fact_6866_arccos__cos2,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X4 )
% 0.23/0.52 => ( ( arccos @ ( cos_real @ X4 ) )
% 0.23/0.52 = ( uminus_uminus_real @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_cos2
% 0.23/0.52 thf(fact_6867_arccos__minus,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.52 => ( ( arccos @ ( uminus_uminus_real @ X4 ) )
% 0.23/0.52 = ( minus_minus_real @ pi @ ( arccos @ X4 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_minus
% 0.23/0.52 thf(fact_6868_fact__num__eq__if,axiom,
% 0.23/0.52 ( semiri5044797733671781792omplex
% 0.23/0.52 = ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_num_eq_if
% 0.23/0.52 thf(fact_6869_fact__num__eq__if,axiom,
% 0.23/0.52 ( semiri773545260158071498ct_rat
% 0.23/0.52 = ( ^ [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 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_num_eq_if
% 0.23/0.52 thf(fact_6870_fact__num__eq__if,axiom,
% 0.23/0.52 ( semiri1406184849735516958ct_int
% 0.23/0.52 = ( ^ [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 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_num_eq_if
% 0.23/0.52 thf(fact_6871_fact__num__eq__if,axiom,
% 0.23/0.52 ( semiri1408675320244567234ct_nat
% 0.23/0.52 = ( ^ [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 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_num_eq_if
% 0.23/0.52 thf(fact_6872_fact__num__eq__if,axiom,
% 0.23/0.52 ( semiri2265585572941072030t_real
% 0.23/0.52 = ( ^ [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 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_num_eq_if
% 0.23/0.52 thf(fact_6873_fact__reduce,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.52 => ( ( semiri5044797733671781792omplex @ N )
% 0.23/0.52 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_reduce
% 0.23/0.52 thf(fact_6874_fact__reduce,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.52 => ( ( semiri773545260158071498ct_rat @ N )
% 0.23/0.52 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_reduce
% 0.23/0.52 thf(fact_6875_fact__reduce,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.52 => ( ( semiri1406184849735516958ct_int @ N )
% 0.23/0.52 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_reduce
% 0.23/0.52 thf(fact_6876_fact__reduce,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.52 => ( ( semiri1408675320244567234ct_nat @ N )
% 0.23/0.52 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_reduce
% 0.23/0.52 thf(fact_6877_fact__reduce,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.52 => ( ( semiri2265585572941072030t_real @ N )
% 0.23/0.52 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_reduce
% 0.23/0.52 thf(fact_6878_pochhammer__same,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
% 0.23/0.52 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % pochhammer_same
% 0.23/0.52 thf(fact_6879_pochhammer__same,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
% 0.23/0.52 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % pochhammer_same
% 0.23/0.52 thf(fact_6880_pochhammer__same,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
% 0.23/0.52 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % pochhammer_same
% 0.23/0.52 thf(fact_6881_pochhammer__same,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
% 0.23/0.52 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % pochhammer_same
% 0.23/0.52 thf(fact_6882_pochhammer__same,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
% 0.23/0.52 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % pochhammer_same
% 0.23/0.52 thf(fact_6883_binomial__fact,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % binomial_fact
% 0.23/0.52 thf(fact_6884_binomial__fact,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 0.23/0.52 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % binomial_fact
% 0.23/0.52 thf(fact_6885_binomial__fact,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 0.23/0.52 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % binomial_fact
% 0.23/0.52 thf(fact_6886_fact__binomial,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_binomial
% 0.23/0.52 thf(fact_6887_fact__binomial,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 0.23/0.52 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_binomial
% 0.23/0.52 thf(fact_6888_fact__binomial,axiom,
% 0.23/0.52 ! [K: nat,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.52 => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 0.23/0.52 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_binomial
% 0.23/0.52 thf(fact_6889_arccos,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.23/0.52 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
% 0.23/0.52 & ( ( cos_real @ ( arccos @ Y3 ) )
% 0.23/0.52 = Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos
% 0.23/0.52 thf(fact_6890_arccos__minus__abs,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.52 => ( ( arccos @ ( uminus_uminus_real @ X4 ) )
% 0.23/0.52 = ( minus_minus_real @ pi @ ( arccos @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_minus_abs
% 0.23/0.52 thf(fact_6891_div__noneq__sgn__abs,axiom,
% 0.23/0.52 ! [L2: int,K: int] :
% 0.23/0.52 ( ( L2 != zero_zero_int )
% 0.23/0.52 => ( ( ( sgn_sgn_int @ K )
% 0.23/0.52 != ( sgn_sgn_int @ L2 ) )
% 0.23/0.52 => ( ( divide_divide_int @ K @ L2 )
% 0.23/0.52 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) )
% 0.23/0.52 @ ( zero_n2684676970156552555ol_int
% 0.23/0.52 @ ~ ( dvd_dvd_int @ L2 @ K ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % div_noneq_sgn_abs
% 0.23/0.52 thf(fact_6892_arccos__le__pi2,axiom,
% 0.23/0.52 ! [Y3: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.52 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.23/0.52 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arccos_le_pi2
% 0.23/0.52 thf(fact_6893_divide__int__unfold,axiom,
% 0.23/0.52 ! [L2: int,K: int,N: nat,M: nat] :
% 0.23/0.52 ( ( ( ( ( sgn_sgn_int @ L2 )
% 0.23/0.52 = zero_zero_int )
% 0.23/0.52 | ( ( sgn_sgn_int @ K )
% 0.23/0.52 = zero_zero_int )
% 0.23/0.52 | ( N = zero_zero_nat ) )
% 0.23/0.52 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.23/0.52 = zero_zero_int ) )
% 0.23/0.52 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 0.23/0.52 = zero_zero_int )
% 0.23/0.52 | ( ( sgn_sgn_int @ K )
% 0.23/0.52 = zero_zero_int )
% 0.23/0.52 | ( N = zero_zero_nat ) )
% 0.23/0.52 => ( ( ( ( sgn_sgn_int @ K )
% 0.23/0.52 = ( sgn_sgn_int @ L2 ) )
% 0.23/0.52 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.23/0.52 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 0.23/0.52 & ( ( ( sgn_sgn_int @ K )
% 0.23/0.52 != ( sgn_sgn_int @ L2 ) )
% 0.23/0.52 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.23/0.52 = ( uminus_uminus_int
% 0.23/0.52 @ ( semiri1314217659103216013at_int
% 0.23/0.52 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 0.23/0.52 @ ( zero_n2687167440665602831ol_nat
% 0.23/0.52 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % divide_int_unfold
% 0.23/0.52 thf(fact_6894_sin__coeff__def,axiom,
% 0.23/0.52 ( sin_coeff
% 0.23/0.52 = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_coeff_def
% 0.23/0.52 thf(fact_6895_binomial__code,axiom,
% 0.23/0.52 ( binomial
% 0.23/0.52 = ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % binomial_code
% 0.23/0.52 thf(fact_6896_cos__coeff__def,axiom,
% 0.23/0.52 ( cos_coeff
% 0.23/0.52 = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_coeff_def
% 0.23/0.52 thf(fact_6897_exp__two__pi__i,axiom,
% 0.23/0.52 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 0.23/0.52 = one_one_complex ) ).
% 0.23/0.52
% 0.23/0.52 % exp_two_pi_i
% 0.23/0.52 thf(fact_6898_exp__two__pi__i_H,axiom,
% 0.23/0.52 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 0.23/0.52 = one_one_complex ) ).
% 0.23/0.52
% 0.23/0.52 % exp_two_pi_i'
% 0.23/0.52 thf(fact_6899_zero__le__sgn__iff,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X4 ) )
% 0.23/0.52 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.52
% 0.23/0.52 % zero_le_sgn_iff
% 0.23/0.52 thf(fact_6900_sgn__le__0__iff,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X4 ) @ zero_zero_real )
% 0.23/0.52 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_le_0_iff
% 0.23/0.52 thf(fact_6901_norm__ii,axiom,
% 0.23/0.52 ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % norm_ii
% 0.23/0.52 thf(fact_6902_cos__coeff__0,axiom,
% 0.23/0.52 ( ( cos_coeff @ zero_zero_nat )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % cos_coeff_0
% 0.23/0.52 thf(fact_6903_divide__i,axiom,
% 0.23/0.52 ! [X4: complex] :
% 0.23/0.52 ( ( divide1717551699836669952omplex @ X4 @ imaginary_unit )
% 0.23/0.52 = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X4 ) ) ).
% 0.23/0.52
% 0.23/0.52 % divide_i
% 0.23/0.52 thf(fact_6904_complex__i__mult__minus,axiom,
% 0.23/0.52 ! [X4: complex] :
% 0.23/0.52 ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X4 ) )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ X4 ) ) ).
% 0.23/0.52
% 0.23/0.52 % complex_i_mult_minus
% 0.23/0.52 thf(fact_6905_i__squared,axiom,
% 0.23/0.52 ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.52
% 0.23/0.52 % i_squared
% 0.23/0.52 thf(fact_6906_divide__numeral__i,axiom,
% 0.23/0.52 ! [Z: complex,N: num] :
% 0.23/0.52 ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % divide_numeral_i
% 0.23/0.52 thf(fact_6907_power2__i,axiom,
% 0.23/0.52 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.52
% 0.23/0.52 % power2_i
% 0.23/0.52 thf(fact_6908_i__even__power,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.52 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % i_even_power
% 0.23/0.52 thf(fact_6909_exp__pi__i_H,axiom,
% 0.23/0.52 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.52
% 0.23/0.52 % exp_pi_i'
% 0.23/0.52 thf(fact_6910_exp__pi__i,axiom,
% 0.23/0.52 ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.23/0.52
% 0.23/0.52 % exp_pi_i
% 0.23/0.52 thf(fact_6911_real__sgn__eq,axiom,
% 0.23/0.52 ( sgn_sgn_real
% 0.23/0.52 = ( ^ [X: real] : ( divide_divide_real @ X @ ( abs_abs_real @ X ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % real_sgn_eq
% 0.23/0.52 thf(fact_6912_sgn__eq,axiom,
% 0.23/0.52 ( sgn_sgn_complex
% 0.23/0.52 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ Z5 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z5 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_eq
% 0.23/0.52 thf(fact_6913_sin__coeff__Suc,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( sin_coeff @ ( suc @ N ) )
% 0.23/0.52 = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sin_coeff_Suc
% 0.23/0.52 thf(fact_6914_i__times__eq__iff,axiom,
% 0.23/0.52 ! [W: complex,Z: complex] :
% 0.23/0.52 ( ( ( times_times_complex @ imaginary_unit @ W )
% 0.23/0.52 = Z )
% 0.23/0.52 = ( W
% 0.23/0.52 = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % i_times_eq_iff
% 0.23/0.52 thf(fact_6915_cos__coeff__Suc,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( cos_coeff @ ( suc @ N ) )
% 0.23/0.52 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cos_coeff_Suc
% 0.23/0.52 thf(fact_6916_imaginary__unit_Ocode,axiom,
% 0.23/0.52 ( imaginary_unit
% 0.23/0.52 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 0.23/0.52
% 0.23/0.52 % imaginary_unit.code
% 0.23/0.52 thf(fact_6917_Complex__eq__i,axiom,
% 0.23/0.52 ! [X4: real,Y3: real] :
% 0.23/0.52 ( ( ( complex2 @ X4 @ Y3 )
% 0.23/0.52 = imaginary_unit )
% 0.23/0.52 = ( ( X4 = zero_zero_real )
% 0.23/0.52 & ( Y3 = one_one_real ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Complex_eq_i
% 0.23/0.52 thf(fact_6918_i__mult__Complex,axiom,
% 0.23/0.52 ! [A: real,B2: real] :
% 0.23/0.52 ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B2 ) )
% 0.23/0.52 = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % i_mult_Complex
% 0.23/0.52 thf(fact_6919_Complex__mult__i,axiom,
% 0.23/0.52 ! [A: real,B2: real] :
% 0.23/0.52 ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ imaginary_unit )
% 0.23/0.52 = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % Complex_mult_i
% 0.23/0.52 thf(fact_6920_sgn__real__def,axiom,
% 0.23/0.52 ( sgn_sgn_real
% 0.23/0.52 = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_real_def
% 0.23/0.52 thf(fact_6921_sgn__power__injE,axiom,
% 0.23/0.52 ! [A: real,N: nat,X4: real,B2: real] :
% 0.23/0.52 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.23/0.52 = X4 )
% 0.23/0.52 => ( ( X4
% 0.23/0.52 = ( times_times_real @ ( sgn_sgn_real @ B2 ) @ ( power_power_real @ ( abs_abs_real @ B2 ) @ N ) ) )
% 0.23/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.52 => ( A = B2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % sgn_power_injE
% 0.23/0.52 thf(fact_6922_fold__atLeastAtMost__nat_Osimps,axiom,
% 0.23/0.52 ( set_fo2584398358068434914at_nat
% 0.23/0.52 = ( ^ [F2: nat > nat > nat,A4: nat,B3: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B3 @ A4 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B3 @ ( F2 @ A4 @ Acc ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fold_atLeastAtMost_nat.simps
% 0.23/0.52 thf(fact_6923_fold__atLeastAtMost__nat_Oelims,axiom,
% 0.23/0.52 ! [X4: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y3: nat] :
% 0.23/0.52 ( ( ( set_fo2584398358068434914at_nat @ X4 @ Xa @ Xb @ Xc )
% 0.23/0.52 = Y3 )
% 0.23/0.52 => ( ( ( ord_less_nat @ Xb @ Xa )
% 0.23/0.52 => ( Y3 = Xc ) )
% 0.23/0.52 & ( ~ ( ord_less_nat @ Xb @ Xa )
% 0.23/0.52 => ( Y3
% 0.23/0.52 = ( set_fo2584398358068434914at_nat @ X4 @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X4 @ Xa @ Xc ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fold_atLeastAtMost_nat.elims
% 0.23/0.52 thf(fact_6924_complex__of__real__i,axiom,
% 0.23/0.52 ! [R2: real] :
% 0.23/0.52 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
% 0.23/0.52 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 0.23/0.52
% 0.23/0.52 % complex_of_real_i
% 0.23/0.52 thf(fact_6925_i__complex__of__real,axiom,
% 0.23/0.52 ! [R2: real] :
% 0.23/0.52 ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
% 0.23/0.52 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 0.23/0.52
% 0.23/0.52 % i_complex_of_real
% 0.23/0.52 thf(fact_6926_Complex__eq,axiom,
% 0.23/0.52 ( complex2
% 0.23/0.52 = ( ^ [A4: real,B3: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A4 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Complex_eq
% 0.23/0.52 thf(fact_6927_complex__split__polar,axiom,
% 0.23/0.52 ! [Z: complex] :
% 0.23/0.52 ? [R3: real,A5: real] :
% 0.23/0.52 ( Z
% 0.23/0.52 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A5 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A5 ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % complex_split_polar
% 0.23/0.52 thf(fact_6928_cmod__unit__one,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % cmod_unit_one
% 0.23/0.52 thf(fact_6929_cmod__complex__polar,axiom,
% 0.23/0.52 ! [R2: real,A: real] :
% 0.23/0.52 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
% 0.23/0.52 = ( abs_abs_real @ R2 ) ) ).
% 0.23/0.52
% 0.23/0.52 % cmod_complex_polar
% 0.23/0.52 thf(fact_6930_arctan__inverse,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( X4 != zero_zero_real )
% 0.23/0.52 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X4 ) )
% 0.23/0.52 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X4 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % arctan_inverse
% 0.23/0.52 thf(fact_6931_fact__code,axiom,
% 0.23/0.52 ( semiri5044797733671781792omplex
% 0.23/0.52 = ( ^ [N2: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_code
% 0.23/0.52 thf(fact_6932_fact__code,axiom,
% 0.23/0.52 ( semiri773545260158071498ct_rat
% 0.23/0.52 = ( ^ [N2: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_code
% 0.23/0.52 thf(fact_6933_fact__code,axiom,
% 0.23/0.52 ( semiri1406184849735516958ct_int
% 0.23/0.52 = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_code
% 0.23/0.52 thf(fact_6934_fact__code,axiom,
% 0.23/0.52 ( semiri1408675320244567234ct_nat
% 0.23/0.52 = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_code
% 0.23/0.52 thf(fact_6935_fact__code,axiom,
% 0.23/0.52 ( semiri2265585572941072030t_real
% 0.23/0.52 = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % fact_code
% 0.23/0.52 thf(fact_6936_Arg__minus__ii,axiom,
% 0.23/0.52 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 0.23/0.52 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Arg_minus_ii
% 0.23/0.52 thf(fact_6937_csqrt__ii,axiom,
% 0.23/0.52 ( ( csqrt @ imaginary_unit )
% 0.23/0.52 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % csqrt_ii
% 0.23/0.52 thf(fact_6938_Arg__ii,axiom,
% 0.23/0.52 ( ( arg @ imaginary_unit )
% 0.23/0.52 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Arg_ii
% 0.23/0.52 thf(fact_6939_cis__minus__pi__half,axiom,
% 0.23/0.52 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.23/0.52 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 0.23/0.52
% 0.23/0.52 % cis_minus_pi_half
% 0.23/0.52 thf(fact_6940_modulo__int__def,axiom,
% 0.23/0.52 ( modulo_modulo_int
% 0.23/0.52 = ( ^ [K3: int,L: int] :
% 0.23/0.52 ( if_int @ ( L = zero_zero_int ) @ K3
% 0.23/0.52 @ ( if_int
% 0.23/0.52 @ ( ( sgn_sgn_int @ K3 )
% 0.23/0.52 = ( sgn_sgn_int @ L ) )
% 0.23/0.52 @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
% 0.23/0.52 @ ( times_times_int @ ( sgn_sgn_int @ L )
% 0.23/0.52 @ ( minus_minus_int
% 0.23/0.52 @ ( times_times_int @ ( abs_abs_int @ L )
% 0.23/0.52 @ ( zero_n2684676970156552555ol_int
% 0.23/0.52 @ ~ ( dvd_dvd_int @ L @ K3 ) ) )
% 0.23/0.52 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % modulo_int_def
% 0.23/0.52 thf(fact_6941_nat__numeral,axiom,
% 0.23/0.52 ! [K: num] :
% 0.23/0.52 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 0.23/0.52 = ( numeral_numeral_nat @ K ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_numeral
% 0.23/0.52 thf(fact_6942_norm__cis,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 0.23/0.52 = one_one_real ) ).
% 0.23/0.52
% 0.23/0.52 % norm_cis
% 0.23/0.52 thf(fact_6943_nat__1,axiom,
% 0.23/0.52 ( ( nat2 @ one_one_int )
% 0.23/0.52 = ( suc @ zero_zero_nat ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_1
% 0.23/0.52 thf(fact_6944_zless__nat__conj,axiom,
% 0.23/0.52 ! [W: int,Z: int] :
% 0.23/0.52 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.23/0.52 = ( ( ord_less_int @ zero_zero_int @ Z )
% 0.23/0.52 & ( ord_less_int @ W @ Z ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % zless_nat_conj
% 0.23/0.52 thf(fact_6945_of__nat__nat__take__bit__eq,axiom,
% 0.23/0.52 ! [N: nat,K: int] :
% 0.23/0.52 ( ( semiri8010041392384452111omplex @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.23/0.52 = ( ring_17405671764205052669omplex @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_nat_nat_take_bit_eq
% 0.23/0.52 thf(fact_6946_of__nat__nat__take__bit__eq,axiom,
% 0.23/0.52 ! [N: nat,K: int] :
% 0.23/0.52 ( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.23/0.52 = ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_nat_nat_take_bit_eq
% 0.23/0.52 thf(fact_6947_of__nat__nat__take__bit__eq,axiom,
% 0.23/0.52 ! [N: nat,K: int] :
% 0.23/0.52 ( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.23/0.52 = ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_nat_nat_take_bit_eq
% 0.23/0.52 thf(fact_6948_of__nat__nat__take__bit__eq,axiom,
% 0.23/0.52 ! [N: nat,K: int] :
% 0.23/0.52 ( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.23/0.52 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_nat_nat_take_bit_eq
% 0.23/0.52 thf(fact_6949_zero__less__nat__eq,axiom,
% 0.23/0.52 ! [Z: int] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 0.23/0.52 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.23/0.52
% 0.23/0.52 % zero_less_nat_eq
% 0.23/0.52 thf(fact_6950_diff__nat__numeral,axiom,
% 0.23/0.52 ! [V: num,V2: num] :
% 0.23/0.52 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V2 ) )
% 0.23/0.52 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % diff_nat_numeral
% 0.23/0.52 thf(fact_6951_numeral__power__eq__nat__cancel__iff,axiom,
% 0.23/0.52 ! [X4: num,N: nat,Y3: int] :
% 0.23/0.52 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 0.23/0.52 = ( nat2 @ Y3 ) )
% 0.23/0.52 = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 0.23/0.52 = Y3 ) ) ).
% 0.23/0.52
% 0.23/0.52 % numeral_power_eq_nat_cancel_iff
% 0.23/0.52 thf(fact_6952_nat__eq__numeral__power__cancel__iff,axiom,
% 0.23/0.52 ! [Y3: int,X4: num,N: nat] :
% 0.23/0.52 ( ( ( nat2 @ Y3 )
% 0.23/0.52 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
% 0.23/0.52 = ( Y3
% 0.23/0.52 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_eq_numeral_power_cancel_iff
% 0.23/0.52 thf(fact_6953_power2__csqrt,axiom,
% 0.23/0.52 ! [Z: complex] :
% 0.23/0.52 ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.52 = Z ) ).
% 0.23/0.52
% 0.23/0.52 % power2_csqrt
% 0.23/0.52 thf(fact_6954_nat__ceiling__le__eq,axiom,
% 0.23/0.52 ! [X4: real,A: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X4 ) ) @ A )
% 0.23/0.52 = ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_ceiling_le_eq
% 0.23/0.52 thf(fact_6955_one__less__nat__eq,axiom,
% 0.23/0.52 ! [Z: int] :
% 0.23/0.52 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 0.23/0.52 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.23/0.52
% 0.23/0.52 % one_less_nat_eq
% 0.23/0.52 thf(fact_6956_nat__numeral__diff__1,axiom,
% 0.23/0.52 ! [V: num] :
% 0.23/0.52 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 0.23/0.52 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_numeral_diff_1
% 0.23/0.52 thf(fact_6957_numeral__power__less__nat__cancel__iff,axiom,
% 0.23/0.52 ! [X4: num,N: nat,A: int] :
% 0.23/0.52 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) @ ( nat2 @ A ) )
% 0.23/0.52 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % numeral_power_less_nat_cancel_iff
% 0.23/0.52 thf(fact_6958_nat__less__numeral__power__cancel__iff,axiom,
% 0.23/0.52 ! [A: int,X4: num,N: nat] :
% 0.23/0.52 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
% 0.23/0.52 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_less_numeral_power_cancel_iff
% 0.23/0.52 thf(fact_6959_numeral__power__le__nat__cancel__iff,axiom,
% 0.23/0.52 ! [X4: num,N: nat,A: int] :
% 0.23/0.52 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) @ ( nat2 @ A ) )
% 0.23/0.52 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 0.23/0.52
% 0.23/0.52 % numeral_power_le_nat_cancel_iff
% 0.23/0.52 thf(fact_6960_nat__le__numeral__power__cancel__iff,axiom,
% 0.23/0.52 ! [A: int,X4: num,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
% 0.23/0.52 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_le_numeral_power_cancel_iff
% 0.23/0.52 thf(fact_6961_cis__pi__half,axiom,
% 0.23/0.52 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.52 = imaginary_unit ) ).
% 0.23/0.52
% 0.23/0.52 % cis_pi_half
% 0.23/0.52 thf(fact_6962_cis__2pi,axiom,
% 0.23/0.52 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.23/0.52 = one_one_complex ) ).
% 0.23/0.52
% 0.23/0.52 % cis_2pi
% 0.23/0.52 thf(fact_6963_cis__divide,axiom,
% 0.23/0.52 ! [A: real,B2: real] :
% 0.23/0.52 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B2 ) )
% 0.23/0.52 = ( cis @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cis_divide
% 0.23/0.52 thf(fact_6964_nat__zero__as__int,axiom,
% 0.23/0.52 ( zero_zero_nat
% 0.23/0.52 = ( nat2 @ zero_zero_int ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_zero_as_int
% 0.23/0.52 thf(fact_6965_nat__numeral__as__int,axiom,
% 0.23/0.52 ( numeral_numeral_nat
% 0.23/0.52 = ( ^ [I2: num] : ( nat2 @ ( numeral_numeral_int @ I2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_numeral_as_int
% 0.23/0.52 thf(fact_6966_nat__mono,axiom,
% 0.23/0.52 ! [X4: int,Y3: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.52 => ( ord_less_eq_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_mono
% 0.23/0.52 thf(fact_6967_nat__one__as__int,axiom,
% 0.23/0.52 ( one_one_nat
% 0.23/0.52 = ( nat2 @ one_one_int ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_one_as_int
% 0.23/0.52 thf(fact_6968_unset__bit__nat__def,axiom,
% 0.23/0.52 ( bit_se4205575877204974255it_nat
% 0.23/0.52 = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M6 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % unset_bit_nat_def
% 0.23/0.52 thf(fact_6969_DeMoivre,axiom,
% 0.23/0.52 ! [A: real,N: nat] :
% 0.23/0.52 ( ( power_power_complex @ ( cis @ A ) @ N )
% 0.23/0.52 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % DeMoivre
% 0.23/0.52 thf(fact_6970_nat__mask__eq,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.23/0.52 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_mask_eq
% 0.23/0.52 thf(fact_6971_Arg__correct,axiom,
% 0.23/0.52 ! [Z: complex] :
% 0.23/0.52 ( ( Z != zero_zero_complex )
% 0.23/0.52 => ( ( ( sgn_sgn_complex @ Z )
% 0.23/0.52 = ( cis @ ( arg @ Z ) ) )
% 0.23/0.52 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.23/0.52 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Arg_correct
% 0.23/0.52 thf(fact_6972_cis__Arg__unique,axiom,
% 0.23/0.52 ! [Z: complex,X4: real] :
% 0.23/0.52 ( ( ( sgn_sgn_complex @ Z )
% 0.23/0.52 = ( cis @ X4 ) )
% 0.23/0.52 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_real @ X4 @ pi )
% 0.23/0.52 => ( ( arg @ Z )
% 0.23/0.52 = X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cis_Arg_unique
% 0.23/0.52 thf(fact_6973_cis__mult,axiom,
% 0.23/0.52 ! [A: real,B2: real] :
% 0.23/0.52 ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B2 ) )
% 0.23/0.52 = ( cis @ ( plus_plus_real @ A @ B2 ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cis_mult
% 0.23/0.52 thf(fact_6974_nat__mono__iff,axiom,
% 0.23/0.52 ! [Z: int,W: int] :
% 0.23/0.52 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.23/0.52 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.23/0.52 = ( ord_less_int @ W @ Z ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_mono_iff
% 0.23/0.52 thf(fact_6975_of__nat__ceiling,axiom,
% 0.23/0.52 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_nat_ceiling
% 0.23/0.52 thf(fact_6976_of__nat__ceiling,axiom,
% 0.23/0.52 ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_nat_ceiling
% 0.23/0.52 thf(fact_6977_zless__nat__eq__int__zless,axiom,
% 0.23/0.52 ! [M: nat,Z: int] :
% 0.23/0.52 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 0.23/0.52 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 0.23/0.52
% 0.23/0.52 % zless_nat_eq_int_zless
% 0.23/0.52 thf(fact_6978_nat__le__iff,axiom,
% 0.23/0.52 ! [X4: int,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_nat @ ( nat2 @ X4 ) @ N )
% 0.23/0.52 = ( ord_less_eq_int @ X4 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_le_iff
% 0.23/0.52 thf(fact_6979_nat__int__add,axiom,
% 0.23/0.52 ! [A: nat,B2: nat] :
% 0.23/0.52 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
% 0.23/0.52 = ( plus_plus_nat @ A @ B2 ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_int_add
% 0.23/0.52 thf(fact_6980_int__minus,axiom,
% 0.23/0.52 ! [N: nat,M: nat] :
% 0.23/0.52 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 0.23/0.52 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % int_minus
% 0.23/0.52 thf(fact_6981_nat__abs__mult__distrib,axiom,
% 0.23/0.52 ! [W: int,Z: int] :
% 0.23/0.52 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 0.23/0.52 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_abs_mult_distrib
% 0.23/0.52 thf(fact_6982_and__nat__def,axiom,
% 0.23/0.52 ( bit_se727722235901077358nd_nat
% 0.23/0.52 = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % and_nat_def
% 0.23/0.52 thf(fact_6983_real__nat__ceiling__ge,axiom,
% 0.23/0.52 ! [X4: real] : ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % real_nat_ceiling_ge
% 0.23/0.52 thf(fact_6984_of__nat__floor,axiom,
% 0.23/0.52 ! [R2: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.23/0.52 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_nat_floor
% 0.23/0.52 thf(fact_6985_of__nat__floor,axiom,
% 0.23/0.52 ! [R2: rat] :
% 0.23/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 0.23/0.52 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_nat_floor
% 0.23/0.52 thf(fact_6986_nat__less__eq__zless,axiom,
% 0.23/0.52 ! [W: int,Z: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.23/0.52 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.23/0.52 = ( ord_less_int @ W @ Z ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_less_eq_zless
% 0.23/0.52 thf(fact_6987_nat__le__eq__zle,axiom,
% 0.23/0.52 ! [W: int,Z: int] :
% 0.23/0.52 ( ( ( ord_less_int @ zero_zero_int @ W )
% 0.23/0.52 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 0.23/0.52 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.23/0.52 = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_le_eq_zle
% 0.23/0.52 thf(fact_6988_split__nat,axiom,
% 0.23/0.52 ! [P: nat > $o,I: int] :
% 0.23/0.52 ( ( P @ ( nat2 @ I ) )
% 0.23/0.52 = ( ! [N2: nat] :
% 0.23/0.52 ( ( I
% 0.23/0.52 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.23/0.52 => ( P @ N2 ) )
% 0.23/0.52 & ( ( ord_less_int @ I @ zero_zero_int )
% 0.23/0.52 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % split_nat
% 0.23/0.52 thf(fact_6989_le__mult__nat__floor,axiom,
% 0.23/0.52 ! [A: real,B2: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % le_mult_nat_floor
% 0.23/0.52 thf(fact_6990_le__mult__nat__floor,axiom,
% 0.23/0.52 ! [A: rat,B2: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % le_mult_nat_floor
% 0.23/0.52 thf(fact_6991_le__nat__iff,axiom,
% 0.23/0.52 ! [K: int,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.52 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 0.23/0.52 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % le_nat_iff
% 0.23/0.52 thf(fact_6992_nat__add__distrib,axiom,
% 0.23/0.52 ! [Z: int,Z6: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.23/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.23/0.52 => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
% 0.23/0.52 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_add_distrib
% 0.23/0.52 thf(fact_6993_nat__mult__distrib,axiom,
% 0.23/0.52 ! [Z: int,Z6: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.23/0.52 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 0.23/0.52 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_mult_distrib
% 0.23/0.52 thf(fact_6994_Suc__as__int,axiom,
% 0.23/0.52 ( suc
% 0.23/0.52 = ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Suc_as_int
% 0.23/0.52 thf(fact_6995_nat__diff__distrib,axiom,
% 0.23/0.52 ! [Z6: int,Z: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.23/0.52 => ( ( ord_less_eq_int @ Z6 @ Z )
% 0.23/0.52 => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
% 0.23/0.52 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_diff_distrib
% 0.23/0.52 thf(fact_6996_nat__diff__distrib_H,axiom,
% 0.23/0.52 ! [X4: int,Y3: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.52 => ( ( nat2 @ ( minus_minus_int @ X4 @ Y3 ) )
% 0.23/0.52 = ( minus_minus_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_diff_distrib'
% 0.23/0.52 thf(fact_6997_nat__abs__triangle__ineq,axiom,
% 0.23/0.52 ! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_abs_triangle_ineq
% 0.23/0.52 thf(fact_6998_nat__div__distrib,axiom,
% 0.23/0.52 ! [X4: int,Y3: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.52 => ( ( nat2 @ ( divide_divide_int @ X4 @ Y3 ) )
% 0.23/0.52 = ( divide_divide_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_div_distrib
% 0.23/0.52 thf(fact_6999_nat__div__distrib_H,axiom,
% 0.23/0.52 ! [Y3: int,X4: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.52 => ( ( nat2 @ ( divide_divide_int @ X4 @ Y3 ) )
% 0.23/0.52 = ( divide_divide_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_div_distrib'
% 0.23/0.52 thf(fact_7000_nat__power__eq,axiom,
% 0.23/0.52 ! [Z: int,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.23/0.52 => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 0.23/0.52 = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_power_eq
% 0.23/0.52 thf(fact_7001_nat__floor__neg,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.23/0.52 => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.52 = zero_zero_nat ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_floor_neg
% 0.23/0.52 thf(fact_7002_nat__mod__distrib,axiom,
% 0.23/0.52 ! [X4: int,Y3: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.52 => ( ( nat2 @ ( modulo_modulo_int @ X4 @ Y3 ) )
% 0.23/0.52 = ( modulo_modulo_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_mod_distrib
% 0.23/0.52 thf(fact_7003_div__abs__eq__div__nat,axiom,
% 0.23/0.52 ! [K: int,L2: int] :
% 0.23/0.52 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
% 0.23/0.52 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % div_abs_eq_div_nat
% 0.23/0.52 thf(fact_7004_floor__eq3,axiom,
% 0.23/0.52 ! [N: nat,X4: real] :
% 0.23/0.52 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.23/0.52 => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.52 = N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % floor_eq3
% 0.23/0.52 thf(fact_7005_le__nat__floor,axiom,
% 0.23/0.52 ! [X4: nat,A: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ A )
% 0.23/0.52 => ( ord_less_eq_nat @ X4 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % le_nat_floor
% 0.23/0.52 thf(fact_7006_mod__abs__eq__div__nat,axiom,
% 0.23/0.52 ! [K: int,L2: int] :
% 0.23/0.52 ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
% 0.23/0.52 = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % mod_abs_eq_div_nat
% 0.23/0.52 thf(fact_7007_nat__take__bit__eq,axiom,
% 0.23/0.52 ! [K: int,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.52 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.23/0.52 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_take_bit_eq
% 0.23/0.52 thf(fact_7008_take__bit__nat__eq,axiom,
% 0.23/0.52 ! [K: int,N: nat] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.52 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 0.23/0.52 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % take_bit_nat_eq
% 0.23/0.52 thf(fact_7009_bit__nat__iff,axiom,
% 0.23/0.52 ! [K: int,N: nat] :
% 0.23/0.52 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 0.23/0.52 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.52 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % bit_nat_iff
% 0.23/0.52 thf(fact_7010_nat__2,axiom,
% 0.23/0.52 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.52 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_2
% 0.23/0.52 thf(fact_7011_Suc__nat__eq__nat__zadd1,axiom,
% 0.23/0.52 ! [Z: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.23/0.52 => ( ( suc @ ( nat2 @ Z ) )
% 0.23/0.52 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Suc_nat_eq_nat_zadd1
% 0.23/0.52 thf(fact_7012_nat__less__iff,axiom,
% 0.23/0.52 ! [W: int,M: nat] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.23/0.52 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 0.23/0.52 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_less_iff
% 0.23/0.52 thf(fact_7013_nat__mult__distrib__neg,axiom,
% 0.23/0.52 ! [Z: int,Z6: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.23/0.52 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 0.23/0.52 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_mult_distrib_neg
% 0.23/0.52 thf(fact_7014_nat__abs__int__diff,axiom,
% 0.23/0.52 ! [A: nat,B2: nat] :
% 0.23/0.52 ( ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.52 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
% 0.23/0.52 = ( minus_minus_nat @ B2 @ A ) ) )
% 0.23/0.52 & ( ~ ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.52 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
% 0.23/0.52 = ( minus_minus_nat @ A @ B2 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % nat_abs_int_diff
% 0.23/0.52 thf(fact_7015_floor__eq4,axiom,
% 0.23/0.52 ! [N: nat,X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X4 )
% 0.23/0.52 => ( ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.23/0.52 => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
% 0.23/0.52 = N ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % floor_eq4
% 0.23/0.52 thf(fact_7016_cis__conv__exp,axiom,
% 0.23/0.52 ( cis
% 0.23/0.52 = ( ^ [B3: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % cis_conv_exp
% 0.23/0.52 thf(fact_7017_of__int__of__nat,axiom,
% 0.23/0.52 ( ring_18347121197199848620nteger
% 0.23/0.52 = ( ^ [K3: int] : ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_int_of_nat
% 0.23/0.52 thf(fact_7018_of__int__of__nat,axiom,
% 0.23/0.52 ( ring_17405671764205052669omplex
% 0.23/0.52 = ( ^ [K3: int] : ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_int_of_nat
% 0.23/0.52 thf(fact_7019_of__int__of__nat,axiom,
% 0.23/0.52 ( ring_1_of_int_real
% 0.23/0.52 = ( ^ [K3: int] : ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_int_of_nat
% 0.23/0.52 thf(fact_7020_of__int__of__nat,axiom,
% 0.23/0.52 ( ring_1_of_int_rat
% 0.23/0.52 = ( ^ [K3: int] : ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_int_of_nat
% 0.23/0.52 thf(fact_7021_of__int__of__nat,axiom,
% 0.23/0.52 ( ring_1_of_int_int
% 0.23/0.52 = ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_int_of_nat
% 0.23/0.52 thf(fact_7022_of__real__sqrt,axiom,
% 0.23/0.52 ! [X4: real] :
% 0.23/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X4 ) )
% 0.23/0.52 = ( csqrt @ ( real_V4546457046886955230omplex @ X4 ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % of_real_sqrt
% 0.23/0.52 thf(fact_7023_Arg__bounded,axiom,
% 0.23/0.52 ! [Z: complex] :
% 0.23/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.23/0.52 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 0.23/0.52
% 0.23/0.52 % Arg_bounded
% 0.23/0.52 thf(fact_7024_even__nat__iff,axiom,
% 0.23/0.52 ! [K: int] :
% 0.23/0.52 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.52 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 0.23/0.52 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % even_nat_iff
% 0.23/0.52 thf(fact_7025_powr__int,axiom,
% 0.23/0.52 ! [X4: real,I: int] :
% 0.23/0.52 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.23/0.52 => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ I ) )
% 0.23/0.52 = ( power_power_real @ X4 @ ( nat2 @ I ) ) ) )
% 0.23/0.52 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 0.23/0.52 => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ I ) )
% 0.23/0.52 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X4 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % powr_int
% 0.23/0.52 thf(fact_7026_divide__int__def,axiom,
% 0.23/0.52 ( divide_divide_int
% 0.23/0.52 = ( ^ [K3: int,L: int] :
% 0.23/0.52 ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
% 0.23/0.52 @ ( if_int
% 0.23/0.52 @ ( ( sgn_sgn_int @ K3 )
% 0.23/0.52 = ( sgn_sgn_int @ L ) )
% 0.23/0.52 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
% 0.23/0.52 @ ( uminus_uminus_int
% 0.23/0.52 @ ( semiri1314217659103216013at_int
% 0.23/0.52 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
% 0.23/0.52 @ ( zero_n2687167440665602831ol_nat
% 0.23/0.52 @ ~ ( dvd_dvd_int @ L @ K3 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % divide_int_def
% 0.23/0.52 thf(fact_7027_cis__multiple__2pi,axiom,
% 0.23/0.52 ! [N: real] :
% 0.23/0.52 ( ( member_real @ N @ ring_1_Ints_real )
% 0.23/0.52 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.23/0.52 = one_one_complex ) ) ).
% 0.23/0.52
% 0.23/0.52 % cis_multiple_2pi
% 0.23/0.52 thf(fact_7028_powr__real__of__int,axiom,
% 0.23/0.52 ! [X4: real,N: int] :
% 0.23/0.52 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.52 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.23/0.52 => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N ) )
% 0.23/0.52 = ( power_power_real @ X4 @ ( nat2 @ N ) ) ) )
% 0.23/0.52 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 0.23/0.52 => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N ) )
% 0.23/0.52 = ( inverse_inverse_real @ ( power_power_real @ X4 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % powr_real_of_int
% 0.23/0.52 thf(fact_7029_Suc__0__xor__eq,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.23/0.52 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.52 @ ( zero_n2687167440665602831ol_nat
% 0.23/0.52 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % Suc_0_xor_eq
% 0.23/0.52 thf(fact_7030_xor__Suc__0__eq,axiom,
% 0.23/0.52 ! [N: nat] :
% 0.23/0.52 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.23/0.52 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.52 @ ( zero_n2687167440665602831ol_nat
% 0.23/0.52 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % xor_Suc_0_eq
% 0.23/0.52 thf(fact_7031_gbinomial__absorption_H,axiom,
% 0.23/0.52 ! [K: nat,A: complex] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.23/0.52 => ( ( gbinomial_complex @ A @ K )
% 0.23/0.52 = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % gbinomial_absorption'
% 0.23/0.52 thf(fact_7032_gbinomial__absorption_H,axiom,
% 0.23/0.52 ! [K: nat,A: real] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.23/0.52 => ( ( gbinomial_real @ A @ K )
% 0.23/0.52 = ( 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 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % gbinomial_absorption'
% 0.23/0.52 thf(fact_7033_gbinomial__absorption_H,axiom,
% 0.23/0.52 ! [K: nat,A: rat] :
% 0.23/0.52 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.23/0.52 => ( ( gbinomial_rat @ A @ K )
% 0.23/0.52 = ( 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 ) ) ) ) ) ).
% 0.23/0.52
% 0.23/0.52 % gbinomial_absorption'
% 0.23/0.52 thf(fact_7034_inverse__inverse__eq,axiom,
% 0.23/0.52 ! [A: real] :
% 0.23/0.52 ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % inverse_inverse_eq
% 0.23/0.52 thf(fact_7035_inverse__inverse__eq,axiom,
% 0.23/0.52 ! [A: complex] :
% 0.23/0.52 ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % inverse_inverse_eq
% 0.23/0.52 thf(fact_7036_inverse__inverse__eq,axiom,
% 0.23/0.52 ! [A: rat] :
% 0.23/0.52 ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 0.23/0.52 = A ) ).
% 0.23/0.52
% 0.23/0.52 % inverse_inverse_eq
% 0.23/0.52 thf(fact_7037_inverse__eq__iff__eq,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ( inverse_inverse_real @ A )
% 0.23/0.53 = ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 = ( A = B2 ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_iff_eq
% 0.23/0.53 thf(fact_7038_inverse__eq__iff__eq,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( ( invers8013647133539491842omplex @ A )
% 0.23/0.53 = ( invers8013647133539491842omplex @ B2 ) )
% 0.23/0.53 = ( A = B2 ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_iff_eq
% 0.23/0.53 thf(fact_7039_inverse__eq__iff__eq,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ( inverse_inverse_rat @ A )
% 0.23/0.53 = ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 = ( A = B2 ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_iff_eq
% 0.23/0.53 thf(fact_7040_bit_Oxor__left__self,axiom,
% 0.23/0.53 ! [X4: int,Y3: int] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) )
% 0.23/0.53 = Y3 ) ).
% 0.23/0.53
% 0.23/0.53 % bit.xor_left_self
% 0.23/0.53 thf(fact_7041_inverse__nonzero__iff__nonzero,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ( inverse_inverse_real @ A )
% 0.23/0.53 = zero_zero_real )
% 0.23/0.53 = ( A = zero_zero_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_nonzero_iff_nonzero
% 0.23/0.53 thf(fact_7042_inverse__nonzero__iff__nonzero,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( ( invers8013647133539491842omplex @ A )
% 0.23/0.53 = zero_zero_complex )
% 0.23/0.53 = ( A = zero_zero_complex ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_nonzero_iff_nonzero
% 0.23/0.53 thf(fact_7043_inverse__nonzero__iff__nonzero,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ( inverse_inverse_rat @ A )
% 0.23/0.53 = zero_zero_rat )
% 0.23/0.53 = ( A = zero_zero_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_nonzero_iff_nonzero
% 0.23/0.53 thf(fact_7044_inverse__zero,axiom,
% 0.23/0.53 ( ( inverse_inverse_real @ zero_zero_real )
% 0.23/0.53 = zero_zero_real ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_zero
% 0.23/0.53 thf(fact_7045_inverse__zero,axiom,
% 0.23/0.53 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 0.23/0.53 = zero_zero_complex ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_zero
% 0.23/0.53 thf(fact_7046_inverse__zero,axiom,
% 0.23/0.53 ( ( inverse_inverse_rat @ zero_zero_rat )
% 0.23/0.53 = zero_zero_rat ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_zero
% 0.23/0.53 thf(fact_7047_inverse__mult__distrib,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( inverse_inverse_real @ ( times_times_real @ A @ B2 ) )
% 0.23/0.53 = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_mult_distrib
% 0.23/0.53 thf(fact_7048_inverse__mult__distrib,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B2 ) )
% 0.23/0.53 = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_mult_distrib
% 0.23/0.53 thf(fact_7049_inverse__mult__distrib,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B2 ) )
% 0.23/0.53 = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_mult_distrib
% 0.23/0.53 thf(fact_7050_inverse__eq__1__iff,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ( inverse_inverse_real @ X4 )
% 0.23/0.53 = one_one_real )
% 0.23/0.53 = ( X4 = one_one_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_1_iff
% 0.23/0.53 thf(fact_7051_inverse__eq__1__iff,axiom,
% 0.23/0.53 ! [X4: complex] :
% 0.23/0.53 ( ( ( invers8013647133539491842omplex @ X4 )
% 0.23/0.53 = one_one_complex )
% 0.23/0.53 = ( X4 = one_one_complex ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_1_iff
% 0.23/0.53 thf(fact_7052_inverse__eq__1__iff,axiom,
% 0.23/0.53 ! [X4: rat] :
% 0.23/0.53 ( ( ( inverse_inverse_rat @ X4 )
% 0.23/0.53 = one_one_rat )
% 0.23/0.53 = ( X4 = one_one_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_1_iff
% 0.23/0.53 thf(fact_7053_inverse__1,axiom,
% 0.23/0.53 ( ( inverse_inverse_real @ one_one_real )
% 0.23/0.53 = one_one_real ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_1
% 0.23/0.53 thf(fact_7054_inverse__1,axiom,
% 0.23/0.53 ( ( invers8013647133539491842omplex @ one_one_complex )
% 0.23/0.53 = one_one_complex ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_1
% 0.23/0.53 thf(fact_7055_inverse__1,axiom,
% 0.23/0.53 ( ( inverse_inverse_rat @ one_one_rat )
% 0.23/0.53 = one_one_rat ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_1
% 0.23/0.53 thf(fact_7056_inverse__divide,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B2 ) )
% 0.23/0.53 = ( divide_divide_real @ B2 @ A ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_divide
% 0.23/0.53 thf(fact_7057_inverse__divide,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.23/0.53 = ( divide1717551699836669952omplex @ B2 @ A ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_divide
% 0.23/0.53 thf(fact_7058_inverse__divide,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.23/0.53 = ( divide_divide_rat @ B2 @ A ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_divide
% 0.23/0.53 thf(fact_7059_inverse__minus__eq,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 0.23/0.53 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_minus_eq
% 0.23/0.53 thf(fact_7060_inverse__minus__eq,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.23/0.53 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_minus_eq
% 0.23/0.53 thf(fact_7061_inverse__minus__eq,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 0.23/0.53 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_minus_eq
% 0.23/0.53 thf(fact_7062_xor_Oright__neutral,axiom,
% 0.23/0.53 ! [A: nat] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 0.23/0.53 = A ) ).
% 0.23/0.53
% 0.23/0.53 % xor.right_neutral
% 0.23/0.53 thf(fact_7063_xor_Oright__neutral,axiom,
% 0.23/0.53 ! [A: int] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 0.23/0.53 = A ) ).
% 0.23/0.53
% 0.23/0.53 % xor.right_neutral
% 0.23/0.53 thf(fact_7064_xor_Oleft__neutral,axiom,
% 0.23/0.53 ! [A: nat] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 0.23/0.53 = A ) ).
% 0.23/0.53
% 0.23/0.53 % xor.left_neutral
% 0.23/0.53 thf(fact_7065_xor_Oleft__neutral,axiom,
% 0.23/0.53 ! [A: int] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 0.23/0.53 = A ) ).
% 0.23/0.53
% 0.23/0.53 % xor.left_neutral
% 0.23/0.53 thf(fact_7066_xor__self__eq,axiom,
% 0.23/0.53 ! [A: nat] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ A @ A )
% 0.23/0.53 = zero_zero_nat ) ).
% 0.23/0.53
% 0.23/0.53 % xor_self_eq
% 0.23/0.53 thf(fact_7067_xor__self__eq,axiom,
% 0.23/0.53 ! [A: int] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ A @ A )
% 0.23/0.53 = zero_zero_int ) ).
% 0.23/0.53
% 0.23/0.53 % xor_self_eq
% 0.23/0.53 thf(fact_7068_bit_Oxor__self,axiom,
% 0.23/0.53 ! [X4: int] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ X4 @ X4 )
% 0.23/0.53 = zero_zero_int ) ).
% 0.23/0.53
% 0.23/0.53 % bit.xor_self
% 0.23/0.53 thf(fact_7069_abs__inverse,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 0.23/0.53 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % abs_inverse
% 0.23/0.53 thf(fact_7070_abs__inverse,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( abs_abs_complex @ ( invers8013647133539491842omplex @ A ) )
% 0.23/0.53 = ( invers8013647133539491842omplex @ ( abs_abs_complex @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % abs_inverse
% 0.23/0.53 thf(fact_7071_abs__inverse,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 0.23/0.53 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % abs_inverse
% 0.23/0.53 thf(fact_7072_inverse__sgn,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( inverse_inverse_real @ ( sgn_sgn_real @ A ) )
% 0.23/0.53 = ( sgn_sgn_real @ A ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_sgn
% 0.23/0.53 thf(fact_7073_inverse__sgn,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) )
% 0.23/0.53 = ( sgn_sgn_rat @ A ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_sgn
% 0.23/0.53 thf(fact_7074_sgn__inverse,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( sgn_sgn_real @ ( inverse_inverse_real @ A ) )
% 0.23/0.53 = ( inverse_inverse_real @ ( sgn_sgn_real @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % sgn_inverse
% 0.23/0.53 thf(fact_7075_sgn__inverse,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( sgn_sgn_complex @ ( invers8013647133539491842omplex @ A ) )
% 0.23/0.53 = ( invers8013647133539491842omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % sgn_inverse
% 0.23/0.53 thf(fact_7076_sgn__inverse,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( sgn_sgn_rat @ ( inverse_inverse_rat @ A ) )
% 0.23/0.53 = ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % sgn_inverse
% 0.23/0.53 thf(fact_7077_take__bit__xor,axiom,
% 0.23/0.53 ! [N: nat,A: int,B2: int] :
% 0.23/0.53 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B2 ) )
% 0.23/0.53 = ( bit_se6526347334894502574or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B2 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % take_bit_xor
% 0.23/0.53 thf(fact_7078_take__bit__xor,axiom,
% 0.23/0.53 ! [N: nat,A: nat,B2: nat] :
% 0.23/0.53 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B2 ) )
% 0.23/0.53 = ( bit_se6528837805403552850or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B2 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % take_bit_xor
% 0.23/0.53 thf(fact_7079_inverse__nonpositive__iff__nonpositive,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.23/0.53 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_nonpositive_iff_nonpositive
% 0.23/0.53 thf(fact_7080_inverse__nonpositive__iff__nonpositive,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.23/0.53 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_nonpositive_iff_nonpositive
% 0.23/0.53 thf(fact_7081_inverse__nonnegative__iff__nonnegative,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.23/0.53 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_nonnegative_iff_nonnegative
% 0.23/0.53 thf(fact_7082_inverse__nonnegative__iff__nonnegative,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.23/0.53 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_nonnegative_iff_nonnegative
% 0.23/0.53 thf(fact_7083_inverse__positive__iff__positive,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.23/0.53 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_positive_iff_positive
% 0.23/0.53 thf(fact_7084_inverse__positive__iff__positive,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.23/0.53 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_positive_iff_positive
% 0.23/0.53 thf(fact_7085_inverse__negative__iff__negative,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.23/0.53 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_negative_iff_negative
% 0.23/0.53 thf(fact_7086_inverse__negative__iff__negative,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.23/0.53 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_negative_iff_negative
% 0.23/0.53 thf(fact_7087_inverse__less__iff__less__neg,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_real @ A @ zero_zero_real )
% 0.23/0.53 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.23/0.53 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 = ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_iff_less_neg
% 0.23/0.53 thf(fact_7088_inverse__less__iff__less__neg,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.23/0.53 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.23/0.53 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 = ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_iff_less_neg
% 0.23/0.53 thf(fact_7089_inverse__less__iff__less,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.53 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.23/0.53 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 = ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_iff_less
% 0.23/0.53 thf(fact_7090_inverse__less__iff__less,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.53 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.23/0.53 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 = ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_iff_less
% 0.23/0.53 thf(fact_7091_gbinomial__0_I2_J,axiom,
% 0.23/0.53 ! [K: nat] :
% 0.23/0.53 ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 0.23/0.53 = zero_zero_real ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_0(2)
% 0.23/0.53 thf(fact_7092_gbinomial__0_I2_J,axiom,
% 0.23/0.53 ! [K: nat] :
% 0.23/0.53 ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 0.23/0.53 = zero_zero_rat ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_0(2)
% 0.23/0.53 thf(fact_7093_gbinomial__0_I2_J,axiom,
% 0.23/0.53 ! [K: nat] :
% 0.23/0.53 ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 0.23/0.53 = zero_zero_nat ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_0(2)
% 0.23/0.53 thf(fact_7094_gbinomial__0_I2_J,axiom,
% 0.23/0.53 ! [K: nat] :
% 0.23/0.53 ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 0.23/0.53 = zero_zero_int ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_0(2)
% 0.23/0.53 thf(fact_7095_gbinomial__0_I1_J,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( gbinomial_complex @ A @ zero_zero_nat )
% 0.23/0.53 = one_one_complex ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_0(1)
% 0.23/0.53 thf(fact_7096_gbinomial__0_I1_J,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( gbinomial_real @ A @ zero_zero_nat )
% 0.23/0.53 = one_one_real ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_0(1)
% 0.23/0.53 thf(fact_7097_gbinomial__0_I1_J,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( gbinomial_rat @ A @ zero_zero_nat )
% 0.23/0.53 = one_one_rat ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_0(1)
% 0.23/0.53 thf(fact_7098_gbinomial__0_I1_J,axiom,
% 0.23/0.53 ! [A: nat] :
% 0.23/0.53 ( ( gbinomial_nat @ A @ zero_zero_nat )
% 0.23/0.53 = one_one_nat ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_0(1)
% 0.23/0.53 thf(fact_7099_gbinomial__0_I1_J,axiom,
% 0.23/0.53 ! [A: int] :
% 0.23/0.53 ( ( gbinomial_int @ A @ zero_zero_nat )
% 0.23/0.53 = one_one_int ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_0(1)
% 0.23/0.53 thf(fact_7100_inverse__le__iff__le,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.53 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.23/0.53 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 = ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_iff_le
% 0.23/0.53 thf(fact_7101_inverse__le__iff__le,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.53 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.23/0.53 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 = ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_iff_le
% 0.23/0.53 thf(fact_7102_inverse__le__iff__le__neg,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_real @ A @ zero_zero_real )
% 0.23/0.53 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.23/0.53 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 = ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_iff_le_neg
% 0.23/0.53 thf(fact_7103_inverse__le__iff__le__neg,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.23/0.53 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.23/0.53 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 = ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_iff_le_neg
% 0.23/0.53 thf(fact_7104_right__inverse,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
% 0.23/0.53 = one_one_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % right_inverse
% 0.23/0.53 thf(fact_7105_right__inverse,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
% 0.23/0.53 = one_one_complex ) ) ).
% 0.23/0.53
% 0.23/0.53 % right_inverse
% 0.23/0.53 thf(fact_7106_right__inverse,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
% 0.23/0.53 = one_one_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % right_inverse
% 0.23/0.53 thf(fact_7107_left__inverse,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 0.23/0.53 = one_one_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % left_inverse
% 0.23/0.53 thf(fact_7108_left__inverse,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 0.23/0.53 = one_one_complex ) ) ).
% 0.23/0.53
% 0.23/0.53 % left_inverse
% 0.23/0.53 thf(fact_7109_left__inverse,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 0.23/0.53 = one_one_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % left_inverse
% 0.23/0.53 thf(fact_7110_inverse__eq__divide__numeral,axiom,
% 0.23/0.53 ! [W: num] :
% 0.23/0.53 ( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
% 0.23/0.53 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_divide_numeral
% 0.23/0.53 thf(fact_7111_inverse__eq__divide__numeral,axiom,
% 0.23/0.53 ! [W: num] :
% 0.23/0.53 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.23/0.53 = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_divide_numeral
% 0.23/0.53 thf(fact_7112_inverse__eq__divide__numeral,axiom,
% 0.23/0.53 ! [W: num] :
% 0.23/0.53 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
% 0.23/0.53 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_divide_numeral
% 0.23/0.53 thf(fact_7113_floor__add2,axiom,
% 0.23/0.53 ! [X4: real,Y3: real] :
% 0.23/0.53 ( ( ( member_real @ X4 @ ring_1_Ints_real )
% 0.23/0.53 | ( member_real @ Y3 @ ring_1_Ints_real ) )
% 0.23/0.53 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.53 = ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % floor_add2
% 0.23/0.53 thf(fact_7114_floor__add2,axiom,
% 0.23/0.53 ! [X4: rat,Y3: rat] :
% 0.23/0.53 ( ( ( member_rat @ X4 @ ring_1_Ints_rat )
% 0.23/0.53 | ( member_rat @ Y3 @ ring_1_Ints_rat ) )
% 0.23/0.53 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
% 0.23/0.53 = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % floor_add2
% 0.23/0.53 thf(fact_7115_frac__gt__0__iff,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X4 ) )
% 0.23/0.53 = ( ~ ( member_real @ X4 @ ring_1_Ints_real ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % frac_gt_0_iff
% 0.23/0.53 thf(fact_7116_frac__gt__0__iff,axiom,
% 0.23/0.53 ! [X4: rat] :
% 0.23/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X4 ) )
% 0.23/0.53 = ( ~ ( member_rat @ X4 @ ring_1_Ints_rat ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % frac_gt_0_iff
% 0.23/0.53 thf(fact_7117_inverse__eq__divide__neg__numeral,axiom,
% 0.23/0.53 ! [W: num] :
% 0.23/0.53 ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.23/0.53 = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_divide_neg_numeral
% 0.23/0.53 thf(fact_7118_inverse__eq__divide__neg__numeral,axiom,
% 0.23/0.53 ! [W: num] :
% 0.23/0.53 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.23/0.53 = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_divide_neg_numeral
% 0.23/0.53 thf(fact_7119_inverse__eq__divide__neg__numeral,axiom,
% 0.23/0.53 ! [W: num] :
% 0.23/0.53 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.23/0.53 = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_divide_neg_numeral
% 0.23/0.53 thf(fact_7120_xor__numerals_I3_J,axiom,
% 0.23/0.53 ! [X4: num,Y3: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.23/0.53 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(3)
% 0.23/0.53 thf(fact_7121_xor__numerals_I3_J,axiom,
% 0.23/0.53 ! [X4: num,Y3: num] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 0.23/0.53 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(3)
% 0.23/0.53 thf(fact_7122_xor__numerals_I1_J,axiom,
% 0.23/0.53 ! [Y3: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.23/0.53 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(1)
% 0.23/0.53 thf(fact_7123_xor__numerals_I1_J,axiom,
% 0.23/0.53 ! [Y3: num] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 0.23/0.53 = ( numeral_numeral_int @ ( bit1 @ Y3 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(1)
% 0.23/0.53 thf(fact_7124_xor__numerals_I2_J,axiom,
% 0.23/0.53 ! [Y3: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.23/0.53 = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(2)
% 0.23/0.53 thf(fact_7125_xor__numerals_I2_J,axiom,
% 0.23/0.53 ! [Y3: num] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 0.23/0.53 = ( numeral_numeral_int @ ( bit0 @ Y3 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(2)
% 0.23/0.53 thf(fact_7126_xor__numerals_I5_J,axiom,
% 0.23/0.53 ! [X4: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ one_one_nat )
% 0.23/0.53 = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(5)
% 0.23/0.53 thf(fact_7127_xor__numerals_I5_J,axiom,
% 0.23/0.53 ! [X4: num] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ one_one_int )
% 0.23/0.53 = ( numeral_numeral_int @ ( bit1 @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(5)
% 0.23/0.53 thf(fact_7128_xor__numerals_I8_J,axiom,
% 0.23/0.53 ! [X4: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ one_one_nat )
% 0.23/0.53 = ( numeral_numeral_nat @ ( bit0 @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(8)
% 0.23/0.53 thf(fact_7129_xor__numerals_I8_J,axiom,
% 0.23/0.53 ! [X4: num] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ one_one_int )
% 0.23/0.53 = ( numeral_numeral_int @ ( bit0 @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(8)
% 0.23/0.53 thf(fact_7130_xor__numerals_I7_J,axiom,
% 0.23/0.53 ! [X4: num,Y3: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.23/0.53 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(7)
% 0.23/0.53 thf(fact_7131_xor__numerals_I7_J,axiom,
% 0.23/0.53 ! [X4: num,Y3: num] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 0.23/0.53 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(7)
% 0.23/0.53 thf(fact_7132_xor__nat__numerals_I1_J,axiom,
% 0.23/0.53 ! [Y3: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.23/0.53 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_nat_numerals(1)
% 0.23/0.53 thf(fact_7133_xor__nat__numerals_I2_J,axiom,
% 0.23/0.53 ! [Y3: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.23/0.53 = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_nat_numerals(2)
% 0.23/0.53 thf(fact_7134_xor__nat__numerals_I3_J,axiom,
% 0.23/0.53 ! [X4: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 0.23/0.53 = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_nat_numerals(3)
% 0.23/0.53 thf(fact_7135_xor__nat__numerals_I4_J,axiom,
% 0.23/0.53 ! [X4: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 0.23/0.53 = ( numeral_numeral_nat @ ( bit0 @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_nat_numerals(4)
% 0.23/0.53 thf(fact_7136_xor__numerals_I4_J,axiom,
% 0.23/0.53 ! [X4: num,Y3: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.23/0.53 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(4)
% 0.23/0.53 thf(fact_7137_xor__numerals_I4_J,axiom,
% 0.23/0.53 ! [X4: num,Y3: num] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 0.23/0.53 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(4)
% 0.23/0.53 thf(fact_7138_xor__numerals_I6_J,axiom,
% 0.23/0.53 ! [X4: num,Y3: num] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.23/0.53 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(6)
% 0.23/0.53 thf(fact_7139_xor__numerals_I6_J,axiom,
% 0.23/0.53 ! [X4: num,Y3: num] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 0.23/0.53 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor_numerals(6)
% 0.23/0.53 thf(fact_7140_power__inverse,axiom,
% 0.23/0.53 ! [A: real,N: nat] :
% 0.23/0.53 ( ( power_power_real @ ( inverse_inverse_real @ A ) @ N )
% 0.23/0.53 = ( inverse_inverse_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_inverse
% 0.23/0.53 thf(fact_7141_power__inverse,axiom,
% 0.23/0.53 ! [A: complex,N: nat] :
% 0.23/0.53 ( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N )
% 0.23/0.53 = ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_inverse
% 0.23/0.53 thf(fact_7142_power__inverse,axiom,
% 0.23/0.53 ! [A: rat,N: nat] :
% 0.23/0.53 ( ( power_power_rat @ ( inverse_inverse_rat @ A ) @ N )
% 0.23/0.53 = ( inverse_inverse_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_inverse
% 0.23/0.53 thf(fact_7143_mult__commute__imp__mult__inverse__commute,axiom,
% 0.23/0.53 ! [Y3: real,X4: real] :
% 0.23/0.53 ( ( ( times_times_real @ Y3 @ X4 )
% 0.23/0.53 = ( times_times_real @ X4 @ Y3 ) )
% 0.23/0.53 => ( ( times_times_real @ ( inverse_inverse_real @ Y3 ) @ X4 )
% 0.23/0.53 = ( times_times_real @ X4 @ ( inverse_inverse_real @ Y3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % mult_commute_imp_mult_inverse_commute
% 0.23/0.53 thf(fact_7144_mult__commute__imp__mult__inverse__commute,axiom,
% 0.23/0.53 ! [Y3: complex,X4: complex] :
% 0.23/0.53 ( ( ( times_times_complex @ Y3 @ X4 )
% 0.23/0.53 = ( times_times_complex @ X4 @ Y3 ) )
% 0.23/0.53 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y3 ) @ X4 )
% 0.23/0.53 = ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ Y3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % mult_commute_imp_mult_inverse_commute
% 0.23/0.53 thf(fact_7145_mult__commute__imp__mult__inverse__commute,axiom,
% 0.23/0.53 ! [Y3: rat,X4: rat] :
% 0.23/0.53 ( ( ( times_times_rat @ Y3 @ X4 )
% 0.23/0.53 = ( times_times_rat @ X4 @ Y3 ) )
% 0.23/0.53 => ( ( times_times_rat @ ( inverse_inverse_rat @ Y3 ) @ X4 )
% 0.23/0.53 = ( times_times_rat @ X4 @ ( inverse_inverse_rat @ Y3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % mult_commute_imp_mult_inverse_commute
% 0.23/0.53 thf(fact_7146_of__nat__xor__eq,axiom,
% 0.23/0.53 ! [M: nat,N: nat] :
% 0.23/0.53 ( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 0.23/0.53 = ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % of_nat_xor_eq
% 0.23/0.53 thf(fact_7147_of__nat__xor__eq,axiom,
% 0.23/0.53 ! [M: nat,N: nat] :
% 0.23/0.53 ( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 0.23/0.53 = ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % of_nat_xor_eq
% 0.23/0.53 thf(fact_7148_of__int__xor__eq,axiom,
% 0.23/0.53 ! [K: int,L2: int] :
% 0.23/0.53 ( ( ring_1_of_int_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
% 0.23/0.53 = ( bit_se6526347334894502574or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L2 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % of_int_xor_eq
% 0.23/0.53 thf(fact_7149_xor_Oleft__commute,axiom,
% 0.23/0.53 ! [B2: nat,A: nat,C: nat] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ B2 @ ( bit_se6528837805403552850or_nat @ A @ C ) )
% 0.23/0.53 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B2 @ C ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor.left_commute
% 0.23/0.53 thf(fact_7150_xor_Oleft__commute,axiom,
% 0.23/0.53 ! [B2: int,A: int,C: int] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ B2 @ ( bit_se6526347334894502574or_int @ A @ C ) )
% 0.23/0.53 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B2 @ C ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor.left_commute
% 0.23/0.53 thf(fact_7151_xor_Ocommute,axiom,
% 0.23/0.53 ( bit_se6528837805403552850or_nat
% 0.23/0.53 = ( ^ [A4: nat,B3: nat] : ( bit_se6528837805403552850or_nat @ B3 @ A4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor.commute
% 0.23/0.53 thf(fact_7152_xor_Ocommute,axiom,
% 0.23/0.53 ( bit_se6526347334894502574or_int
% 0.23/0.53 = ( ^ [A4: int,B3: int] : ( bit_se6526347334894502574or_int @ B3 @ A4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor.commute
% 0.23/0.53 thf(fact_7153_xor_Oassoc,axiom,
% 0.23/0.53 ! [A: nat,B2: nat,C: nat] :
% 0.23/0.53 ( ( bit_se6528837805403552850or_nat @ ( bit_se6528837805403552850or_nat @ A @ B2 ) @ C )
% 0.23/0.53 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B2 @ C ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor.assoc
% 0.23/0.53 thf(fact_7154_xor_Oassoc,axiom,
% 0.23/0.53 ! [A: int,B2: int,C: int] :
% 0.23/0.53 ( ( bit_se6526347334894502574or_int @ ( bit_se6526347334894502574or_int @ A @ B2 ) @ C )
% 0.23/0.53 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B2 @ C ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % xor.assoc
% 0.23/0.53 thf(fact_7155_inverse__eq__imp__eq,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ( inverse_inverse_real @ A )
% 0.23/0.53 = ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 => ( A = B2 ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_imp_eq
% 0.23/0.53 thf(fact_7156_inverse__eq__imp__eq,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( ( invers8013647133539491842omplex @ A )
% 0.23/0.53 = ( invers8013647133539491842omplex @ B2 ) )
% 0.23/0.53 => ( A = B2 ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_imp_eq
% 0.23/0.53 thf(fact_7157_inverse__eq__imp__eq,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ( inverse_inverse_rat @ A )
% 0.23/0.53 = ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 => ( A = B2 ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_imp_eq
% 0.23/0.53 thf(fact_7158_field__class_Ofield__inverse__zero,axiom,
% 0.23/0.53 ( ( inverse_inverse_real @ zero_zero_real )
% 0.23/0.53 = zero_zero_real ) ).
% 0.23/0.53
% 0.23/0.53 % field_class.field_inverse_zero
% 0.23/0.53 thf(fact_7159_field__class_Ofield__inverse__zero,axiom,
% 0.23/0.53 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 0.23/0.53 = zero_zero_complex ) ).
% 0.23/0.53
% 0.23/0.53 % field_class.field_inverse_zero
% 0.23/0.53 thf(fact_7160_field__class_Ofield__inverse__zero,axiom,
% 0.23/0.53 ( ( inverse_inverse_rat @ zero_zero_rat )
% 0.23/0.53 = zero_zero_rat ) ).
% 0.23/0.53
% 0.23/0.53 % field_class.field_inverse_zero
% 0.23/0.53 thf(fact_7161_inverse__zero__imp__zero,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ( inverse_inverse_real @ A )
% 0.23/0.53 = zero_zero_real )
% 0.23/0.53 => ( A = zero_zero_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_zero_imp_zero
% 0.23/0.53 thf(fact_7162_inverse__zero__imp__zero,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( ( invers8013647133539491842omplex @ A )
% 0.23/0.53 = zero_zero_complex )
% 0.23/0.53 => ( A = zero_zero_complex ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_zero_imp_zero
% 0.23/0.53 thf(fact_7163_inverse__zero__imp__zero,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ( inverse_inverse_rat @ A )
% 0.23/0.53 = zero_zero_rat )
% 0.23/0.53 => ( A = zero_zero_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_zero_imp_zero
% 0.23/0.53 thf(fact_7164_nonzero__inverse__eq__imp__eq,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ( inverse_inverse_real @ A )
% 0.23/0.53 = ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 => ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( B2 != zero_zero_real )
% 0.23/0.53 => ( A = B2 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_eq_imp_eq
% 0.23/0.53 thf(fact_7165_nonzero__inverse__eq__imp__eq,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( ( invers8013647133539491842omplex @ A )
% 0.23/0.53 = ( invers8013647133539491842omplex @ B2 ) )
% 0.23/0.53 => ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( B2 != zero_zero_complex )
% 0.23/0.53 => ( A = B2 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_eq_imp_eq
% 0.23/0.53 thf(fact_7166_nonzero__inverse__eq__imp__eq,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ( inverse_inverse_rat @ A )
% 0.23/0.53 = ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 => ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( B2 != zero_zero_rat )
% 0.23/0.53 => ( A = B2 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_eq_imp_eq
% 0.23/0.53 thf(fact_7167_nonzero__inverse__inverse__eq,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 0.23/0.53 = A ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_inverse_eq
% 0.23/0.53 thf(fact_7168_nonzero__inverse__inverse__eq,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.23/0.53 = A ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_inverse_eq
% 0.23/0.53 thf(fact_7169_nonzero__inverse__inverse__eq,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 0.23/0.53 = A ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_inverse_eq
% 0.23/0.53 thf(fact_7170_nonzero__imp__inverse__nonzero,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( inverse_inverse_real @ A )
% 0.23/0.53 != zero_zero_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_imp_inverse_nonzero
% 0.23/0.53 thf(fact_7171_nonzero__imp__inverse__nonzero,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( invers8013647133539491842omplex @ A )
% 0.23/0.53 != zero_zero_complex ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_imp_inverse_nonzero
% 0.23/0.53 thf(fact_7172_nonzero__imp__inverse__nonzero,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( inverse_inverse_rat @ A )
% 0.23/0.53 != zero_zero_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_imp_inverse_nonzero
% 0.23/0.53 thf(fact_7173_Ints__numeral,axiom,
% 0.23/0.53 ! [N: num] : ( member_complex @ ( numera6690914467698888265omplex @ N ) @ ring_1_Ints_complex ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_numeral
% 0.23/0.53 thf(fact_7174_Ints__numeral,axiom,
% 0.23/0.53 ! [N: num] : ( member_real @ ( numeral_numeral_real @ N ) @ ring_1_Ints_real ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_numeral
% 0.23/0.53 thf(fact_7175_Ints__numeral,axiom,
% 0.23/0.53 ! [N: num] : ( member_rat @ ( numeral_numeral_rat @ N ) @ ring_1_Ints_rat ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_numeral
% 0.23/0.53 thf(fact_7176_Ints__numeral,axiom,
% 0.23/0.53 ! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_numeral
% 0.23/0.53 thf(fact_7177_Ints__mult,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.23/0.53 => ( ( member_complex @ B2 @ ring_1_Ints_complex )
% 0.23/0.53 => ( member_complex @ ( times_times_complex @ A @ B2 ) @ ring_1_Ints_complex ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_mult
% 0.23/0.53 thf(fact_7178_Ints__mult,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.53 => ( ( member_real @ B2 @ ring_1_Ints_real )
% 0.23/0.53 => ( member_real @ ( times_times_real @ A @ B2 ) @ ring_1_Ints_real ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_mult
% 0.23/0.53 thf(fact_7179_Ints__mult,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( member_rat @ B2 @ ring_1_Ints_rat )
% 0.23/0.53 => ( member_rat @ ( times_times_rat @ A @ B2 ) @ ring_1_Ints_rat ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_mult
% 0.23/0.53 thf(fact_7180_Ints__mult,axiom,
% 0.23/0.53 ! [A: int,B2: int] :
% 0.23/0.53 ( ( member_int @ A @ ring_1_Ints_int )
% 0.23/0.53 => ( ( member_int @ B2 @ ring_1_Ints_int )
% 0.23/0.53 => ( member_int @ ( times_times_int @ A @ B2 ) @ ring_1_Ints_int ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_mult
% 0.23/0.53 thf(fact_7181_Ints__1,axiom,
% 0.23/0.53 member_complex @ one_one_complex @ ring_1_Ints_complex ).
% 0.23/0.53
% 0.23/0.53 % Ints_1
% 0.23/0.53 thf(fact_7182_Ints__1,axiom,
% 0.23/0.53 member_rat @ one_one_rat @ ring_1_Ints_rat ).
% 0.23/0.53
% 0.23/0.53 % Ints_1
% 0.23/0.53 thf(fact_7183_Ints__1,axiom,
% 0.23/0.53 member_int @ one_one_int @ ring_1_Ints_int ).
% 0.23/0.53
% 0.23/0.53 % Ints_1
% 0.23/0.53 thf(fact_7184_Ints__1,axiom,
% 0.23/0.53 member_real @ one_one_real @ ring_1_Ints_real ).
% 0.23/0.53
% 0.23/0.53 % Ints_1
% 0.23/0.53 thf(fact_7185_Ints__add,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.23/0.53 => ( ( member_complex @ B2 @ ring_1_Ints_complex )
% 0.23/0.53 => ( member_complex @ ( plus_plus_complex @ A @ B2 ) @ ring_1_Ints_complex ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_add
% 0.23/0.53 thf(fact_7186_Ints__add,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.53 => ( ( member_real @ B2 @ ring_1_Ints_real )
% 0.23/0.53 => ( member_real @ ( plus_plus_real @ A @ B2 ) @ ring_1_Ints_real ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_add
% 0.23/0.53 thf(fact_7187_Ints__add,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( member_rat @ B2 @ ring_1_Ints_rat )
% 0.23/0.53 => ( member_rat @ ( plus_plus_rat @ A @ B2 ) @ ring_1_Ints_rat ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_add
% 0.23/0.53 thf(fact_7188_Ints__add,axiom,
% 0.23/0.53 ! [A: int,B2: int] :
% 0.23/0.53 ( ( member_int @ A @ ring_1_Ints_int )
% 0.23/0.53 => ( ( member_int @ B2 @ ring_1_Ints_int )
% 0.23/0.53 => ( member_int @ ( plus_plus_int @ A @ B2 ) @ ring_1_Ints_int ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_add
% 0.23/0.53 thf(fact_7189_real__sqrt__inverse,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( sqrt @ ( inverse_inverse_real @ X4 ) )
% 0.23/0.53 = ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % real_sqrt_inverse
% 0.23/0.53 thf(fact_7190_Ints__diff,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.23/0.53 => ( ( member_complex @ B2 @ ring_1_Ints_complex )
% 0.23/0.53 => ( member_complex @ ( minus_minus_complex @ A @ B2 ) @ ring_1_Ints_complex ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_diff
% 0.23/0.53 thf(fact_7191_Ints__diff,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.53 => ( ( member_real @ B2 @ ring_1_Ints_real )
% 0.23/0.53 => ( member_real @ ( minus_minus_real @ A @ B2 ) @ ring_1_Ints_real ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_diff
% 0.23/0.53 thf(fact_7192_Ints__diff,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( member_rat @ B2 @ ring_1_Ints_rat )
% 0.23/0.53 => ( member_rat @ ( minus_minus_rat @ A @ B2 ) @ ring_1_Ints_rat ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_diff
% 0.23/0.53 thf(fact_7193_Ints__diff,axiom,
% 0.23/0.53 ! [A: int,B2: int] :
% 0.23/0.53 ( ( member_int @ A @ ring_1_Ints_int )
% 0.23/0.53 => ( ( member_int @ B2 @ ring_1_Ints_int )
% 0.23/0.53 => ( member_int @ ( minus_minus_int @ A @ B2 ) @ ring_1_Ints_int ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_diff
% 0.23/0.53 thf(fact_7194_Ints__power,axiom,
% 0.23/0.53 ! [A: real,N: nat] :
% 0.23/0.53 ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.53 => ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_power
% 0.23/0.53 thf(fact_7195_Ints__power,axiom,
% 0.23/0.53 ! [A: int,N: nat] :
% 0.23/0.53 ( ( member_int @ A @ ring_1_Ints_int )
% 0.23/0.53 => ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_power
% 0.23/0.53 thf(fact_7196_Ints__power,axiom,
% 0.23/0.53 ! [A: complex,N: nat] :
% 0.23/0.53 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.23/0.53 => ( member_complex @ ( power_power_complex @ A @ N ) @ ring_1_Ints_complex ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_power
% 0.23/0.53 thf(fact_7197_bit_Oconj__xor__distrib2,axiom,
% 0.23/0.53 ! [Y3: int,Z: int,X4: int] :
% 0.23/0.53 ( ( bit_se725231765392027082nd_int @ ( bit_se6526347334894502574or_int @ Y3 @ Z ) @ X4 )
% 0.23/0.53 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ Y3 @ X4 ) @ ( bit_se725231765392027082nd_int @ Z @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % bit.conj_xor_distrib2
% 0.23/0.53 thf(fact_7198_bit_Oconj__xor__distrib,axiom,
% 0.23/0.53 ! [X4: int,Y3: int,Z: int] :
% 0.23/0.53 ( ( bit_se725231765392027082nd_int @ X4 @ ( bit_se6526347334894502574or_int @ Y3 @ Z ) )
% 0.23/0.53 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ ( bit_se725231765392027082nd_int @ X4 @ Z ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % bit.conj_xor_distrib
% 0.23/0.53 thf(fact_7199_bit__xor__iff,axiom,
% 0.23/0.53 ! [A: nat,B2: nat,N: nat] :
% 0.23/0.53 ( ( bit_se1148574629649215175it_nat @ ( bit_se6528837805403552850or_nat @ A @ B2 ) @ N )
% 0.23/0.53 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.23/0.53 != ( bit_se1148574629649215175it_nat @ B2 @ N ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % bit_xor_iff
% 0.23/0.53 thf(fact_7200_bit__xor__iff,axiom,
% 0.23/0.53 ! [A: int,B2: int,N: nat] :
% 0.23/0.53 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ A @ B2 ) @ N )
% 0.23/0.53 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.23/0.53 != ( bit_se1146084159140164899it_int @ B2 @ N ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % bit_xor_iff
% 0.23/0.53 thf(fact_7201_norm__inverse__le__norm,axiom,
% 0.23/0.53 ! [R2: real,X4: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X4 ) )
% 0.23/0.53 => ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.23/0.53 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X4 ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % norm_inverse_le_norm
% 0.23/0.53 thf(fact_7202_norm__inverse__le__norm,axiom,
% 0.23/0.53 ! [R2: real,X4: complex] :
% 0.23/0.53 ( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X4 ) )
% 0.23/0.53 => ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.23/0.53 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X4 ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % norm_inverse_le_norm
% 0.23/0.53 thf(fact_7203_positive__imp__inverse__positive,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.53 => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % positive_imp_inverse_positive
% 0.23/0.53 thf(fact_7204_positive__imp__inverse__positive,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.53 => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % positive_imp_inverse_positive
% 0.23/0.53 thf(fact_7205_negative__imp__inverse__negative,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ord_less_real @ A @ zero_zero_real )
% 0.23/0.53 => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % negative_imp_inverse_negative
% 0.23/0.53 thf(fact_7206_negative__imp__inverse__negative,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.23/0.53 => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % negative_imp_inverse_negative
% 0.23/0.53 thf(fact_7207_inverse__positive__imp__positive,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.23/0.53 => ( ( A != zero_zero_real )
% 0.23/0.53 => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_positive_imp_positive
% 0.23/0.53 thf(fact_7208_inverse__positive__imp__positive,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.23/0.53 => ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_positive_imp_positive
% 0.23/0.53 thf(fact_7209_inverse__negative__imp__negative,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.23/0.53 => ( ( A != zero_zero_real )
% 0.23/0.53 => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_negative_imp_negative
% 0.23/0.53 thf(fact_7210_inverse__negative__imp__negative,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.23/0.53 => ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_negative_imp_negative
% 0.23/0.53 thf(fact_7211_less__imp__inverse__less__neg,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_real @ A @ B2 )
% 0.23/0.53 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.23/0.53 => ( ord_less_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % less_imp_inverse_less_neg
% 0.23/0.53 thf(fact_7212_less__imp__inverse__less__neg,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.53 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.23/0.53 => ( ord_less_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % less_imp_inverse_less_neg
% 0.23/0.53 thf(fact_7213_inverse__less__imp__less__neg,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.23/0.53 => ( ord_less_real @ B2 @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_imp_less_neg
% 0.23/0.53 thf(fact_7214_inverse__less__imp__less__neg,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.23/0.53 => ( ord_less_rat @ B2 @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_imp_less_neg
% 0.23/0.53 thf(fact_7215_less__imp__inverse__less,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_real @ A @ B2 )
% 0.23/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.53 => ( ord_less_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % less_imp_inverse_less
% 0.23/0.53 thf(fact_7216_less__imp__inverse__less,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.53 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.53 => ( ord_less_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % less_imp_inverse_less
% 0.23/0.53 thf(fact_7217_inverse__less__imp__less,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.53 => ( ord_less_real @ B2 @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_imp_less
% 0.23/0.53 thf(fact_7218_inverse__less__imp__less,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.53 => ( ord_less_rat @ B2 @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_imp_less
% 0.23/0.53 thf(fact_7219_nonzero__inverse__mult__distrib,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( B2 != zero_zero_real )
% 0.23/0.53 => ( ( inverse_inverse_real @ ( times_times_real @ A @ B2 ) )
% 0.23/0.53 = ( times_times_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_mult_distrib
% 0.23/0.53 thf(fact_7220_nonzero__inverse__mult__distrib,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( B2 != zero_zero_complex )
% 0.23/0.53 => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B2 ) )
% 0.23/0.53 = ( times_times_complex @ ( invers8013647133539491842omplex @ B2 ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_mult_distrib
% 0.23/0.53 thf(fact_7221_nonzero__inverse__mult__distrib,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( B2 != zero_zero_rat )
% 0.23/0.53 => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B2 ) )
% 0.23/0.53 = ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_mult_distrib
% 0.23/0.53 thf(fact_7222_nonzero__inverse__minus__eq,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 0.23/0.53 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_minus_eq
% 0.23/0.53 thf(fact_7223_nonzero__inverse__minus__eq,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.23/0.53 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_minus_eq
% 0.23/0.53 thf(fact_7224_nonzero__inverse__minus__eq,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 0.23/0.53 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_minus_eq
% 0.23/0.53 thf(fact_7225_inverse__numeral__1,axiom,
% 0.23/0.53 ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
% 0.23/0.53 = ( numeral_numeral_real @ one ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_numeral_1
% 0.23/0.53 thf(fact_7226_inverse__numeral__1,axiom,
% 0.23/0.53 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.23/0.53 = ( numera6690914467698888265omplex @ one ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_numeral_1
% 0.23/0.53 thf(fact_7227_inverse__numeral__1,axiom,
% 0.23/0.53 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
% 0.23/0.53 = ( numeral_numeral_rat @ one ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_numeral_1
% 0.23/0.53 thf(fact_7228_inverse__unique,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ( times_times_real @ A @ B2 )
% 0.23/0.53 = one_one_real )
% 0.23/0.53 => ( ( inverse_inverse_real @ A )
% 0.23/0.53 = B2 ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_unique
% 0.23/0.53 thf(fact_7229_inverse__unique,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( ( times_times_complex @ A @ B2 )
% 0.23/0.53 = one_one_complex )
% 0.23/0.53 => ( ( invers8013647133539491842omplex @ A )
% 0.23/0.53 = B2 ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_unique
% 0.23/0.53 thf(fact_7230_inverse__unique,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ( times_times_rat @ A @ B2 )
% 0.23/0.53 = one_one_rat )
% 0.23/0.53 => ( ( inverse_inverse_rat @ A )
% 0.23/0.53 = B2 ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_unique
% 0.23/0.53 thf(fact_7231_divide__inverse__commute,axiom,
% 0.23/0.53 ( divide_divide_real
% 0.23/0.53 = ( ^ [A4: real,B3: real] : ( times_times_real @ ( inverse_inverse_real @ B3 ) @ A4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % divide_inverse_commute
% 0.23/0.53 thf(fact_7232_divide__inverse__commute,axiom,
% 0.23/0.53 ( divide1717551699836669952omplex
% 0.23/0.53 = ( ^ [A4: complex,B3: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B3 ) @ A4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % divide_inverse_commute
% 0.23/0.53 thf(fact_7233_divide__inverse__commute,axiom,
% 0.23/0.53 ( divide_divide_rat
% 0.23/0.53 = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B3 ) @ A4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % divide_inverse_commute
% 0.23/0.53 thf(fact_7234_divide__inverse,axiom,
% 0.23/0.53 ( divide_divide_real
% 0.23/0.53 = ( ^ [A4: real,B3: real] : ( times_times_real @ A4 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % divide_inverse
% 0.23/0.53 thf(fact_7235_divide__inverse,axiom,
% 0.23/0.53 ( divide1717551699836669952omplex
% 0.23/0.53 = ( ^ [A4: complex,B3: complex] : ( times_times_complex @ A4 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % divide_inverse
% 0.23/0.53 thf(fact_7236_divide__inverse,axiom,
% 0.23/0.53 ( divide_divide_rat
% 0.23/0.53 = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ A4 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % divide_inverse
% 0.23/0.53 thf(fact_7237_field__class_Ofield__divide__inverse,axiom,
% 0.23/0.53 ( divide_divide_real
% 0.23/0.53 = ( ^ [A4: real,B3: real] : ( times_times_real @ A4 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % field_class.field_divide_inverse
% 0.23/0.53 thf(fact_7238_field__class_Ofield__divide__inverse,axiom,
% 0.23/0.53 ( divide1717551699836669952omplex
% 0.23/0.53 = ( ^ [A4: complex,B3: complex] : ( times_times_complex @ A4 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % field_class.field_divide_inverse
% 0.23/0.53 thf(fact_7239_field__class_Ofield__divide__inverse,axiom,
% 0.23/0.53 ( divide_divide_rat
% 0.23/0.53 = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ A4 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % field_class.field_divide_inverse
% 0.23/0.53 thf(fact_7240_inverse__eq__divide,axiom,
% 0.23/0.53 ( inverse_inverse_real
% 0.23/0.53 = ( divide_divide_real @ one_one_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_divide
% 0.23/0.53 thf(fact_7241_inverse__eq__divide,axiom,
% 0.23/0.53 ( invers8013647133539491842omplex
% 0.23/0.53 = ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_divide
% 0.23/0.53 thf(fact_7242_inverse__eq__divide,axiom,
% 0.23/0.53 ( inverse_inverse_rat
% 0.23/0.53 = ( divide_divide_rat @ one_one_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_eq_divide
% 0.23/0.53 thf(fact_7243_power__mult__power__inverse__commute,axiom,
% 0.23/0.53 ! [X4: real,M: nat,N: nat] :
% 0.23/0.53 ( ( times_times_real @ ( power_power_real @ X4 @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ N ) )
% 0.23/0.53 = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ N ) @ ( power_power_real @ X4 @ M ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_mult_power_inverse_commute
% 0.23/0.53 thf(fact_7244_power__mult__power__inverse__commute,axiom,
% 0.23/0.53 ! [X4: complex,M: nat,N: nat] :
% 0.23/0.53 ( ( times_times_complex @ ( power_power_complex @ X4 @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ N ) )
% 0.23/0.53 = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ N ) @ ( power_power_complex @ X4 @ M ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_mult_power_inverse_commute
% 0.23/0.53 thf(fact_7245_power__mult__power__inverse__commute,axiom,
% 0.23/0.53 ! [X4: rat,M: nat,N: nat] :
% 0.23/0.53 ( ( times_times_rat @ ( power_power_rat @ X4 @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X4 ) @ N ) )
% 0.23/0.53 = ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X4 ) @ N ) @ ( power_power_rat @ X4 @ M ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_mult_power_inverse_commute
% 0.23/0.53 thf(fact_7246_power__mult__inverse__distrib,axiom,
% 0.23/0.53 ! [X4: real,M: nat] :
% 0.23/0.53 ( ( times_times_real @ ( power_power_real @ X4 @ M ) @ ( inverse_inverse_real @ X4 ) )
% 0.23/0.53 = ( times_times_real @ ( inverse_inverse_real @ X4 ) @ ( power_power_real @ X4 @ M ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_mult_inverse_distrib
% 0.23/0.53 thf(fact_7247_power__mult__inverse__distrib,axiom,
% 0.23/0.53 ! [X4: complex,M: nat] :
% 0.23/0.53 ( ( times_times_complex @ ( power_power_complex @ X4 @ M ) @ ( invers8013647133539491842omplex @ X4 ) )
% 0.23/0.53 = ( times_times_complex @ ( invers8013647133539491842omplex @ X4 ) @ ( power_power_complex @ X4 @ M ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_mult_inverse_distrib
% 0.23/0.53 thf(fact_7248_power__mult__inverse__distrib,axiom,
% 0.23/0.53 ! [X4: rat,M: nat] :
% 0.23/0.53 ( ( times_times_rat @ ( power_power_rat @ X4 @ M ) @ ( inverse_inverse_rat @ X4 ) )
% 0.23/0.53 = ( times_times_rat @ ( inverse_inverse_rat @ X4 ) @ ( power_power_rat @ X4 @ M ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_mult_inverse_distrib
% 0.23/0.53 thf(fact_7249_mult__inverse__of__nat__commute,axiom,
% 0.23/0.53 ! [Xa: nat,X4: real] :
% 0.23/0.53 ( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X4 )
% 0.23/0.53 = ( times_times_real @ X4 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % mult_inverse_of_nat_commute
% 0.23/0.53 thf(fact_7250_mult__inverse__of__nat__commute,axiom,
% 0.23/0.53 ! [Xa: nat,X4: complex] :
% 0.23/0.53 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X4 )
% 0.23/0.53 = ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % mult_inverse_of_nat_commute
% 0.23/0.53 thf(fact_7251_mult__inverse__of__nat__commute,axiom,
% 0.23/0.53 ! [Xa: nat,X4: rat] :
% 0.23/0.53 ( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) @ X4 )
% 0.23/0.53 = ( times_times_rat @ X4 @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % mult_inverse_of_nat_commute
% 0.23/0.53 thf(fact_7252_nonzero__abs__inverse,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 0.23/0.53 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_abs_inverse
% 0.23/0.53 thf(fact_7253_nonzero__abs__inverse,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 0.23/0.53 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_abs_inverse
% 0.23/0.53 thf(fact_7254_mult__inverse__of__int__commute,axiom,
% 0.23/0.53 ! [Xa: int,X4: real] :
% 0.23/0.53 ( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) @ X4 )
% 0.23/0.53 = ( times_times_real @ X4 @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % mult_inverse_of_int_commute
% 0.23/0.53 thf(fact_7255_mult__inverse__of__int__commute,axiom,
% 0.23/0.53 ! [Xa: int,X4: complex] :
% 0.23/0.53 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) @ X4 )
% 0.23/0.53 = ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % mult_inverse_of_int_commute
% 0.23/0.53 thf(fact_7256_mult__inverse__of__int__commute,axiom,
% 0.23/0.53 ! [Xa: int,X4: rat] :
% 0.23/0.53 ( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) @ X4 )
% 0.23/0.53 = ( times_times_rat @ X4 @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % mult_inverse_of_int_commute
% 0.23/0.53 thf(fact_7257_divide__real__def,axiom,
% 0.23/0.53 ( divide_divide_real
% 0.23/0.53 = ( ^ [X: real,Y: real] : ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % divide_real_def
% 0.23/0.53 thf(fact_7258_Ints__double__eq__0__iff,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.23/0.53 => ( ( ( plus_plus_complex @ A @ A )
% 0.23/0.53 = zero_zero_complex )
% 0.23/0.53 = ( A = zero_zero_complex ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_double_eq_0_iff
% 0.23/0.53 thf(fact_7259_Ints__double__eq__0__iff,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.53 => ( ( ( plus_plus_real @ A @ A )
% 0.23/0.53 = zero_zero_real )
% 0.23/0.53 = ( A = zero_zero_real ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_double_eq_0_iff
% 0.23/0.53 thf(fact_7260_Ints__double__eq__0__iff,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( ( plus_plus_rat @ A @ A )
% 0.23/0.53 = zero_zero_rat )
% 0.23/0.53 = ( A = zero_zero_rat ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_double_eq_0_iff
% 0.23/0.53 thf(fact_7261_Ints__double__eq__0__iff,axiom,
% 0.23/0.53 ! [A: int] :
% 0.23/0.53 ( ( member_int @ A @ ring_1_Ints_int )
% 0.23/0.53 => ( ( ( plus_plus_int @ A @ A )
% 0.23/0.53 = zero_zero_int )
% 0.23/0.53 = ( A = zero_zero_int ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_double_eq_0_iff
% 0.23/0.53 thf(fact_7262_inverse__le__imp__le,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.53 => ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_imp_le
% 0.23/0.53 thf(fact_7263_inverse__le__imp__le,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.53 => ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_imp_le
% 0.23/0.53 thf(fact_7264_le__imp__inverse__le,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.53 => ( ord_less_eq_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % le_imp_inverse_le
% 0.23/0.53 thf(fact_7265_le__imp__inverse__le,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.53 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.53 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % le_imp_inverse_le
% 0.23/0.53 thf(fact_7266_inverse__le__imp__le__neg,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.23/0.53 => ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_imp_le_neg
% 0.23/0.53 thf(fact_7267_inverse__le__imp__le__neg,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.23/0.53 => ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_imp_le_neg
% 0.23/0.53 thf(fact_7268_le__imp__inverse__le__neg,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.53 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.23/0.53 => ( ord_less_eq_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % le_imp_inverse_le_neg
% 0.23/0.53 thf(fact_7269_le__imp__inverse__le__neg,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.53 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.23/0.53 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % le_imp_inverse_le_neg
% 0.23/0.53 thf(fact_7270_inverse__le__1__iff,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ ( inverse_inverse_real @ X4 ) @ one_one_real )
% 0.23/0.53 = ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.23/0.53 | ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_1_iff
% 0.23/0.53 thf(fact_7271_inverse__le__1__iff,axiom,
% 0.23/0.53 ! [X4: rat] :
% 0.23/0.53 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X4 ) @ one_one_rat )
% 0.23/0.53 = ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 0.23/0.53 | ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_1_iff
% 0.23/0.53 thf(fact_7272_one__less__inverse__iff,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X4 ) )
% 0.23/0.53 = ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.53 & ( ord_less_real @ X4 @ one_one_real ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % one_less_inverse_iff
% 0.23/0.53 thf(fact_7273_one__less__inverse__iff,axiom,
% 0.23/0.53 ! [X4: rat] :
% 0.23/0.53 ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X4 ) )
% 0.23/0.53 = ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 0.23/0.53 & ( ord_less_rat @ X4 @ one_one_rat ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % one_less_inverse_iff
% 0.23/0.53 thf(fact_7274_one__less__inverse,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.53 => ( ( ord_less_real @ A @ one_one_real )
% 0.23/0.53 => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % one_less_inverse
% 0.23/0.53 thf(fact_7275_one__less__inverse,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.53 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.23/0.53 => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % one_less_inverse
% 0.23/0.53 thf(fact_7276_field__class_Ofield__inverse,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 0.23/0.53 = one_one_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % field_class.field_inverse
% 0.23/0.53 thf(fact_7277_field__class_Ofield__inverse,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 0.23/0.53 = one_one_complex ) ) ).
% 0.23/0.53
% 0.23/0.53 % field_class.field_inverse
% 0.23/0.53 thf(fact_7278_field__class_Ofield__inverse,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 0.23/0.53 = one_one_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % field_class.field_inverse
% 0.23/0.53 thf(fact_7279_division__ring__inverse__add,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( B2 != zero_zero_real )
% 0.23/0.53 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B2 ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % division_ring_inverse_add
% 0.23/0.53 thf(fact_7280_division__ring__inverse__add,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( B2 != zero_zero_complex )
% 0.23/0.53 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
% 0.23/0.53 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B2 ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % division_ring_inverse_add
% 0.23/0.53 thf(fact_7281_division__ring__inverse__add,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( B2 != zero_zero_rat )
% 0.23/0.53 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B2 ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % division_ring_inverse_add
% 0.23/0.53 thf(fact_7282_inverse__add,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( B2 != zero_zero_real )
% 0.23/0.53 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_add
% 0.23/0.53 thf(fact_7283_inverse__add,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( B2 != zero_zero_complex )
% 0.23/0.53 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
% 0.23/0.53 = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_add
% 0.23/0.53 thf(fact_7284_inverse__add,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( B2 != zero_zero_rat )
% 0.23/0.53 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_add
% 0.23/0.53 thf(fact_7285_division__ring__inverse__diff,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( B2 != zero_zero_real )
% 0.23/0.53 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B2 @ A ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % division_ring_inverse_diff
% 0.23/0.53 thf(fact_7286_division__ring__inverse__diff,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( B2 != zero_zero_complex )
% 0.23/0.53 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
% 0.23/0.53 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B2 @ A ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % division_ring_inverse_diff
% 0.23/0.53 thf(fact_7287_division__ring__inverse__diff,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( B2 != zero_zero_rat )
% 0.23/0.53 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B2 @ A ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % division_ring_inverse_diff
% 0.23/0.53 thf(fact_7288_nonzero__inverse__eq__divide,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( inverse_inverse_real @ A )
% 0.23/0.53 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_eq_divide
% 0.23/0.53 thf(fact_7289_nonzero__inverse__eq__divide,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( invers8013647133539491842omplex @ A )
% 0.23/0.53 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_eq_divide
% 0.23/0.53 thf(fact_7290_nonzero__inverse__eq__divide,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( inverse_inverse_rat @ A )
% 0.23/0.53 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % nonzero_inverse_eq_divide
% 0.23/0.53 thf(fact_7291_gbinomial__Suc__Suc,axiom,
% 0.23/0.53 ! [A: complex,K: nat] :
% 0.23/0.53 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.23/0.53 = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_Suc_Suc
% 0.23/0.53 thf(fact_7292_gbinomial__Suc__Suc,axiom,
% 0.23/0.53 ! [A: real,K: nat] :
% 0.23/0.53 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.23/0.53 = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_Suc_Suc
% 0.23/0.53 thf(fact_7293_gbinomial__Suc__Suc,axiom,
% 0.23/0.53 ! [A: rat,K: nat] :
% 0.23/0.53 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.23/0.53 = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_Suc_Suc
% 0.23/0.53 thf(fact_7294_gbinomial__of__nat__symmetric,axiom,
% 0.23/0.53 ! [K: nat,N: nat] :
% 0.23/0.53 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.53 => ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K )
% 0.23/0.53 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_of_nat_symmetric
% 0.23/0.53 thf(fact_7295_gbinomial__of__nat__symmetric,axiom,
% 0.23/0.53 ! [K: nat,N: nat] :
% 0.23/0.53 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.53 => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 0.23/0.53 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_of_nat_symmetric
% 0.23/0.53 thf(fact_7296_gbinomial__of__nat__symmetric,axiom,
% 0.23/0.53 ! [K: nat,N: nat] :
% 0.23/0.53 ( ( ord_less_eq_nat @ K @ N )
% 0.23/0.53 => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
% 0.23/0.53 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_of_nat_symmetric
% 0.23/0.53 thf(fact_7297_inverse__powr,axiom,
% 0.23/0.53 ! [Y3: real,A: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.53 => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A )
% 0.23/0.53 = ( inverse_inverse_real @ ( powr_real @ Y3 @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_powr
% 0.23/0.53 thf(fact_7298_Ints__odd__nonzero,axiom,
% 0.23/0.53 ! [A: complex] :
% 0.23/0.53 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.23/0.53 => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
% 0.23/0.53 != zero_zero_complex ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_odd_nonzero
% 0.23/0.53 thf(fact_7299_Ints__odd__nonzero,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.53 => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
% 0.23/0.53 != zero_zero_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_odd_nonzero
% 0.23/0.53 thf(fact_7300_Ints__odd__nonzero,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
% 0.23/0.53 != zero_zero_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_odd_nonzero
% 0.23/0.53 thf(fact_7301_Ints__odd__nonzero,axiom,
% 0.23/0.53 ! [A: int] :
% 0.23/0.53 ( ( member_int @ A @ ring_1_Ints_int )
% 0.23/0.53 => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
% 0.23/0.53 != zero_zero_int ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_odd_nonzero
% 0.23/0.53 thf(fact_7302_of__int__divide__in__Ints,axiom,
% 0.23/0.53 ! [B2: int,A: int] :
% 0.23/0.53 ( ( dvd_dvd_int @ B2 @ A )
% 0.23/0.53 => ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B2 ) ) @ ring_1_Ints_complex ) ) ).
% 0.23/0.53
% 0.23/0.53 % of_int_divide_in_Ints
% 0.23/0.53 thf(fact_7303_of__int__divide__in__Ints,axiom,
% 0.23/0.53 ! [B2: int,A: int] :
% 0.23/0.53 ( ( dvd_dvd_int @ B2 @ A )
% 0.23/0.53 => ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B2 ) ) @ ring_1_Ints_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % of_int_divide_in_Ints
% 0.23/0.53 thf(fact_7304_of__int__divide__in__Ints,axiom,
% 0.23/0.53 ! [B2: int,A: int] :
% 0.23/0.53 ( ( dvd_dvd_int @ B2 @ A )
% 0.23/0.53 => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B2 ) ) @ ring_1_Ints_rat ) ) ).
% 0.23/0.53
% 0.23/0.53 % of_int_divide_in_Ints
% 0.23/0.53 thf(fact_7305_of__int__divide__in__Ints,axiom,
% 0.23/0.53 ! [B2: int,A: int] :
% 0.23/0.53 ( ( dvd_dvd_int @ B2 @ A )
% 0.23/0.53 => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B2 ) ) @ ring_1_Ints_int ) ) ).
% 0.23/0.53
% 0.23/0.53 % of_int_divide_in_Ints
% 0.23/0.53 thf(fact_7306_inverse__le__iff,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.23/0.53 => ( ord_less_eq_real @ B2 @ A ) )
% 0.23/0.53 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 0.23/0.53 => ( ord_less_eq_real @ A @ B2 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_iff
% 0.23/0.53 thf(fact_7307_inverse__le__iff,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.23/0.53 => ( ord_less_eq_rat @ B2 @ A ) )
% 0.23/0.53 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 0.23/0.53 => ( ord_less_eq_rat @ A @ B2 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_le_iff
% 0.23/0.53 thf(fact_7308_inverse__less__iff,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.23/0.53 => ( ord_less_real @ B2 @ A ) )
% 0.23/0.53 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 0.23/0.53 => ( ord_less_real @ A @ B2 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_iff
% 0.23/0.53 thf(fact_7309_inverse__less__iff,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.23/0.53 => ( ord_less_rat @ B2 @ A ) )
% 0.23/0.53 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 0.23/0.53 => ( ord_less_rat @ A @ B2 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_iff
% 0.23/0.53 thf(fact_7310_one__le__inverse,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.53 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.23/0.53 => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % one_le_inverse
% 0.23/0.53 thf(fact_7311_one__le__inverse,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.23/0.53 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.23/0.53 => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % one_le_inverse
% 0.23/0.53 thf(fact_7312_inverse__less__1__iff,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ord_less_real @ ( inverse_inverse_real @ X4 ) @ one_one_real )
% 0.23/0.53 = ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.23/0.53 | ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_1_iff
% 0.23/0.53 thf(fact_7313_inverse__less__1__iff,axiom,
% 0.23/0.53 ! [X4: rat] :
% 0.23/0.53 ( ( ord_less_rat @ ( inverse_inverse_rat @ X4 ) @ one_one_rat )
% 0.23/0.53 = ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 0.23/0.53 | ( ord_less_rat @ one_one_rat @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_less_1_iff
% 0.23/0.53 thf(fact_7314_one__le__inverse__iff,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X4 ) )
% 0.23/0.53 = ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.53 & ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % one_le_inverse_iff
% 0.23/0.53 thf(fact_7315_one__le__inverse__iff,axiom,
% 0.23/0.53 ! [X4: rat] :
% 0.23/0.53 ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X4 ) )
% 0.23/0.53 = ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 0.23/0.53 & ( ord_less_eq_rat @ X4 @ one_one_rat ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % one_le_inverse_iff
% 0.23/0.53 thf(fact_7316_inverse__diff__inverse,axiom,
% 0.23/0.53 ! [A: real,B2: real] :
% 0.23/0.53 ( ( A != zero_zero_real )
% 0.23/0.53 => ( ( B2 != zero_zero_real )
% 0.23/0.53 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.23/0.53 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B2 ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_diff_inverse
% 0.23/0.53 thf(fact_7317_inverse__diff__inverse,axiom,
% 0.23/0.53 ! [A: complex,B2: complex] :
% 0.23/0.53 ( ( A != zero_zero_complex )
% 0.23/0.53 => ( ( B2 != zero_zero_complex )
% 0.23/0.53 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
% 0.23/0.53 = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B2 ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_diff_inverse
% 0.23/0.53 thf(fact_7318_inverse__diff__inverse,axiom,
% 0.23/0.53 ! [A: rat,B2: rat] :
% 0.23/0.53 ( ( A != zero_zero_rat )
% 0.23/0.53 => ( ( B2 != zero_zero_rat )
% 0.23/0.53 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.23/0.53 = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B2 ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % inverse_diff_inverse
% 0.23/0.53 thf(fact_7319_reals__Archimedean,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.53 => ? [N4: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ X4 ) ) ).
% 0.23/0.53
% 0.23/0.53 % reals_Archimedean
% 0.23/0.53 thf(fact_7320_reals__Archimedean,axiom,
% 0.23/0.53 ! [X4: rat] :
% 0.23/0.53 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 0.23/0.53 => ? [N4: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ X4 ) ) ).
% 0.23/0.53
% 0.23/0.53 % reals_Archimedean
% 0.23/0.53 thf(fact_7321_gbinomial__addition__formula,axiom,
% 0.23/0.53 ! [A: complex,K: nat] :
% 0.23/0.53 ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 0.23/0.53 = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_addition_formula
% 0.23/0.53 thf(fact_7322_gbinomial__addition__formula,axiom,
% 0.23/0.53 ! [A: real,K: nat] :
% 0.23/0.53 ( ( gbinomial_real @ A @ ( suc @ K ) )
% 0.23/0.53 = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_addition_formula
% 0.23/0.53 thf(fact_7323_gbinomial__addition__formula,axiom,
% 0.23/0.53 ! [A: rat,K: nat] :
% 0.23/0.53 ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 0.23/0.53 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_addition_formula
% 0.23/0.53 thf(fact_7324_gbinomial__absorb__comp,axiom,
% 0.23/0.53 ! [A: complex,K: nat] :
% 0.23/0.53 ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
% 0.23/0.53 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_absorb_comp
% 0.23/0.53 thf(fact_7325_gbinomial__absorb__comp,axiom,
% 0.23/0.53 ! [A: real,K: nat] :
% 0.23/0.53 ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
% 0.23/0.53 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_absorb_comp
% 0.23/0.53 thf(fact_7326_gbinomial__absorb__comp,axiom,
% 0.23/0.53 ! [A: rat,K: nat] :
% 0.23/0.53 ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
% 0.23/0.53 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_absorb_comp
% 0.23/0.53 thf(fact_7327_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.23/0.53 ! [K: nat,A: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 0.23/0.53 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_ge_n_over_k_pow_k
% 0.23/0.53 thf(fact_7328_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.23/0.53 ! [K: nat,A: rat] :
% 0.23/0.53 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 0.23/0.53 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_ge_n_over_k_pow_k
% 0.23/0.53 thf(fact_7329_gbinomial__mult__1_H,axiom,
% 0.23/0.53 ! [A: complex,K: nat] :
% 0.23/0.53 ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
% 0.23/0.53 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_mult_1'
% 0.23/0.53 thf(fact_7330_gbinomial__mult__1_H,axiom,
% 0.23/0.53 ! [A: real,K: nat] :
% 0.23/0.53 ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 0.23/0.53 = ( 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 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_mult_1'
% 0.23/0.53 thf(fact_7331_gbinomial__mult__1_H,axiom,
% 0.23/0.53 ! [A: rat,K: nat] :
% 0.23/0.53 ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 0.23/0.53 = ( 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 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_mult_1'
% 0.23/0.53 thf(fact_7332_gbinomial__mult__1,axiom,
% 0.23/0.53 ! [A: complex,K: nat] :
% 0.23/0.53 ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
% 0.23/0.53 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_mult_1
% 0.23/0.53 thf(fact_7333_gbinomial__mult__1,axiom,
% 0.23/0.53 ! [A: real,K: nat] :
% 0.23/0.53 ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 0.23/0.53 = ( 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 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_mult_1
% 0.23/0.53 thf(fact_7334_gbinomial__mult__1,axiom,
% 0.23/0.53 ! [A: rat,K: nat] :
% 0.23/0.53 ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 0.23/0.53 = ( 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 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_mult_1
% 0.23/0.53 thf(fact_7335_forall__pos__mono__1,axiom,
% 0.23/0.53 ! [P: real > $o,E: real] :
% 0.23/0.53 ( ! [D3: real,E2: real] :
% 0.23/0.53 ( ( ord_less_real @ D3 @ E2 )
% 0.23/0.53 => ( ( P @ D3 )
% 0.23/0.53 => ( P @ E2 ) ) )
% 0.23/0.53 => ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
% 0.23/0.53 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.23/0.53 => ( P @ E ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % forall_pos_mono_1
% 0.23/0.53 thf(fact_7336_forall__pos__mono,axiom,
% 0.23/0.53 ! [P: real > $o,E: real] :
% 0.23/0.53 ( ! [D3: real,E2: real] :
% 0.23/0.53 ( ( ord_less_real @ D3 @ E2 )
% 0.23/0.53 => ( ( P @ D3 )
% 0.23/0.53 => ( P @ E2 ) ) )
% 0.23/0.53 => ( ! [N4: nat] :
% 0.23/0.53 ( ( N4 != zero_zero_nat )
% 0.23/0.53 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
% 0.23/0.53 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.23/0.53 => ( P @ E ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % forall_pos_mono
% 0.23/0.53 thf(fact_7337_real__arch__inverse,axiom,
% 0.23/0.53 ! [E: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ E )
% 0.23/0.53 = ( ? [N2: nat] :
% 0.23/0.53 ( ( N2 != zero_zero_nat )
% 0.23/0.53 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.23/0.53 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % real_arch_inverse
% 0.23/0.53 thf(fact_7338_even__xor__iff,axiom,
% 0.23/0.53 ! [A: code_integer,B2: code_integer] :
% 0.23/0.53 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ A @ B2 ) )
% 0.23/0.53 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.23/0.53 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % even_xor_iff
% 0.23/0.53 thf(fact_7339_even__xor__iff,axiom,
% 0.23/0.53 ! [A: nat,B2: nat] :
% 0.23/0.53 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B2 ) )
% 0.23/0.53 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.23/0.53 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % even_xor_iff
% 0.23/0.53 thf(fact_7340_even__xor__iff,axiom,
% 0.23/0.53 ! [A: int,B2: int] :
% 0.23/0.53 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B2 ) )
% 0.23/0.53 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.23/0.53 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % even_xor_iff
% 0.23/0.53 thf(fact_7341_sqrt__divide__self__eq,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.53 => ( ( divide_divide_real @ ( sqrt @ X4 ) @ X4 )
% 0.23/0.53 = ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % sqrt_divide_self_eq
% 0.23/0.53 thf(fact_7342_ln__inverse,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.53 => ( ( ln_ln_real @ ( inverse_inverse_real @ X4 ) )
% 0.23/0.53 = ( uminus_uminus_real @ ( ln_ln_real @ X4 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % ln_inverse
% 0.23/0.53 thf(fact_7343_Ints__odd__less__0,axiom,
% 0.23/0.53 ! [A: real] :
% 0.23/0.53 ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.53 => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
% 0.23/0.53 = ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_odd_less_0
% 0.23/0.53 thf(fact_7344_Ints__odd__less__0,axiom,
% 0.23/0.53 ! [A: rat] :
% 0.23/0.53 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
% 0.23/0.53 = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_odd_less_0
% 0.23/0.53 thf(fact_7345_Ints__odd__less__0,axiom,
% 0.23/0.53 ! [A: int] :
% 0.23/0.53 ( ( member_int @ A @ ring_1_Ints_int )
% 0.23/0.53 => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
% 0.23/0.53 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_odd_less_0
% 0.23/0.53 thf(fact_7346_Ints__nonzero__abs__ge1,axiom,
% 0.23/0.53 ! [X4: code_integer] :
% 0.23/0.53 ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
% 0.23/0.53 => ( ( X4 != zero_z3403309356797280102nteger )
% 0.23/0.53 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X4 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_nonzero_abs_ge1
% 0.23/0.53 thf(fact_7347_Ints__nonzero__abs__ge1,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( member_real @ X4 @ ring_1_Ints_real )
% 0.23/0.53 => ( ( X4 != zero_zero_real )
% 0.23/0.53 => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X4 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_nonzero_abs_ge1
% 0.23/0.53 thf(fact_7348_Ints__nonzero__abs__ge1,axiom,
% 0.23/0.53 ! [X4: rat] :
% 0.23/0.53 ( ( member_rat @ X4 @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( X4 != zero_zero_rat )
% 0.23/0.53 => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X4 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_nonzero_abs_ge1
% 0.23/0.53 thf(fact_7349_Ints__nonzero__abs__ge1,axiom,
% 0.23/0.53 ! [X4: int] :
% 0.23/0.53 ( ( member_int @ X4 @ ring_1_Ints_int )
% 0.23/0.53 => ( ( X4 != zero_zero_int )
% 0.23/0.53 => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X4 ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_nonzero_abs_ge1
% 0.23/0.53 thf(fact_7350_Ints__nonzero__abs__less1,axiom,
% 0.23/0.53 ! [X4: code_integer] :
% 0.23/0.53 ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
% 0.23/0.53 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer )
% 0.23/0.53 => ( X4 = zero_z3403309356797280102nteger ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_nonzero_abs_less1
% 0.23/0.53 thf(fact_7351_Ints__nonzero__abs__less1,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( member_real @ X4 @ ring_1_Ints_real )
% 0.23/0.53 => ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.53 => ( X4 = zero_zero_real ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_nonzero_abs_less1
% 0.23/0.53 thf(fact_7352_Ints__nonzero__abs__less1,axiom,
% 0.23/0.53 ! [X4: rat] :
% 0.23/0.53 ( ( member_rat @ X4 @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( ord_less_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat )
% 0.23/0.53 => ( X4 = zero_zero_rat ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_nonzero_abs_less1
% 0.23/0.53 thf(fact_7353_Ints__nonzero__abs__less1,axiom,
% 0.23/0.53 ! [X4: int] :
% 0.23/0.53 ( ( member_int @ X4 @ ring_1_Ints_int )
% 0.23/0.53 => ( ( ord_less_int @ ( abs_abs_int @ X4 ) @ one_one_int )
% 0.23/0.53 => ( X4 = zero_zero_int ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_nonzero_abs_less1
% 0.23/0.53 thf(fact_7354_Ints__eq__abs__less1,axiom,
% 0.23/0.53 ! [X4: code_integer,Y3: code_integer] :
% 0.23/0.53 ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
% 0.23/0.53 => ( ( member_Code_integer @ Y3 @ ring_11222124179247155820nteger )
% 0.23/0.53 => ( ( X4 = Y3 )
% 0.23/0.53 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ Y3 ) ) @ one_one_Code_integer ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_eq_abs_less1
% 0.23/0.53 thf(fact_7355_Ints__eq__abs__less1,axiom,
% 0.23/0.53 ! [X4: real,Y3: real] :
% 0.23/0.53 ( ( member_real @ X4 @ ring_1_Ints_real )
% 0.23/0.53 => ( ( member_real @ Y3 @ ring_1_Ints_real )
% 0.23/0.53 => ( ( X4 = Y3 )
% 0.23/0.53 = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ one_one_real ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_eq_abs_less1
% 0.23/0.53 thf(fact_7356_Ints__eq__abs__less1,axiom,
% 0.23/0.53 ! [X4: rat,Y3: rat] :
% 0.23/0.53 ( ( member_rat @ X4 @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( member_rat @ Y3 @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( X4 = Y3 )
% 0.23/0.53 = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ Y3 ) ) @ one_one_rat ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_eq_abs_less1
% 0.23/0.53 thf(fact_7357_Ints__eq__abs__less1,axiom,
% 0.23/0.53 ! [X4: int,Y3: int] :
% 0.23/0.53 ( ( member_int @ X4 @ ring_1_Ints_int )
% 0.23/0.53 => ( ( member_int @ Y3 @ ring_1_Ints_int )
% 0.23/0.53 => ( ( X4 = Y3 )
% 0.23/0.53 = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Ints_eq_abs_less1
% 0.23/0.53 thf(fact_7358_sin__times__pi__eq__0,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ( sin_real @ ( times_times_real @ X4 @ pi ) )
% 0.23/0.53 = zero_zero_real )
% 0.23/0.53 = ( member_real @ X4 @ ring_1_Ints_real ) ) ).
% 0.23/0.53
% 0.23/0.53 % sin_times_pi_eq_0
% 0.23/0.53 thf(fact_7359_ex__inverse__of__nat__less,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.53 => ? [N4: nat] :
% 0.23/0.53 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.23/0.53 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % ex_inverse_of_nat_less
% 0.23/0.53 thf(fact_7360_ex__inverse__of__nat__less,axiom,
% 0.23/0.53 ! [X4: rat] :
% 0.23/0.53 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 0.23/0.53 => ? [N4: nat] :
% 0.23/0.53 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.23/0.53 & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N4 ) ) @ X4 ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % ex_inverse_of_nat_less
% 0.23/0.53 thf(fact_7361_power__diff__conv__inverse,axiom,
% 0.23/0.53 ! [X4: real,M: nat,N: nat] :
% 0.23/0.53 ( ( X4 != zero_zero_real )
% 0.23/0.53 => ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.53 => ( ( power_power_real @ X4 @ ( minus_minus_nat @ N @ M ) )
% 0.23/0.53 = ( times_times_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ M ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_diff_conv_inverse
% 0.23/0.53 thf(fact_7362_power__diff__conv__inverse,axiom,
% 0.23/0.53 ! [X4: complex,M: nat,N: nat] :
% 0.23/0.53 ( ( X4 != zero_zero_complex )
% 0.23/0.53 => ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.53 => ( ( power_power_complex @ X4 @ ( minus_minus_nat @ N @ M ) )
% 0.23/0.53 = ( times_times_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ M ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_diff_conv_inverse
% 0.23/0.53 thf(fact_7363_power__diff__conv__inverse,axiom,
% 0.23/0.53 ! [X4: rat,M: nat,N: nat] :
% 0.23/0.53 ( ( X4 != zero_zero_rat )
% 0.23/0.53 => ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.53 => ( ( power_power_rat @ X4 @ ( minus_minus_nat @ N @ M ) )
% 0.23/0.53 = ( times_times_rat @ ( power_power_rat @ X4 @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X4 ) @ M ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % power_diff_conv_inverse
% 0.23/0.53 thf(fact_7364_Suc__times__gbinomial,axiom,
% 0.23/0.53 ! [K: nat,A: complex] :
% 0.23/0.53 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 0.23/0.53 = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Suc_times_gbinomial
% 0.23/0.53 thf(fact_7365_Suc__times__gbinomial,axiom,
% 0.23/0.53 ! [K: nat,A: real] :
% 0.23/0.53 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 0.23/0.53 = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Suc_times_gbinomial
% 0.23/0.53 thf(fact_7366_Suc__times__gbinomial,axiom,
% 0.23/0.53 ! [K: nat,A: rat] :
% 0.23/0.53 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 0.23/0.53 = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % Suc_times_gbinomial
% 0.23/0.53 thf(fact_7367_gbinomial__absorption,axiom,
% 0.23/0.53 ! [K: nat,A: complex] :
% 0.23/0.53 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 0.23/0.53 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_absorption
% 0.23/0.53 thf(fact_7368_gbinomial__absorption,axiom,
% 0.23/0.53 ! [K: nat,A: real] :
% 0.23/0.53 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 0.23/0.53 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_absorption
% 0.23/0.53 thf(fact_7369_gbinomial__absorption,axiom,
% 0.23/0.53 ! [K: nat,A: rat] :
% 0.23/0.53 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 0.23/0.53 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_absorption
% 0.23/0.53 thf(fact_7370_gbinomial__trinomial__revision,axiom,
% 0.23/0.53 ! [K: nat,M: nat,A: complex] :
% 0.23/0.53 ( ( ord_less_eq_nat @ K @ M )
% 0.23/0.53 => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 0.23/0.53 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_trinomial_revision
% 0.23/0.53 thf(fact_7371_gbinomial__trinomial__revision,axiom,
% 0.23/0.53 ! [K: nat,M: nat,A: real] :
% 0.23/0.53 ( ( ord_less_eq_nat @ K @ M )
% 0.23/0.53 => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 0.23/0.53 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_trinomial_revision
% 0.23/0.53 thf(fact_7372_gbinomial__trinomial__revision,axiom,
% 0.23/0.53 ! [K: nat,M: nat,A: rat] :
% 0.23/0.53 ( ( ord_less_eq_nat @ K @ M )
% 0.23/0.53 => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 0.23/0.53 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_trinomial_revision
% 0.23/0.53 thf(fact_7373_log__inverse,axiom,
% 0.23/0.53 ! [A: real,X4: real] :
% 0.23/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.53 => ( ( A != one_one_real )
% 0.23/0.53 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.53 => ( ( log @ A @ ( inverse_inverse_real @ X4 ) )
% 0.23/0.53 = ( uminus_uminus_real @ ( log @ A @ X4 ) ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % log_inverse
% 0.23/0.53 thf(fact_7374_frac__neg,axiom,
% 0.23/0.53 ! [X4: real] :
% 0.23/0.53 ( ( ( member_real @ X4 @ ring_1_Ints_real )
% 0.23/0.53 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X4 ) )
% 0.23/0.53 = zero_zero_real ) )
% 0.23/0.53 & ( ~ ( member_real @ X4 @ ring_1_Ints_real )
% 0.23/0.53 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X4 ) )
% 0.23/0.53 = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X4 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % frac_neg
% 0.23/0.53 thf(fact_7375_frac__neg,axiom,
% 0.23/0.53 ! [X4: rat] :
% 0.23/0.53 ( ( ( member_rat @ X4 @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X4 ) )
% 0.23/0.53 = zero_zero_rat ) )
% 0.23/0.53 & ( ~ ( member_rat @ X4 @ ring_1_Ints_rat )
% 0.23/0.53 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X4 ) )
% 0.23/0.53 = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X4 ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % frac_neg
% 0.23/0.53 thf(fact_7376_gbinomial__factors,axiom,
% 0.23/0.53 ! [A: complex,K: nat] :
% 0.23/0.53 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.23/0.53 = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_factors
% 0.23/0.53 thf(fact_7377_gbinomial__factors,axiom,
% 0.23/0.53 ! [A: real,K: nat] :
% 0.23/0.53 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.23/0.53 = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_factors
% 0.23/0.53 thf(fact_7378_gbinomial__factors,axiom,
% 0.23/0.53 ! [A: rat,K: nat] :
% 0.23/0.53 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.23/0.53 = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_factors
% 0.23/0.53 thf(fact_7379_gbinomial__rec,axiom,
% 0.23/0.53 ! [A: complex,K: nat] :
% 0.23/0.53 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.23/0.53 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_rec
% 0.23/0.53 thf(fact_7380_gbinomial__rec,axiom,
% 0.23/0.53 ! [A: real,K: nat] :
% 0.23/0.53 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.23/0.53 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 0.23/0.53
% 0.23/0.53 % gbinomial_rec
% 0.23/0.54 thf(fact_7381_gbinomial__rec,axiom,
% 0.23/0.54 ! [A: rat,K: nat] :
% 0.23/0.54 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.23/0.54 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_rec
% 0.23/0.54 thf(fact_7382_gbinomial__negated__upper,axiom,
% 0.23/0.54 ( gbinomial_complex
% 0.23/0.54 = ( ^ [A4: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A4 ) @ one_one_complex ) @ K3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_negated_upper
% 0.23/0.54 thf(fact_7383_gbinomial__negated__upper,axiom,
% 0.23/0.54 ( gbinomial_real
% 0.23/0.54 = ( ^ [A4: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A4 ) @ one_one_real ) @ K3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_negated_upper
% 0.23/0.54 thf(fact_7384_gbinomial__negated__upper,axiom,
% 0.23/0.54 ( gbinomial_rat
% 0.23/0.54 = ( ^ [A4: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A4 ) @ one_one_rat ) @ K3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_negated_upper
% 0.23/0.54 thf(fact_7385_gbinomial__index__swap,axiom,
% 0.23/0.54 ! [K: nat,N: nat] :
% 0.23/0.54 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
% 0.23/0.54 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_index_swap
% 0.23/0.54 thf(fact_7386_gbinomial__index__swap,axiom,
% 0.23/0.54 ! [K: nat,N: nat] :
% 0.23/0.54 ( ( 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 ) )
% 0.23/0.54 = ( 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 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_index_swap
% 0.23/0.54 thf(fact_7387_gbinomial__index__swap,axiom,
% 0.23/0.54 ! [K: nat,N: nat] :
% 0.23/0.54 ( ( 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 ) )
% 0.23/0.54 = ( 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 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_index_swap
% 0.23/0.54 thf(fact_7388_exp__plus__inverse__exp,axiom,
% 0.23/0.54 ! [X4: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % exp_plus_inverse_exp
% 0.23/0.54 thf(fact_7389_le__mult__floor__Ints,axiom,
% 0.23/0.54 ! [A: real,B2: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.54 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.54 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % le_mult_floor_Ints
% 0.23/0.54 thf(fact_7390_le__mult__floor__Ints,axiom,
% 0.23/0.54 ! [A: real,B2: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.54 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.54 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % le_mult_floor_Ints
% 0.23/0.54 thf(fact_7391_le__mult__floor__Ints,axiom,
% 0.23/0.54 ! [A: real,B2: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.54 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.54 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % le_mult_floor_Ints
% 0.23/0.54 thf(fact_7392_le__mult__floor__Ints,axiom,
% 0.23/0.54 ! [A: rat,B2: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.54 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.54 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % le_mult_floor_Ints
% 0.23/0.54 thf(fact_7393_le__mult__floor__Ints,axiom,
% 0.23/0.54 ! [A: rat,B2: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.54 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.54 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % le_mult_floor_Ints
% 0.23/0.54 thf(fact_7394_le__mult__floor__Ints,axiom,
% 0.23/0.54 ! [A: rat,B2: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.54 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.54 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % le_mult_floor_Ints
% 0.23/0.54 thf(fact_7395_frac__unique__iff,axiom,
% 0.23/0.54 ! [X4: real,A: real] :
% 0.23/0.54 ( ( ( archim2898591450579166408c_real @ X4 )
% 0.23/0.54 = A )
% 0.23/0.54 = ( ( member_real @ ( minus_minus_real @ X4 @ A ) @ ring_1_Ints_real )
% 0.23/0.54 & ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.54 & ( ord_less_real @ A @ one_one_real ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % frac_unique_iff
% 0.23/0.54 thf(fact_7396_frac__unique__iff,axiom,
% 0.23/0.54 ! [X4: rat,A: rat] :
% 0.23/0.54 ( ( ( archimedean_frac_rat @ X4 )
% 0.23/0.54 = A )
% 0.23/0.54 = ( ( member_rat @ ( minus_minus_rat @ X4 @ A ) @ ring_1_Ints_rat )
% 0.23/0.54 & ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.54 & ( ord_less_rat @ A @ one_one_rat ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % frac_unique_iff
% 0.23/0.54 thf(fact_7397_mult__ceiling__le__Ints,axiom,
% 0.23/0.54 ! [A: real,B2: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.54 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.54 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % mult_ceiling_le_Ints
% 0.23/0.54 thf(fact_7398_mult__ceiling__le__Ints,axiom,
% 0.23/0.54 ! [A: real,B2: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.54 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.54 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % mult_ceiling_le_Ints
% 0.23/0.54 thf(fact_7399_mult__ceiling__le__Ints,axiom,
% 0.23/0.54 ! [A: real,B2: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.23/0.54 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.23/0.54 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % mult_ceiling_le_Ints
% 0.23/0.54 thf(fact_7400_mult__ceiling__le__Ints,axiom,
% 0.23/0.54 ! [A: rat,B2: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.54 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.54 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % mult_ceiling_le_Ints
% 0.23/0.54 thf(fact_7401_mult__ceiling__le__Ints,axiom,
% 0.23/0.54 ! [A: rat,B2: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.54 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.54 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % mult_ceiling_le_Ints
% 0.23/0.54 thf(fact_7402_mult__ceiling__le__Ints,axiom,
% 0.23/0.54 ! [A: rat,B2: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.23/0.54 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.23/0.54 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % mult_ceiling_le_Ints
% 0.23/0.54 thf(fact_7403_gbinomial__minus,axiom,
% 0.23/0.54 ! [A: complex,K: nat] :
% 0.23/0.54 ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 0.23/0.54 = ( 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 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_minus
% 0.23/0.54 thf(fact_7404_gbinomial__minus,axiom,
% 0.23/0.54 ! [A: real,K: nat] :
% 0.23/0.54 ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 0.23/0.54 = ( 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 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_minus
% 0.23/0.54 thf(fact_7405_gbinomial__minus,axiom,
% 0.23/0.54 ! [A: rat,K: nat] :
% 0.23/0.54 ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 0.23/0.54 = ( 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 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_minus
% 0.23/0.54 thf(fact_7406_plus__inverse__ge__2,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.54 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % plus_inverse_ge_2
% 0.23/0.54 thf(fact_7407_real__inv__sqrt__pow2,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.54 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.54 = ( inverse_inverse_real @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % real_inv_sqrt_pow2
% 0.23/0.54 thf(fact_7408_gbinomial__reduce__nat,axiom,
% 0.23/0.54 ! [K: nat,A: complex] :
% 0.23/0.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.23/0.54 => ( ( gbinomial_complex @ A @ K )
% 0.23/0.54 = ( 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 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_reduce_nat
% 0.23/0.54 thf(fact_7409_gbinomial__reduce__nat,axiom,
% 0.23/0.54 ! [K: nat,A: real] :
% 0.23/0.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.23/0.54 => ( ( gbinomial_real @ A @ K )
% 0.23/0.54 = ( 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 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_reduce_nat
% 0.23/0.54 thf(fact_7410_gbinomial__reduce__nat,axiom,
% 0.23/0.54 ! [K: nat,A: rat] :
% 0.23/0.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.23/0.54 => ( ( gbinomial_rat @ A @ K )
% 0.23/0.54 = ( 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 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_reduce_nat
% 0.23/0.54 thf(fact_7411_gbinomial__pochhammer,axiom,
% 0.23/0.54 ( gbinomial_complex
% 0.23/0.54 = ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A4 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_pochhammer
% 0.23/0.54 thf(fact_7412_gbinomial__pochhammer,axiom,
% 0.23/0.54 ( gbinomial_rat
% 0.23/0.54 = ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A4 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_pochhammer
% 0.23/0.54 thf(fact_7413_gbinomial__pochhammer,axiom,
% 0.23/0.54 ( gbinomial_real
% 0.23/0.54 = ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A4 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_pochhammer
% 0.23/0.54 thf(fact_7414_gbinomial__pochhammer_H,axiom,
% 0.23/0.54 ( gbinomial_complex
% 0.23/0.54 = ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_pochhammer'
% 0.23/0.54 thf(fact_7415_gbinomial__pochhammer_H,axiom,
% 0.23/0.54 ( gbinomial_rat
% 0.23/0.54 = ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_pochhammer'
% 0.23/0.54 thf(fact_7416_gbinomial__pochhammer_H,axiom,
% 0.23/0.54 ( gbinomial_real
% 0.23/0.54 = ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % gbinomial_pochhammer'
% 0.23/0.54 thf(fact_7417_tan__cot,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) )
% 0.23/0.54 = ( inverse_inverse_real @ ( tan_real @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % tan_cot
% 0.23/0.54 thf(fact_7418_sin__integer__2pi,axiom,
% 0.23/0.54 ! [N: real] :
% 0.23/0.54 ( ( member_real @ N @ ring_1_Ints_real )
% 0.23/0.54 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.23/0.54 = zero_zero_real ) ) ).
% 0.23/0.54
% 0.23/0.54 % sin_integer_2pi
% 0.23/0.54 thf(fact_7419_cos__integer__2pi,axiom,
% 0.23/0.54 ! [N: real] :
% 0.23/0.54 ( ( member_real @ N @ ring_1_Ints_real )
% 0.23/0.54 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.23/0.54 = one_one_real ) ) ).
% 0.23/0.54
% 0.23/0.54 % cos_integer_2pi
% 0.23/0.54 thf(fact_7420_xor__nat__unfold,axiom,
% 0.23/0.54 ( bit_se6528837805403552850or_nat
% 0.23/0.54 = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = 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 @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % xor_nat_unfold
% 0.23/0.54 thf(fact_7421_xor__nat__rec,axiom,
% 0.23/0.54 ( bit_se6528837805403552850or_nat
% 0.23/0.54 = ( ^ [M6: nat,N2: nat] :
% 0.23/0.54 ( plus_plus_nat
% 0.23/0.54 @ ( zero_n2687167440665602831ol_nat
% 0.23/0.54 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
% 0.23/0.54 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.23/0.54 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % xor_nat_rec
% 0.23/0.54 thf(fact_7422_real__le__x__sinh,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.54 => ( ord_less_eq_real @ X4 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % real_le_x_sinh
% 0.23/0.54 thf(fact_7423_xor__one__eq,axiom,
% 0.23/0.54 ! [A: code_integer] :
% 0.23/0.54 ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
% 0.23/0.54 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.23/0.54 @ ( zero_n356916108424825756nteger
% 0.23/0.54 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % xor_one_eq
% 0.23/0.54 thf(fact_7424_xor__one__eq,axiom,
% 0.23/0.54 ! [A: nat] :
% 0.23/0.54 ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
% 0.23/0.54 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.23/0.54 @ ( zero_n2687167440665602831ol_nat
% 0.23/0.54 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % xor_one_eq
% 0.23/0.54 thf(fact_7425_xor__one__eq,axiom,
% 0.23/0.54 ! [A: int] :
% 0.23/0.54 ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
% 0.23/0.54 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.23/0.54 @ ( zero_n2684676970156552555ol_int
% 0.23/0.54 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % xor_one_eq
% 0.23/0.54 thf(fact_7426_one__xor__eq,axiom,
% 0.23/0.54 ! [A: code_integer] :
% 0.23/0.54 ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
% 0.23/0.54 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.23/0.54 @ ( zero_n356916108424825756nteger
% 0.23/0.54 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % one_xor_eq
% 0.23/0.54 thf(fact_7427_one__xor__eq,axiom,
% 0.23/0.54 ! [A: nat] :
% 0.23/0.54 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
% 0.23/0.54 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.23/0.54 @ ( zero_n2687167440665602831ol_nat
% 0.23/0.54 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % one_xor_eq
% 0.23/0.54 thf(fact_7428_one__xor__eq,axiom,
% 0.23/0.54 ! [A: int] :
% 0.23/0.54 ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
% 0.23/0.54 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.23/0.54 @ ( zero_n2684676970156552555ol_int
% 0.23/0.54 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % one_xor_eq
% 0.23/0.54 thf(fact_7429_real__le__abs__sinh,axiom,
% 0.23/0.54 ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % real_le_abs_sinh
% 0.23/0.54 thf(fact_7430_tan__sec,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ( cos_real @ X4 )
% 0.23/0.54 != zero_zero_real )
% 0.23/0.54 => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.54 = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % tan_sec
% 0.23/0.54 thf(fact_7431_tan__sec,axiom,
% 0.23/0.54 ! [X4: complex] :
% 0.23/0.54 ( ( ( cos_complex @ X4 )
% 0.23/0.54 != zero_zero_complex )
% 0.23/0.54 => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.54 = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % tan_sec
% 0.23/0.54 thf(fact_7432_sinh__ln__real,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.54 => ( ( sinh_real @ ( ln_ln_real @ X4 ) )
% 0.23/0.54 = ( divide_divide_real @ ( minus_minus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_ln_real
% 0.23/0.54 thf(fact_7433_cosh__ln__real,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.54 => ( ( cosh_real @ ( ln_ln_real @ X4 ) )
% 0.23/0.54 = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_ln_real
% 0.23/0.54 thf(fact_7434_cosh__zero__iff,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ( cosh_real @ X4 )
% 0.23/0.54 = zero_zero_real )
% 0.23/0.54 = ( ( power_power_real @ ( exp_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.54 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_zero_iff
% 0.23/0.54 thf(fact_7435_cosh__zero__iff,axiom,
% 0.23/0.54 ! [X4: complex] :
% 0.23/0.54 ( ( ( cosh_complex @ X4 )
% 0.23/0.54 = zero_zero_complex )
% 0.23/0.54 = ( ( power_power_complex @ ( exp_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.54 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_zero_iff
% 0.23/0.54 thf(fact_7436_psubsetI,axiom,
% 0.23/0.54 ! [A3: set_nat,B4: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.23/0.54 => ( ( A3 != B4 )
% 0.23/0.54 => ( ord_less_set_nat @ A3 @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubsetI
% 0.23/0.54 thf(fact_7437_push__bit__numeral__minus__1,axiom,
% 0.23/0.54 ! [N: num] :
% 0.23/0.54 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.54 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_numeral_minus_1
% 0.23/0.54 thf(fact_7438_push__bit__numeral__minus__1,axiom,
% 0.23/0.54 ! [N: num] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.54 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_numeral_minus_1
% 0.23/0.54 thf(fact_7439_push__bit__nonnegative__int__iff,axiom,
% 0.23/0.54 ! [N: nat,K: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.23/0.54 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_nonnegative_int_iff
% 0.23/0.54 thf(fact_7440_push__bit__negative__int__iff,axiom,
% 0.23/0.54 ! [N: nat,K: int] :
% 0.23/0.54 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 0.23/0.54 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_negative_int_iff
% 0.23/0.54 thf(fact_7441_push__bit__of__0,axiom,
% 0.23/0.54 ! [N: nat] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
% 0.23/0.54 = zero_zero_int ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_of_0
% 0.23/0.54 thf(fact_7442_push__bit__of__0,axiom,
% 0.23/0.54 ! [N: nat] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
% 0.23/0.54 = zero_zero_nat ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_of_0
% 0.23/0.54 thf(fact_7443_push__bit__eq__0__iff,axiom,
% 0.23/0.54 ! [N: nat,A: int] :
% 0.23/0.54 ( ( ( bit_se545348938243370406it_int @ N @ A )
% 0.23/0.54 = zero_zero_int )
% 0.23/0.54 = ( A = zero_zero_int ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_eq_0_iff
% 0.23/0.54 thf(fact_7444_push__bit__eq__0__iff,axiom,
% 0.23/0.54 ! [N: nat,A: nat] :
% 0.23/0.54 ( ( ( bit_se547839408752420682it_nat @ N @ A )
% 0.23/0.54 = zero_zero_nat )
% 0.23/0.54 = ( A = zero_zero_nat ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_eq_0_iff
% 0.23/0.54 thf(fact_7445_push__bit__push__bit,axiom,
% 0.23/0.54 ! [M: nat,N: nat,A: int] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 0.23/0.54 = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_push_bit
% 0.23/0.54 thf(fact_7446_push__bit__push__bit,axiom,
% 0.23/0.54 ! [M: nat,N: nat,A: nat] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 0.23/0.54 = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_push_bit
% 0.23/0.54 thf(fact_7447_push__bit__and,axiom,
% 0.23/0.54 ! [N: nat,A: int,B2: int] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
% 0.23/0.54 = ( bit_se725231765392027082nd_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_and
% 0.23/0.54 thf(fact_7448_push__bit__and,axiom,
% 0.23/0.54 ! [N: nat,A: nat,B2: nat] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
% 0.23/0.54 = ( bit_se727722235901077358nd_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_and
% 0.23/0.54 thf(fact_7449_push__bit__xor,axiom,
% 0.23/0.54 ! [N: nat,A: int,B2: int] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B2 ) )
% 0.23/0.54 = ( bit_se6526347334894502574or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_xor
% 0.23/0.54 thf(fact_7450_push__bit__xor,axiom,
% 0.23/0.54 ! [N: nat,A: nat,B2: nat] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B2 ) )
% 0.23/0.54 = ( bit_se6528837805403552850or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_xor
% 0.23/0.54 thf(fact_7451_concat__bit__of__zero__1,axiom,
% 0.23/0.54 ! [N: nat,L2: int] :
% 0.23/0.54 ( ( bit_concat_bit @ N @ zero_zero_int @ L2 )
% 0.23/0.54 = ( bit_se545348938243370406it_int @ N @ L2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % concat_bit_of_zero_1
% 0.23/0.54 thf(fact_7452_sinh__real__less__iff,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ord_less_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) )
% 0.23/0.54 = ( ord_less_real @ X4 @ Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_real_less_iff
% 0.23/0.54 thf(fact_7453_sinh__real__le__iff,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) )
% 0.23/0.54 = ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_real_le_iff
% 0.23/0.54 thf(fact_7454_cosh__0,axiom,
% 0.23/0.54 ( ( cosh_complex @ zero_zero_complex )
% 0.23/0.54 = one_one_complex ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_0
% 0.23/0.54 thf(fact_7455_cosh__0,axiom,
% 0.23/0.54 ( ( cosh_real @ zero_zero_real )
% 0.23/0.54 = one_one_real ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_0
% 0.23/0.54 thf(fact_7456_sinh__real__pos__iff,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X4 ) )
% 0.23/0.54 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_real_pos_iff
% 0.23/0.54 thf(fact_7457_sinh__real__neg__iff,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ord_less_real @ ( sinh_real @ X4 ) @ zero_zero_real )
% 0.23/0.54 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_real_neg_iff
% 0.23/0.54 thf(fact_7458_sinh__real__nonneg__iff,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X4 ) )
% 0.23/0.54 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_real_nonneg_iff
% 0.23/0.54 thf(fact_7459_sinh__real__nonpos__iff,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ ( sinh_real @ X4 ) @ zero_zero_real )
% 0.23/0.54 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_real_nonpos_iff
% 0.23/0.54 thf(fact_7460_xor__nonnegative__int__iff,axiom,
% 0.23/0.54 ! [K: int,L2: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
% 0.23/0.54 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.54 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % xor_nonnegative_int_iff
% 0.23/0.54 thf(fact_7461_xor__negative__int__iff,axiom,
% 0.23/0.54 ! [K: int,L2: int] :
% 0.23/0.54 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ zero_zero_int )
% 0.23/0.54 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.23/0.54 != ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % xor_negative_int_iff
% 0.23/0.54 thf(fact_7462_push__bit__Suc__numeral,axiom,
% 0.23/0.54 ! [N: nat,K: num] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( numeral_numeral_int @ K ) )
% 0.23/0.54 = ( bit_se545348938243370406it_int @ N @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_Suc_numeral
% 0.23/0.54 thf(fact_7463_push__bit__Suc__numeral,axiom,
% 0.23/0.54 ! [N: nat,K: num] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.23/0.54 = ( bit_se547839408752420682it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_Suc_numeral
% 0.23/0.54 thf(fact_7464_push__bit__Suc__minus__numeral,axiom,
% 0.23/0.54 ! [N: nat,K: num] :
% 0.23/0.54 ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.23/0.54 = ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_Suc_minus_numeral
% 0.23/0.54 thf(fact_7465_push__bit__Suc__minus__numeral,axiom,
% 0.23/0.54 ! [N: nat,K: num] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.54 = ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_Suc_minus_numeral
% 0.23/0.54 thf(fact_7466_push__bit__numeral,axiom,
% 0.23/0.54 ! [L2: num,K: num] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ K ) )
% 0.23/0.54 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_numeral
% 0.23/0.54 thf(fact_7467_push__bit__numeral,axiom,
% 0.23/0.54 ! [L2: num,K: num] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ K ) )
% 0.23/0.54 = ( bit_se547839408752420682it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_numeral
% 0.23/0.54 thf(fact_7468_push__bit__of__Suc__0,axiom,
% 0.23/0.54 ! [N: nat] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.23/0.54 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_of_Suc_0
% 0.23/0.54 thf(fact_7469_push__bit__Suc,axiom,
% 0.23/0.54 ! [N: nat,A: int] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
% 0.23/0.54 = ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_Suc
% 0.23/0.54 thf(fact_7470_push__bit__Suc,axiom,
% 0.23/0.54 ! [N: nat,A: nat] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
% 0.23/0.54 = ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_Suc
% 0.23/0.54 thf(fact_7471_push__bit__of__1,axiom,
% 0.23/0.54 ! [N: nat] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ N @ one_one_int )
% 0.23/0.54 = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_of_1
% 0.23/0.54 thf(fact_7472_push__bit__of__1,axiom,
% 0.23/0.54 ! [N: nat] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
% 0.23/0.54 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_of_1
% 0.23/0.54 thf(fact_7473_even__push__bit__iff,axiom,
% 0.23/0.54 ! [N: nat,A: code_integer] :
% 0.23/0.54 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 0.23/0.54 = ( ( N != zero_zero_nat )
% 0.23/0.54 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % even_push_bit_iff
% 0.23/0.54 thf(fact_7474_even__push__bit__iff,axiom,
% 0.23/0.54 ! [N: nat,A: int] :
% 0.23/0.54 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
% 0.23/0.54 = ( ( N != zero_zero_nat )
% 0.23/0.54 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % even_push_bit_iff
% 0.23/0.54 thf(fact_7475_even__push__bit__iff,axiom,
% 0.23/0.54 ! [N: nat,A: nat] :
% 0.23/0.54 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 0.23/0.54 = ( ( N != zero_zero_nat )
% 0.23/0.54 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % even_push_bit_iff
% 0.23/0.54 thf(fact_7476_push__bit__minus__numeral,axiom,
% 0.23/0.54 ! [L2: num,K: num] :
% 0.23/0.54 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.23/0.54 = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_minus_numeral
% 0.23/0.54 thf(fact_7477_push__bit__minus__numeral,axiom,
% 0.23/0.54 ! [L2: num,K: num] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.54 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_minus_numeral
% 0.23/0.54 thf(fact_7478_flip__bit__int__def,axiom,
% 0.23/0.54 ( bit_se2159334234014336723it_int
% 0.23/0.54 = ( ^ [N2: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % flip_bit_int_def
% 0.23/0.54 thf(fact_7479_push__bit__add,axiom,
% 0.23/0.54 ! [N: nat,A: int,B2: int] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ N @ ( plus_plus_int @ A @ B2 ) )
% 0.23/0.54 = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_add
% 0.23/0.54 thf(fact_7480_push__bit__add,axiom,
% 0.23/0.54 ! [N: nat,A: nat,B2: nat] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ N @ ( plus_plus_nat @ A @ B2 ) )
% 0.23/0.54 = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_add
% 0.23/0.54 thf(fact_7481_cosh__plus__sinh,axiom,
% 0.23/0.54 ! [X4: complex] :
% 0.23/0.54 ( ( plus_plus_complex @ ( cosh_complex @ X4 ) @ ( sinh_complex @ X4 ) )
% 0.23/0.54 = ( exp_complex @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_plus_sinh
% 0.23/0.54 thf(fact_7482_cosh__plus__sinh,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( plus_plus_real @ ( cosh_real @ X4 ) @ ( sinh_real @ X4 ) )
% 0.23/0.54 = ( exp_real @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_plus_sinh
% 0.23/0.54 thf(fact_7483_sinh__plus__cosh,axiom,
% 0.23/0.54 ! [X4: complex] :
% 0.23/0.54 ( ( plus_plus_complex @ ( sinh_complex @ X4 ) @ ( cosh_complex @ X4 ) )
% 0.23/0.54 = ( exp_complex @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_plus_cosh
% 0.23/0.54 thf(fact_7484_sinh__plus__cosh,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( plus_plus_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) )
% 0.23/0.54 = ( exp_real @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_plus_cosh
% 0.23/0.54 thf(fact_7485_psubsetD,axiom,
% 0.23/0.54 ! [A3: set_Extended_enat,B4: set_Extended_enat,C: extended_enat] :
% 0.23/0.54 ( ( ord_le2529575680413868914d_enat @ A3 @ B4 )
% 0.23/0.54 => ( ( member_Extended_enat @ C @ A3 )
% 0.23/0.54 => ( member_Extended_enat @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubsetD
% 0.23/0.54 thf(fact_7486_psubsetD,axiom,
% 0.23/0.54 ! [A3: set_complex,B4: set_complex,C: complex] :
% 0.23/0.54 ( ( ord_less_set_complex @ A3 @ B4 )
% 0.23/0.54 => ( ( member_complex @ C @ A3 )
% 0.23/0.54 => ( member_complex @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubsetD
% 0.23/0.54 thf(fact_7487_psubsetD,axiom,
% 0.23/0.54 ! [A3: set_real,B4: set_real,C: real] :
% 0.23/0.54 ( ( ord_less_set_real @ A3 @ B4 )
% 0.23/0.54 => ( ( member_real @ C @ A3 )
% 0.23/0.54 => ( member_real @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubsetD
% 0.23/0.54 thf(fact_7488_psubsetD,axiom,
% 0.23/0.54 ! [A3: set_nat,B4: set_nat,C: nat] :
% 0.23/0.54 ( ( ord_less_set_nat @ A3 @ B4 )
% 0.23/0.54 => ( ( member_nat @ C @ A3 )
% 0.23/0.54 => ( member_nat @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubsetD
% 0.23/0.54 thf(fact_7489_psubsetD,axiom,
% 0.23/0.54 ! [A3: set_int,B4: set_int,C: int] :
% 0.23/0.54 ( ( ord_less_set_int @ A3 @ B4 )
% 0.23/0.54 => ( ( member_int @ C @ A3 )
% 0.23/0.54 => ( member_int @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubsetD
% 0.23/0.54 thf(fact_7490_sinh__le__cosh__real,axiom,
% 0.23/0.54 ! [X4: real] : ( ord_less_eq_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_le_cosh_real
% 0.23/0.54 thf(fact_7491_sinh__less__cosh__real,axiom,
% 0.23/0.54 ! [X4: real] : ( ord_less_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_less_cosh_real
% 0.23/0.54 thf(fact_7492_push__bit__of__int,axiom,
% 0.23/0.54 ! [N: nat,K: int] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ N @ ( ring_1_of_int_int @ K ) )
% 0.23/0.54 = ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_of_int
% 0.23/0.54 thf(fact_7493_of__nat__push__bit,axiom,
% 0.23/0.54 ! [M: nat,N: nat] :
% 0.23/0.54 ( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 0.23/0.54 = ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % of_nat_push_bit
% 0.23/0.54 thf(fact_7494_of__nat__push__bit,axiom,
% 0.23/0.54 ! [M: nat,N: nat] :
% 0.23/0.54 ( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 0.23/0.54 = ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % of_nat_push_bit
% 0.23/0.54 thf(fact_7495_push__bit__of__nat,axiom,
% 0.23/0.54 ! [N: nat,M: nat] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 0.23/0.54 = ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_of_nat
% 0.23/0.54 thf(fact_7496_push__bit__of__nat,axiom,
% 0.23/0.54 ! [N: nat,M: nat] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 0.23/0.54 = ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_of_nat
% 0.23/0.54 thf(fact_7497_psubset__imp__ex__mem,axiom,
% 0.23/0.54 ! [A3: set_Extended_enat,B4: set_Extended_enat] :
% 0.23/0.54 ( ( ord_le2529575680413868914d_enat @ A3 @ B4 )
% 0.23/0.54 => ? [B5: extended_enat] : ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ B4 @ A3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubset_imp_ex_mem
% 0.23/0.54 thf(fact_7498_psubset__imp__ex__mem,axiom,
% 0.23/0.54 ! [A3: set_complex,B4: set_complex] :
% 0.23/0.54 ( ( ord_less_set_complex @ A3 @ B4 )
% 0.23/0.54 => ? [B5: complex] : ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B4 @ A3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubset_imp_ex_mem
% 0.23/0.54 thf(fact_7499_psubset__imp__ex__mem,axiom,
% 0.23/0.54 ! [A3: set_real,B4: set_real] :
% 0.23/0.54 ( ( ord_less_set_real @ A3 @ B4 )
% 0.23/0.54 => ? [B5: real] : ( member_real @ B5 @ ( minus_minus_set_real @ B4 @ A3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubset_imp_ex_mem
% 0.23/0.54 thf(fact_7500_psubset__imp__ex__mem,axiom,
% 0.23/0.54 ! [A3: set_int,B4: set_int] :
% 0.23/0.54 ( ( ord_less_set_int @ A3 @ B4 )
% 0.23/0.54 => ? [B5: int] : ( member_int @ B5 @ ( minus_minus_set_int @ B4 @ A3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubset_imp_ex_mem
% 0.23/0.54 thf(fact_7501_psubset__imp__ex__mem,axiom,
% 0.23/0.54 ! [A3: set_nat,B4: set_nat] :
% 0.23/0.54 ( ( ord_less_set_nat @ A3 @ B4 )
% 0.23/0.54 => ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B4 @ A3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubset_imp_ex_mem
% 0.23/0.54 thf(fact_7502_sinh__diff,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( sinh_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 0.23/0.54 = ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( cosh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_diff
% 0.23/0.54 thf(fact_7503_cosh__diff,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( cosh_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 0.23/0.54 = ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_diff
% 0.23/0.54 thf(fact_7504_sinh__add,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( sinh_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.54 = ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( cosh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_add
% 0.23/0.54 thf(fact_7505_cosh__add,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( cosh_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.54 = ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_add
% 0.23/0.54 thf(fact_7506_push__bit__nat__eq,axiom,
% 0.23/0.54 ! [N: nat,K: int] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 0.23/0.54 = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_nat_eq
% 0.23/0.54 thf(fact_7507_push__bit__minus,axiom,
% 0.23/0.54 ! [N: nat,A: code_integer] :
% 0.23/0.54 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ A ) )
% 0.23/0.54 = ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_minus
% 0.23/0.54 thf(fact_7508_push__bit__minus,axiom,
% 0.23/0.54 ! [N: nat,A: int] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ A ) )
% 0.23/0.54 = ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_minus
% 0.23/0.54 thf(fact_7509_tanh__def,axiom,
% 0.23/0.54 ( tanh_complex
% 0.23/0.54 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % tanh_def
% 0.23/0.54 thf(fact_7510_tanh__def,axiom,
% 0.23/0.54 ( tanh_real
% 0.23/0.54 = ( ^ [X: real] : ( divide_divide_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % tanh_def
% 0.23/0.54 thf(fact_7511_bit__xor__int__iff,axiom,
% 0.23/0.54 ! [K: int,L2: int,N: nat] :
% 0.23/0.54 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N )
% 0.23/0.54 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.23/0.54 != ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit_xor_int_iff
% 0.23/0.54 thf(fact_7512_sinh__minus__cosh,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( minus_minus_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) )
% 0.23/0.54 = ( uminus_uminus_real @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_minus_cosh
% 0.23/0.54 thf(fact_7513_sinh__minus__cosh,axiom,
% 0.23/0.54 ! [X4: complex] :
% 0.23/0.54 ( ( minus_minus_complex @ ( sinh_complex @ X4 ) @ ( cosh_complex @ X4 ) )
% 0.23/0.54 = ( uminus1482373934393186551omplex @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_minus_cosh
% 0.23/0.54 thf(fact_7514_cosh__minus__sinh,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( minus_minus_real @ ( cosh_real @ X4 ) @ ( sinh_real @ X4 ) )
% 0.23/0.54 = ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_minus_sinh
% 0.23/0.54 thf(fact_7515_cosh__minus__sinh,axiom,
% 0.23/0.54 ! [X4: complex] :
% 0.23/0.54 ( ( minus_minus_complex @ ( cosh_complex @ X4 ) @ ( sinh_complex @ X4 ) )
% 0.23/0.54 = ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_minus_sinh
% 0.23/0.54 thf(fact_7516_XOR__lower,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.54 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % XOR_lower
% 0.23/0.54 thf(fact_7517_cosh__real__pos,axiom,
% 0.23/0.54 ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_real_pos
% 0.23/0.54 thf(fact_7518_cosh__real__nonneg,axiom,
% 0.23/0.54 ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_real_nonneg
% 0.23/0.54 thf(fact_7519_cosh__real__nonneg__le__iff,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.54 => ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
% 0.23/0.54 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_real_nonneg_le_iff
% 0.23/0.54 thf(fact_7520_cosh__real__nonpos__le__iff,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.23/0.54 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.23/0.54 => ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
% 0.23/0.54 = ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_real_nonpos_le_iff
% 0.23/0.54 thf(fact_7521_cosh__real__ge__1,axiom,
% 0.23/0.54 ! [X4: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_real_ge_1
% 0.23/0.54 thf(fact_7522_sinh__double,axiom,
% 0.23/0.54 ! [X4: complex] :
% 0.23/0.54 ( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.54 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X4 ) ) @ ( cosh_complex @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_double
% 0.23/0.54 thf(fact_7523_sinh__double,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.54 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X4 ) ) @ ( cosh_real @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_double
% 0.23/0.54 thf(fact_7524_push__bit__take__bit,axiom,
% 0.23/0.54 ! [M: nat,N: nat,A: int] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.23/0.54 = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_take_bit
% 0.23/0.54 thf(fact_7525_push__bit__take__bit,axiom,
% 0.23/0.54 ! [M: nat,N: nat,A: nat] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.23/0.54 = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_take_bit
% 0.23/0.54 thf(fact_7526_take__bit__push__bit,axiom,
% 0.23/0.54 ! [M: nat,N: nat,A: int] :
% 0.23/0.54 ( ( bit_se2923211474154528505it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 0.23/0.54 = ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % take_bit_push_bit
% 0.23/0.54 thf(fact_7527_take__bit__push__bit,axiom,
% 0.23/0.54 ! [M: nat,N: nat,A: nat] :
% 0.23/0.54 ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 0.23/0.54 = ( bit_se547839408752420682it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % take_bit_push_bit
% 0.23/0.54 thf(fact_7528_divide__complex__def,axiom,
% 0.23/0.54 ( divide1717551699836669952omplex
% 0.23/0.54 = ( ^ [X: complex,Y: complex] : ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % divide_complex_def
% 0.23/0.54 thf(fact_7529_flip__bit__nat__def,axiom,
% 0.23/0.54 ( bit_se2161824704523386999it_nat
% 0.23/0.54 = ( ^ [M6: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % flip_bit_nat_def
% 0.23/0.54 thf(fact_7530_cosh__real__strict__mono,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.54 => ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.54 => ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_real_strict_mono
% 0.23/0.54 thf(fact_7531_cosh__real__nonneg__less__iff,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.54 => ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
% 0.23/0.54 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_real_nonneg_less_iff
% 0.23/0.54 thf(fact_7532_cosh__real__nonpos__less__iff,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.23/0.54 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.23/0.54 => ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
% 0.23/0.54 = ( ord_less_real @ Y3 @ X4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_real_nonpos_less_iff
% 0.23/0.54 thf(fact_7533_bit__push__bit__iff__int,axiom,
% 0.23/0.54 ! [M: nat,K: int,N: nat] :
% 0.23/0.54 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 0.23/0.54 = ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.54 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit_push_bit_iff_int
% 0.23/0.54 thf(fact_7534_cosh__square__eq,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.54 = ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_square_eq
% 0.23/0.54 thf(fact_7535_cosh__square__eq,axiom,
% 0.23/0.54 ! [X4: complex] :
% 0.23/0.54 ( ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.54 = ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_square_eq
% 0.23/0.54 thf(fact_7536_sinh__square__eq,axiom,
% 0.23/0.54 ! [X4: complex] :
% 0.23/0.54 ( ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.54 = ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_square_eq
% 0.23/0.54 thf(fact_7537_sinh__square__eq,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.54 = ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_square_eq
% 0.23/0.54 thf(fact_7538_hyperbolic__pythagoras,axiom,
% 0.23/0.54 ! [X4: complex] :
% 0.23/0.54 ( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.54 = one_one_complex ) ).
% 0.23/0.54
% 0.23/0.54 % hyperbolic_pythagoras
% 0.23/0.54 thf(fact_7539_hyperbolic__pythagoras,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.54 = one_one_real ) ).
% 0.23/0.54
% 0.23/0.54 % hyperbolic_pythagoras
% 0.23/0.54 thf(fact_7540_xor__nat__def,axiom,
% 0.23/0.54 ( bit_se6528837805403552850or_nat
% 0.23/0.54 = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % xor_nat_def
% 0.23/0.54 thf(fact_7541_psubsetE,axiom,
% 0.23/0.54 ! [A3: set_nat,B4: set_nat] :
% 0.23/0.54 ( ( ord_less_set_nat @ A3 @ B4 )
% 0.23/0.54 => ~ ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.23/0.54 => ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubsetE
% 0.23/0.54 thf(fact_7542_psubset__eq,axiom,
% 0.23/0.54 ( ord_less_set_nat
% 0.23/0.54 = ( ^ [A6: set_nat,B7: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ A6 @ B7 )
% 0.23/0.54 & ( A6 != B7 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubset_eq
% 0.23/0.54 thf(fact_7543_psubset__imp__subset,axiom,
% 0.23/0.54 ! [A3: set_nat,B4: set_nat] :
% 0.23/0.54 ( ( ord_less_set_nat @ A3 @ B4 )
% 0.23/0.54 => ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubset_imp_subset
% 0.23/0.54 thf(fact_7544_psubset__subset__trans,axiom,
% 0.23/0.54 ! [A3: set_nat,B4: set_nat,C5: set_nat] :
% 0.23/0.54 ( ( ord_less_set_nat @ A3 @ B4 )
% 0.23/0.54 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 0.23/0.54 => ( ord_less_set_nat @ A3 @ C5 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % psubset_subset_trans
% 0.23/0.54 thf(fact_7545_subset__not__subset__eq,axiom,
% 0.23/0.54 ( ord_less_set_nat
% 0.23/0.54 = ( ^ [A6: set_nat,B7: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ A6 @ B7 )
% 0.23/0.54 & ~ ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % subset_not_subset_eq
% 0.23/0.54 thf(fact_7546_subset__psubset__trans,axiom,
% 0.23/0.54 ! [A3: set_nat,B4: set_nat,C5: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.23/0.54 => ( ( ord_less_set_nat @ B4 @ C5 )
% 0.23/0.54 => ( ord_less_set_nat @ A3 @ C5 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % subset_psubset_trans
% 0.23/0.54 thf(fact_7547_subset__iff__psubset__eq,axiom,
% 0.23/0.54 ( ord_less_eq_set_nat
% 0.23/0.54 = ( ^ [A6: set_nat,B7: set_nat] :
% 0.23/0.54 ( ( ord_less_set_nat @ A6 @ B7 )
% 0.23/0.54 | ( A6 = B7 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % subset_iff_psubset_eq
% 0.23/0.54 thf(fact_7548_double__diff,axiom,
% 0.23/0.54 ! [A3: set_nat,B4: set_nat,C5: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 0.23/0.54 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 0.23/0.54 => ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C5 @ A3 ) )
% 0.23/0.54 = A3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % double_diff
% 0.23/0.54 thf(fact_7549_Diff__subset,axiom,
% 0.23/0.54 ! [A3: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ A3 ) ).
% 0.23/0.54
% 0.23/0.54 % Diff_subset
% 0.23/0.54 thf(fact_7550_Diff__mono,axiom,
% 0.23/0.54 ! [A3: set_nat,C5: set_nat,D4: set_nat,B4: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ A3 @ C5 )
% 0.23/0.54 => ( ( ord_less_eq_set_nat @ D4 @ B4 )
% 0.23/0.54 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ ( minus_minus_set_nat @ C5 @ D4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Diff_mono
% 0.23/0.54 thf(fact_7551_bit__push__bit__iff__nat,axiom,
% 0.23/0.54 ! [M: nat,Q2: nat,N: nat] :
% 0.23/0.54 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
% 0.23/0.54 = ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.54 & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit_push_bit_iff_nat
% 0.23/0.54 thf(fact_7552_concat__bit__eq,axiom,
% 0.23/0.54 ( bit_concat_bit
% 0.23/0.54 = ( ^ [N2: nat,K3: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % concat_bit_eq
% 0.23/0.54 thf(fact_7553_arcosh__cosh__real,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.54 => ( ( arcosh_real @ ( cosh_real @ X4 ) )
% 0.23/0.54 = X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % arcosh_cosh_real
% 0.23/0.54 thf(fact_7554_flip__bit__eq__xor,axiom,
% 0.23/0.54 ( bit_se2159334234014336723it_int
% 0.23/0.54 = ( ^ [N2: nat,A4: int] : ( bit_se6526347334894502574or_int @ A4 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % flip_bit_eq_xor
% 0.23/0.54 thf(fact_7555_flip__bit__eq__xor,axiom,
% 0.23/0.54 ( bit_se2161824704523386999it_nat
% 0.23/0.54 = ( ^ [N2: nat,A4: nat] : ( bit_se6528837805403552850or_nat @ A4 @ ( bit_se547839408752420682it_nat @ N2 @ one_one_nat ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % flip_bit_eq_xor
% 0.23/0.54 thf(fact_7556_cosh__double,axiom,
% 0.23/0.54 ! [X4: complex] :
% 0.23/0.54 ( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.54 = ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_double
% 0.23/0.54 thf(fact_7557_cosh__double,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 0.23/0.54 = ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_double
% 0.23/0.54 thf(fact_7558_push__bit__double,axiom,
% 0.23/0.54 ! [N: nat,A: int] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.23/0.54 = ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_double
% 0.23/0.54 thf(fact_7559_push__bit__double,axiom,
% 0.23/0.54 ! [N: nat,A: nat] :
% 0.23/0.54 ( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.54 = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_double
% 0.23/0.54 thf(fact_7560_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.23/0.54 ( bit_se1146084159140164899it_int
% 0.23/0.54 = ( ^ [A4: int,N2: nat] :
% 0.23/0.54 ( ( bit_se725231765392027082nd_int @ A4 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) )
% 0.23/0.54 != zero_zero_int ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit_iff_and_push_bit_not_eq_0
% 0.23/0.54 thf(fact_7561_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.23/0.54 ( bit_se1148574629649215175it_nat
% 0.23/0.54 = ( ^ [A4: nat,N2: nat] :
% 0.23/0.54 ( ( bit_se727722235901077358nd_nat @ A4 @ ( bit_se547839408752420682it_nat @ N2 @ one_one_nat ) )
% 0.23/0.54 != zero_zero_nat ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit_iff_and_push_bit_not_eq_0
% 0.23/0.54 thf(fact_7562_push__bit__int__def,axiom,
% 0.23/0.54 ( bit_se545348938243370406it_int
% 0.23/0.54 = ( ^ [N2: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_int_def
% 0.23/0.54 thf(fact_7563_push__bit__nat__def,axiom,
% 0.23/0.54 ( bit_se547839408752420682it_nat
% 0.23/0.54 = ( ^ [N2: nat,M6: nat] : ( times_times_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_nat_def
% 0.23/0.54 thf(fact_7564_push__bit__eq__mult,axiom,
% 0.23/0.54 ( bit_se545348938243370406it_int
% 0.23/0.54 = ( ^ [N2: nat,A4: int] : ( times_times_int @ A4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_eq_mult
% 0.23/0.54 thf(fact_7565_push__bit__eq__mult,axiom,
% 0.23/0.54 ( bit_se547839408752420682it_nat
% 0.23/0.54 = ( ^ [N2: nat,A4: nat] : ( times_times_nat @ A4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_eq_mult
% 0.23/0.54 thf(fact_7566_exp__dvdE,axiom,
% 0.23/0.54 ! [N: nat,A: code_integer] :
% 0.23/0.54 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A )
% 0.23/0.54 => ~ ! [B5: code_integer] :
% 0.23/0.54 ( A
% 0.23/0.54 != ( bit_se7788150548672797655nteger @ N @ B5 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % exp_dvdE
% 0.23/0.54 thf(fact_7567_exp__dvdE,axiom,
% 0.23/0.54 ! [N: nat,A: int] :
% 0.23/0.54 ( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A )
% 0.23/0.54 => ~ ! [B5: int] :
% 0.23/0.54 ( A
% 0.23/0.54 != ( bit_se545348938243370406it_int @ N @ B5 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % exp_dvdE
% 0.23/0.54 thf(fact_7568_exp__dvdE,axiom,
% 0.23/0.54 ! [N: nat,A: nat] :
% 0.23/0.54 ( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A )
% 0.23/0.54 => ~ ! [B5: nat] :
% 0.23/0.54 ( A
% 0.23/0.54 != ( bit_se547839408752420682it_nat @ N @ B5 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % exp_dvdE
% 0.23/0.54 thf(fact_7569_push__bit__minus__one,axiom,
% 0.23/0.54 ! [N: nat] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.54 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_minus_one
% 0.23/0.54 thf(fact_7570_tanh__add,axiom,
% 0.23/0.54 ! [X4: complex,Y3: complex] :
% 0.23/0.54 ( ( ( cosh_complex @ X4 )
% 0.23/0.54 != zero_zero_complex )
% 0.23/0.54 => ( ( ( cosh_complex @ Y3 )
% 0.23/0.54 != zero_zero_complex )
% 0.23/0.54 => ( ( tanh_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
% 0.23/0.54 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X4 ) @ ( tanh_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X4 ) @ ( tanh_complex @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % tanh_add
% 0.23/0.54 thf(fact_7571_tanh__add,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ( cosh_real @ X4 )
% 0.23/0.54 != zero_zero_real )
% 0.23/0.54 => ( ( ( cosh_real @ Y3 )
% 0.23/0.54 != zero_zero_real )
% 0.23/0.54 => ( ( tanh_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 0.23/0.54 = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % tanh_add
% 0.23/0.54 thf(fact_7572_XOR__upper,axiom,
% 0.23/0.54 ! [X4: int,N: nat,Y3: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.54 => ( ( ord_less_int @ X4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.54 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.54 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % XOR_upper
% 0.23/0.54 thf(fact_7573_cosh__field__def,axiom,
% 0.23/0.54 ( cosh_real
% 0.23/0.54 = ( ^ [Z5: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_field_def
% 0.23/0.54 thf(fact_7574_cosh__field__def,axiom,
% 0.23/0.54 ( cosh_complex
% 0.23/0.54 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % cosh_field_def
% 0.23/0.54 thf(fact_7575_complex__inverse,axiom,
% 0.23/0.54 ! [A: real,B2: real] :
% 0.23/0.54 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B2 ) )
% 0.23/0.54 = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B2 ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % complex_inverse
% 0.23/0.54 thf(fact_7576_xor__int__rec,axiom,
% 0.23/0.54 ( bit_se6526347334894502574or_int
% 0.23/0.54 = ( ^ [K3: int,L: int] :
% 0.23/0.54 ( plus_plus_int
% 0.23/0.54 @ ( zero_n2684676970156552555ol_int
% 0.23/0.54 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
% 0.23/0.54 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 0.23/0.54 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % xor_int_rec
% 0.23/0.54 thf(fact_7577_sinh__field__def,axiom,
% 0.23/0.54 ( sinh_real
% 0.23/0.54 = ( ^ [Z5: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_field_def
% 0.23/0.54 thf(fact_7578_sinh__field__def,axiom,
% 0.23/0.54 ( sinh_complex
% 0.23/0.54 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % sinh_field_def
% 0.23/0.54 thf(fact_7579_xor__int__unfold,axiom,
% 0.23/0.54 ( bit_se6526347334894502574or_int
% 0.23/0.54 = ( ^ [K3: int,L: int] :
% 0.23/0.54 ( if_int
% 0.23/0.54 @ ( K3
% 0.23/0.54 = ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.54 @ ( bit_ri7919022796975470100ot_int @ L )
% 0.23/0.54 @ ( if_int
% 0.23/0.54 @ ( L
% 0.23/0.54 = ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.54 @ ( bit_ri7919022796975470100ot_int @ K3 )
% 0.23/0.54 @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % xor_int_unfold
% 0.23/0.54 thf(fact_7580_True,axiom,
% 0.23/0.54 ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% 0.23/0.54
% 0.23/0.54 % True
% 0.23/0.54 thf(fact_7581_Cauchy__iff2,axiom,
% 0.23/0.54 ( topolo4055970368930404560y_real
% 0.23/0.54 = ( ^ [X6: nat > real] :
% 0.23/0.54 ! [J3: nat] :
% 0.23/0.54 ? [M8: nat] :
% 0.23/0.54 ! [M6: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ M8 @ M6 )
% 0.23/0.54 => ! [N2: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ M8 @ N2 )
% 0.23/0.54 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Cauchy_iff2
% 0.23/0.54 thf(fact_7582_dual__order_Orefl,axiom,
% 0.23/0.54 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.refl
% 0.23/0.54 thf(fact_7583_dual__order_Orefl,axiom,
% 0.23/0.54 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.refl
% 0.23/0.54 thf(fact_7584_dual__order_Orefl,axiom,
% 0.23/0.54 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.refl
% 0.23/0.54 thf(fact_7585_dual__order_Orefl,axiom,
% 0.23/0.54 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.refl
% 0.23/0.54 thf(fact_7586_dual__order_Orefl,axiom,
% 0.23/0.54 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.refl
% 0.23/0.54 thf(fact_7587_order__refl,axiom,
% 0.23/0.54 ! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).
% 0.23/0.54
% 0.23/0.54 % order_refl
% 0.23/0.54 thf(fact_7588_order__refl,axiom,
% 0.23/0.54 ! [X4: rat] : ( ord_less_eq_rat @ X4 @ X4 ) ).
% 0.23/0.54
% 0.23/0.54 % order_refl
% 0.23/0.54 thf(fact_7589_order__refl,axiom,
% 0.23/0.54 ! [X4: num] : ( ord_less_eq_num @ X4 @ X4 ) ).
% 0.23/0.54
% 0.23/0.54 % order_refl
% 0.23/0.54 thf(fact_7590_order__refl,axiom,
% 0.23/0.54 ! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% 0.23/0.54
% 0.23/0.54 % order_refl
% 0.23/0.54 thf(fact_7591_order__refl,axiom,
% 0.23/0.54 ! [X4: int] : ( ord_less_eq_int @ X4 @ X4 ) ).
% 0.23/0.54
% 0.23/0.54 % order_refl
% 0.23/0.54 thf(fact_7592__C5_Ohyps_C_I4_J,axiom,
% 0.23/0.54 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.23/0.54 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 0.23/0.54
% 0.23/0.54 % "5.hyps"(4)
% 0.23/0.54 thf(fact_7593_DiffI,axiom,
% 0.23/0.54 ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
% 0.23/0.54 ( ( member_Extended_enat @ C @ A3 )
% 0.23/0.54 => ( ~ ( member_Extended_enat @ C @ B4 )
% 0.23/0.54 => ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffI
% 0.23/0.54 thf(fact_7594_DiffI,axiom,
% 0.23/0.54 ! [C: complex,A3: set_complex,B4: set_complex] :
% 0.23/0.54 ( ( member_complex @ C @ A3 )
% 0.23/0.54 => ( ~ ( member_complex @ C @ B4 )
% 0.23/0.54 => ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffI
% 0.23/0.54 thf(fact_7595_DiffI,axiom,
% 0.23/0.54 ! [C: real,A3: set_real,B4: set_real] :
% 0.23/0.54 ( ( member_real @ C @ A3 )
% 0.23/0.54 => ( ~ ( member_real @ C @ B4 )
% 0.23/0.54 => ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffI
% 0.23/0.54 thf(fact_7596_DiffI,axiom,
% 0.23/0.54 ! [C: int,A3: set_int,B4: set_int] :
% 0.23/0.54 ( ( member_int @ C @ A3 )
% 0.23/0.54 => ( ~ ( member_int @ C @ B4 )
% 0.23/0.54 => ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffI
% 0.23/0.54 thf(fact_7597_DiffI,axiom,
% 0.23/0.54 ! [C: nat,A3: set_nat,B4: set_nat] :
% 0.23/0.54 ( ( member_nat @ C @ A3 )
% 0.23/0.54 => ( ~ ( member_nat @ C @ B4 )
% 0.23/0.54 => ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffI
% 0.23/0.54 thf(fact_7598_Diff__iff,axiom,
% 0.23/0.54 ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
% 0.23/0.54 ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
% 0.23/0.54 = ( ( member_Extended_enat @ C @ A3 )
% 0.23/0.54 & ~ ( member_Extended_enat @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Diff_iff
% 0.23/0.54 thf(fact_7599_Diff__iff,axiom,
% 0.23/0.54 ! [C: complex,A3: set_complex,B4: set_complex] :
% 0.23/0.54 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
% 0.23/0.54 = ( ( member_complex @ C @ A3 )
% 0.23/0.54 & ~ ( member_complex @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Diff_iff
% 0.23/0.54 thf(fact_7600_Diff__iff,axiom,
% 0.23/0.54 ! [C: real,A3: set_real,B4: set_real] :
% 0.23/0.54 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
% 0.23/0.54 = ( ( member_real @ C @ A3 )
% 0.23/0.54 & ~ ( member_real @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Diff_iff
% 0.23/0.54 thf(fact_7601_Diff__iff,axiom,
% 0.23/0.54 ! [C: int,A3: set_int,B4: set_int] :
% 0.23/0.54 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
% 0.23/0.54 = ( ( member_int @ C @ A3 )
% 0.23/0.54 & ~ ( member_int @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Diff_iff
% 0.23/0.54 thf(fact_7602_Diff__iff,axiom,
% 0.23/0.54 ! [C: nat,A3: set_nat,B4: set_nat] :
% 0.23/0.54 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
% 0.23/0.54 = ( ( member_nat @ C @ A3 )
% 0.23/0.54 & ~ ( member_nat @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Diff_iff
% 0.23/0.54 thf(fact_7603_Diff__idemp,axiom,
% 0.23/0.54 ! [A3: set_nat,B4: set_nat] :
% 0.23/0.54 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ B4 )
% 0.23/0.54 = ( minus_minus_set_nat @ A3 @ B4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % Diff_idemp
% 0.23/0.54 thf(fact_7604_bit_Ocompl__eq__compl__iff,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( ( bit_ri7919022796975470100ot_int @ X4 )
% 0.23/0.54 = ( bit_ri7919022796975470100ot_int @ Y3 ) )
% 0.23/0.54 = ( X4 = Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.compl_eq_compl_iff
% 0.23/0.54 thf(fact_7605_bit_Odouble__compl,axiom,
% 0.23/0.54 ! [X4: int] :
% 0.23/0.54 ( ( bit_ri7919022796975470100ot_int @ ( bit_ri7919022796975470100ot_int @ X4 ) )
% 0.23/0.54 = X4 ) ).
% 0.23/0.54
% 0.23/0.54 % bit.double_compl
% 0.23/0.54 thf(fact_7606_bit_Oxor__compl__right,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_ri7919022796975470100ot_int @ Y3 ) )
% 0.23/0.54 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.xor_compl_right
% 0.23/0.54 thf(fact_7607_bit_Oxor__compl__left,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ Y3 )
% 0.23/0.54 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.xor_compl_left
% 0.23/0.54 thf(fact_7608_bit_Oconj__cancel__right,axiom,
% 0.23/0.54 ! [X4: int] :
% 0.23/0.54 ( ( bit_se725231765392027082nd_int @ X4 @ ( bit_ri7919022796975470100ot_int @ X4 ) )
% 0.23/0.54 = zero_zero_int ) ).
% 0.23/0.54
% 0.23/0.54 % bit.conj_cancel_right
% 0.23/0.54 thf(fact_7609_bit_Oconj__cancel__left,axiom,
% 0.23/0.54 ! [X4: int] :
% 0.23/0.54 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ X4 )
% 0.23/0.54 = zero_zero_int ) ).
% 0.23/0.54
% 0.23/0.54 % bit.conj_cancel_left
% 0.23/0.54 thf(fact_7610_bit_Ocompl__zero,axiom,
% 0.23/0.54 ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
% 0.23/0.54 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.compl_zero
% 0.23/0.54 thf(fact_7611_bit_Ocompl__zero,axiom,
% 0.23/0.54 ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
% 0.23/0.54 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.compl_zero
% 0.23/0.54 thf(fact_7612_bit_Ocompl__one,axiom,
% 0.23/0.54 ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.54 = zero_z3403309356797280102nteger ) ).
% 0.23/0.54
% 0.23/0.54 % bit.compl_one
% 0.23/0.54 thf(fact_7613_bit_Ocompl__one,axiom,
% 0.23/0.54 ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.54 = zero_zero_int ) ).
% 0.23/0.54
% 0.23/0.54 % bit.compl_one
% 0.23/0.54 thf(fact_7614_bit_Oxor__one__left,axiom,
% 0.23/0.54 ! [X4: code_integer] :
% 0.23/0.54 ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X4 )
% 0.23/0.54 = ( bit_ri7632146776885996613nteger @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.xor_one_left
% 0.23/0.54 thf(fact_7615_bit_Oxor__one__left,axiom,
% 0.23/0.54 ! [X4: int] :
% 0.23/0.54 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X4 )
% 0.23/0.54 = ( bit_ri7919022796975470100ot_int @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.xor_one_left
% 0.23/0.54 thf(fact_7616_bit_Oxor__one__right,axiom,
% 0.23/0.54 ! [X4: code_integer] :
% 0.23/0.54 ( ( bit_se3222712562003087583nteger @ X4 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.54 = ( bit_ri7632146776885996613nteger @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.xor_one_right
% 0.23/0.54 thf(fact_7617_bit_Oxor__one__right,axiom,
% 0.23/0.54 ! [X4: int] :
% 0.23/0.54 ( ( bit_se6526347334894502574or_int @ X4 @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.54 = ( bit_ri7919022796975470100ot_int @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.xor_one_right
% 0.23/0.54 thf(fact_7618_bit_Oxor__cancel__left,axiom,
% 0.23/0.54 ! [X4: code_integer] :
% 0.23/0.54 ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X4 ) @ X4 )
% 0.23/0.54 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.xor_cancel_left
% 0.23/0.54 thf(fact_7619_bit_Oxor__cancel__left,axiom,
% 0.23/0.54 ! [X4: int] :
% 0.23/0.54 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ X4 )
% 0.23/0.54 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.xor_cancel_left
% 0.23/0.54 thf(fact_7620_bit_Oxor__cancel__right,axiom,
% 0.23/0.54 ! [X4: code_integer] :
% 0.23/0.54 ( ( bit_se3222712562003087583nteger @ X4 @ ( bit_ri7632146776885996613nteger @ X4 ) )
% 0.23/0.54 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.xor_cancel_right
% 0.23/0.54 thf(fact_7621_bit_Oxor__cancel__right,axiom,
% 0.23/0.54 ! [X4: int] :
% 0.23/0.54 ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_ri7919022796975470100ot_int @ X4 ) )
% 0.23/0.54 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit.xor_cancel_right
% 0.23/0.54 thf(fact_7622_not__negative__int__iff,axiom,
% 0.23/0.54 ! [K: int] :
% 0.23/0.54 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 0.23/0.54 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_negative_int_iff
% 0.23/0.54 thf(fact_7623_not__nonnegative__int__iff,axiom,
% 0.23/0.54 ! [K: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.23/0.54 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_nonnegative_int_iff
% 0.23/0.54 thf(fact_7624_minus__not__numeral__eq,axiom,
% 0.23/0.54 ! [N: num] :
% 0.23/0.54 ( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.23/0.54 = ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % minus_not_numeral_eq
% 0.23/0.54 thf(fact_7625_minus__not__numeral__eq,axiom,
% 0.23/0.54 ! [N: num] :
% 0.23/0.54 ( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.54 = ( numeral_numeral_int @ ( inc @ N ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % minus_not_numeral_eq
% 0.23/0.54 thf(fact_7626_even__not__iff,axiom,
% 0.23/0.54 ! [A: code_integer] :
% 0.23/0.54 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri7632146776885996613nteger @ A ) )
% 0.23/0.54 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % even_not_iff
% 0.23/0.54 thf(fact_7627_even__not__iff,axiom,
% 0.23/0.54 ! [A: int] :
% 0.23/0.54 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ A ) )
% 0.23/0.54 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % even_not_iff
% 0.23/0.54 thf(fact_7628_push__bit__minus__one__eq__not__mask,axiom,
% 0.23/0.54 ! [N: nat] :
% 0.23/0.54 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.23/0.54 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_minus_one_eq_not_mask
% 0.23/0.54 thf(fact_7629_push__bit__minus__one__eq__not__mask,axiom,
% 0.23/0.54 ! [N: nat] :
% 0.23/0.54 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.54 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % push_bit_minus_one_eq_not_mask
% 0.23/0.54 thf(fact_7630_not__one__eq,axiom,
% 0.23/0.54 ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
% 0.23/0.54 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_one_eq
% 0.23/0.54 thf(fact_7631_not__one__eq,axiom,
% 0.23/0.54 ( ( bit_ri7919022796975470100ot_int @ one_one_int )
% 0.23/0.54 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_one_eq
% 0.23/0.54 thf(fact_7632_length__induct,axiom,
% 0.23/0.54 ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
% 0.23/0.54 ( ! [Xs3: list_VEBT_VEBT] :
% 0.23/0.54 ( ! [Ys: list_VEBT_VEBT] :
% 0.23/0.54 ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 0.23/0.54 => ( P @ Ys ) )
% 0.23/0.54 => ( P @ Xs3 ) )
% 0.23/0.54 => ( P @ Xs2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % length_induct
% 0.23/0.54 thf(fact_7633_length__induct,axiom,
% 0.23/0.54 ! [P: list_o > $o,Xs2: list_o] :
% 0.23/0.54 ( ! [Xs3: list_o] :
% 0.23/0.54 ( ! [Ys: list_o] :
% 0.23/0.54 ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs3 ) )
% 0.23/0.54 => ( P @ Ys ) )
% 0.23/0.54 => ( P @ Xs3 ) )
% 0.23/0.54 => ( P @ Xs2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % length_induct
% 0.23/0.54 thf(fact_7634_length__induct,axiom,
% 0.23/0.54 ! [P: list_nat > $o,Xs2: list_nat] :
% 0.23/0.54 ( ! [Xs3: list_nat] :
% 0.23/0.54 ( ! [Ys: list_nat] :
% 0.23/0.54 ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs3 ) )
% 0.23/0.54 => ( P @ Ys ) )
% 0.23/0.54 => ( P @ Xs3 ) )
% 0.23/0.54 => ( P @ Xs2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % length_induct
% 0.23/0.54 thf(fact_7635_length__induct,axiom,
% 0.23/0.54 ! [P: list_int > $o,Xs2: list_int] :
% 0.23/0.54 ( ! [Xs3: list_int] :
% 0.23/0.54 ( ! [Ys: list_int] :
% 0.23/0.54 ( ( ord_less_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs3 ) )
% 0.23/0.54 => ( P @ Ys ) )
% 0.23/0.54 => ( P @ Xs3 ) )
% 0.23/0.54 => ( P @ Xs2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % length_induct
% 0.23/0.54 thf(fact_7636_DiffE,axiom,
% 0.23/0.54 ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
% 0.23/0.54 ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
% 0.23/0.54 => ~ ( ( member_Extended_enat @ C @ A3 )
% 0.23/0.54 => ( member_Extended_enat @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffE
% 0.23/0.54 thf(fact_7637_DiffE,axiom,
% 0.23/0.54 ! [C: complex,A3: set_complex,B4: set_complex] :
% 0.23/0.54 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
% 0.23/0.54 => ~ ( ( member_complex @ C @ A3 )
% 0.23/0.54 => ( member_complex @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffE
% 0.23/0.54 thf(fact_7638_DiffE,axiom,
% 0.23/0.54 ! [C: real,A3: set_real,B4: set_real] :
% 0.23/0.54 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
% 0.23/0.54 => ~ ( ( member_real @ C @ A3 )
% 0.23/0.54 => ( member_real @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffE
% 0.23/0.54 thf(fact_7639_DiffE,axiom,
% 0.23/0.54 ! [C: int,A3: set_int,B4: set_int] :
% 0.23/0.54 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
% 0.23/0.54 => ~ ( ( member_int @ C @ A3 )
% 0.23/0.54 => ( member_int @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffE
% 0.23/0.54 thf(fact_7640_DiffE,axiom,
% 0.23/0.54 ! [C: nat,A3: set_nat,B4: set_nat] :
% 0.23/0.54 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
% 0.23/0.54 => ~ ( ( member_nat @ C @ A3 )
% 0.23/0.54 => ( member_nat @ C @ B4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffE
% 0.23/0.54 thf(fact_7641_DiffD1,axiom,
% 0.23/0.54 ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
% 0.23/0.54 ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
% 0.23/0.54 => ( member_Extended_enat @ C @ A3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffD1
% 0.23/0.54 thf(fact_7642_DiffD1,axiom,
% 0.23/0.54 ! [C: complex,A3: set_complex,B4: set_complex] :
% 0.23/0.54 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
% 0.23/0.54 => ( member_complex @ C @ A3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffD1
% 0.23/0.54 thf(fact_7643_DiffD1,axiom,
% 0.23/0.54 ! [C: real,A3: set_real,B4: set_real] :
% 0.23/0.54 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
% 0.23/0.54 => ( member_real @ C @ A3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffD1
% 0.23/0.54 thf(fact_7644_DiffD1,axiom,
% 0.23/0.54 ! [C: int,A3: set_int,B4: set_int] :
% 0.23/0.54 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
% 0.23/0.54 => ( member_int @ C @ A3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffD1
% 0.23/0.54 thf(fact_7645_DiffD1,axiom,
% 0.23/0.54 ! [C: nat,A3: set_nat,B4: set_nat] :
% 0.23/0.54 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
% 0.23/0.54 => ( member_nat @ C @ A3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffD1
% 0.23/0.54 thf(fact_7646_DiffD2,axiom,
% 0.23/0.54 ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
% 0.23/0.54 ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
% 0.23/0.54 => ~ ( member_Extended_enat @ C @ B4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffD2
% 0.23/0.54 thf(fact_7647_DiffD2,axiom,
% 0.23/0.54 ! [C: complex,A3: set_complex,B4: set_complex] :
% 0.23/0.54 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
% 0.23/0.54 => ~ ( member_complex @ C @ B4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffD2
% 0.23/0.54 thf(fact_7648_DiffD2,axiom,
% 0.23/0.54 ! [C: real,A3: set_real,B4: set_real] :
% 0.23/0.54 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
% 0.23/0.54 => ~ ( member_real @ C @ B4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffD2
% 0.23/0.54 thf(fact_7649_DiffD2,axiom,
% 0.23/0.54 ! [C: int,A3: set_int,B4: set_int] :
% 0.23/0.54 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
% 0.23/0.54 => ~ ( member_int @ C @ B4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffD2
% 0.23/0.54 thf(fact_7650_DiffD2,axiom,
% 0.23/0.54 ! [C: nat,A3: set_nat,B4: set_nat] :
% 0.23/0.54 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
% 0.23/0.54 => ~ ( member_nat @ C @ B4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % DiffD2
% 0.23/0.54 thf(fact_7651_take__bit__not__iff,axiom,
% 0.23/0.54 ! [N: nat,A: int,B2: int] :
% 0.23/0.54 ( ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 0.23/0.54 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ B2 ) ) )
% 0.23/0.54 = ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.23/0.54 = ( bit_se2923211474154528505it_int @ N @ B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % take_bit_not_iff
% 0.23/0.54 thf(fact_7652_take__bit__not__take__bit,axiom,
% 0.23/0.54 ! [N: nat,A: int] :
% 0.23/0.54 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.23/0.54 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % take_bit_not_take_bit
% 0.23/0.54 thf(fact_7653_bit__not__int__iff,axiom,
% 0.23/0.54 ! [K: int,N: nat] :
% 0.23/0.54 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 0.23/0.54 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % bit_not_int_iff
% 0.23/0.54 thf(fact_7654_of__int__not__eq,axiom,
% 0.23/0.54 ! [K: int] :
% 0.23/0.54 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.23/0.54 = ( bit_ri7919022796975470100ot_int @ ( ring_1_of_int_int @ K ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % of_int_not_eq
% 0.23/0.54 thf(fact_7655_size__neq__size__imp__neq,axiom,
% 0.23/0.54 ! [X4: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 0.23/0.54 ( ( ( size_s6755466524823107622T_VEBT @ X4 )
% 0.23/0.54 != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 0.23/0.54 => ( X4 != Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % size_neq_size_imp_neq
% 0.23/0.54 thf(fact_7656_size__neq__size__imp__neq,axiom,
% 0.23/0.54 ! [X4: list_o,Y3: list_o] :
% 0.23/0.54 ( ( ( size_size_list_o @ X4 )
% 0.23/0.54 != ( size_size_list_o @ Y3 ) )
% 0.23/0.54 => ( X4 != Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % size_neq_size_imp_neq
% 0.23/0.54 thf(fact_7657_size__neq__size__imp__neq,axiom,
% 0.23/0.54 ! [X4: list_nat,Y3: list_nat] :
% 0.23/0.54 ( ( ( size_size_list_nat @ X4 )
% 0.23/0.54 != ( size_size_list_nat @ Y3 ) )
% 0.23/0.54 => ( X4 != Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % size_neq_size_imp_neq
% 0.23/0.54 thf(fact_7658_size__neq__size__imp__neq,axiom,
% 0.23/0.54 ! [X4: list_int,Y3: list_int] :
% 0.23/0.54 ( ( ( size_size_list_int @ X4 )
% 0.23/0.54 != ( size_size_list_int @ Y3 ) )
% 0.23/0.54 => ( X4 != Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % size_neq_size_imp_neq
% 0.23/0.54 thf(fact_7659_size__neq__size__imp__neq,axiom,
% 0.23/0.54 ! [X4: num,Y3: num] :
% 0.23/0.54 ( ( ( size_size_num @ X4 )
% 0.23/0.54 != ( size_size_num @ Y3 ) )
% 0.23/0.54 => ( X4 != Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % size_neq_size_imp_neq
% 0.23/0.54 thf(fact_7660_of__int__not__numeral,axiom,
% 0.23/0.54 ! [K: num] :
% 0.23/0.54 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.54 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % of_int_not_numeral
% 0.23/0.54 thf(fact_7661_not__add__distrib,axiom,
% 0.23/0.54 ! [A: int,B2: int] :
% 0.23/0.54 ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B2 ) )
% 0.23/0.54 = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_add_distrib
% 0.23/0.54 thf(fact_7662_not__diff__distrib,axiom,
% 0.23/0.54 ! [A: int,B2: int] :
% 0.23/0.54 ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B2 ) )
% 0.23/0.54 = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_diff_distrib
% 0.23/0.54 thf(fact_7663_minus__eq__not__plus__1,axiom,
% 0.23/0.54 ( uminus1351360451143612070nteger
% 0.23/0.54 = ( ^ [A4: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A4 ) @ one_one_Code_integer ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % minus_eq_not_plus_1
% 0.23/0.54 thf(fact_7664_minus__eq__not__plus__1,axiom,
% 0.23/0.54 ( uminus_uminus_int
% 0.23/0.54 = ( ^ [A4: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A4 ) @ one_one_int ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % minus_eq_not_plus_1
% 0.23/0.54 thf(fact_7665_not__eq__complement,axiom,
% 0.23/0.54 ( bit_ri7632146776885996613nteger
% 0.23/0.54 = ( ^ [A4: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A4 ) @ one_one_Code_integer ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_eq_complement
% 0.23/0.54 thf(fact_7666_not__eq__complement,axiom,
% 0.23/0.54 ( bit_ri7919022796975470100ot_int
% 0.23/0.54 = ( ^ [A4: int] : ( minus_minus_int @ ( uminus_uminus_int @ A4 ) @ one_one_int ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_eq_complement
% 0.23/0.54 thf(fact_7667_minus__eq__not__minus__1,axiom,
% 0.23/0.54 ( uminus1351360451143612070nteger
% 0.23/0.54 = ( ^ [A4: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A4 @ one_one_Code_integer ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % minus_eq_not_minus_1
% 0.23/0.54 thf(fact_7668_minus__eq__not__minus__1,axiom,
% 0.23/0.54 ( uminus_uminus_int
% 0.23/0.54 = ( ^ [A4: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A4 @ one_one_int ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % minus_eq_not_minus_1
% 0.23/0.54 thf(fact_7669_not__int__def,axiom,
% 0.23/0.54 ( bit_ri7919022796975470100ot_int
% 0.23/0.54 = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_int_def
% 0.23/0.54 thf(fact_7670_and__not__numerals_I1_J,axiom,
% 0.23/0.54 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.23/0.54 = zero_zero_int ) ).
% 0.23/0.54
% 0.23/0.54 % and_not_numerals(1)
% 0.23/0.54 thf(fact_7671_disjunctive__diff,axiom,
% 0.23/0.54 ! [B2: int,A: int] :
% 0.23/0.54 ( ! [N4: nat] :
% 0.23/0.54 ( ( bit_se1146084159140164899it_int @ B2 @ N4 )
% 0.23/0.54 => ( bit_se1146084159140164899it_int @ A @ N4 ) )
% 0.23/0.54 => ( ( minus_minus_int @ A @ B2 )
% 0.23/0.54 = ( bit_se725231765392027082nd_int @ A @ ( bit_ri7919022796975470100ot_int @ B2 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % disjunctive_diff
% 0.23/0.54 thf(fact_7672_nle__le,axiom,
% 0.23/0.54 ! [A: rat,B2: rat] :
% 0.23/0.54 ( ( ~ ( ord_less_eq_rat @ A @ B2 ) )
% 0.23/0.54 = ( ( ord_less_eq_rat @ B2 @ A )
% 0.23/0.54 & ( B2 != A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % nle_le
% 0.23/0.54 thf(fact_7673_nle__le,axiom,
% 0.23/0.54 ! [A: num,B2: num] :
% 0.23/0.54 ( ( ~ ( ord_less_eq_num @ A @ B2 ) )
% 0.23/0.54 = ( ( ord_less_eq_num @ B2 @ A )
% 0.23/0.54 & ( B2 != A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % nle_le
% 0.23/0.54 thf(fact_7674_nle__le,axiom,
% 0.23/0.54 ! [A: nat,B2: nat] :
% 0.23/0.54 ( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
% 0.23/0.54 = ( ( ord_less_eq_nat @ B2 @ A )
% 0.23/0.54 & ( B2 != A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % nle_le
% 0.23/0.54 thf(fact_7675_nle__le,axiom,
% 0.23/0.54 ! [A: int,B2: int] :
% 0.23/0.54 ( ( ~ ( ord_less_eq_int @ A @ B2 ) )
% 0.23/0.54 = ( ( ord_less_eq_int @ B2 @ A )
% 0.23/0.54 & ( B2 != A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % nle_le
% 0.23/0.54 thf(fact_7676_le__cases3,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat,Z: rat] :
% 0.23/0.54 ( ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.54 => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
% 0.23/0.54 => ( ( ( ord_less_eq_rat @ Y3 @ X4 )
% 0.23/0.54 => ~ ( ord_less_eq_rat @ X4 @ Z ) )
% 0.23/0.54 => ( ( ( ord_less_eq_rat @ X4 @ Z )
% 0.23/0.54 => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
% 0.23/0.54 => ( ( ( ord_less_eq_rat @ Z @ Y3 )
% 0.23/0.54 => ~ ( ord_less_eq_rat @ Y3 @ X4 ) )
% 0.23/0.54 => ( ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.23/0.54 => ~ ( ord_less_eq_rat @ Z @ X4 ) )
% 0.23/0.54 => ~ ( ( ord_less_eq_rat @ Z @ X4 )
% 0.23/0.54 => ~ ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % le_cases3
% 0.23/0.54 thf(fact_7677_le__cases3,axiom,
% 0.23/0.54 ! [X4: num,Y3: num,Z: num] :
% 0.23/0.54 ( ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.23/0.54 => ~ ( ord_less_eq_num @ Y3 @ Z ) )
% 0.23/0.54 => ( ( ( ord_less_eq_num @ Y3 @ X4 )
% 0.23/0.54 => ~ ( ord_less_eq_num @ X4 @ Z ) )
% 0.23/0.54 => ( ( ( ord_less_eq_num @ X4 @ Z )
% 0.23/0.54 => ~ ( ord_less_eq_num @ Z @ Y3 ) )
% 0.23/0.54 => ( ( ( ord_less_eq_num @ Z @ Y3 )
% 0.23/0.54 => ~ ( ord_less_eq_num @ Y3 @ X4 ) )
% 0.23/0.54 => ( ( ( ord_less_eq_num @ Y3 @ Z )
% 0.23/0.54 => ~ ( ord_less_eq_num @ Z @ X4 ) )
% 0.23/0.54 => ~ ( ( ord_less_eq_num @ Z @ X4 )
% 0.23/0.54 => ~ ( ord_less_eq_num @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % le_cases3
% 0.23/0.54 thf(fact_7678_le__cases3,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat,Z: nat] :
% 0.23/0.54 ( ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.23/0.54 => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 0.23/0.54 => ( ( ( ord_less_eq_nat @ Y3 @ X4 )
% 0.23/0.54 => ~ ( ord_less_eq_nat @ X4 @ Z ) )
% 0.23/0.54 => ( ( ( ord_less_eq_nat @ X4 @ Z )
% 0.23/0.54 => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 0.23/0.54 => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 0.23/0.54 => ~ ( ord_less_eq_nat @ Y3 @ X4 ) )
% 0.23/0.54 => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.23/0.54 => ~ ( ord_less_eq_nat @ Z @ X4 ) )
% 0.23/0.54 => ~ ( ( ord_less_eq_nat @ Z @ X4 )
% 0.23/0.54 => ~ ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % le_cases3
% 0.23/0.54 thf(fact_7679_le__cases3,axiom,
% 0.23/0.54 ! [X4: int,Y3: int,Z: int] :
% 0.23/0.54 ( ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.54 => ~ ( ord_less_eq_int @ Y3 @ Z ) )
% 0.23/0.54 => ( ( ( ord_less_eq_int @ Y3 @ X4 )
% 0.23/0.54 => ~ ( ord_less_eq_int @ X4 @ Z ) )
% 0.23/0.54 => ( ( ( ord_less_eq_int @ X4 @ Z )
% 0.23/0.54 => ~ ( ord_less_eq_int @ Z @ Y3 ) )
% 0.23/0.54 => ( ( ( ord_less_eq_int @ Z @ Y3 )
% 0.23/0.54 => ~ ( ord_less_eq_int @ Y3 @ X4 ) )
% 0.23/0.54 => ( ( ( ord_less_eq_int @ Y3 @ Z )
% 0.23/0.54 => ~ ( ord_less_eq_int @ Z @ X4 ) )
% 0.23/0.54 => ~ ( ( ord_less_eq_int @ Z @ X4 )
% 0.23/0.54 => ~ ( ord_less_eq_int @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % le_cases3
% 0.23/0.54 thf(fact_7680_order__class_Oorder__eq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: set_nat,Z4: set_nat] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [X: set_nat,Y: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.23/0.54 & ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_class.order_eq_iff
% 0.23/0.54 thf(fact_7681_order__class_Oorder__eq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [X: rat,Y: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X @ Y )
% 0.23/0.54 & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_class.order_eq_iff
% 0.23/0.54 thf(fact_7682_order__class_Oorder__eq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [X: num,Y: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X @ Y )
% 0.23/0.54 & ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_class.order_eq_iff
% 0.23/0.54 thf(fact_7683_order__class_Oorder__eq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [X: nat,Y: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X @ Y )
% 0.23/0.54 & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_class.order_eq_iff
% 0.23/0.54 thf(fact_7684_order__class_Oorder__eq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [X: int,Y: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ X @ Y )
% 0.23/0.54 & ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_class.order_eq_iff
% 0.23/0.54 thf(fact_7685_ord__eq__le__trans,axiom,
% 0.23/0.54 ! [A: set_nat,B2: set_nat,C: set_nat] :
% 0.23/0.54 ( ( A = B2 )
% 0.23/0.54 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 0.23/0.54 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_trans
% 0.23/0.54 thf(fact_7686_ord__eq__le__trans,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.54 ( ( A = B2 )
% 0.23/0.54 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.54 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_trans
% 0.23/0.54 thf(fact_7687_ord__eq__le__trans,axiom,
% 0.23/0.54 ! [A: num,B2: num,C: num] :
% 0.23/0.54 ( ( A = B2 )
% 0.23/0.54 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.54 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_trans
% 0.23/0.54 thf(fact_7688_ord__eq__le__trans,axiom,
% 0.23/0.54 ! [A: nat,B2: nat,C: nat] :
% 0.23/0.54 ( ( A = B2 )
% 0.23/0.54 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.23/0.54 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_trans
% 0.23/0.54 thf(fact_7689_ord__eq__le__trans,axiom,
% 0.23/0.54 ! [A: int,B2: int,C: int] :
% 0.23/0.54 ( ( A = B2 )
% 0.23/0.54 => ( ( ord_less_eq_int @ B2 @ C )
% 0.23/0.54 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_trans
% 0.23/0.54 thf(fact_7690_ord__le__eq__trans,axiom,
% 0.23/0.54 ! [A: set_nat,B2: set_nat,C: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ A @ B2 )
% 0.23/0.54 => ( ( B2 = C )
% 0.23/0.54 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_trans
% 0.23/0.54 thf(fact_7691_ord__le__eq__trans,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( ( B2 = C )
% 0.23/0.54 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_trans
% 0.23/0.54 thf(fact_7692_ord__le__eq__trans,axiom,
% 0.23/0.54 ! [A: num,B2: num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( ( B2 = C )
% 0.23/0.54 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_trans
% 0.23/0.54 thf(fact_7693_ord__le__eq__trans,axiom,
% 0.23/0.54 ! [A: nat,B2: nat,C: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.54 => ( ( B2 = C )
% 0.23/0.54 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_trans
% 0.23/0.54 thf(fact_7694_ord__le__eq__trans,axiom,
% 0.23/0.54 ! [A: int,B2: int,C: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ A @ B2 )
% 0.23/0.54 => ( ( B2 = C )
% 0.23/0.54 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_trans
% 0.23/0.54 thf(fact_7695_order__antisym,axiom,
% 0.23/0.54 ! [X4: set_nat,Y3: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_eq_set_nat @ Y3 @ X4 )
% 0.23/0.54 => ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_antisym
% 0.23/0.54 thf(fact_7696_order__antisym,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_eq_rat @ Y3 @ X4 )
% 0.23/0.54 => ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_antisym
% 0.23/0.54 thf(fact_7697_order__antisym,axiom,
% 0.23/0.54 ! [X4: num,Y3: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_eq_num @ Y3 @ X4 )
% 0.23/0.54 => ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_antisym
% 0.23/0.54 thf(fact_7698_order__antisym,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_eq_nat @ Y3 @ X4 )
% 0.23/0.54 => ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_antisym
% 0.23/0.54 thf(fact_7699_order__antisym,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_eq_int @ Y3 @ X4 )
% 0.23/0.54 => ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_antisym
% 0.23/0.54 thf(fact_7700_order_Otrans,axiom,
% 0.23/0.54 ! [A: set_nat,B2: set_nat,C: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 0.23/0.54 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.trans
% 0.23/0.54 thf(fact_7701_order_Otrans,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.54 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.trans
% 0.23/0.54 thf(fact_7702_order_Otrans,axiom,
% 0.23/0.54 ! [A: num,B2: num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.54 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.trans
% 0.23/0.54 thf(fact_7703_order_Otrans,axiom,
% 0.23/0.54 ! [A: nat,B2: nat,C: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.23/0.54 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.trans
% 0.23/0.54 thf(fact_7704_order_Otrans,axiom,
% 0.23/0.54 ! [A: int,B2: int,C: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_int @ B2 @ C )
% 0.23/0.54 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.trans
% 0.23/0.54 thf(fact_7705_order__trans,axiom,
% 0.23/0.54 ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_eq_set_nat @ Y3 @ Z )
% 0.23/0.54 => ( ord_less_eq_set_nat @ X4 @ Z ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_trans
% 0.23/0.54 thf(fact_7706_order__trans,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat,Z: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.23/0.54 => ( ord_less_eq_rat @ X4 @ Z ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_trans
% 0.23/0.54 thf(fact_7707_order__trans,axiom,
% 0.23/0.54 ! [X4: num,Y3: num,Z: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_eq_num @ Y3 @ Z )
% 0.23/0.54 => ( ord_less_eq_num @ X4 @ Z ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_trans
% 0.23/0.54 thf(fact_7708_order__trans,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat,Z: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.23/0.54 => ( ord_less_eq_nat @ X4 @ Z ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_trans
% 0.23/0.54 thf(fact_7709_order__trans,axiom,
% 0.23/0.54 ! [X4: int,Y3: int,Z: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.23/0.54 => ( ord_less_eq_int @ X4 @ Z ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_trans
% 0.23/0.54 thf(fact_7710_linorder__wlog,axiom,
% 0.23/0.54 ! [P: rat > rat > $o,A: rat,B2: rat] :
% 0.23/0.54 ( ! [A5: rat,B5: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A5 @ B5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( ! [A5: rat,B5: rat] :
% 0.23/0.54 ( ( P @ B5 @ A5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( P @ A @ B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_wlog
% 0.23/0.54 thf(fact_7711_linorder__wlog,axiom,
% 0.23/0.54 ! [P: num > num > $o,A: num,B2: num] :
% 0.23/0.54 ( ! [A5: num,B5: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ A5 @ B5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( ! [A5: num,B5: num] :
% 0.23/0.54 ( ( P @ B5 @ A5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( P @ A @ B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_wlog
% 0.23/0.54 thf(fact_7712_linorder__wlog,axiom,
% 0.23/0.54 ! [P: nat > nat > $o,A: nat,B2: nat] :
% 0.23/0.54 ( ! [A5: nat,B5: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ A5 @ B5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( ! [A5: nat,B5: nat] :
% 0.23/0.54 ( ( P @ B5 @ A5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( P @ A @ B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_wlog
% 0.23/0.54 thf(fact_7713_linorder__wlog,axiom,
% 0.23/0.54 ! [P: int > int > $o,A: int,B2: int] :
% 0.23/0.54 ( ! [A5: int,B5: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ A5 @ B5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( ! [A5: int,B5: int] :
% 0.23/0.54 ( ( P @ B5 @ A5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( P @ A @ B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_wlog
% 0.23/0.54 thf(fact_7714_dual__order_Oeq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: set_nat,Z4: set_nat] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [A4: set_nat,B3: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ B3 @ A4 )
% 0.23/0.54 & ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.eq_iff
% 0.23/0.54 thf(fact_7715_dual__order_Oeq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [A4: rat,B3: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ B3 @ A4 )
% 0.23/0.54 & ( ord_less_eq_rat @ A4 @ B3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.eq_iff
% 0.23/0.54 thf(fact_7716_dual__order_Oeq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [A4: num,B3: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ B3 @ A4 )
% 0.23/0.54 & ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.eq_iff
% 0.23/0.54 thf(fact_7717_dual__order_Oeq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [A4: nat,B3: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ B3 @ A4 )
% 0.23/0.54 & ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.eq_iff
% 0.23/0.54 thf(fact_7718_dual__order_Oeq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [A4: int,B3: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ B3 @ A4 )
% 0.23/0.54 & ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.eq_iff
% 0.23/0.54 thf(fact_7719_dual__order_Oantisym,axiom,
% 0.23/0.54 ! [B2: set_nat,A: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_eq_set_nat @ A @ B2 )
% 0.23/0.54 => ( A = B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.antisym
% 0.23/0.54 thf(fact_7720_dual__order_Oantisym,axiom,
% 0.23/0.54 ! [B2: rat,A: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( A = B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.antisym
% 0.23/0.54 thf(fact_7721_dual__order_Oantisym,axiom,
% 0.23/0.54 ! [B2: num,A: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( A = B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.antisym
% 0.23/0.54 thf(fact_7722_dual__order_Oantisym,axiom,
% 0.23/0.54 ! [B2: nat,A: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.54 => ( A = B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.antisym
% 0.23/0.54 thf(fact_7723_dual__order_Oantisym,axiom,
% 0.23/0.54 ! [B2: int,A: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_eq_int @ A @ B2 )
% 0.23/0.54 => ( A = B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.antisym
% 0.23/0.54 thf(fact_7724_dual__order_Otrans,axiom,
% 0.23/0.54 ! [B2: set_nat,A: set_nat,C: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_eq_set_nat @ C @ B2 )
% 0.23/0.54 => ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.trans
% 0.23/0.54 thf(fact_7725_dual__order_Otrans,axiom,
% 0.23/0.54 ! [B2: rat,A: rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_eq_rat @ C @ B2 )
% 0.23/0.54 => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.trans
% 0.23/0.54 thf(fact_7726_dual__order_Otrans,axiom,
% 0.23/0.54 ! [B2: num,A: num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_eq_num @ C @ B2 )
% 0.23/0.54 => ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.trans
% 0.23/0.54 thf(fact_7727_dual__order_Otrans,axiom,
% 0.23/0.54 ! [B2: nat,A: nat,C: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_eq_nat @ C @ B2 )
% 0.23/0.54 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.trans
% 0.23/0.54 thf(fact_7728_dual__order_Otrans,axiom,
% 0.23/0.54 ! [B2: int,A: int,C: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_eq_int @ C @ B2 )
% 0.23/0.54 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.trans
% 0.23/0.54 thf(fact_7729_antisym,axiom,
% 0.23/0.54 ! [A: set_nat,B2: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_set_nat @ B2 @ A )
% 0.23/0.54 => ( A = B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % antisym
% 0.23/0.54 thf(fact_7730_antisym,axiom,
% 0.23/0.54 ! [A: rat,B2: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_rat @ B2 @ A )
% 0.23/0.54 => ( A = B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % antisym
% 0.23/0.54 thf(fact_7731_antisym,axiom,
% 0.23/0.54 ! [A: num,B2: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_num @ B2 @ A )
% 0.23/0.54 => ( A = B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % antisym
% 0.23/0.54 thf(fact_7732_antisym,axiom,
% 0.23/0.54 ! [A: nat,B2: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_nat @ B2 @ A )
% 0.23/0.54 => ( A = B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % antisym
% 0.23/0.54 thf(fact_7733_antisym,axiom,
% 0.23/0.54 ! [A: int,B2: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_int @ B2 @ A )
% 0.23/0.54 => ( A = B2 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % antisym
% 0.23/0.54 thf(fact_7734_Orderings_Oorder__eq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: set_nat,Z4: set_nat] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [A4: set_nat,B3: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ A4 @ B3 )
% 0.23/0.54 & ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Orderings.order_eq_iff
% 0.23/0.54 thf(fact_7735_Orderings_Oorder__eq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [A4: rat,B3: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A4 @ B3 )
% 0.23/0.54 & ( ord_less_eq_rat @ B3 @ A4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Orderings.order_eq_iff
% 0.23/0.54 thf(fact_7736_Orderings_Oorder__eq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [A4: num,B3: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ A4 @ B3 )
% 0.23/0.54 & ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Orderings.order_eq_iff
% 0.23/0.54 thf(fact_7737_Orderings_Oorder__eq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [A4: nat,B3: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ A4 @ B3 )
% 0.23/0.54 & ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Orderings.order_eq_iff
% 0.23/0.54 thf(fact_7738_Orderings_Oorder__eq__iff,axiom,
% 0.23/0.54 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.23/0.54 = ( ^ [A4: int,B3: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ A4 @ B3 )
% 0.23/0.54 & ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % Orderings.order_eq_iff
% 0.23/0.54 thf(fact_7739_order__subst1,axiom,
% 0.23/0.54 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst1
% 0.23/0.54 thf(fact_7740_order__subst1,axiom,
% 0.23/0.54 ! [A: rat,F: num > rat,B2: num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst1
% 0.23/0.54 thf(fact_7741_order__subst1,axiom,
% 0.23/0.54 ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst1
% 0.23/0.54 thf(fact_7742_order__subst1,axiom,
% 0.23/0.54 ! [A: rat,F: int > rat,B2: int,C: int] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_int @ B2 @ C )
% 0.23/0.54 => ( ! [X3: int,Y4: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst1
% 0.23/0.54 thf(fact_7743_order__subst1,axiom,
% 0.23/0.54 ! [A: num,F: rat > num,B2: rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst1
% 0.23/0.54 thf(fact_7744_order__subst1,axiom,
% 0.23/0.54 ! [A: num,F: num > num,B2: num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst1
% 0.23/0.54 thf(fact_7745_order__subst1,axiom,
% 0.23/0.54 ! [A: num,F: nat > num,B2: nat,C: nat] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst1
% 0.23/0.54 thf(fact_7746_order__subst1,axiom,
% 0.23/0.54 ! [A: num,F: int > num,B2: int,C: int] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_int @ B2 @ C )
% 0.23/0.54 => ( ! [X3: int,Y4: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst1
% 0.23/0.54 thf(fact_7747_order__subst1,axiom,
% 0.23/0.54 ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst1
% 0.23/0.54 thf(fact_7748_order__subst1,axiom,
% 0.23/0.54 ! [A: nat,F: num > nat,B2: num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst1
% 0.23/0.54 thf(fact_7749_order__subst2,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst2
% 0.23/0.54 thf(fact_7750_order__subst2,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,F: rat > num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst2
% 0.23/0.54 thf(fact_7751_order__subst2,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst2
% 0.23/0.54 thf(fact_7752_order__subst2,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,F: rat > int,C: int] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst2
% 0.23/0.54 thf(fact_7753_order__subst2,axiom,
% 0.23/0.54 ! [A: num,B2: num,F: num > rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst2
% 0.23/0.54 thf(fact_7754_order__subst2,axiom,
% 0.23/0.54 ! [A: num,B2: num,F: num > num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst2
% 0.23/0.54 thf(fact_7755_order__subst2,axiom,
% 0.23/0.54 ! [A: num,B2: num,F: num > nat,C: nat] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst2
% 0.23/0.54 thf(fact_7756_order__subst2,axiom,
% 0.23/0.54 ! [A: num,B2: num,F: num > int,C: int] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst2
% 0.23/0.54 thf(fact_7757_order__subst2,axiom,
% 0.23/0.54 ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.23/0.54 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst2
% 0.23/0.54 thf(fact_7758_order__subst2,axiom,
% 0.23/0.54 ! [A: nat,B2: nat,F: nat > num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 0.23/0.54 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_subst2
% 0.23/0.54 thf(fact_7759_order__eq__refl,axiom,
% 0.23/0.54 ! [X4: set_nat,Y3: set_nat] :
% 0.23/0.54 ( ( X4 = Y3 )
% 0.23/0.54 => ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_eq_refl
% 0.23/0.54 thf(fact_7760_order__eq__refl,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat] :
% 0.23/0.54 ( ( X4 = Y3 )
% 0.23/0.54 => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_eq_refl
% 0.23/0.54 thf(fact_7761_order__eq__refl,axiom,
% 0.23/0.54 ! [X4: num,Y3: num] :
% 0.23/0.54 ( ( X4 = Y3 )
% 0.23/0.54 => ( ord_less_eq_num @ X4 @ Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_eq_refl
% 0.23/0.54 thf(fact_7762_order__eq__refl,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat] :
% 0.23/0.54 ( ( X4 = Y3 )
% 0.23/0.54 => ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_eq_refl
% 0.23/0.54 thf(fact_7763_order__eq__refl,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( X4 = Y3 )
% 0.23/0.54 => ( ord_less_eq_int @ X4 @ Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_eq_refl
% 0.23/0.54 thf(fact_7764_linorder__linear,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.54 | ( ord_less_eq_rat @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_linear
% 0.23/0.54 thf(fact_7765_linorder__linear,axiom,
% 0.23/0.54 ! [X4: num,Y3: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.23/0.54 | ( ord_less_eq_num @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_linear
% 0.23/0.54 thf(fact_7766_linorder__linear,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.23/0.54 | ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_linear
% 0.23/0.54 thf(fact_7767_linorder__linear,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.54 | ( ord_less_eq_int @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_linear
% 0.23/0.54 thf(fact_7768_ord__eq__le__subst,axiom,
% 0.23/0.54 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_subst
% 0.23/0.54 thf(fact_7769_ord__eq__le__subst,axiom,
% 0.23/0.54 ! [A: num,F: rat > num,B2: rat,C: rat] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_subst
% 0.23/0.54 thf(fact_7770_ord__eq__le__subst,axiom,
% 0.23/0.54 ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_subst
% 0.23/0.54 thf(fact_7771_ord__eq__le__subst,axiom,
% 0.23/0.54 ! [A: int,F: rat > int,B2: rat,C: rat] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_subst
% 0.23/0.54 thf(fact_7772_ord__eq__le__subst,axiom,
% 0.23/0.54 ! [A: rat,F: num > rat,B2: num,C: num] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_subst
% 0.23/0.54 thf(fact_7773_ord__eq__le__subst,axiom,
% 0.23/0.54 ! [A: num,F: num > num,B2: num,C: num] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_subst
% 0.23/0.54 thf(fact_7774_ord__eq__le__subst,axiom,
% 0.23/0.54 ! [A: nat,F: num > nat,B2: num,C: num] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_subst
% 0.23/0.54 thf(fact_7775_ord__eq__le__subst,axiom,
% 0.23/0.54 ! [A: int,F: num > int,B2: num,C: num] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_subst
% 0.23/0.54 thf(fact_7776_ord__eq__le__subst,axiom,
% 0.23/0.54 ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_subst
% 0.23/0.54 thf(fact_7777_ord__eq__le__subst,axiom,
% 0.23/0.54 ! [A: num,F: nat > num,B2: nat,C: nat] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_le_subst
% 0.23/0.54 thf(fact_7778_ord__le__eq__subst,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( ( ( F @ B2 )
% 0.23/0.54 = C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_subst
% 0.23/0.54 thf(fact_7779_ord__le__eq__subst,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,F: rat > num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( ( ( F @ B2 )
% 0.23/0.54 = C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_subst
% 0.23/0.54 thf(fact_7780_ord__le__eq__subst,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( ( ( F @ B2 )
% 0.23/0.54 = C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_subst
% 0.23/0.54 thf(fact_7781_ord__le__eq__subst,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,F: rat > int,C: int] :
% 0.23/0.54 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.54 => ( ( ( F @ B2 )
% 0.23/0.54 = C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_subst
% 0.23/0.54 thf(fact_7782_ord__le__eq__subst,axiom,
% 0.23/0.54 ! [A: num,B2: num,F: num > rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( ( ( F @ B2 )
% 0.23/0.54 = C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_subst
% 0.23/0.54 thf(fact_7783_ord__le__eq__subst,axiom,
% 0.23/0.54 ! [A: num,B2: num,F: num > num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( ( ( F @ B2 )
% 0.23/0.54 = C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_subst
% 0.23/0.54 thf(fact_7784_ord__le__eq__subst,axiom,
% 0.23/0.54 ! [A: num,B2: num,F: num > nat,C: nat] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( ( ( F @ B2 )
% 0.23/0.54 = C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_subst
% 0.23/0.54 thf(fact_7785_ord__le__eq__subst,axiom,
% 0.23/0.54 ! [A: num,B2: num,F: num > int,C: int] :
% 0.23/0.54 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.54 => ( ( ( F @ B2 )
% 0.23/0.54 = C )
% 0.23/0.54 => ( ! [X3: num,Y4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_subst
% 0.23/0.54 thf(fact_7786_ord__le__eq__subst,axiom,
% 0.23/0.54 ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.54 => ( ( ( F @ B2 )
% 0.23/0.54 = C )
% 0.23/0.54 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_subst
% 0.23/0.54 thf(fact_7787_ord__le__eq__subst,axiom,
% 0.23/0.54 ! [A: nat,B2: nat,F: nat > num,C: num] :
% 0.23/0.54 ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.54 => ( ( ( F @ B2 )
% 0.23/0.54 = C )
% 0.23/0.54 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_le_eq_subst
% 0.23/0.54 thf(fact_7788_linorder__le__cases,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat] :
% 0.23/0.54 ( ~ ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.54 => ( ord_less_eq_rat @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_le_cases
% 0.23/0.54 thf(fact_7789_linorder__le__cases,axiom,
% 0.23/0.54 ! [X4: num,Y3: num] :
% 0.23/0.54 ( ~ ( ord_less_eq_num @ X4 @ Y3 )
% 0.23/0.54 => ( ord_less_eq_num @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_le_cases
% 0.23/0.54 thf(fact_7790_linorder__le__cases,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat] :
% 0.23/0.54 ( ~ ( ord_less_eq_nat @ X4 @ Y3 )
% 0.23/0.54 => ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_le_cases
% 0.23/0.54 thf(fact_7791_linorder__le__cases,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ~ ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.54 => ( ord_less_eq_int @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_le_cases
% 0.23/0.54 thf(fact_7792_order__antisym__conv,axiom,
% 0.23/0.54 ! [Y3: set_nat,X4: set_nat] :
% 0.23/0.54 ( ( ord_less_eq_set_nat @ Y3 @ X4 )
% 0.23/0.54 => ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 0.23/0.54 = ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_antisym_conv
% 0.23/0.54 thf(fact_7793_order__antisym__conv,axiom,
% 0.23/0.54 ! [Y3: rat,X4: rat] :
% 0.23/0.54 ( ( ord_less_eq_rat @ Y3 @ X4 )
% 0.23/0.54 => ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.54 = ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_antisym_conv
% 0.23/0.54 thf(fact_7794_order__antisym__conv,axiom,
% 0.23/0.54 ! [Y3: num,X4: num] :
% 0.23/0.54 ( ( ord_less_eq_num @ Y3 @ X4 )
% 0.23/0.54 => ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.23/0.54 = ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_antisym_conv
% 0.23/0.54 thf(fact_7795_order__antisym__conv,axiom,
% 0.23/0.54 ! [Y3: nat,X4: nat] :
% 0.23/0.54 ( ( ord_less_eq_nat @ Y3 @ X4 )
% 0.23/0.54 => ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.23/0.54 = ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_antisym_conv
% 0.23/0.54 thf(fact_7796_order__antisym__conv,axiom,
% 0.23/0.54 ! [Y3: int,X4: int] :
% 0.23/0.54 ( ( ord_less_eq_int @ Y3 @ X4 )
% 0.23/0.54 => ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.54 = ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_antisym_conv
% 0.23/0.54 thf(fact_7797_lt__ex,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ? [Y4: real] : ( ord_less_real @ Y4 @ X4 ) ).
% 0.23/0.54
% 0.23/0.54 % lt_ex
% 0.23/0.54 thf(fact_7798_lt__ex,axiom,
% 0.23/0.54 ! [X4: rat] :
% 0.23/0.54 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X4 ) ).
% 0.23/0.54
% 0.23/0.54 % lt_ex
% 0.23/0.54 thf(fact_7799_lt__ex,axiom,
% 0.23/0.54 ! [X4: int] :
% 0.23/0.54 ? [Y4: int] : ( ord_less_int @ Y4 @ X4 ) ).
% 0.23/0.54
% 0.23/0.54 % lt_ex
% 0.23/0.54 thf(fact_7800_gt__ex,axiom,
% 0.23/0.54 ! [X4: real] :
% 0.23/0.54 ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% 0.23/0.54
% 0.23/0.54 % gt_ex
% 0.23/0.54 thf(fact_7801_gt__ex,axiom,
% 0.23/0.54 ! [X4: rat] :
% 0.23/0.54 ? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).
% 0.23/0.54
% 0.23/0.54 % gt_ex
% 0.23/0.54 thf(fact_7802_gt__ex,axiom,
% 0.23/0.54 ! [X4: nat] :
% 0.23/0.54 ? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% 0.23/0.54
% 0.23/0.54 % gt_ex
% 0.23/0.54 thf(fact_7803_gt__ex,axiom,
% 0.23/0.54 ! [X4: int] :
% 0.23/0.54 ? [X_1: int] : ( ord_less_int @ X4 @ X_1 ) ).
% 0.23/0.54
% 0.23/0.54 % gt_ex
% 0.23/0.54 thf(fact_7804_dense,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.54 => ? [Z2: real] :
% 0.23/0.54 ( ( ord_less_real @ X4 @ Z2 )
% 0.23/0.54 & ( ord_less_real @ Z2 @ Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dense
% 0.23/0.54 thf(fact_7805_dense,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat] :
% 0.23/0.54 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.54 => ? [Z2: rat] :
% 0.23/0.54 ( ( ord_less_rat @ X4 @ Z2 )
% 0.23/0.54 & ( ord_less_rat @ Z2 @ Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dense
% 0.23/0.54 thf(fact_7806_less__imp__neq,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.54 => ( X4 != Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % less_imp_neq
% 0.23/0.54 thf(fact_7807_less__imp__neq,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat] :
% 0.23/0.54 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.54 => ( X4 != Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % less_imp_neq
% 0.23/0.54 thf(fact_7808_less__imp__neq,axiom,
% 0.23/0.54 ! [X4: num,Y3: num] :
% 0.23/0.54 ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.54 => ( X4 != Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % less_imp_neq
% 0.23/0.54 thf(fact_7809_less__imp__neq,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat] :
% 0.23/0.54 ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.54 => ( X4 != Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % less_imp_neq
% 0.23/0.54 thf(fact_7810_less__imp__neq,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.54 => ( X4 != Y3 ) ) ).
% 0.23/0.54
% 0.23/0.54 % less_imp_neq
% 0.23/0.54 thf(fact_7811_order_Oasym,axiom,
% 0.23/0.54 ! [A: real,B2: real] :
% 0.23/0.54 ( ( ord_less_real @ A @ B2 )
% 0.23/0.54 => ~ ( ord_less_real @ B2 @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.asym
% 0.23/0.54 thf(fact_7812_order_Oasym,axiom,
% 0.23/0.54 ! [A: rat,B2: rat] :
% 0.23/0.54 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.54 => ~ ( ord_less_rat @ B2 @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.asym
% 0.23/0.54 thf(fact_7813_order_Oasym,axiom,
% 0.23/0.54 ! [A: num,B2: num] :
% 0.23/0.54 ( ( ord_less_num @ A @ B2 )
% 0.23/0.54 => ~ ( ord_less_num @ B2 @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.asym
% 0.23/0.54 thf(fact_7814_order_Oasym,axiom,
% 0.23/0.54 ! [A: nat,B2: nat] :
% 0.23/0.54 ( ( ord_less_nat @ A @ B2 )
% 0.23/0.54 => ~ ( ord_less_nat @ B2 @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.asym
% 0.23/0.54 thf(fact_7815_order_Oasym,axiom,
% 0.23/0.54 ! [A: int,B2: int] :
% 0.23/0.54 ( ( ord_less_int @ A @ B2 )
% 0.23/0.54 => ~ ( ord_less_int @ B2 @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.asym
% 0.23/0.54 thf(fact_7816_ord__eq__less__trans,axiom,
% 0.23/0.54 ! [A: real,B2: real,C: real] :
% 0.23/0.54 ( ( A = B2 )
% 0.23/0.54 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.54 => ( ord_less_real @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_trans
% 0.23/0.54 thf(fact_7817_ord__eq__less__trans,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.54 ( ( A = B2 )
% 0.23/0.54 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.54 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_trans
% 0.23/0.54 thf(fact_7818_ord__eq__less__trans,axiom,
% 0.23/0.54 ! [A: num,B2: num,C: num] :
% 0.23/0.54 ( ( A = B2 )
% 0.23/0.54 => ( ( ord_less_num @ B2 @ C )
% 0.23/0.54 => ( ord_less_num @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_trans
% 0.23/0.54 thf(fact_7819_ord__eq__less__trans,axiom,
% 0.23/0.54 ! [A: nat,B2: nat,C: nat] :
% 0.23/0.54 ( ( A = B2 )
% 0.23/0.54 => ( ( ord_less_nat @ B2 @ C )
% 0.23/0.54 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_trans
% 0.23/0.54 thf(fact_7820_ord__eq__less__trans,axiom,
% 0.23/0.54 ! [A: int,B2: int,C: int] :
% 0.23/0.54 ( ( A = B2 )
% 0.23/0.54 => ( ( ord_less_int @ B2 @ C )
% 0.23/0.54 => ( ord_less_int @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_trans
% 0.23/0.54 thf(fact_7821_ord__less__eq__trans,axiom,
% 0.23/0.54 ! [A: real,B2: real,C: real] :
% 0.23/0.54 ( ( ord_less_real @ A @ B2 )
% 0.23/0.54 => ( ( B2 = C )
% 0.23/0.54 => ( ord_less_real @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_less_eq_trans
% 0.23/0.54 thf(fact_7822_ord__less__eq__trans,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.54 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.54 => ( ( B2 = C )
% 0.23/0.54 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_less_eq_trans
% 0.23/0.54 thf(fact_7823_ord__less__eq__trans,axiom,
% 0.23/0.54 ! [A: num,B2: num,C: num] :
% 0.23/0.54 ( ( ord_less_num @ A @ B2 )
% 0.23/0.54 => ( ( B2 = C )
% 0.23/0.54 => ( ord_less_num @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_less_eq_trans
% 0.23/0.54 thf(fact_7824_ord__less__eq__trans,axiom,
% 0.23/0.54 ! [A: nat,B2: nat,C: nat] :
% 0.23/0.54 ( ( ord_less_nat @ A @ B2 )
% 0.23/0.54 => ( ( B2 = C )
% 0.23/0.54 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_less_eq_trans
% 0.23/0.54 thf(fact_7825_ord__less__eq__trans,axiom,
% 0.23/0.54 ! [A: int,B2: int,C: int] :
% 0.23/0.54 ( ( ord_less_int @ A @ B2 )
% 0.23/0.54 => ( ( B2 = C )
% 0.23/0.54 => ( ord_less_int @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_less_eq_trans
% 0.23/0.54 thf(fact_7826_less__induct,axiom,
% 0.23/0.54 ! [P: nat > $o,A: nat] :
% 0.23/0.54 ( ! [X3: nat] :
% 0.23/0.54 ( ! [Y5: nat] :
% 0.23/0.54 ( ( ord_less_nat @ Y5 @ X3 )
% 0.23/0.54 => ( P @ Y5 ) )
% 0.23/0.54 => ( P @ X3 ) )
% 0.23/0.54 => ( P @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % less_induct
% 0.23/0.54 thf(fact_7827_antisym__conv3,axiom,
% 0.23/0.54 ! [Y3: real,X4: real] :
% 0.23/0.54 ( ~ ( ord_less_real @ Y3 @ X4 )
% 0.23/0.54 => ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
% 0.23/0.54 = ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % antisym_conv3
% 0.23/0.54 thf(fact_7828_antisym__conv3,axiom,
% 0.23/0.54 ! [Y3: rat,X4: rat] :
% 0.23/0.54 ( ~ ( ord_less_rat @ Y3 @ X4 )
% 0.23/0.54 => ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
% 0.23/0.54 = ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % antisym_conv3
% 0.23/0.54 thf(fact_7829_antisym__conv3,axiom,
% 0.23/0.54 ! [Y3: num,X4: num] :
% 0.23/0.54 ( ~ ( ord_less_num @ Y3 @ X4 )
% 0.23/0.54 => ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
% 0.23/0.54 = ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % antisym_conv3
% 0.23/0.54 thf(fact_7830_antisym__conv3,axiom,
% 0.23/0.54 ! [Y3: nat,X4: nat] :
% 0.23/0.54 ( ~ ( ord_less_nat @ Y3 @ X4 )
% 0.23/0.54 => ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 0.23/0.54 = ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % antisym_conv3
% 0.23/0.54 thf(fact_7831_antisym__conv3,axiom,
% 0.23/0.54 ! [Y3: int,X4: int] :
% 0.23/0.54 ( ~ ( ord_less_int @ Y3 @ X4 )
% 0.23/0.54 => ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
% 0.23/0.54 = ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % antisym_conv3
% 0.23/0.54 thf(fact_7832_linorder__cases,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ~ ( ord_less_real @ X4 @ Y3 )
% 0.23/0.54 => ( ( X4 != Y3 )
% 0.23/0.54 => ( ord_less_real @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_cases
% 0.23/0.54 thf(fact_7833_linorder__cases,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat] :
% 0.23/0.54 ( ~ ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.54 => ( ( X4 != Y3 )
% 0.23/0.54 => ( ord_less_rat @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_cases
% 0.23/0.54 thf(fact_7834_linorder__cases,axiom,
% 0.23/0.54 ! [X4: num,Y3: num] :
% 0.23/0.54 ( ~ ( ord_less_num @ X4 @ Y3 )
% 0.23/0.54 => ( ( X4 != Y3 )
% 0.23/0.54 => ( ord_less_num @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_cases
% 0.23/0.54 thf(fact_7835_linorder__cases,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat] :
% 0.23/0.54 ( ~ ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.54 => ( ( X4 != Y3 )
% 0.23/0.54 => ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_cases
% 0.23/0.54 thf(fact_7836_linorder__cases,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ~ ( ord_less_int @ X4 @ Y3 )
% 0.23/0.54 => ( ( X4 != Y3 )
% 0.23/0.54 => ( ord_less_int @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_cases
% 0.23/0.54 thf(fact_7837_dual__order_Oasym,axiom,
% 0.23/0.54 ! [B2: real,A: real] :
% 0.23/0.54 ( ( ord_less_real @ B2 @ A )
% 0.23/0.54 => ~ ( ord_less_real @ A @ B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.asym
% 0.23/0.54 thf(fact_7838_dual__order_Oasym,axiom,
% 0.23/0.54 ! [B2: rat,A: rat] :
% 0.23/0.54 ( ( ord_less_rat @ B2 @ A )
% 0.23/0.54 => ~ ( ord_less_rat @ A @ B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.asym
% 0.23/0.54 thf(fact_7839_dual__order_Oasym,axiom,
% 0.23/0.54 ! [B2: num,A: num] :
% 0.23/0.54 ( ( ord_less_num @ B2 @ A )
% 0.23/0.54 => ~ ( ord_less_num @ A @ B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.asym
% 0.23/0.54 thf(fact_7840_dual__order_Oasym,axiom,
% 0.23/0.54 ! [B2: nat,A: nat] :
% 0.23/0.54 ( ( ord_less_nat @ B2 @ A )
% 0.23/0.54 => ~ ( ord_less_nat @ A @ B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.asym
% 0.23/0.54 thf(fact_7841_dual__order_Oasym,axiom,
% 0.23/0.54 ! [B2: int,A: int] :
% 0.23/0.54 ( ( ord_less_int @ B2 @ A )
% 0.23/0.54 => ~ ( ord_less_int @ A @ B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.asym
% 0.23/0.54 thf(fact_7842_dual__order_Oirrefl,axiom,
% 0.23/0.54 ! [A: real] :
% 0.23/0.54 ~ ( ord_less_real @ A @ A ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.irrefl
% 0.23/0.54 thf(fact_7843_dual__order_Oirrefl,axiom,
% 0.23/0.54 ! [A: rat] :
% 0.23/0.54 ~ ( ord_less_rat @ A @ A ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.irrefl
% 0.23/0.54 thf(fact_7844_dual__order_Oirrefl,axiom,
% 0.23/0.54 ! [A: num] :
% 0.23/0.54 ~ ( ord_less_num @ A @ A ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.irrefl
% 0.23/0.54 thf(fact_7845_dual__order_Oirrefl,axiom,
% 0.23/0.54 ! [A: nat] :
% 0.23/0.54 ~ ( ord_less_nat @ A @ A ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.irrefl
% 0.23/0.54 thf(fact_7846_dual__order_Oirrefl,axiom,
% 0.23/0.54 ! [A: int] :
% 0.23/0.54 ~ ( ord_less_int @ A @ A ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.irrefl
% 0.23/0.54 thf(fact_7847_exists__least__iff,axiom,
% 0.23/0.54 ( ( ^ [P5: nat > $o] :
% 0.23/0.54 ? [X7: nat] : ( P5 @ X7 ) )
% 0.23/0.54 = ( ^ [P6: nat > $o] :
% 0.23/0.54 ? [N2: nat] :
% 0.23/0.54 ( ( P6 @ N2 )
% 0.23/0.54 & ! [M6: nat] :
% 0.23/0.54 ( ( ord_less_nat @ M6 @ N2 )
% 0.23/0.54 => ~ ( P6 @ M6 ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % exists_least_iff
% 0.23/0.54 thf(fact_7848_linorder__less__wlog,axiom,
% 0.23/0.54 ! [P: real > real > $o,A: real,B2: real] :
% 0.23/0.54 ( ! [A5: real,B5: real] :
% 0.23/0.54 ( ( ord_less_real @ A5 @ B5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( ! [A5: real] : ( P @ A5 @ A5 )
% 0.23/0.54 => ( ! [A5: real,B5: real] :
% 0.23/0.54 ( ( P @ B5 @ A5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( P @ A @ B2 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_less_wlog
% 0.23/0.54 thf(fact_7849_linorder__less__wlog,axiom,
% 0.23/0.54 ! [P: rat > rat > $o,A: rat,B2: rat] :
% 0.23/0.54 ( ! [A5: rat,B5: rat] :
% 0.23/0.54 ( ( ord_less_rat @ A5 @ B5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( ! [A5: rat] : ( P @ A5 @ A5 )
% 0.23/0.54 => ( ! [A5: rat,B5: rat] :
% 0.23/0.54 ( ( P @ B5 @ A5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( P @ A @ B2 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_less_wlog
% 0.23/0.54 thf(fact_7850_linorder__less__wlog,axiom,
% 0.23/0.54 ! [P: num > num > $o,A: num,B2: num] :
% 0.23/0.54 ( ! [A5: num,B5: num] :
% 0.23/0.54 ( ( ord_less_num @ A5 @ B5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( ! [A5: num] : ( P @ A5 @ A5 )
% 0.23/0.54 => ( ! [A5: num,B5: num] :
% 0.23/0.54 ( ( P @ B5 @ A5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( P @ A @ B2 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_less_wlog
% 0.23/0.54 thf(fact_7851_linorder__less__wlog,axiom,
% 0.23/0.54 ! [P: nat > nat > $o,A: nat,B2: nat] :
% 0.23/0.54 ( ! [A5: nat,B5: nat] :
% 0.23/0.54 ( ( ord_less_nat @ A5 @ B5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( ! [A5: nat] : ( P @ A5 @ A5 )
% 0.23/0.54 => ( ! [A5: nat,B5: nat] :
% 0.23/0.54 ( ( P @ B5 @ A5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( P @ A @ B2 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_less_wlog
% 0.23/0.54 thf(fact_7852_linorder__less__wlog,axiom,
% 0.23/0.54 ! [P: int > int > $o,A: int,B2: int] :
% 0.23/0.54 ( ! [A5: int,B5: int] :
% 0.23/0.54 ( ( ord_less_int @ A5 @ B5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( ! [A5: int] : ( P @ A5 @ A5 )
% 0.23/0.54 => ( ! [A5: int,B5: int] :
% 0.23/0.54 ( ( P @ B5 @ A5 )
% 0.23/0.54 => ( P @ A5 @ B5 ) )
% 0.23/0.54 => ( P @ A @ B2 ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_less_wlog
% 0.23/0.54 thf(fact_7853_order_Ostrict__trans,axiom,
% 0.23/0.54 ! [A: real,B2: real,C: real] :
% 0.23/0.54 ( ( ord_less_real @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.54 => ( ord_less_real @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.strict_trans
% 0.23/0.54 thf(fact_7854_order_Ostrict__trans,axiom,
% 0.23/0.54 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.54 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.54 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.strict_trans
% 0.23/0.54 thf(fact_7855_order_Ostrict__trans,axiom,
% 0.23/0.54 ! [A: num,B2: num,C: num] :
% 0.23/0.54 ( ( ord_less_num @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_num @ B2 @ C )
% 0.23/0.54 => ( ord_less_num @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.strict_trans
% 0.23/0.54 thf(fact_7856_order_Ostrict__trans,axiom,
% 0.23/0.54 ! [A: nat,B2: nat,C: nat] :
% 0.23/0.54 ( ( ord_less_nat @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_nat @ B2 @ C )
% 0.23/0.54 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.strict_trans
% 0.23/0.54 thf(fact_7857_order_Ostrict__trans,axiom,
% 0.23/0.54 ! [A: int,B2: int,C: int] :
% 0.23/0.54 ( ( ord_less_int @ A @ B2 )
% 0.23/0.54 => ( ( ord_less_int @ B2 @ C )
% 0.23/0.54 => ( ord_less_int @ A @ C ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.strict_trans
% 0.23/0.54 thf(fact_7858_not__less__iff__gr__or__eq,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
% 0.23/0.54 = ( ( ord_less_real @ Y3 @ X4 )
% 0.23/0.54 | ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_less_iff_gr_or_eq
% 0.23/0.54 thf(fact_7859_not__less__iff__gr__or__eq,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat] :
% 0.23/0.54 ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
% 0.23/0.54 = ( ( ord_less_rat @ Y3 @ X4 )
% 0.23/0.54 | ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_less_iff_gr_or_eq
% 0.23/0.54 thf(fact_7860_not__less__iff__gr__or__eq,axiom,
% 0.23/0.54 ! [X4: num,Y3: num] :
% 0.23/0.54 ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
% 0.23/0.54 = ( ( ord_less_num @ Y3 @ X4 )
% 0.23/0.54 | ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_less_iff_gr_or_eq
% 0.23/0.54 thf(fact_7861_not__less__iff__gr__or__eq,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat] :
% 0.23/0.54 ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 0.23/0.54 = ( ( ord_less_nat @ Y3 @ X4 )
% 0.23/0.54 | ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_less_iff_gr_or_eq
% 0.23/0.54 thf(fact_7862_not__less__iff__gr__or__eq,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
% 0.23/0.54 = ( ( ord_less_int @ Y3 @ X4 )
% 0.23/0.54 | ( X4 = Y3 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % not_less_iff_gr_or_eq
% 0.23/0.54 thf(fact_7863_dual__order_Ostrict__trans,axiom,
% 0.23/0.54 ! [B2: real,A: real,C: real] :
% 0.23/0.54 ( ( ord_less_real @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_real @ C @ B2 )
% 0.23/0.54 => ( ord_less_real @ C @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.strict_trans
% 0.23/0.54 thf(fact_7864_dual__order_Ostrict__trans,axiom,
% 0.23/0.54 ! [B2: rat,A: rat,C: rat] :
% 0.23/0.54 ( ( ord_less_rat @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_rat @ C @ B2 )
% 0.23/0.54 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.strict_trans
% 0.23/0.54 thf(fact_7865_dual__order_Ostrict__trans,axiom,
% 0.23/0.54 ! [B2: num,A: num,C: num] :
% 0.23/0.54 ( ( ord_less_num @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_num @ C @ B2 )
% 0.23/0.54 => ( ord_less_num @ C @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.strict_trans
% 0.23/0.54 thf(fact_7866_dual__order_Ostrict__trans,axiom,
% 0.23/0.54 ! [B2: nat,A: nat,C: nat] :
% 0.23/0.54 ( ( ord_less_nat @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_nat @ C @ B2 )
% 0.23/0.54 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.strict_trans
% 0.23/0.54 thf(fact_7867_dual__order_Ostrict__trans,axiom,
% 0.23/0.54 ! [B2: int,A: int,C: int] :
% 0.23/0.54 ( ( ord_less_int @ B2 @ A )
% 0.23/0.54 => ( ( ord_less_int @ C @ B2 )
% 0.23/0.54 => ( ord_less_int @ C @ A ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.strict_trans
% 0.23/0.54 thf(fact_7868_order_Ostrict__implies__not__eq,axiom,
% 0.23/0.54 ! [A: real,B2: real] :
% 0.23/0.54 ( ( ord_less_real @ A @ B2 )
% 0.23/0.54 => ( A != B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.strict_implies_not_eq
% 0.23/0.54 thf(fact_7869_order_Ostrict__implies__not__eq,axiom,
% 0.23/0.54 ! [A: rat,B2: rat] :
% 0.23/0.54 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.54 => ( A != B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.strict_implies_not_eq
% 0.23/0.54 thf(fact_7870_order_Ostrict__implies__not__eq,axiom,
% 0.23/0.54 ! [A: num,B2: num] :
% 0.23/0.54 ( ( ord_less_num @ A @ B2 )
% 0.23/0.54 => ( A != B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.strict_implies_not_eq
% 0.23/0.54 thf(fact_7871_order_Ostrict__implies__not__eq,axiom,
% 0.23/0.54 ! [A: nat,B2: nat] :
% 0.23/0.54 ( ( ord_less_nat @ A @ B2 )
% 0.23/0.54 => ( A != B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.strict_implies_not_eq
% 0.23/0.54 thf(fact_7872_order_Ostrict__implies__not__eq,axiom,
% 0.23/0.54 ! [A: int,B2: int] :
% 0.23/0.54 ( ( ord_less_int @ A @ B2 )
% 0.23/0.54 => ( A != B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order.strict_implies_not_eq
% 0.23/0.54 thf(fact_7873_dual__order_Ostrict__implies__not__eq,axiom,
% 0.23/0.54 ! [B2: real,A: real] :
% 0.23/0.54 ( ( ord_less_real @ B2 @ A )
% 0.23/0.54 => ( A != B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.strict_implies_not_eq
% 0.23/0.54 thf(fact_7874_dual__order_Ostrict__implies__not__eq,axiom,
% 0.23/0.54 ! [B2: rat,A: rat] :
% 0.23/0.54 ( ( ord_less_rat @ B2 @ A )
% 0.23/0.54 => ( A != B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.strict_implies_not_eq
% 0.23/0.54 thf(fact_7875_dual__order_Ostrict__implies__not__eq,axiom,
% 0.23/0.54 ! [B2: num,A: num] :
% 0.23/0.54 ( ( ord_less_num @ B2 @ A )
% 0.23/0.54 => ( A != B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.strict_implies_not_eq
% 0.23/0.54 thf(fact_7876_dual__order_Ostrict__implies__not__eq,axiom,
% 0.23/0.54 ! [B2: nat,A: nat] :
% 0.23/0.54 ( ( ord_less_nat @ B2 @ A )
% 0.23/0.54 => ( A != B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.strict_implies_not_eq
% 0.23/0.54 thf(fact_7877_dual__order_Ostrict__implies__not__eq,axiom,
% 0.23/0.54 ! [B2: int,A: int] :
% 0.23/0.54 ( ( ord_less_int @ B2 @ A )
% 0.23/0.54 => ( A != B2 ) ) ).
% 0.23/0.54
% 0.23/0.54 % dual_order.strict_implies_not_eq
% 0.23/0.54 thf(fact_7878_linorder__neqE,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( X4 != Y3 )
% 0.23/0.54 => ( ~ ( ord_less_real @ X4 @ Y3 )
% 0.23/0.54 => ( ord_less_real @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_neqE
% 0.23/0.54 thf(fact_7879_linorder__neqE,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat] :
% 0.23/0.54 ( ( X4 != Y3 )
% 0.23/0.54 => ( ~ ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.54 => ( ord_less_rat @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_neqE
% 0.23/0.54 thf(fact_7880_linorder__neqE,axiom,
% 0.23/0.54 ! [X4: num,Y3: num] :
% 0.23/0.54 ( ( X4 != Y3 )
% 0.23/0.54 => ( ~ ( ord_less_num @ X4 @ Y3 )
% 0.23/0.54 => ( ord_less_num @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_neqE
% 0.23/0.54 thf(fact_7881_linorder__neqE,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat] :
% 0.23/0.54 ( ( X4 != Y3 )
% 0.23/0.54 => ( ~ ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.54 => ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_neqE
% 0.23/0.54 thf(fact_7882_linorder__neqE,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( X4 != Y3 )
% 0.23/0.54 => ( ~ ( ord_less_int @ X4 @ Y3 )
% 0.23/0.54 => ( ord_less_int @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_neqE
% 0.23/0.54 thf(fact_7883_order__less__asym,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.54 => ~ ( ord_less_real @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_asym
% 0.23/0.54 thf(fact_7884_order__less__asym,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat] :
% 0.23/0.54 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.54 => ~ ( ord_less_rat @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_asym
% 0.23/0.54 thf(fact_7885_order__less__asym,axiom,
% 0.23/0.54 ! [X4: num,Y3: num] :
% 0.23/0.54 ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.54 => ~ ( ord_less_num @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_asym
% 0.23/0.54 thf(fact_7886_order__less__asym,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat] :
% 0.23/0.54 ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.54 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_asym
% 0.23/0.54 thf(fact_7887_order__less__asym,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.54 => ~ ( ord_less_int @ Y3 @ X4 ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_asym
% 0.23/0.54 thf(fact_7888_linorder__neq__iff,axiom,
% 0.23/0.54 ! [X4: real,Y3: real] :
% 0.23/0.54 ( ( X4 != Y3 )
% 0.23/0.54 = ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.54 | ( ord_less_real @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_neq_iff
% 0.23/0.54 thf(fact_7889_linorder__neq__iff,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat] :
% 0.23/0.54 ( ( X4 != Y3 )
% 0.23/0.54 = ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.54 | ( ord_less_rat @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_neq_iff
% 0.23/0.54 thf(fact_7890_linorder__neq__iff,axiom,
% 0.23/0.54 ! [X4: num,Y3: num] :
% 0.23/0.54 ( ( X4 != Y3 )
% 0.23/0.54 = ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.54 | ( ord_less_num @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_neq_iff
% 0.23/0.54 thf(fact_7891_linorder__neq__iff,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat] :
% 0.23/0.54 ( ( X4 != Y3 )
% 0.23/0.54 = ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.54 | ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_neq_iff
% 0.23/0.54 thf(fact_7892_linorder__neq__iff,axiom,
% 0.23/0.54 ! [X4: int,Y3: int] :
% 0.23/0.54 ( ( X4 != Y3 )
% 0.23/0.54 = ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.54 | ( ord_less_int @ Y3 @ X4 ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % linorder_neq_iff
% 0.23/0.54 thf(fact_7893_order__less__asym_H,axiom,
% 0.23/0.54 ! [A: real,B2: real] :
% 0.23/0.54 ( ( ord_less_real @ A @ B2 )
% 0.23/0.54 => ~ ( ord_less_real @ B2 @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_asym'
% 0.23/0.54 thf(fact_7894_order__less__asym_H,axiom,
% 0.23/0.54 ! [A: rat,B2: rat] :
% 0.23/0.54 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.54 => ~ ( ord_less_rat @ B2 @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_asym'
% 0.23/0.54 thf(fact_7895_order__less__asym_H,axiom,
% 0.23/0.54 ! [A: num,B2: num] :
% 0.23/0.54 ( ( ord_less_num @ A @ B2 )
% 0.23/0.54 => ~ ( ord_less_num @ B2 @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_asym'
% 0.23/0.54 thf(fact_7896_order__less__asym_H,axiom,
% 0.23/0.54 ! [A: nat,B2: nat] :
% 0.23/0.54 ( ( ord_less_nat @ A @ B2 )
% 0.23/0.54 => ~ ( ord_less_nat @ B2 @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_asym'
% 0.23/0.54 thf(fact_7897_order__less__asym_H,axiom,
% 0.23/0.54 ! [A: int,B2: int] :
% 0.23/0.54 ( ( ord_less_int @ A @ B2 )
% 0.23/0.54 => ~ ( ord_less_int @ B2 @ A ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_asym'
% 0.23/0.54 thf(fact_7898_order__less__trans,axiom,
% 0.23/0.54 ! [X4: real,Y3: real,Z: real] :
% 0.23/0.54 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_real @ Y3 @ Z )
% 0.23/0.54 => ( ord_less_real @ X4 @ Z ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_trans
% 0.23/0.54 thf(fact_7899_order__less__trans,axiom,
% 0.23/0.54 ! [X4: rat,Y3: rat,Z: rat] :
% 0.23/0.54 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_rat @ Y3 @ Z )
% 0.23/0.54 => ( ord_less_rat @ X4 @ Z ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_trans
% 0.23/0.54 thf(fact_7900_order__less__trans,axiom,
% 0.23/0.54 ! [X4: num,Y3: num,Z: num] :
% 0.23/0.54 ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_num @ Y3 @ Z )
% 0.23/0.54 => ( ord_less_num @ X4 @ Z ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_trans
% 0.23/0.54 thf(fact_7901_order__less__trans,axiom,
% 0.23/0.54 ! [X4: nat,Y3: nat,Z: nat] :
% 0.23/0.54 ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_nat @ Y3 @ Z )
% 0.23/0.54 => ( ord_less_nat @ X4 @ Z ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_trans
% 0.23/0.54 thf(fact_7902_order__less__trans,axiom,
% 0.23/0.54 ! [X4: int,Y3: int,Z: int] :
% 0.23/0.54 ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.54 => ( ( ord_less_int @ Y3 @ Z )
% 0.23/0.54 => ( ord_less_int @ X4 @ Z ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % order_less_trans
% 0.23/0.54 thf(fact_7903_ord__eq__less__subst,axiom,
% 0.23/0.54 ! [A: real,F: real > real,B2: real,C: real] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.54 => ( ! [X3: real,Y4: real] :
% 0.23/0.54 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_subst
% 0.23/0.54 thf(fact_7904_ord__eq__less__subst,axiom,
% 0.23/0.54 ! [A: rat,F: real > rat,B2: real,C: real] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.54 => ( ! [X3: real,Y4: real] :
% 0.23/0.54 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_subst
% 0.23/0.54 thf(fact_7905_ord__eq__less__subst,axiom,
% 0.23/0.54 ! [A: num,F: real > num,B2: real,C: real] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.54 => ( ! [X3: real,Y4: real] :
% 0.23/0.54 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_subst
% 0.23/0.54 thf(fact_7906_ord__eq__less__subst,axiom,
% 0.23/0.54 ! [A: nat,F: real > nat,B2: real,C: real] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.54 => ( ! [X3: real,Y4: real] :
% 0.23/0.54 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_subst
% 0.23/0.54 thf(fact_7907_ord__eq__less__subst,axiom,
% 0.23/0.54 ! [A: int,F: real > int,B2: real,C: real] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.54 => ( ! [X3: real,Y4: real] :
% 0.23/0.54 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_subst
% 0.23/0.54 thf(fact_7908_ord__eq__less__subst,axiom,
% 0.23/0.54 ! [A: real,F: rat > real,B2: rat,C: rat] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_subst
% 0.23/0.54 thf(fact_7909_ord__eq__less__subst,axiom,
% 0.23/0.54 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_subst
% 0.23/0.54 thf(fact_7910_ord__eq__less__subst,axiom,
% 0.23/0.54 ! [A: num,F: rat > num,B2: rat,C: rat] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.54
% 0.23/0.54 % ord_eq_less_subst
% 0.23/0.54 thf(fact_7911_ord__eq__less__subst,axiom,
% 0.23/0.54 ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 0.23/0.54 ( ( A
% 0.23/0.54 = ( F @ B2 ) )
% 0.23/0.54 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.54 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.54 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.54 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.54 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_eq_less_subst
% 0.23/0.55 thf(fact_7912_ord__eq__less__subst,axiom,
% 0.23/0.55 ! [A: int,F: rat > int,B2: rat,C: rat] :
% 0.23/0.55 ( ( A
% 0.23/0.55 = ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_eq_less_subst
% 0.23/0.55 thf(fact_7913_ord__less__eq__subst,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > real,C: real] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ( F @ B2 )
% 0.23/0.55 = C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_less_eq_subst
% 0.23/0.55 thf(fact_7914_ord__less__eq__subst,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > rat,C: rat] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ( F @ B2 )
% 0.23/0.55 = C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_less_eq_subst
% 0.23/0.55 thf(fact_7915_ord__less__eq__subst,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > num,C: num] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ( F @ B2 )
% 0.23/0.55 = C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_less_eq_subst
% 0.23/0.55 thf(fact_7916_ord__less__eq__subst,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > nat,C: nat] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ( F @ B2 )
% 0.23/0.55 = C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_less_eq_subst
% 0.23/0.55 thf(fact_7917_ord__less__eq__subst,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > int,C: int] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ( F @ B2 )
% 0.23/0.55 = C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_less_eq_subst
% 0.23/0.55 thf(fact_7918_ord__less__eq__subst,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > real,C: real] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ( F @ B2 )
% 0.23/0.55 = C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_less_eq_subst
% 0.23/0.55 thf(fact_7919_ord__less__eq__subst,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ( F @ B2 )
% 0.23/0.55 = C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_less_eq_subst
% 0.23/0.55 thf(fact_7920_ord__less__eq__subst,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > num,C: num] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ( F @ B2 )
% 0.23/0.55 = C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_less_eq_subst
% 0.23/0.55 thf(fact_7921_ord__less__eq__subst,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ( F @ B2 )
% 0.23/0.55 = C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_less_eq_subst
% 0.23/0.55 thf(fact_7922_ord__less__eq__subst,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > int,C: int] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ( F @ B2 )
% 0.23/0.55 = C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % ord_less_eq_subst
% 0.23/0.55 thf(fact_7923_order__less__irrefl,axiom,
% 0.23/0.55 ! [X4: real] :
% 0.23/0.55 ~ ( ord_less_real @ X4 @ X4 ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_irrefl
% 0.23/0.55 thf(fact_7924_order__less__irrefl,axiom,
% 0.23/0.55 ! [X4: rat] :
% 0.23/0.55 ~ ( ord_less_rat @ X4 @ X4 ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_irrefl
% 0.23/0.55 thf(fact_7925_order__less__irrefl,axiom,
% 0.23/0.55 ! [X4: num] :
% 0.23/0.55 ~ ( ord_less_num @ X4 @ X4 ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_irrefl
% 0.23/0.55 thf(fact_7926_order__less__irrefl,axiom,
% 0.23/0.55 ! [X4: nat] :
% 0.23/0.55 ~ ( ord_less_nat @ X4 @ X4 ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_irrefl
% 0.23/0.55 thf(fact_7927_order__less__irrefl,axiom,
% 0.23/0.55 ! [X4: int] :
% 0.23/0.55 ~ ( ord_less_int @ X4 @ X4 ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_irrefl
% 0.23/0.55 thf(fact_7928_order__less__subst1,axiom,
% 0.23/0.55 ! [A: real,F: real > real,B2: real,C: real] :
% 0.23/0.55 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst1
% 0.23/0.55 thf(fact_7929_order__less__subst1,axiom,
% 0.23/0.55 ! [A: real,F: rat > real,B2: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst1
% 0.23/0.55 thf(fact_7930_order__less__subst1,axiom,
% 0.23/0.55 ! [A: real,F: num > real,B2: num,C: num] :
% 0.23/0.55 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_num @ B2 @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst1
% 0.23/0.55 thf(fact_7931_order__less__subst1,axiom,
% 0.23/0.55 ! [A: real,F: nat > real,B2: nat,C: nat] :
% 0.23/0.55 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_nat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst1
% 0.23/0.55 thf(fact_7932_order__less__subst1,axiom,
% 0.23/0.55 ! [A: real,F: int > real,B2: int,C: int] :
% 0.23/0.55 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_int @ B2 @ C )
% 0.23/0.55 => ( ! [X3: int,Y4: int] :
% 0.23/0.55 ( ( ord_less_int @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst1
% 0.23/0.55 thf(fact_7933_order__less__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: real > rat,B2: real,C: real] :
% 0.23/0.55 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst1
% 0.23/0.55 thf(fact_7934_order__less__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst1
% 0.23/0.55 thf(fact_7935_order__less__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: num > rat,B2: num,C: num] :
% 0.23/0.55 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_num @ B2 @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst1
% 0.23/0.55 thf(fact_7936_order__less__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 0.23/0.55 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_nat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst1
% 0.23/0.55 thf(fact_7937_order__less__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: int > rat,B2: int,C: int] :
% 0.23/0.55 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_int @ B2 @ C )
% 0.23/0.55 => ( ! [X3: int,Y4: int] :
% 0.23/0.55 ( ( ord_less_int @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst1
% 0.23/0.55 thf(fact_7938_order__less__subst2,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > real,C: real] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst2
% 0.23/0.55 thf(fact_7939_order__less__subst2,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > rat,C: rat] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst2
% 0.23/0.55 thf(fact_7940_order__less__subst2,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > num,C: num] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst2
% 0.23/0.55 thf(fact_7941_order__less__subst2,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > nat,C: nat] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst2
% 0.23/0.55 thf(fact_7942_order__less__subst2,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > int,C: int] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst2
% 0.23/0.55 thf(fact_7943_order__less__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > real,C: real] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst2
% 0.23/0.55 thf(fact_7944_order__less__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst2
% 0.23/0.55 thf(fact_7945_order__less__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > num,C: num] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst2
% 0.23/0.55 thf(fact_7946_order__less__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst2
% 0.23/0.55 thf(fact_7947_order__less__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > int,C: int] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_subst2
% 0.23/0.55 thf(fact_7948_order__less__not__sym,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 => ~ ( ord_less_real @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_not_sym
% 0.23/0.55 thf(fact_7949_order__less__not__sym,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 => ~ ( ord_less_rat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_not_sym
% 0.23/0.55 thf(fact_7950_order__less__not__sym,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.55 => ~ ( ord_less_num @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_not_sym
% 0.23/0.55 thf(fact_7951_order__less__not__sym,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_not_sym
% 0.23/0.55 thf(fact_7952_order__less__not__sym,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.55 => ~ ( ord_less_int @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_not_sym
% 0.23/0.55 thf(fact_7953_order__less__imp__triv,axiom,
% 0.23/0.55 ! [X4: real,Y3: real,P: $o] :
% 0.23/0.55 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_real @ Y3 @ X4 )
% 0.23/0.55 => P ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_triv
% 0.23/0.55 thf(fact_7954_order__less__imp__triv,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat,P: $o] :
% 0.23/0.55 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_rat @ Y3 @ X4 )
% 0.23/0.55 => P ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_triv
% 0.23/0.55 thf(fact_7955_order__less__imp__triv,axiom,
% 0.23/0.55 ! [X4: num,Y3: num,P: $o] :
% 0.23/0.55 ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_num @ Y3 @ X4 )
% 0.23/0.55 => P ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_triv
% 0.23/0.55 thf(fact_7956_order__less__imp__triv,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat,P: $o] :
% 0.23/0.55 ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_nat @ Y3 @ X4 )
% 0.23/0.55 => P ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_triv
% 0.23/0.55 thf(fact_7957_order__less__imp__triv,axiom,
% 0.23/0.55 ! [X4: int,Y3: int,P: $o] :
% 0.23/0.55 ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_int @ Y3 @ X4 )
% 0.23/0.55 => P ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_triv
% 0.23/0.55 thf(fact_7958_linorder__less__linear,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 | ( X4 = Y3 )
% 0.23/0.55 | ( ord_less_real @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_less_linear
% 0.23/0.55 thf(fact_7959_linorder__less__linear,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 | ( X4 = Y3 )
% 0.23/0.55 | ( ord_less_rat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_less_linear
% 0.23/0.55 thf(fact_7960_linorder__less__linear,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.55 | ( X4 = Y3 )
% 0.23/0.55 | ( ord_less_num @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_less_linear
% 0.23/0.55 thf(fact_7961_linorder__less__linear,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 | ( X4 = Y3 )
% 0.23/0.55 | ( ord_less_nat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_less_linear
% 0.23/0.55 thf(fact_7962_linorder__less__linear,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.55 | ( X4 = Y3 )
% 0.23/0.55 | ( ord_less_int @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_less_linear
% 0.23/0.55 thf(fact_7963_order__less__imp__not__eq,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 => ( X4 != Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_eq
% 0.23/0.55 thf(fact_7964_order__less__imp__not__eq,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 => ( X4 != Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_eq
% 0.23/0.55 thf(fact_7965_order__less__imp__not__eq,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.55 => ( X4 != Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_eq
% 0.23/0.55 thf(fact_7966_order__less__imp__not__eq,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 => ( X4 != Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_eq
% 0.23/0.55 thf(fact_7967_order__less__imp__not__eq,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.55 => ( X4 != Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_eq
% 0.23/0.55 thf(fact_7968_order__less__imp__not__eq2,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 => ( Y3 != X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_eq2
% 0.23/0.55 thf(fact_7969_order__less__imp__not__eq2,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 => ( Y3 != X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_eq2
% 0.23/0.55 thf(fact_7970_order__less__imp__not__eq2,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.55 => ( Y3 != X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_eq2
% 0.23/0.55 thf(fact_7971_order__less__imp__not__eq2,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 => ( Y3 != X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_eq2
% 0.23/0.55 thf(fact_7972_order__less__imp__not__eq2,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.55 => ( Y3 != X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_eq2
% 0.23/0.55 thf(fact_7973_order__less__imp__not__less,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 => ~ ( ord_less_real @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_less
% 0.23/0.55 thf(fact_7974_order__less__imp__not__less,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 => ~ ( ord_less_rat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_less
% 0.23/0.55 thf(fact_7975_order__less__imp__not__less,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.55 => ~ ( ord_less_num @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_less
% 0.23/0.55 thf(fact_7976_order__less__imp__not__less,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_less
% 0.23/0.55 thf(fact_7977_order__less__imp__not__less,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.55 => ~ ( ord_less_int @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_not_less
% 0.23/0.55 thf(fact_7978_take__bit__not__eq__mask__diff,axiom,
% 0.23/0.55 ! [N: nat,A: int] :
% 0.23/0.55 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 0.23/0.55 = ( minus_minus_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % take_bit_not_eq_mask_diff
% 0.23/0.55 thf(fact_7979_minus__numeral__inc__eq,axiom,
% 0.23/0.55 ! [N: num] :
% 0.23/0.55 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
% 0.23/0.55 = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % minus_numeral_inc_eq
% 0.23/0.55 thf(fact_7980_minus__numeral__inc__eq,axiom,
% 0.23/0.55 ! [N: num] :
% 0.23/0.55 ( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
% 0.23/0.55 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % minus_numeral_inc_eq
% 0.23/0.55 thf(fact_7981_unset__bit__int__def,axiom,
% 0.23/0.55 ( bit_se4203085406695923979it_int
% 0.23/0.55 = ( ^ [N2: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % unset_bit_int_def
% 0.23/0.55 thf(fact_7982_not__int__div__2,axiom,
% 0.23/0.55 ! [K: int] :
% 0.23/0.55 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.55 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % not_int_div_2
% 0.23/0.55 thf(fact_7983_even__not__iff__int,axiom,
% 0.23/0.55 ! [K: int] :
% 0.23/0.55 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.23/0.55 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % even_not_iff_int
% 0.23/0.55 thf(fact_7984_not__numeral__Bit0__eq,axiom,
% 0.23/0.55 ! [N: num] :
% 0.23/0.55 ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
% 0.23/0.55 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % not_numeral_Bit0_eq
% 0.23/0.55 thf(fact_7985_not__numeral__Bit0__eq,axiom,
% 0.23/0.55 ! [N: num] :
% 0.23/0.55 ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
% 0.23/0.55 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % not_numeral_Bit0_eq
% 0.23/0.55 thf(fact_7986_and__not__numerals_I2_J,axiom,
% 0.23/0.55 ! [N: num] :
% 0.23/0.55 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.23/0.55 = one_one_int ) ).
% 0.23/0.55
% 0.23/0.55 % and_not_numerals(2)
% 0.23/0.55 thf(fact_7987_and__not__numerals_I4_J,axiom,
% 0.23/0.55 ! [M: num] :
% 0.23/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.23/0.55 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % and_not_numerals(4)
% 0.23/0.55 thf(fact_7988_bit__minus__int__iff,axiom,
% 0.23/0.55 ! [K: int,N: nat] :
% 0.23/0.55 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 0.23/0.55 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 0.23/0.55
% 0.23/0.55 % bit_minus_int_iff
% 0.23/0.55 thf(fact_7989_take__bit__not__mask__eq__0,axiom,
% 0.23/0.55 ! [M: nat,N: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.55 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) )
% 0.23/0.55 = zero_zero_int ) ) ).
% 0.23/0.55
% 0.23/0.55 % take_bit_not_mask_eq_0
% 0.23/0.55 thf(fact_7990_push__bit__mask__eq,axiom,
% 0.23/0.55 ! [M: nat,N: nat] :
% 0.23/0.55 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.23/0.55 = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % push_bit_mask_eq
% 0.23/0.55 thf(fact_7991_unset__bit__eq__and__not,axiom,
% 0.23/0.55 ( bit_se4203085406695923979it_int
% 0.23/0.55 = ( ^ [N2: nat,A4: int] : ( bit_se725231765392027082nd_int @ A4 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % unset_bit_eq_and_not
% 0.23/0.55 thf(fact_7992_and__not__numerals_I5_J,axiom,
% 0.23/0.55 ! [M: num,N: num] :
% 0.23/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.23/0.55 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % and_not_numerals(5)
% 0.23/0.55 thf(fact_7993_and__not__numerals_I7_J,axiom,
% 0.23/0.55 ! [M: num] :
% 0.23/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.23/0.55 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % and_not_numerals(7)
% 0.23/0.55 thf(fact_7994_and__not__numerals_I3_J,axiom,
% 0.23/0.55 ! [N: num] :
% 0.23/0.55 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.23/0.55 = zero_zero_int ) ).
% 0.23/0.55
% 0.23/0.55 % and_not_numerals(3)
% 0.23/0.55 thf(fact_7995_and__not__numerals_I9_J,axiom,
% 0.23/0.55 ! [M: num,N: num] :
% 0.23/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.23/0.55 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % and_not_numerals(9)
% 0.23/0.55 thf(fact_7996_and__not__numerals_I6_J,axiom,
% 0.23/0.55 ! [M: num,N: num] :
% 0.23/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.23/0.55 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % and_not_numerals(6)
% 0.23/0.55 thf(fact_7997_bit__not__iff__eq,axiom,
% 0.23/0.55 ! [A: int,N: nat] :
% 0.23/0.55 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N )
% 0.23/0.55 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.23/0.55 != zero_zero_int )
% 0.23/0.55 & ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % bit_not_iff_eq
% 0.23/0.55 thf(fact_7998_minus__exp__eq__not__mask,axiom,
% 0.23/0.55 ! [N: nat] :
% 0.23/0.55 ( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.55 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % minus_exp_eq_not_mask
% 0.23/0.55 thf(fact_7999_minus__exp__eq__not__mask,axiom,
% 0.23/0.55 ! [N: nat] :
% 0.23/0.55 ( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.55 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % minus_exp_eq_not_mask
% 0.23/0.55 thf(fact_8000_CauchyD,axiom,
% 0.23/0.55 ! [X8: nat > complex,E: real] :
% 0.23/0.55 ( ( topolo6517432010174082258omplex @ X8 )
% 0.23/0.55 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.23/0.55 => ? [M9: nat] :
% 0.23/0.55 ! [M5: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M9 @ M5 )
% 0.23/0.55 => ! [N6: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M9 @ N6 )
% 0.23/0.55 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ E ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % CauchyD
% 0.23/0.55 thf(fact_8001_CauchyD,axiom,
% 0.23/0.55 ! [X8: nat > real,E: real] :
% 0.23/0.55 ( ( topolo4055970368930404560y_real @ X8 )
% 0.23/0.55 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.23/0.55 => ? [M9: nat] :
% 0.23/0.55 ! [M5: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M9 @ M5 )
% 0.23/0.55 => ! [N6: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M9 @ N6 )
% 0.23/0.55 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ E ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % CauchyD
% 0.23/0.55 thf(fact_8002_CauchyI,axiom,
% 0.23/0.55 ! [X8: nat > complex] :
% 0.23/0.55 ( ! [E2: real] :
% 0.23/0.55 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.23/0.55 => ? [M10: nat] :
% 0.23/0.55 ! [M4: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M10 @ M4 )
% 0.23/0.55 => ! [N4: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M10 @ N4 )
% 0.23/0.55 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ E2 ) ) ) )
% 0.23/0.55 => ( topolo6517432010174082258omplex @ X8 ) ) ).
% 0.23/0.55
% 0.23/0.55 % CauchyI
% 0.23/0.55 thf(fact_8003_CauchyI,axiom,
% 0.23/0.55 ! [X8: nat > real] :
% 0.23/0.55 ( ! [E2: real] :
% 0.23/0.55 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.23/0.55 => ? [M10: nat] :
% 0.23/0.55 ! [M4: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M10 @ M4 )
% 0.23/0.55 => ! [N4: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M10 @ N4 )
% 0.23/0.55 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ E2 ) ) ) )
% 0.23/0.55 => ( topolo4055970368930404560y_real @ X8 ) ) ).
% 0.23/0.55
% 0.23/0.55 % CauchyI
% 0.23/0.55 thf(fact_8004_Cauchy__iff,axiom,
% 0.23/0.55 ( topolo6517432010174082258omplex
% 0.23/0.55 = ( ^ [X6: nat > complex] :
% 0.23/0.55 ! [E3: real] :
% 0.23/0.55 ( ( ord_less_real @ zero_zero_real @ E3 )
% 0.23/0.55 => ? [M8: nat] :
% 0.23/0.55 ! [M6: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M8 @ M6 )
% 0.23/0.55 => ! [N2: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M8 @ N2 )
% 0.23/0.55 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Cauchy_iff
% 0.23/0.55 thf(fact_8005_Cauchy__iff,axiom,
% 0.23/0.55 ( topolo4055970368930404560y_real
% 0.23/0.55 = ( ^ [X6: nat > real] :
% 0.23/0.55 ! [E3: real] :
% 0.23/0.55 ( ( ord_less_real @ zero_zero_real @ E3 )
% 0.23/0.55 => ? [M8: nat] :
% 0.23/0.55 ! [M6: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M8 @ M6 )
% 0.23/0.55 => ! [N2: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ M8 @ N2 )
% 0.23/0.55 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Cauchy_iff
% 0.23/0.55 thf(fact_8006_leD,axiom,
% 0.23/0.55 ! [Y3: real,X4: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ Y3 @ X4 )
% 0.23/0.55 => ~ ( ord_less_real @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % leD
% 0.23/0.55 thf(fact_8007_leD,axiom,
% 0.23/0.55 ! [Y3: set_nat,X4: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ Y3 @ X4 )
% 0.23/0.55 => ~ ( ord_less_set_nat @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % leD
% 0.23/0.55 thf(fact_8008_leD,axiom,
% 0.23/0.55 ! [Y3: rat,X4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ Y3 @ X4 )
% 0.23/0.55 => ~ ( ord_less_rat @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % leD
% 0.23/0.55 thf(fact_8009_leD,axiom,
% 0.23/0.55 ! [Y3: num,X4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ Y3 @ X4 )
% 0.23/0.55 => ~ ( ord_less_num @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % leD
% 0.23/0.55 thf(fact_8010_leD,axiom,
% 0.23/0.55 ! [Y3: nat,X4: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ Y3 @ X4 )
% 0.23/0.55 => ~ ( ord_less_nat @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % leD
% 0.23/0.55 thf(fact_8011_leD,axiom,
% 0.23/0.55 ! [Y3: int,X4: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ Y3 @ X4 )
% 0.23/0.55 => ~ ( ord_less_int @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % leD
% 0.23/0.55 thf(fact_8012_leI,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ~ ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_real @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % leI
% 0.23/0.55 thf(fact_8013_leI,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ~ ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_rat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % leI
% 0.23/0.55 thf(fact_8014_leI,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ~ ( ord_less_num @ X4 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_num @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % leI
% 0.23/0.55 thf(fact_8015_leI,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ~ ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % leI
% 0.23/0.55 thf(fact_8016_leI,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ~ ( ord_less_int @ X4 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_int @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % leI
% 0.23/0.55 thf(fact_8017_nless__le,axiom,
% 0.23/0.55 ! [A: real,B2: real] :
% 0.23/0.55 ( ( ~ ( ord_less_real @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_real @ A @ B2 )
% 0.23/0.55 | ( A = B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nless_le
% 0.23/0.55 thf(fact_8018_nless__le,axiom,
% 0.23/0.55 ! [A: set_nat,B2: set_nat] :
% 0.23/0.55 ( ( ~ ( ord_less_set_nat @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_set_nat @ A @ B2 )
% 0.23/0.55 | ( A = B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nless_le
% 0.23/0.55 thf(fact_8019_nless__le,axiom,
% 0.23/0.55 ! [A: rat,B2: rat] :
% 0.23/0.55 ( ( ~ ( ord_less_rat @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.55 | ( A = B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nless_le
% 0.23/0.55 thf(fact_8020_nless__le,axiom,
% 0.23/0.55 ! [A: num,B2: num] :
% 0.23/0.55 ( ( ~ ( ord_less_num @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_num @ A @ B2 )
% 0.23/0.55 | ( A = B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nless_le
% 0.23/0.55 thf(fact_8021_nless__le,axiom,
% 0.23/0.55 ! [A: nat,B2: nat] :
% 0.23/0.55 ( ( ~ ( ord_less_nat @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.55 | ( A = B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nless_le
% 0.23/0.55 thf(fact_8022_nless__le,axiom,
% 0.23/0.55 ! [A: int,B2: int] :
% 0.23/0.55 ( ( ~ ( ord_less_int @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_int @ A @ B2 )
% 0.23/0.55 | ( A = B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nless_le
% 0.23/0.55 thf(fact_8023_antisym__conv1,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ~ ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv1
% 0.23/0.55 thf(fact_8024_antisym__conv1,axiom,
% 0.23/0.55 ! [X4: set_nat,Y3: set_nat] :
% 0.23/0.55 ( ~ ( ord_less_set_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv1
% 0.23/0.55 thf(fact_8025_antisym__conv1,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ~ ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv1
% 0.23/0.55 thf(fact_8026_antisym__conv1,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ~ ( ord_less_num @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv1
% 0.23/0.55 thf(fact_8027_antisym__conv1,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ~ ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv1
% 0.23/0.55 thf(fact_8028_antisym__conv1,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ~ ( ord_less_int @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv1
% 0.23/0.55 thf(fact_8029_antisym__conv2,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.55 => ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv2
% 0.23/0.55 thf(fact_8030_antisym__conv2,axiom,
% 0.23/0.55 ! [X4: set_nat,Y3: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ~ ( ord_less_set_nat @ X4 @ Y3 ) )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv2
% 0.23/0.55 thf(fact_8031_antisym__conv2,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv2
% 0.23/0.55 thf(fact_8032_antisym__conv2,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.23/0.55 => ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv2
% 0.23/0.55 thf(fact_8033_antisym__conv2,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv2
% 0.23/0.55 thf(fact_8034_antisym__conv2,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.55 => ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
% 0.23/0.55 = ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % antisym_conv2
% 0.23/0.55 thf(fact_8035_dense__ge,axiom,
% 0.23/0.55 ! [Z: real,Y3: real] :
% 0.23/0.55 ( ! [X3: real] :
% 0.23/0.55 ( ( ord_less_real @ Z @ X3 )
% 0.23/0.55 => ( ord_less_eq_real @ Y3 @ X3 ) )
% 0.23/0.55 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 0.23/0.55
% 0.23/0.55 % dense_ge
% 0.23/0.55 thf(fact_8036_dense__ge,axiom,
% 0.23/0.55 ! [Z: rat,Y3: rat] :
% 0.23/0.55 ( ! [X3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ Z @ X3 )
% 0.23/0.55 => ( ord_less_eq_rat @ Y3 @ X3 ) )
% 0.23/0.55 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 0.23/0.55
% 0.23/0.55 % dense_ge
% 0.23/0.55 thf(fact_8037_dense__le,axiom,
% 0.23/0.55 ! [Y3: real,Z: real] :
% 0.23/0.55 ( ! [X3: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_real @ X3 @ Z ) )
% 0.23/0.55 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 0.23/0.55
% 0.23/0.55 % dense_le
% 0.23/0.55 thf(fact_8038_dense__le,axiom,
% 0.23/0.55 ! [Y3: rat,Z: rat] :
% 0.23/0.55 ( ! [X3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_rat @ X3 @ Z ) )
% 0.23/0.55 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 0.23/0.55
% 0.23/0.55 % dense_le
% 0.23/0.55 thf(fact_8039_less__le__not__le,axiom,
% 0.23/0.55 ( ord_less_real
% 0.23/0.55 = ( ^ [X: real,Y: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ X @ Y )
% 0.23/0.55 & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % less_le_not_le
% 0.23/0.55 thf(fact_8040_less__le__not__le,axiom,
% 0.23/0.55 ( ord_less_set_nat
% 0.23/0.55 = ( ^ [X: set_nat,Y: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.23/0.55 & ~ ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % less_le_not_le
% 0.23/0.55 thf(fact_8041_less__le__not__le,axiom,
% 0.23/0.55 ( ord_less_rat
% 0.23/0.55 = ( ^ [X: rat,Y: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X @ Y )
% 0.23/0.55 & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % less_le_not_le
% 0.23/0.55 thf(fact_8042_less__le__not__le,axiom,
% 0.23/0.55 ( ord_less_num
% 0.23/0.55 = ( ^ [X: num,Y: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X @ Y )
% 0.23/0.55 & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % less_le_not_le
% 0.23/0.55 thf(fact_8043_less__le__not__le,axiom,
% 0.23/0.55 ( ord_less_nat
% 0.23/0.55 = ( ^ [X: nat,Y: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ X @ Y )
% 0.23/0.55 & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % less_le_not_le
% 0.23/0.55 thf(fact_8044_less__le__not__le,axiom,
% 0.23/0.55 ( ord_less_int
% 0.23/0.55 = ( ^ [X: int,Y: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ X @ Y )
% 0.23/0.55 & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % less_le_not_le
% 0.23/0.55 thf(fact_8045_not__le__imp__less,axiom,
% 0.23/0.55 ! [Y3: real,X4: real] :
% 0.23/0.55 ( ~ ( ord_less_eq_real @ Y3 @ X4 )
% 0.23/0.55 => ( ord_less_real @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % not_le_imp_less
% 0.23/0.55 thf(fact_8046_not__le__imp__less,axiom,
% 0.23/0.55 ! [Y3: rat,X4: rat] :
% 0.23/0.55 ( ~ ( ord_less_eq_rat @ Y3 @ X4 )
% 0.23/0.55 => ( ord_less_rat @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % not_le_imp_less
% 0.23/0.55 thf(fact_8047_not__le__imp__less,axiom,
% 0.23/0.55 ! [Y3: num,X4: num] :
% 0.23/0.55 ( ~ ( ord_less_eq_num @ Y3 @ X4 )
% 0.23/0.55 => ( ord_less_num @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % not_le_imp_less
% 0.23/0.55 thf(fact_8048_not__le__imp__less,axiom,
% 0.23/0.55 ! [Y3: nat,X4: nat] :
% 0.23/0.55 ( ~ ( ord_less_eq_nat @ Y3 @ X4 )
% 0.23/0.55 => ( ord_less_nat @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % not_le_imp_less
% 0.23/0.55 thf(fact_8049_not__le__imp__less,axiom,
% 0.23/0.55 ! [Y3: int,X4: int] :
% 0.23/0.55 ( ~ ( ord_less_eq_int @ Y3 @ X4 )
% 0.23/0.55 => ( ord_less_int @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % not_le_imp_less
% 0.23/0.55 thf(fact_8050_order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_real
% 0.23/0.55 = ( ^ [A4: real,B3: real] :
% 0.23/0.55 ( ( ord_less_real @ A4 @ B3 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.order_iff_strict
% 0.23/0.55 thf(fact_8051_order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_set_nat
% 0.23/0.55 = ( ^ [A4: set_nat,B3: set_nat] :
% 0.23/0.55 ( ( ord_less_set_nat @ A4 @ B3 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.order_iff_strict
% 0.23/0.55 thf(fact_8052_order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_rat
% 0.23/0.55 = ( ^ [A4: rat,B3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ A4 @ B3 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.order_iff_strict
% 0.23/0.55 thf(fact_8053_order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_num
% 0.23/0.55 = ( ^ [A4: num,B3: num] :
% 0.23/0.55 ( ( ord_less_num @ A4 @ B3 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.order_iff_strict
% 0.23/0.55 thf(fact_8054_order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_nat
% 0.23/0.55 = ( ^ [A4: nat,B3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ A4 @ B3 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.order_iff_strict
% 0.23/0.55 thf(fact_8055_order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_int
% 0.23/0.55 = ( ^ [A4: int,B3: int] :
% 0.23/0.55 ( ( ord_less_int @ A4 @ B3 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.order_iff_strict
% 0.23/0.55 thf(fact_8056_order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_real
% 0.23/0.55 = ( ^ [A4: real,B3: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ A4 @ B3 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_order
% 0.23/0.55 thf(fact_8057_order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_set_nat
% 0.23/0.55 = ( ^ [A4: set_nat,B3: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ A4 @ B3 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_order
% 0.23/0.55 thf(fact_8058_order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_rat
% 0.23/0.55 = ( ^ [A4: rat,B3: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A4 @ B3 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_order
% 0.23/0.55 thf(fact_8059_order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_num
% 0.23/0.55 = ( ^ [A4: num,B3: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ A4 @ B3 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_order
% 0.23/0.55 thf(fact_8060_order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_nat
% 0.23/0.55 = ( ^ [A4: nat,B3: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ A4 @ B3 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_order
% 0.23/0.55 thf(fact_8061_order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_int
% 0.23/0.55 = ( ^ [A4: int,B3: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ A4 @ B3 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_order
% 0.23/0.55 thf(fact_8062_order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [A: real,B2: real,C: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.55 => ( ord_less_real @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans1
% 0.23/0.55 thf(fact_8063_order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [A: set_nat,B2: set_nat,C: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_set_nat @ B2 @ C )
% 0.23/0.55 => ( ord_less_set_nat @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans1
% 0.23/0.55 thf(fact_8064_order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.55 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans1
% 0.23/0.55 thf(fact_8065_order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [A: num,B2: num,C: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_num @ B2 @ C )
% 0.23/0.55 => ( ord_less_num @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans1
% 0.23/0.55 thf(fact_8066_order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [A: nat,B2: nat,C: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_nat @ B2 @ C )
% 0.23/0.55 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans1
% 0.23/0.55 thf(fact_8067_order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [A: int,B2: int,C: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_int @ B2 @ C )
% 0.23/0.55 => ( ord_less_int @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans1
% 0.23/0.55 thf(fact_8068_order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [A: real,B2: real,C: real] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_real @ B2 @ C )
% 0.23/0.55 => ( ord_less_real @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans2
% 0.23/0.55 thf(fact_8069_order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [A: set_nat,B2: set_nat,C: set_nat] :
% 0.23/0.55 ( ( ord_less_set_nat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 0.23/0.55 => ( ord_less_set_nat @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans2
% 0.23/0.55 thf(fact_8070_order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.55 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans2
% 0.23/0.55 thf(fact_8071_order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [A: num,B2: num,C: num] :
% 0.23/0.55 ( ( ord_less_num @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.55 => ( ord_less_num @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans2
% 0.23/0.55 thf(fact_8072_order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [A: nat,B2: nat,C: nat] :
% 0.23/0.55 ( ( ord_less_nat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.23/0.55 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans2
% 0.23/0.55 thf(fact_8073_order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [A: int,B2: int,C: int] :
% 0.23/0.55 ( ( ord_less_int @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_int @ B2 @ C )
% 0.23/0.55 => ( ord_less_int @ A @ C ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_trans2
% 0.23/0.55 thf(fact_8074_order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_real
% 0.23/0.55 = ( ^ [A4: real,B3: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ A4 @ B3 )
% 0.23/0.55 & ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_not
% 0.23/0.55 thf(fact_8075_order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_set_nat
% 0.23/0.55 = ( ^ [A4: set_nat,B3: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ A4 @ B3 )
% 0.23/0.55 & ~ ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_not
% 0.23/0.55 thf(fact_8076_order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_rat
% 0.23/0.55 = ( ^ [A4: rat,B3: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A4 @ B3 )
% 0.23/0.55 & ~ ( ord_less_eq_rat @ B3 @ A4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_not
% 0.23/0.55 thf(fact_8077_order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_num
% 0.23/0.55 = ( ^ [A4: num,B3: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ A4 @ B3 )
% 0.23/0.55 & ~ ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_not
% 0.23/0.55 thf(fact_8078_order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_nat
% 0.23/0.55 = ( ^ [A4: nat,B3: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ A4 @ B3 )
% 0.23/0.55 & ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_not
% 0.23/0.55 thf(fact_8079_order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_int
% 0.23/0.55 = ( ^ [A4: int,B3: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ A4 @ B3 )
% 0.23/0.55 & ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_iff_not
% 0.23/0.55 thf(fact_8080_dense__ge__bounded,axiom,
% 0.23/0.55 ! [Z: real,X4: real,Y3: real] :
% 0.23/0.55 ( ( ord_less_real @ Z @ X4 )
% 0.23/0.55 => ( ! [W3: real] :
% 0.23/0.55 ( ( ord_less_real @ Z @ W3 )
% 0.23/0.55 => ( ( ord_less_real @ W3 @ X4 )
% 0.23/0.55 => ( ord_less_eq_real @ Y3 @ W3 ) ) )
% 0.23/0.55 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dense_ge_bounded
% 0.23/0.55 thf(fact_8081_dense__ge__bounded,axiom,
% 0.23/0.55 ! [Z: rat,X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ Z @ X4 )
% 0.23/0.55 => ( ! [W3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ Z @ W3 )
% 0.23/0.55 => ( ( ord_less_rat @ W3 @ X4 )
% 0.23/0.55 => ( ord_less_eq_rat @ Y3 @ W3 ) ) )
% 0.23/0.55 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dense_ge_bounded
% 0.23/0.55 thf(fact_8082_dense__le__bounded,axiom,
% 0.23/0.55 ! [X4: real,Y3: real,Z: real] :
% 0.23/0.55 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 => ( ! [W3: real] :
% 0.23/0.55 ( ( ord_less_real @ X4 @ W3 )
% 0.23/0.55 => ( ( ord_less_real @ W3 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_real @ W3 @ Z ) ) )
% 0.23/0.55 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dense_le_bounded
% 0.23/0.55 thf(fact_8083_dense__le__bounded,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat,Z: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 => ( ! [W3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X4 @ W3 )
% 0.23/0.55 => ( ( ord_less_rat @ W3 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_rat @ W3 @ Z ) ) )
% 0.23/0.55 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dense_le_bounded
% 0.23/0.55 thf(fact_8084_dual__order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_real
% 0.23/0.55 = ( ^ [B3: real,A4: real] :
% 0.23/0.55 ( ( ord_less_real @ B3 @ A4 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.order_iff_strict
% 0.23/0.55 thf(fact_8085_dual__order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_set_nat
% 0.23/0.55 = ( ^ [B3: set_nat,A4: set_nat] :
% 0.23/0.55 ( ( ord_less_set_nat @ B3 @ A4 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.order_iff_strict
% 0.23/0.55 thf(fact_8086_dual__order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_rat
% 0.23/0.55 = ( ^ [B3: rat,A4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ B3 @ A4 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.order_iff_strict
% 0.23/0.55 thf(fact_8087_dual__order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_num
% 0.23/0.55 = ( ^ [B3: num,A4: num] :
% 0.23/0.55 ( ( ord_less_num @ B3 @ A4 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.order_iff_strict
% 0.23/0.55 thf(fact_8088_dual__order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_nat
% 0.23/0.55 = ( ^ [B3: nat,A4: nat] :
% 0.23/0.55 ( ( ord_less_nat @ B3 @ A4 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.order_iff_strict
% 0.23/0.55 thf(fact_8089_dual__order_Oorder__iff__strict,axiom,
% 0.23/0.55 ( ord_less_eq_int
% 0.23/0.55 = ( ^ [B3: int,A4: int] :
% 0.23/0.55 ( ( ord_less_int @ B3 @ A4 )
% 0.23/0.55 | ( A4 = B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.order_iff_strict
% 0.23/0.55 thf(fact_8090_dual__order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_real
% 0.23/0.55 = ( ^ [B3: real,A4: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ B3 @ A4 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_order
% 0.23/0.55 thf(fact_8091_dual__order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_set_nat
% 0.23/0.55 = ( ^ [B3: set_nat,A4: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ B3 @ A4 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_order
% 0.23/0.55 thf(fact_8092_dual__order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_rat
% 0.23/0.55 = ( ^ [B3: rat,A4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ B3 @ A4 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_order
% 0.23/0.55 thf(fact_8093_dual__order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_num
% 0.23/0.55 = ( ^ [B3: num,A4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ B3 @ A4 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_order
% 0.23/0.55 thf(fact_8094_dual__order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_nat
% 0.23/0.55 = ( ^ [B3: nat,A4: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ B3 @ A4 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_order
% 0.23/0.55 thf(fact_8095_dual__order_Ostrict__iff__order,axiom,
% 0.23/0.55 ( ord_less_int
% 0.23/0.55 = ( ^ [B3: int,A4: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ B3 @ A4 )
% 0.23/0.55 & ( A4 != B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_order
% 0.23/0.55 thf(fact_8096_dual__order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [B2: real,A: real,C: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_real @ C @ B2 )
% 0.23/0.55 => ( ord_less_real @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans1
% 0.23/0.55 thf(fact_8097_dual__order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [B2: set_nat,A: set_nat,C: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_set_nat @ C @ B2 )
% 0.23/0.55 => ( ord_less_set_nat @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans1
% 0.23/0.55 thf(fact_8098_dual__order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [B2: rat,A: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_rat @ C @ B2 )
% 0.23/0.55 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans1
% 0.23/0.55 thf(fact_8099_dual__order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [B2: num,A: num,C: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_num @ C @ B2 )
% 0.23/0.55 => ( ord_less_num @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans1
% 0.23/0.55 thf(fact_8100_dual__order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [B2: nat,A: nat,C: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_nat @ C @ B2 )
% 0.23/0.55 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans1
% 0.23/0.55 thf(fact_8101_dual__order_Ostrict__trans1,axiom,
% 0.23/0.55 ! [B2: int,A: int,C: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_int @ C @ B2 )
% 0.23/0.55 => ( ord_less_int @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans1
% 0.23/0.55 thf(fact_8102_dual__order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [B2: real,A: real,C: real] :
% 0.23/0.55 ( ( ord_less_real @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_eq_real @ C @ B2 )
% 0.23/0.55 => ( ord_less_real @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans2
% 0.23/0.55 thf(fact_8103_dual__order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [B2: set_nat,A: set_nat,C: set_nat] :
% 0.23/0.55 ( ( ord_less_set_nat @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_eq_set_nat @ C @ B2 )
% 0.23/0.55 => ( ord_less_set_nat @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans2
% 0.23/0.55 thf(fact_8104_dual__order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [B2: rat,A: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_rat @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_eq_rat @ C @ B2 )
% 0.23/0.55 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans2
% 0.23/0.55 thf(fact_8105_dual__order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [B2: num,A: num,C: num] :
% 0.23/0.55 ( ( ord_less_num @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_eq_num @ C @ B2 )
% 0.23/0.55 => ( ord_less_num @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans2
% 0.23/0.55 thf(fact_8106_dual__order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [B2: nat,A: nat,C: nat] :
% 0.23/0.55 ( ( ord_less_nat @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_eq_nat @ C @ B2 )
% 0.23/0.55 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans2
% 0.23/0.55 thf(fact_8107_dual__order_Ostrict__trans2,axiom,
% 0.23/0.55 ! [B2: int,A: int,C: int] :
% 0.23/0.55 ( ( ord_less_int @ B2 @ A )
% 0.23/0.55 => ( ( ord_less_eq_int @ C @ B2 )
% 0.23/0.55 => ( ord_less_int @ C @ A ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_trans2
% 0.23/0.55 thf(fact_8108_dual__order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_real
% 0.23/0.55 = ( ^ [B3: real,A4: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ B3 @ A4 )
% 0.23/0.55 & ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_not
% 0.23/0.55 thf(fact_8109_dual__order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_set_nat
% 0.23/0.55 = ( ^ [B3: set_nat,A4: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ B3 @ A4 )
% 0.23/0.55 & ~ ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_not
% 0.23/0.55 thf(fact_8110_dual__order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_rat
% 0.23/0.55 = ( ^ [B3: rat,A4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ B3 @ A4 )
% 0.23/0.55 & ~ ( ord_less_eq_rat @ A4 @ B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_not
% 0.23/0.55 thf(fact_8111_dual__order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_num
% 0.23/0.55 = ( ^ [B3: num,A4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ B3 @ A4 )
% 0.23/0.55 & ~ ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_not
% 0.23/0.55 thf(fact_8112_dual__order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_nat
% 0.23/0.55 = ( ^ [B3: nat,A4: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ B3 @ A4 )
% 0.23/0.55 & ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_not
% 0.23/0.55 thf(fact_8113_dual__order_Ostrict__iff__not,axiom,
% 0.23/0.55 ( ord_less_int
% 0.23/0.55 = ( ^ [B3: int,A4: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ B3 @ A4 )
% 0.23/0.55 & ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_iff_not
% 0.23/0.55 thf(fact_8114_order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [A: real,B2: real] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_implies_order
% 0.23/0.55 thf(fact_8115_order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [A: set_nat,B2: set_nat] :
% 0.23/0.55 ( ( ord_less_set_nat @ A @ B2 )
% 0.23/0.55 => ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_implies_order
% 0.23/0.55 thf(fact_8116_order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [A: rat,B2: rat] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_implies_order
% 0.23/0.55 thf(fact_8117_order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [A: num,B2: num] :
% 0.23/0.55 ( ( ord_less_num @ A @ B2 )
% 0.23/0.55 => ( ord_less_eq_num @ A @ B2 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_implies_order
% 0.23/0.55 thf(fact_8118_order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [A: nat,B2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ A @ B2 )
% 0.23/0.55 => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_implies_order
% 0.23/0.55 thf(fact_8119_order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [A: int,B2: int] :
% 0.23/0.55 ( ( ord_less_int @ A @ B2 )
% 0.23/0.55 => ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order.strict_implies_order
% 0.23/0.55 thf(fact_8120_dual__order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [B2: real,A: real] :
% 0.23/0.55 ( ( ord_less_real @ B2 @ A )
% 0.23/0.55 => ( ord_less_eq_real @ B2 @ A ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_implies_order
% 0.23/0.55 thf(fact_8121_dual__order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [B2: set_nat,A: set_nat] :
% 0.23/0.55 ( ( ord_less_set_nat @ B2 @ A )
% 0.23/0.55 => ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_implies_order
% 0.23/0.55 thf(fact_8122_dual__order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [B2: rat,A: rat] :
% 0.23/0.55 ( ( ord_less_rat @ B2 @ A )
% 0.23/0.55 => ( ord_less_eq_rat @ B2 @ A ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_implies_order
% 0.23/0.55 thf(fact_8123_dual__order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [B2: num,A: num] :
% 0.23/0.55 ( ( ord_less_num @ B2 @ A )
% 0.23/0.55 => ( ord_less_eq_num @ B2 @ A ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_implies_order
% 0.23/0.55 thf(fact_8124_dual__order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [B2: nat,A: nat] :
% 0.23/0.55 ( ( ord_less_nat @ B2 @ A )
% 0.23/0.55 => ( ord_less_eq_nat @ B2 @ A ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_implies_order
% 0.23/0.55 thf(fact_8125_dual__order_Ostrict__implies__order,axiom,
% 0.23/0.55 ! [B2: int,A: int] :
% 0.23/0.55 ( ( ord_less_int @ B2 @ A )
% 0.23/0.55 => ( ord_less_eq_int @ B2 @ A ) ) ).
% 0.23/0.55
% 0.23/0.55 % dual_order.strict_implies_order
% 0.23/0.55 thf(fact_8126_order__le__less,axiom,
% 0.23/0.55 ( ord_less_eq_real
% 0.23/0.55 = ( ^ [X: real,Y: real] :
% 0.23/0.55 ( ( ord_less_real @ X @ Y )
% 0.23/0.55 | ( X = Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less
% 0.23/0.55 thf(fact_8127_order__le__less,axiom,
% 0.23/0.55 ( ord_less_eq_set_nat
% 0.23/0.55 = ( ^ [X: set_nat,Y: set_nat] :
% 0.23/0.55 ( ( ord_less_set_nat @ X @ Y )
% 0.23/0.55 | ( X = Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less
% 0.23/0.55 thf(fact_8128_order__le__less,axiom,
% 0.23/0.55 ( ord_less_eq_rat
% 0.23/0.55 = ( ^ [X: rat,Y: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X @ Y )
% 0.23/0.55 | ( X = Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less
% 0.23/0.55 thf(fact_8129_order__le__less,axiom,
% 0.23/0.55 ( ord_less_eq_num
% 0.23/0.55 = ( ^ [X: num,Y: num] :
% 0.23/0.55 ( ( ord_less_num @ X @ Y )
% 0.23/0.55 | ( X = Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less
% 0.23/0.55 thf(fact_8130_order__le__less,axiom,
% 0.23/0.55 ( ord_less_eq_nat
% 0.23/0.55 = ( ^ [X: nat,Y: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X @ Y )
% 0.23/0.55 | ( X = Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less
% 0.23/0.55 thf(fact_8131_order__le__less,axiom,
% 0.23/0.55 ( ord_less_eq_int
% 0.23/0.55 = ( ^ [X: int,Y: int] :
% 0.23/0.55 ( ( ord_less_int @ X @ Y )
% 0.23/0.55 | ( X = Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less
% 0.23/0.55 thf(fact_8132_order__less__le,axiom,
% 0.23/0.55 ( ord_less_real
% 0.23/0.55 = ( ^ [X: real,Y: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ X @ Y )
% 0.23/0.55 & ( X != Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le
% 0.23/0.55 thf(fact_8133_order__less__le,axiom,
% 0.23/0.55 ( ord_less_set_nat
% 0.23/0.55 = ( ^ [X: set_nat,Y: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.23/0.55 & ( X != Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le
% 0.23/0.55 thf(fact_8134_order__less__le,axiom,
% 0.23/0.55 ( ord_less_rat
% 0.23/0.55 = ( ^ [X: rat,Y: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X @ Y )
% 0.23/0.55 & ( X != Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le
% 0.23/0.55 thf(fact_8135_order__less__le,axiom,
% 0.23/0.55 ( ord_less_num
% 0.23/0.55 = ( ^ [X: num,Y: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X @ Y )
% 0.23/0.55 & ( X != Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le
% 0.23/0.55 thf(fact_8136_order__less__le,axiom,
% 0.23/0.55 ( ord_less_nat
% 0.23/0.55 = ( ^ [X: nat,Y: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ X @ Y )
% 0.23/0.55 & ( X != Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le
% 0.23/0.55 thf(fact_8137_order__less__le,axiom,
% 0.23/0.55 ( ord_less_int
% 0.23/0.55 = ( ^ [X: int,Y: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ X @ Y )
% 0.23/0.55 & ( X != Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le
% 0.23/0.55 thf(fact_8138_linorder__not__le,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ( ~ ( ord_less_eq_real @ X4 @ Y3 ) )
% 0.23/0.55 = ( ord_less_real @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_not_le
% 0.23/0.55 thf(fact_8139_linorder__not__le,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ~ ( ord_less_eq_rat @ X4 @ Y3 ) )
% 0.23/0.55 = ( ord_less_rat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_not_le
% 0.23/0.55 thf(fact_8140_linorder__not__le,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ( ~ ( ord_less_eq_num @ X4 @ Y3 ) )
% 0.23/0.55 = ( ord_less_num @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_not_le
% 0.23/0.55 thf(fact_8141_linorder__not__le,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ( ~ ( ord_less_eq_nat @ X4 @ Y3 ) )
% 0.23/0.55 = ( ord_less_nat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_not_le
% 0.23/0.55 thf(fact_8142_linorder__not__le,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ( ~ ( ord_less_eq_int @ X4 @ Y3 ) )
% 0.23/0.55 = ( ord_less_int @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_not_le
% 0.23/0.55 thf(fact_8143_linorder__not__less,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
% 0.23/0.55 = ( ord_less_eq_real @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_not_less
% 0.23/0.55 thf(fact_8144_linorder__not__less,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
% 0.23/0.55 = ( ord_less_eq_rat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_not_less
% 0.23/0.55 thf(fact_8145_linorder__not__less,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
% 0.23/0.55 = ( ord_less_eq_num @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_not_less
% 0.23/0.55 thf(fact_8146_linorder__not__less,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 0.23/0.55 = ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_not_less
% 0.23/0.55 thf(fact_8147_linorder__not__less,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
% 0.23/0.55 = ( ord_less_eq_int @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_not_less
% 0.23/0.55 thf(fact_8148_order__less__imp__le,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_le
% 0.23/0.55 thf(fact_8149_order__less__imp__le,axiom,
% 0.23/0.55 ! [X4: set_nat,Y3: set_nat] :
% 0.23/0.55 ( ( ord_less_set_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_le
% 0.23/0.55 thf(fact_8150_order__less__imp__le,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_le
% 0.23/0.55 thf(fact_8151_order__less__imp__le,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_num @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_le
% 0.23/0.55 thf(fact_8152_order__less__imp__le,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_le
% 0.23/0.55 thf(fact_8153_order__less__imp__le,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.55 => ( ord_less_eq_int @ X4 @ Y3 ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_imp_le
% 0.23/0.55 thf(fact_8154_order__le__neq__trans,axiom,
% 0.23/0.55 ! [A: real,B2: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.55 => ( ( A != B2 )
% 0.23/0.55 => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_neq_trans
% 0.23/0.55 thf(fact_8155_order__le__neq__trans,axiom,
% 0.23/0.55 ! [A: set_nat,B2: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ A @ B2 )
% 0.23/0.55 => ( ( A != B2 )
% 0.23/0.55 => ( ord_less_set_nat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_neq_trans
% 0.23/0.55 thf(fact_8156_order__le__neq__trans,axiom,
% 0.23/0.55 ! [A: rat,B2: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.55 => ( ( A != B2 )
% 0.23/0.55 => ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_neq_trans
% 0.23/0.55 thf(fact_8157_order__le__neq__trans,axiom,
% 0.23/0.55 ! [A: num,B2: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.55 => ( ( A != B2 )
% 0.23/0.55 => ( ord_less_num @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_neq_trans
% 0.23/0.55 thf(fact_8158_order__le__neq__trans,axiom,
% 0.23/0.55 ! [A: nat,B2: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.55 => ( ( A != B2 )
% 0.23/0.55 => ( ord_less_nat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_neq_trans
% 0.23/0.55 thf(fact_8159_order__le__neq__trans,axiom,
% 0.23/0.55 ! [A: int,B2: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ A @ B2 )
% 0.23/0.55 => ( ( A != B2 )
% 0.23/0.55 => ( ord_less_int @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_neq_trans
% 0.23/0.55 thf(fact_8160_order__neq__le__trans,axiom,
% 0.23/0.55 ! [A: real,B2: real] :
% 0.23/0.55 ( ( A != B2 )
% 0.23/0.55 => ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.55 => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_neq_le_trans
% 0.23/0.55 thf(fact_8161_order__neq__le__trans,axiom,
% 0.23/0.55 ! [A: set_nat,B2: set_nat] :
% 0.23/0.55 ( ( A != B2 )
% 0.23/0.55 => ( ( ord_less_eq_set_nat @ A @ B2 )
% 0.23/0.55 => ( ord_less_set_nat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_neq_le_trans
% 0.23/0.55 thf(fact_8162_order__neq__le__trans,axiom,
% 0.23/0.55 ! [A: rat,B2: rat] :
% 0.23/0.55 ( ( A != B2 )
% 0.23/0.55 => ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.55 => ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_neq_le_trans
% 0.23/0.55 thf(fact_8163_order__neq__le__trans,axiom,
% 0.23/0.55 ! [A: num,B2: num] :
% 0.23/0.55 ( ( A != B2 )
% 0.23/0.55 => ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.55 => ( ord_less_num @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_neq_le_trans
% 0.23/0.55 thf(fact_8164_order__neq__le__trans,axiom,
% 0.23/0.55 ! [A: nat,B2: nat] :
% 0.23/0.55 ( ( A != B2 )
% 0.23/0.55 => ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.55 => ( ord_less_nat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_neq_le_trans
% 0.23/0.55 thf(fact_8165_order__neq__le__trans,axiom,
% 0.23/0.55 ! [A: int,B2: int] :
% 0.23/0.55 ( ( A != B2 )
% 0.23/0.55 => ( ( ord_less_eq_int @ A @ B2 )
% 0.23/0.55 => ( ord_less_int @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_neq_le_trans
% 0.23/0.55 thf(fact_8166_order__le__less__trans,axiom,
% 0.23/0.55 ! [X4: real,Y3: real,Z: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_real @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_real @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_trans
% 0.23/0.55 thf(fact_8167_order__le__less__trans,axiom,
% 0.23/0.55 ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_set_nat @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_set_nat @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_trans
% 0.23/0.55 thf(fact_8168_order__le__less__trans,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat,Z: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_rat @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_rat @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_trans
% 0.23/0.55 thf(fact_8169_order__le__less__trans,axiom,
% 0.23/0.55 ! [X4: num,Y3: num,Z: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_num @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_num @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_trans
% 0.23/0.55 thf(fact_8170_order__le__less__trans,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat,Z: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_nat @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_nat @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_trans
% 0.23/0.55 thf(fact_8171_order__le__less__trans,axiom,
% 0.23/0.55 ! [X4: int,Y3: int,Z: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_int @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_int @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_trans
% 0.23/0.55 thf(fact_8172_order__less__le__trans,axiom,
% 0.23/0.55 ! [X4: real,Y3: real,Z: real] :
% 0.23/0.55 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_real @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_real @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_trans
% 0.23/0.55 thf(fact_8173_order__less__le__trans,axiom,
% 0.23/0.55 ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
% 0.23/0.55 ( ( ord_less_set_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_set_nat @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_set_nat @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_trans
% 0.23/0.55 thf(fact_8174_order__less__le__trans,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat,Z: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_rat @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_trans
% 0.23/0.55 thf(fact_8175_order__less__le__trans,axiom,
% 0.23/0.55 ! [X4: num,Y3: num,Z: num] :
% 0.23/0.55 ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_num @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_num @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_trans
% 0.23/0.55 thf(fact_8176_order__less__le__trans,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat,Z: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_nat @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_trans
% 0.23/0.55 thf(fact_8177_order__less__le__trans,axiom,
% 0.23/0.55 ! [X4: int,Y3: int,Z: int] :
% 0.23/0.55 ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.23/0.55 => ( ord_less_int @ X4 @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_trans
% 0.23/0.55 thf(fact_8178_order__le__less__subst1,axiom,
% 0.23/0.55 ! [A: real,F: real > real,B2: real,C: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst1
% 0.23/0.55 thf(fact_8179_order__le__less__subst1,axiom,
% 0.23/0.55 ! [A: real,F: rat > real,B2: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst1
% 0.23/0.55 thf(fact_8180_order__le__less__subst1,axiom,
% 0.23/0.55 ! [A: real,F: num > real,B2: num,C: num] :
% 0.23/0.55 ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_num @ B2 @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst1
% 0.23/0.55 thf(fact_8181_order__le__less__subst1,axiom,
% 0.23/0.55 ! [A: real,F: nat > real,B2: nat,C: nat] :
% 0.23/0.55 ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_nat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst1
% 0.23/0.55 thf(fact_8182_order__le__less__subst1,axiom,
% 0.23/0.55 ! [A: real,F: int > real,B2: int,C: int] :
% 0.23/0.55 ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_int @ B2 @ C )
% 0.23/0.55 => ( ! [X3: int,Y4: int] :
% 0.23/0.55 ( ( ord_less_int @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst1
% 0.23/0.55 thf(fact_8183_order__le__less__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: real > rat,B2: real,C: real] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_real @ B2 @ C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst1
% 0.23/0.55 thf(fact_8184_order__le__less__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_rat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst1
% 0.23/0.55 thf(fact_8185_order__le__less__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: num > rat,B2: num,C: num] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_num @ B2 @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst1
% 0.23/0.55 thf(fact_8186_order__le__less__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_nat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst1
% 0.23/0.55 thf(fact_8187_order__le__less__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: int > rat,B2: int,C: int] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_int @ B2 @ C )
% 0.23/0.55 => ( ! [X3: int,Y4: int] :
% 0.23/0.55 ( ( ord_less_int @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst1
% 0.23/0.55 thf(fact_8188_order__le__less__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > real,C: real] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst2
% 0.23/0.55 thf(fact_8189_order__le__less__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst2
% 0.23/0.55 thf(fact_8190_order__le__less__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > num,C: num] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst2
% 0.23/0.55 thf(fact_8191_order__le__less__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst2
% 0.23/0.55 thf(fact_8192_order__le__less__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > int,C: int] :
% 0.23/0.55 ( ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst2
% 0.23/0.55 thf(fact_8193_order__le__less__subst2,axiom,
% 0.23/0.55 ! [A: num,B2: num,F: num > real,C: real] :
% 0.23/0.55 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst2
% 0.23/0.55 thf(fact_8194_order__le__less__subst2,axiom,
% 0.23/0.55 ! [A: num,B2: num,F: num > rat,C: rat] :
% 0.23/0.55 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst2
% 0.23/0.55 thf(fact_8195_order__le__less__subst2,axiom,
% 0.23/0.55 ! [A: num,B2: num,F: num > num,C: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst2
% 0.23/0.55 thf(fact_8196_order__le__less__subst2,axiom,
% 0.23/0.55 ! [A: num,B2: num,F: num > nat,C: nat] :
% 0.23/0.55 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst2
% 0.23/0.55 thf(fact_8197_order__le__less__subst2,axiom,
% 0.23/0.55 ! [A: num,B2: num,F: num > int,C: int] :
% 0.23/0.55 ( ( ord_less_eq_num @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_less_subst2
% 0.23/0.55 thf(fact_8198_order__less__le__subst1,axiom,
% 0.23/0.55 ! [A: real,F: rat > real,B2: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst1
% 0.23/0.55 thf(fact_8199_order__less__le__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst1
% 0.23/0.55 thf(fact_8200_order__less__le__subst1,axiom,
% 0.23/0.55 ! [A: num,F: rat > num,B2: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_num @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst1
% 0.23/0.55 thf(fact_8201_order__less__le__subst1,axiom,
% 0.23/0.55 ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_nat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst1
% 0.23/0.55 thf(fact_8202_order__less__le__subst1,axiom,
% 0.23/0.55 ! [A: int,F: rat > int,B2: rat,C: rat] :
% 0.23/0.55 ( ( ord_less_int @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_eq_rat @ B2 @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst1
% 0.23/0.55 thf(fact_8203_order__less__le__subst1,axiom,
% 0.23/0.55 ! [A: real,F: num > real,B2: num,C: num] :
% 0.23/0.55 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst1
% 0.23/0.55 thf(fact_8204_order__less__le__subst1,axiom,
% 0.23/0.55 ! [A: rat,F: num > rat,B2: num,C: num] :
% 0.23/0.55 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst1
% 0.23/0.55 thf(fact_8205_order__less__le__subst1,axiom,
% 0.23/0.55 ! [A: num,F: num > num,B2: num,C: num] :
% 0.23/0.55 ( ( ord_less_num @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst1
% 0.23/0.55 thf(fact_8206_order__less__le__subst1,axiom,
% 0.23/0.55 ! [A: nat,F: num > nat,B2: num,C: num] :
% 0.23/0.55 ( ( ord_less_nat @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst1
% 0.23/0.55 thf(fact_8207_order__less__le__subst1,axiom,
% 0.23/0.55 ! [A: int,F: num > int,B2: num,C: num] :
% 0.23/0.55 ( ( ord_less_int @ A @ ( F @ B2 ) )
% 0.23/0.55 => ( ( ord_less_eq_num @ B2 @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst1
% 0.23/0.55 thf(fact_8208_order__less__le__subst2,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > real,C: real] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst2
% 0.23/0.55 thf(fact_8209_order__less__le__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > real,C: real] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst2
% 0.23/0.55 thf(fact_8210_order__less__le__subst2,axiom,
% 0.23/0.55 ! [A: num,B2: num,F: num > real,C: real] :
% 0.23/0.55 ( ( ord_less_num @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst2
% 0.23/0.55 thf(fact_8211_order__less__le__subst2,axiom,
% 0.23/0.55 ! [A: nat,B2: nat,F: nat > real,C: real] :
% 0.23/0.55 ( ( ord_less_nat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst2
% 0.23/0.55 thf(fact_8212_order__less__le__subst2,axiom,
% 0.23/0.55 ! [A: int,B2: int,F: int > real,C: real] :
% 0.23/0.55 ( ( ord_less_int @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: int,Y4: int] :
% 0.23/0.55 ( ( ord_less_int @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst2
% 0.23/0.55 thf(fact_8213_order__less__le__subst2,axiom,
% 0.23/0.55 ! [A: real,B2: real,F: real > rat,C: rat] :
% 0.23/0.55 ( ( ord_less_real @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: real,Y4: real] :
% 0.23/0.55 ( ( ord_less_real @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst2
% 0.23/0.55 thf(fact_8214_order__less__le__subst2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.23/0.55 ( ( ord_less_rat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: rat,Y4: rat] :
% 0.23/0.55 ( ( ord_less_rat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst2
% 0.23/0.55 thf(fact_8215_order__less__le__subst2,axiom,
% 0.23/0.55 ! [A: num,B2: num,F: num > rat,C: rat] :
% 0.23/0.55 ( ( ord_less_num @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: num,Y4: num] :
% 0.23/0.55 ( ( ord_less_num @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst2
% 0.23/0.55 thf(fact_8216_order__less__le__subst2,axiom,
% 0.23/0.55 ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 0.23/0.55 ( ( ord_less_nat @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: nat,Y4: nat] :
% 0.23/0.55 ( ( ord_less_nat @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst2
% 0.23/0.55 thf(fact_8217_order__less__le__subst2,axiom,
% 0.23/0.55 ! [A: int,B2: int,F: int > rat,C: rat] :
% 0.23/0.55 ( ( ord_less_int @ A @ B2 )
% 0.23/0.55 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.23/0.55 => ( ! [X3: int,Y4: int] :
% 0.23/0.55 ( ( ord_less_int @ X3 @ Y4 )
% 0.23/0.55 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.23/0.55 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_less_le_subst2
% 0.23/0.55 thf(fact_8218_linorder__le__less__linear,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.55 | ( ord_less_real @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_le_less_linear
% 0.23/0.55 thf(fact_8219_linorder__le__less__linear,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.55 | ( ord_less_rat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_le_less_linear
% 0.23/0.55 thf(fact_8220_linorder__le__less__linear,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.23/0.55 | ( ord_less_num @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_le_less_linear
% 0.23/0.55 thf(fact_8221_linorder__le__less__linear,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.23/0.55 | ( ord_less_nat @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_le_less_linear
% 0.23/0.55 thf(fact_8222_linorder__le__less__linear,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.55 | ( ord_less_int @ Y3 @ X4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % linorder_le_less_linear
% 0.23/0.55 thf(fact_8223_order__le__imp__less__or__eq,axiom,
% 0.23/0.55 ! [X4: real,Y3: real] :
% 0.23/0.55 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.55 | ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_imp_less_or_eq
% 0.23/0.55 thf(fact_8224_order__le__imp__less__or__eq,axiom,
% 0.23/0.55 ! [X4: set_nat,Y3: set_nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_set_nat @ X4 @ Y3 )
% 0.23/0.55 | ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_imp_less_or_eq
% 0.23/0.55 thf(fact_8225_order__le__imp__less__or__eq,axiom,
% 0.23/0.55 ! [X4: rat,Y3: rat] :
% 0.23/0.55 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_rat @ X4 @ Y3 )
% 0.23/0.55 | ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_imp_less_or_eq
% 0.23/0.55 thf(fact_8226_order__le__imp__less__or__eq,axiom,
% 0.23/0.55 ! [X4: num,Y3: num] :
% 0.23/0.55 ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_num @ X4 @ Y3 )
% 0.23/0.55 | ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_imp_less_or_eq
% 0.23/0.55 thf(fact_8227_order__le__imp__less__or__eq,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 | ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_imp_less_or_eq
% 0.23/0.55 thf(fact_8228_order__le__imp__less__or__eq,axiom,
% 0.23/0.55 ! [X4: int,Y3: int] :
% 0.23/0.55 ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.23/0.55 => ( ( ord_less_int @ X4 @ Y3 )
% 0.23/0.55 | ( X4 = Y3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % order_le_imp_less_or_eq
% 0.23/0.55 thf(fact_8229_and__not__numerals_I8_J,axiom,
% 0.23/0.55 ! [M: num,N: num] :
% 0.23/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.23/0.55 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % and_not_numerals(8)
% 0.23/0.55 thf(fact_8230_not__int__rec,axiom,
% 0.23/0.55 ( bit_ri7919022796975470100ot_int
% 0.23/0.55 = ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % not_int_rec
% 0.23/0.55 thf(fact_8231__C5_Ohyps_C_I11_J,axiom,
% 0.23/0.55 ( ( mi != ma )
% 0.23/0.55 => ! [I4: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.23/0.55 => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 0.23/0.55 = I4 )
% 0.23/0.55 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 0.23/0.55 & ! [X5: nat] :
% 0.23/0.55 ( ( ( ( vEBT_VEBT_high @ X5 @ na )
% 0.23/0.55 = I4 )
% 0.23/0.55 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
% 0.23/0.55 => ( ( ord_less_nat @ mi @ X5 )
% 0.23/0.55 & ( ord_less_eq_nat @ X5 @ ma ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % "5.hyps"(11)
% 0.23/0.55 thf(fact_8232__C20_C,axiom,
% 0.23/0.55 ( ( za = mi )
% 0.23/0.55 | ( za = ma )
% 0.23/0.55 | ( ( ord_less_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
% 0.23/0.55 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % "20"
% 0.23/0.55 thf(fact_8233_abd,axiom,
% 0.23/0.55 ( ( za = ma )
% 0.23/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % abd
% 0.23/0.55 thf(fact_8234_aaa,axiom,
% 0.23/0.55 ( ( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.55 = ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % aaa
% 0.23/0.55 thf(fact_8235__C33_C,axiom,
% 0.23/0.55 ~ ? [U2: nat] :
% 0.23/0.55 ( ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U2 )
% 0.23/0.55 & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ U2 ) ) ).
% 0.23/0.55
% 0.23/0.55 % "33"
% 0.23/0.55 thf(fact_8236__092_060open_062_092_060exists_062miny_O_Aboth__member__options_A_ItreeList_A_B_Asc_J_Aminy_092_060close_062,axiom,
% 0.23/0.55 ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ sc ) @ X_1 ) ).
% 0.23/0.55
% 0.23/0.55 % \<open>\<exists>miny. both_member_options (treeList ! sc) miny\<close>
% 0.23/0.55 thf(fact_8237_abf,axiom,
% 0.23/0.55 vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % abf
% 0.23/0.55 thf(fact_8238_abe,axiom,
% 0.23/0.55 vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % abe
% 0.23/0.55 thf(fact_8239_aa,axiom,
% 0.23/0.55 ( ( za = ma )
% 0.23/0.55 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % aa
% 0.23/0.55 thf(fact_8240_nth__equalityI,axiom,
% 0.23/0.55 ! [Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 0.23/0.55 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.23/0.55 = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 0.23/0.55 => ( ! [I3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.23/0.55 => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
% 0.23/0.55 = ( nth_VEBT_VEBT @ Ys2 @ I3 ) ) )
% 0.23/0.55 => ( Xs2 = Ys2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nth_equalityI
% 0.23/0.55 thf(fact_8241_nth__equalityI,axiom,
% 0.23/0.55 ! [Xs2: list_o,Ys2: list_o] :
% 0.23/0.55 ( ( ( size_size_list_o @ Xs2 )
% 0.23/0.55 = ( size_size_list_o @ Ys2 ) )
% 0.23/0.55 => ( ! [I3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 0.23/0.55 => ( ( nth_o @ Xs2 @ I3 )
% 0.23/0.55 = ( nth_o @ Ys2 @ I3 ) ) )
% 0.23/0.55 => ( Xs2 = Ys2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nth_equalityI
% 0.23/0.55 thf(fact_8242_nth__equalityI,axiom,
% 0.23/0.55 ! [Xs2: list_nat,Ys2: list_nat] :
% 0.23/0.55 ( ( ( size_size_list_nat @ Xs2 )
% 0.23/0.55 = ( size_size_list_nat @ Ys2 ) )
% 0.23/0.55 => ( ! [I3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 0.23/0.55 => ( ( nth_nat @ Xs2 @ I3 )
% 0.23/0.55 = ( nth_nat @ Ys2 @ I3 ) ) )
% 0.23/0.55 => ( Xs2 = Ys2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nth_equalityI
% 0.23/0.55 thf(fact_8243_nth__equalityI,axiom,
% 0.23/0.55 ! [Xs2: list_int,Ys2: list_int] :
% 0.23/0.55 ( ( ( size_size_list_int @ Xs2 )
% 0.23/0.55 = ( size_size_list_int @ Ys2 ) )
% 0.23/0.55 => ( ! [I3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 0.23/0.55 => ( ( nth_int @ Xs2 @ I3 )
% 0.23/0.55 = ( nth_int @ Ys2 @ I3 ) ) )
% 0.23/0.55 => ( Xs2 = Ys2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nth_equalityI
% 0.23/0.55 thf(fact_8244_Skolem__list__nth,axiom,
% 0.23/0.55 ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 0.23/0.55 ( ( ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ K )
% 0.23/0.55 => ? [X6: vEBT_VEBT] : ( P @ I2 @ X6 ) ) )
% 0.23/0.55 = ( ? [Xs: list_VEBT_VEBT] :
% 0.23/0.55 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.23/0.55 = K )
% 0.23/0.55 & ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ K )
% 0.23/0.55 => ( P @ I2 @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Skolem_list_nth
% 0.23/0.55 thf(fact_8245_Skolem__list__nth,axiom,
% 0.23/0.55 ! [K: nat,P: nat > $o > $o] :
% 0.23/0.55 ( ( ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ K )
% 0.23/0.55 => ? [X6: $o] : ( P @ I2 @ X6 ) ) )
% 0.23/0.55 = ( ? [Xs: list_o] :
% 0.23/0.55 ( ( ( size_size_list_o @ Xs )
% 0.23/0.55 = K )
% 0.23/0.55 & ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ K )
% 0.23/0.55 => ( P @ I2 @ ( nth_o @ Xs @ I2 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Skolem_list_nth
% 0.23/0.55 thf(fact_8246_Skolem__list__nth,axiom,
% 0.23/0.55 ! [K: nat,P: nat > nat > $o] :
% 0.23/0.55 ( ( ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ K )
% 0.23/0.55 => ? [X6: nat] : ( P @ I2 @ X6 ) ) )
% 0.23/0.55 = ( ? [Xs: list_nat] :
% 0.23/0.55 ( ( ( size_size_list_nat @ Xs )
% 0.23/0.55 = K )
% 0.23/0.55 & ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ K )
% 0.23/0.55 => ( P @ I2 @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Skolem_list_nth
% 0.23/0.55 thf(fact_8247_Skolem__list__nth,axiom,
% 0.23/0.55 ! [K: nat,P: nat > int > $o] :
% 0.23/0.55 ( ( ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ K )
% 0.23/0.55 => ? [X6: int] : ( P @ I2 @ X6 ) ) )
% 0.23/0.55 = ( ? [Xs: list_int] :
% 0.23/0.55 ( ( ( size_size_list_int @ Xs )
% 0.23/0.55 = K )
% 0.23/0.55 & ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ K )
% 0.23/0.55 => ( P @ I2 @ ( nth_int @ Xs @ I2 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Skolem_list_nth
% 0.23/0.55 thf(fact_8248_list__eq__iff__nth__eq,axiom,
% 0.23/0.55 ( ( ^ [Y6: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : ( Y6 = Z4 ) )
% 0.23/0.55 = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.23/0.55 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.23/0.55 = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 0.23/0.55 & ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.23/0.55 => ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 0.23/0.55 = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % list_eq_iff_nth_eq
% 0.23/0.55 thf(fact_8249_list__eq__iff__nth__eq,axiom,
% 0.23/0.55 ( ( ^ [Y6: list_o,Z4: list_o] : ( Y6 = Z4 ) )
% 0.23/0.55 = ( ^ [Xs: list_o,Ys3: list_o] :
% 0.23/0.55 ( ( ( size_size_list_o @ Xs )
% 0.23/0.55 = ( size_size_list_o @ Ys3 ) )
% 0.23/0.55 & ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.23/0.55 => ( ( nth_o @ Xs @ I2 )
% 0.23/0.55 = ( nth_o @ Ys3 @ I2 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % list_eq_iff_nth_eq
% 0.23/0.55 thf(fact_8250_list__eq__iff__nth__eq,axiom,
% 0.23/0.55 ( ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) )
% 0.23/0.55 = ( ^ [Xs: list_nat,Ys3: list_nat] :
% 0.23/0.55 ( ( ( size_size_list_nat @ Xs )
% 0.23/0.55 = ( size_size_list_nat @ Ys3 ) )
% 0.23/0.55 & ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.23/0.55 => ( ( nth_nat @ Xs @ I2 )
% 0.23/0.55 = ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % list_eq_iff_nth_eq
% 0.23/0.55 thf(fact_8251_list__eq__iff__nth__eq,axiom,
% 0.23/0.55 ( ( ^ [Y6: list_int,Z4: list_int] : ( Y6 = Z4 ) )
% 0.23/0.55 = ( ^ [Xs: list_int,Ys3: list_int] :
% 0.23/0.55 ( ( ( size_size_list_int @ Xs )
% 0.23/0.55 = ( size_size_list_int @ Ys3 ) )
% 0.23/0.55 & ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.23/0.55 => ( ( nth_int @ Xs @ I2 )
% 0.23/0.55 = ( nth_int @ Ys3 @ I2 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % list_eq_iff_nth_eq
% 0.23/0.55 thf(fact_8252_num_Osize_I4_J,axiom,
% 0.23/0.55 ( ( size_size_num @ one )
% 0.23/0.55 = zero_zero_nat ) ).
% 0.23/0.55
% 0.23/0.55 % num.size(4)
% 0.23/0.55 thf(fact_8253_num_Osize_I5_J,axiom,
% 0.23/0.55 ! [X2: num] :
% 0.23/0.55 ( ( size_size_num @ ( bit0 @ X2 ) )
% 0.23/0.55 = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % num.size(5)
% 0.23/0.55 thf(fact_8254_num_Osize_I6_J,axiom,
% 0.23/0.55 ! [X32: num] :
% 0.23/0.55 ( ( size_size_num @ ( bit1 @ X32 ) )
% 0.23/0.55 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % num.size(6)
% 0.23/0.55 thf(fact_8255_in__children__def,axiom,
% 0.23/0.55 ( vEBT_V5917875025757280293ildren
% 0.23/0.55 = ( ^ [N2: nat,TreeList: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N2 ) ) @ ( vEBT_VEBT_low @ X @ N2 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % in_children_def
% 0.23/0.55 thf(fact_8256_abc,axiom,
% 0.23/0.55 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).
% 0.23/0.55
% 0.23/0.55 % abc
% 0.23/0.55 thf(fact_8257__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
% 0.23/0.55 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).
% 0.23/0.55
% 0.23/0.55 % \<open>invar_vebt (treeList ! high x (deg div 2)) n\<close>
% 0.23/0.55 thf(fact_8258_abh,axiom,
% 0.23/0.55 vEBT_vebt_member @ summary @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % abh
% 0.23/0.55 thf(fact_8259_valid__0__not,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT] :
% 0.23/0.55 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.23/0.55
% 0.23/0.55 % valid_0_not
% 0.23/0.55 thf(fact_8260_valid__tree__deg__neq__0,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT] :
% 0.23/0.55 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.23/0.55
% 0.23/0.55 % valid_tree_deg_neq_0
% 0.23/0.55 thf(fact_8261__C5_Ohyps_C_I2_J,axiom,
% 0.23/0.55 vEBT_invar_vebt @ summary @ m ).
% 0.23/0.55
% 0.23/0.55 % "5.hyps"(2)
% 0.23/0.55 thf(fact_8262_deg__not__0,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT,N: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.55 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.23/0.55
% 0.23/0.55 % deg_not_0
% 0.23/0.55 thf(fact_8263_both__member__options__equiv__member,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.55 => ( ( vEBT_V8194947554948674370ptions @ T @ X4 )
% 0.23/0.55 = ( vEBT_vebt_member @ T @ X4 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % both_member_options_equiv_member
% 0.23/0.55 thf(fact_8264_valid__member__both__member__options,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.55 => ( ( vEBT_V8194947554948674370ptions @ T @ X4 )
% 0.23/0.55 => ( vEBT_vebt_member @ T @ X4 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % valid_member_both_member_options
% 0.23/0.55 thf(fact_8265__092_060open_062both__member__options_Asummary_Asc_092_060close_062,axiom,
% 0.23/0.55 vEBT_V8194947554948674370ptions @ summary @ sc ).
% 0.23/0.55
% 0.23/0.55 % \<open>both_member_options summary sc\<close>
% 0.23/0.55 thf(fact_8266__092_060open_062vebt__member_Asummary_Asc_092_060close_062,axiom,
% 0.23/0.55 vEBT_vebt_member @ summary @ sc ).
% 0.23/0.55
% 0.23/0.55 % \<open>vebt_member summary sc\<close>
% 0.23/0.55 thf(fact_8267_member__correct,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.55 => ( ( vEBT_vebt_member @ T @ X4 )
% 0.23/0.55 = ( member_nat @ X4 @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % member_correct
% 0.23/0.55 thf(fact_8268_member__bound,axiom,
% 0.23/0.55 ! [Tree: vEBT_VEBT,X4: nat,N: nat] :
% 0.23/0.55 ( ( vEBT_vebt_member @ Tree @ X4 )
% 0.23/0.55 => ( ( vEBT_invar_vebt @ Tree @ N )
% 0.23/0.55 => ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % member_bound
% 0.23/0.55 thf(fact_8269_abg,axiom,
% 0.23/0.55 vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % abg
% 0.23/0.55 thf(fact_8270__092_060open_062length_AtreeList_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_Am_092_060close_062,axiom,
% 0.23/0.55 ( ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.23/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.23/0.55 & ( vEBT_invar_vebt @ summary @ m ) ) ).
% 0.23/0.55
% 0.23/0.55 % \<open>length treeList = 2 ^ m \<and> invar_vebt summary m\<close>
% 0.23/0.55 thf(fact_8271__C5_Ohyps_C_I7_J,axiom,
% 0.23/0.55 ! [I4: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.23/0.55 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ X6 ) )
% 0.23/0.55 = ( vEBT_V8194947554948674370ptions @ summary @ I4 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % "5.hyps"(7)
% 0.23/0.55 thf(fact_8272_post__member__pre__member,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT,N: nat,X4: nat,Y3: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.55 => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.55 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.55 => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X4 ) @ Y3 )
% 0.23/0.55 => ( ( vEBT_vebt_member @ T @ Y3 )
% 0.23/0.55 | ( X4 = Y3 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % post_member_pre_member
% 0.23/0.55 thf(fact_8273_valid__insert__both__member__options__pres,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT,N: nat,X4: nat,Y3: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.55 => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.55 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.55 => ( ( vEBT_V8194947554948674370ptions @ T @ X4 )
% 0.23/0.55 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X4 ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % valid_insert_both_member_options_pres
% 0.23/0.55 thf(fact_8274_valid__insert__both__member__options__add,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.55 => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.55 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X4 ) @ X4 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % valid_insert_both_member_options_add
% 0.23/0.55 thf(fact_8275_both__member__options__ding,axiom,
% 0.23/0.55 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X4: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 0.23/0.55 => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.23/0.55 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.55 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % both_member_options_ding
% 0.23/0.55 thf(fact_8276_deg__deg__n,axiom,
% 0.23/0.55 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 0.23/0.55 => ( Deg = N ) ) ).
% 0.23/0.55
% 0.23/0.55 % deg_deg_n
% 0.23/0.55 thf(fact_8277_deg__SUcn__Node,axiom,
% 0.23/0.55 ! [Tree: vEBT_VEBT,N: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 0.23/0.55 => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.23/0.55 ( Tree
% 0.23/0.55 = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S3 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % deg_SUcn_Node
% 0.23/0.55 thf(fact_8278__C5_Ohyps_C_I8_J,axiom,
% 0.23/0.55 ( ( mi = ma )
% 0.23/0.55 => ! [X5: vEBT_VEBT] :
% 0.23/0.55 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.23/0.55 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % "5.hyps"(8)
% 0.23/0.55 thf(fact_8279_set__vebt__set__vebt_H__valid,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT,N: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.55 => ( ( vEBT_set_vebt @ T )
% 0.23/0.55 = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % set_vebt_set_vebt'_valid
% 0.23/0.55 thf(fact_8280_buildup__gives__valid,axiom,
% 0.23/0.55 ! [N: nat] :
% 0.23/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.55 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 0.23/0.55
% 0.23/0.55 % buildup_gives_valid
% 0.23/0.55 thf(fact_8281__092_060open_062is__succ__in__set_A_Iset__vebt_H_Asummary_J_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_Asc_092_060close_062,axiom,
% 0.23/0.55 vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ).
% 0.23/0.55
% 0.23/0.55 % \<open>is_succ_in_set (set_vebt' summary) (high x (deg div 2)) sc\<close>
% 0.23/0.55 thf(fact_8282_set__n__deg__not__0,axiom,
% 0.23/0.55 ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
% 0.23/0.55 ( ! [X3: vEBT_VEBT] :
% 0.23/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.55 => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.23/0.55 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.23/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.55 => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % set_n_deg_not_0
% 0.23/0.55 thf(fact_8283_inthall,axiom,
% 0.23/0.55 ! [Xs2: list_Extended_enat,P: extended_enat > $o,N: nat] :
% 0.23/0.55 ( ! [X3: extended_enat] :
% 0.23/0.55 ( ( member_Extended_enat @ X3 @ ( set_Extended_enat2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X3 ) )
% 0.23/0.55 => ( ( ord_less_nat @ N @ ( size_s3941691890525107288d_enat @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_Extended_enat @ Xs2 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % inthall
% 0.23/0.55 thf(fact_8284_inthall,axiom,
% 0.23/0.55 ! [Xs2: list_complex,P: complex > $o,N: nat] :
% 0.23/0.55 ( ! [X3: complex] :
% 0.23/0.55 ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X3 ) )
% 0.23/0.55 => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % inthall
% 0.23/0.55 thf(fact_8285_inthall,axiom,
% 0.23/0.55 ! [Xs2: list_real,P: real > $o,N: nat] :
% 0.23/0.55 ( ! [X3: real] :
% 0.23/0.55 ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X3 ) )
% 0.23/0.55 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % inthall
% 0.23/0.55 thf(fact_8286_inthall,axiom,
% 0.23/0.55 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 0.23/0.55 ( ! [X3: vEBT_VEBT] :
% 0.23/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X3 ) )
% 0.23/0.55 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % inthall
% 0.23/0.55 thf(fact_8287_inthall,axiom,
% 0.23/0.55 ! [Xs2: list_o,P: $o > $o,N: nat] :
% 0.23/0.55 ( ! [X3: $o] :
% 0.23/0.55 ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X3 ) )
% 0.23/0.55 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % inthall
% 0.23/0.55 thf(fact_8288_inthall,axiom,
% 0.23/0.55 ! [Xs2: list_nat,P: nat > $o,N: nat] :
% 0.23/0.55 ( ! [X3: nat] :
% 0.23/0.55 ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X3 ) )
% 0.23/0.55 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % inthall
% 0.23/0.55 thf(fact_8289_inthall,axiom,
% 0.23/0.55 ! [Xs2: list_int,P: int > $o,N: nat] :
% 0.23/0.55 ( ! [X3: int] :
% 0.23/0.55 ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X3 ) )
% 0.23/0.55 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % inthall
% 0.23/0.55 thf(fact_8290_succ__member,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT,X4: nat,Y3: nat] :
% 0.23/0.55 ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 @ Y3 )
% 0.23/0.55 = ( ( vEBT_vebt_member @ T @ Y3 )
% 0.23/0.55 & ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.55 & ! [Z5: nat] :
% 0.23/0.55 ( ( ( vEBT_vebt_member @ T @ Z5 )
% 0.23/0.55 & ( ord_less_nat @ X4 @ Z5 ) )
% 0.23/0.55 => ( ord_less_eq_nat @ Y3 @ Z5 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % succ_member
% 0.23/0.55 thf(fact_8291_length__pos__if__in__set,axiom,
% 0.23/0.55 ! [X4: extended_enat,Xs2: list_Extended_enat] :
% 0.23/0.55 ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs2 ) )
% 0.23/0.55 => ( ord_less_nat @ zero_zero_nat @ ( size_s3941691890525107288d_enat @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % length_pos_if_in_set
% 0.23/0.55 thf(fact_8292_length__pos__if__in__set,axiom,
% 0.23/0.55 ! [X4: complex,Xs2: list_complex] :
% 0.23/0.55 ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
% 0.23/0.55 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % length_pos_if_in_set
% 0.23/0.55 thf(fact_8293_length__pos__if__in__set,axiom,
% 0.23/0.55 ! [X4: real,Xs2: list_real] :
% 0.23/0.55 ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
% 0.23/0.55 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % length_pos_if_in_set
% 0.23/0.55 thf(fact_8294_length__pos__if__in__set,axiom,
% 0.23/0.55 ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 0.23/0.55 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.23/0.55 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % length_pos_if_in_set
% 0.23/0.55 thf(fact_8295_length__pos__if__in__set,axiom,
% 0.23/0.55 ! [X4: $o,Xs2: list_o] :
% 0.23/0.55 ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
% 0.23/0.55 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % length_pos_if_in_set
% 0.23/0.55 thf(fact_8296_length__pos__if__in__set,axiom,
% 0.23/0.55 ! [X4: nat,Xs2: list_nat] :
% 0.23/0.55 ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
% 0.23/0.55 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % length_pos_if_in_set
% 0.23/0.55 thf(fact_8297_length__pos__if__in__set,axiom,
% 0.23/0.55 ! [X4: int,Xs2: list_int] :
% 0.23/0.55 ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
% 0.23/0.55 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % length_pos_if_in_set
% 0.23/0.55 thf(fact_8298_nth__mem,axiom,
% 0.23/0.55 ! [N: nat,Xs2: list_Extended_enat] :
% 0.23/0.55 ( ( ord_less_nat @ N @ ( size_s3941691890525107288d_enat @ Xs2 ) )
% 0.23/0.55 => ( member_Extended_enat @ ( nth_Extended_enat @ Xs2 @ N ) @ ( set_Extended_enat2 @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nth_mem
% 0.23/0.55 thf(fact_8299_nth__mem,axiom,
% 0.23/0.55 ! [N: nat,Xs2: list_complex] :
% 0.23/0.55 ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.23/0.55 => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nth_mem
% 0.23/0.55 thf(fact_8300_nth__mem,axiom,
% 0.23/0.55 ! [N: nat,Xs2: list_real] :
% 0.23/0.55 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 0.23/0.55 => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nth_mem
% 0.23/0.55 thf(fact_8301_nth__mem,axiom,
% 0.23/0.55 ! [N: nat,Xs2: list_VEBT_VEBT] :
% 0.23/0.55 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.23/0.55 => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nth_mem
% 0.23/0.55 thf(fact_8302_nth__mem,axiom,
% 0.23/0.55 ! [N: nat,Xs2: list_o] :
% 0.23/0.55 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.23/0.55 => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nth_mem
% 0.23/0.55 thf(fact_8303_nth__mem,axiom,
% 0.23/0.55 ! [N: nat,Xs2: list_nat] :
% 0.23/0.55 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.23/0.55 => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nth_mem
% 0.23/0.55 thf(fact_8304_nth__mem,axiom,
% 0.23/0.55 ! [N: nat,Xs2: list_int] :
% 0.23/0.55 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.23/0.55 => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % nth_mem
% 0.23/0.55 thf(fact_8305_list__ball__nth,axiom,
% 0.23/0.55 ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.23/0.55 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.23/0.55 => ( ! [X3: vEBT_VEBT] :
% 0.23/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X3 ) )
% 0.23/0.55 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % list_ball_nth
% 0.23/0.55 thf(fact_8306_list__ball__nth,axiom,
% 0.23/0.55 ! [N: nat,Xs2: list_o,P: $o > $o] :
% 0.23/0.55 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.23/0.55 => ( ! [X3: $o] :
% 0.23/0.55 ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X3 ) )
% 0.23/0.55 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % list_ball_nth
% 0.23/0.55 thf(fact_8307_list__ball__nth,axiom,
% 0.23/0.55 ! [N: nat,Xs2: list_nat,P: nat > $o] :
% 0.23/0.55 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.23/0.55 => ( ! [X3: nat] :
% 0.23/0.55 ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X3 ) )
% 0.23/0.55 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % list_ball_nth
% 0.23/0.55 thf(fact_8308_list__ball__nth,axiom,
% 0.23/0.55 ! [N: nat,Xs2: list_int,P: int > $o] :
% 0.23/0.55 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.23/0.55 => ( ! [X3: int] :
% 0.23/0.55 ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X3 ) )
% 0.23/0.55 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % list_ball_nth
% 0.23/0.55 thf(fact_8309_in__set__conv__nth,axiom,
% 0.23/0.55 ! [X4: extended_enat,Xs2: list_Extended_enat] :
% 0.23/0.55 ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs2 ) )
% 0.23/0.55 = ( ? [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_s3941691890525107288d_enat @ Xs2 ) )
% 0.23/0.55 & ( ( nth_Extended_enat @ Xs2 @ I2 )
% 0.23/0.55 = X4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % in_set_conv_nth
% 0.23/0.55 thf(fact_8310_in__set__conv__nth,axiom,
% 0.23/0.55 ! [X4: complex,Xs2: list_complex] :
% 0.23/0.55 ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
% 0.23/0.55 = ( ? [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.23/0.55 & ( ( nth_complex @ Xs2 @ I2 )
% 0.23/0.55 = X4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % in_set_conv_nth
% 0.23/0.55 thf(fact_8311_in__set__conv__nth,axiom,
% 0.23/0.55 ! [X4: real,Xs2: list_real] :
% 0.23/0.55 ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
% 0.23/0.55 = ( ? [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
% 0.23/0.55 & ( ( nth_real @ Xs2 @ I2 )
% 0.23/0.55 = X4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % in_set_conv_nth
% 0.23/0.55 thf(fact_8312_in__set__conv__nth,axiom,
% 0.23/0.55 ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 0.23/0.55 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.23/0.55 = ( ? [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.23/0.55 & ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
% 0.23/0.55 = X4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % in_set_conv_nth
% 0.23/0.55 thf(fact_8313_in__set__conv__nth,axiom,
% 0.23/0.55 ! [X4: $o,Xs2: list_o] :
% 0.23/0.55 ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
% 0.23/0.55 = ( ? [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 0.23/0.55 & ( ( nth_o @ Xs2 @ I2 )
% 0.23/0.55 = X4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % in_set_conv_nth
% 0.23/0.55 thf(fact_8314_in__set__conv__nth,axiom,
% 0.23/0.55 ! [X4: nat,Xs2: list_nat] :
% 0.23/0.55 ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
% 0.23/0.55 = ( ? [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 0.23/0.55 & ( ( nth_nat @ Xs2 @ I2 )
% 0.23/0.55 = X4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % in_set_conv_nth
% 0.23/0.55 thf(fact_8315_in__set__conv__nth,axiom,
% 0.23/0.55 ! [X4: int,Xs2: list_int] :
% 0.23/0.55 ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
% 0.23/0.55 = ( ? [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 0.23/0.55 & ( ( nth_int @ Xs2 @ I2 )
% 0.23/0.55 = X4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % in_set_conv_nth
% 0.23/0.55 thf(fact_8316_all__nth__imp__all__set,axiom,
% 0.23/0.55 ! [Xs2: list_Extended_enat,P: extended_enat > $o,X4: extended_enat] :
% 0.23/0.55 ( ! [I3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I3 @ ( size_s3941691890525107288d_enat @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_Extended_enat @ Xs2 @ I3 ) ) )
% 0.23/0.55 => ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X4 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % all_nth_imp_all_set
% 0.23/0.55 thf(fact_8317_all__nth__imp__all__set,axiom,
% 0.23/0.55 ! [Xs2: list_complex,P: complex > $o,X4: complex] :
% 0.23/0.55 ( ! [I3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_complex @ Xs2 @ I3 ) ) )
% 0.23/0.55 => ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X4 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % all_nth_imp_all_set
% 0.23/0.55 thf(fact_8318_all__nth__imp__all__set,axiom,
% 0.23/0.55 ! [Xs2: list_real,P: real > $o,X4: real] :
% 0.23/0.55 ( ! [I3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_real @ Xs2 @ I3 ) ) )
% 0.23/0.55 => ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X4 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % all_nth_imp_all_set
% 0.23/0.55 thf(fact_8319_all__nth__imp__all__set,axiom,
% 0.23/0.55 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X4: vEBT_VEBT] :
% 0.23/0.55 ( ! [I3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
% 0.23/0.55 => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X4 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % all_nth_imp_all_set
% 0.23/0.55 thf(fact_8320_all__nth__imp__all__set,axiom,
% 0.23/0.55 ! [Xs2: list_o,P: $o > $o,X4: $o] :
% 0.23/0.55 ( ! [I3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_o @ Xs2 @ I3 ) ) )
% 0.23/0.55 => ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X4 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % all_nth_imp_all_set
% 0.23/0.55 thf(fact_8321_all__nth__imp__all__set,axiom,
% 0.23/0.55 ! [Xs2: list_nat,P: nat > $o,X4: nat] :
% 0.23/0.55 ( ! [I3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
% 0.23/0.55 => ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X4 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % all_nth_imp_all_set
% 0.23/0.55 thf(fact_8322_all__nth__imp__all__set,axiom,
% 0.23/0.55 ! [Xs2: list_int,P: int > $o,X4: int] :
% 0.23/0.55 ( ! [I3: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_int @ Xs2 @ I3 ) ) )
% 0.23/0.55 => ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X4 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % all_nth_imp_all_set
% 0.23/0.55 thf(fact_8323_all__set__conv__all__nth,axiom,
% 0.23/0.55 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.23/0.55 ( ( ! [X: vEBT_VEBT] :
% 0.23/0.55 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X ) ) )
% 0.23/0.55 = ( ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % all_set_conv_all_nth
% 0.23/0.55 thf(fact_8324_all__set__conv__all__nth,axiom,
% 0.23/0.55 ! [Xs2: list_o,P: $o > $o] :
% 0.23/0.55 ( ( ! [X: $o] :
% 0.23/0.55 ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X ) ) )
% 0.23/0.55 = ( ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_o @ Xs2 @ I2 ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % all_set_conv_all_nth
% 0.23/0.55 thf(fact_8325_all__set__conv__all__nth,axiom,
% 0.23/0.55 ! [Xs2: list_nat,P: nat > $o] :
% 0.23/0.55 ( ( ! [X: nat] :
% 0.23/0.55 ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X ) ) )
% 0.23/0.55 = ( ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % all_set_conv_all_nth
% 0.23/0.55 thf(fact_8326_all__set__conv__all__nth,axiom,
% 0.23/0.55 ! [Xs2: list_int,P: int > $o] :
% 0.23/0.55 ( ( ! [X: int] :
% 0.23/0.55 ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 0.23/0.55 => ( P @ X ) ) )
% 0.23/0.55 = ( ! [I2: nat] :
% 0.23/0.55 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 0.23/0.55 => ( P @ ( nth_int @ Xs2 @ I2 ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % all_set_conv_all_nth
% 0.23/0.55 thf(fact_8327_is__succ__in__set__def,axiom,
% 0.23/0.55 ( vEBT_is_succ_in_set
% 0.23/0.55 = ( ^ [Xs: set_nat,X: nat,Y: nat] :
% 0.23/0.55 ( ( member_nat @ Y @ Xs )
% 0.23/0.55 & ( ord_less_nat @ X @ Y )
% 0.23/0.55 & ! [Z5: nat] :
% 0.23/0.55 ( ( member_nat @ Z5 @ Xs )
% 0.23/0.55 => ( ( ord_less_nat @ X @ Z5 )
% 0.23/0.55 => ( ord_less_eq_nat @ Y @ Z5 ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % is_succ_in_set_def
% 0.23/0.55 thf(fact_8328_inrange,axiom,
% 0.23/0.55 ! [T: vEBT_VEBT,N: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.55 => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % inrange
% 0.23/0.55 thf(fact_8329_fgh,axiom,
% 0.23/0.55 ( ( vEBT_VEBT_set_vebt @ ( nth_VEBT_VEBT @ treeList @ sc ) )
% 0.23/0.55 != bot_bot_set_nat ) ).
% 0.23/0.55
% 0.23/0.55 % fgh
% 0.23/0.55 thf(fact_8330__092_060open_062vebt__member_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Az_A_092_060and_062_Ax_A_060_Az_092_060close_062,axiom,
% 0.23/0.55 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ za )
% 0.23/0.55 & ( ord_less_nat @ xa @ za ) ) ).
% 0.23/0.55
% 0.23/0.55 % \<open>vebt_member (Node (Some (mi, ma)) deg treeList summary) z \<and> x < z\<close>
% 0.23/0.55 thf(fact_8331__092_060open_062vebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_092_060close_062,axiom,
% 0.23/0.55 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.55 = ( some_nat @ sc ) ) ).
% 0.23/0.55
% 0.23/0.55 % \<open>vebt_succ summary (high x (deg div 2)) = Some sc\<close>
% 0.23/0.55 thf(fact_8332_buildup__gives__empty,axiom,
% 0.23/0.55 ! [N: nat] :
% 0.23/0.55 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 0.23/0.55 = bot_bot_set_nat ) ).
% 0.23/0.55
% 0.23/0.55 % buildup_gives_empty
% 0.23/0.55 thf(fact_8333_power__shift,axiom,
% 0.23/0.55 ! [X4: nat,Y3: nat,Z: nat] :
% 0.23/0.55 ( ( ( power_power_nat @ X4 @ Y3 )
% 0.23/0.55 = Z )
% 0.23/0.55 = ( ( vEBT_VEBT_power @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) )
% 0.23/0.55 = ( some_nat @ Z ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % power_shift
% 0.23/0.55 thf(fact_8334__C5_Ohyps_C_I1_J,axiom,
% 0.23/0.55 ! [X5: vEBT_VEBT] :
% 0.23/0.55 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.23/0.55 => ( ( vEBT_invar_vebt @ X5 @ na )
% 0.23/0.55 & ! [Xa2: nat,Xb2: nat] :
% 0.23/0.55 ( ( ( vEBT_vebt_succ @ X5 @ Xa2 )
% 0.23/0.55 = ( some_nat @ Xb2 ) )
% 0.23/0.55 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ X5 ) @ Xa2 @ Xb2 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % "5.hyps"(1)
% 0.23/0.55 thf(fact_8335_mi__eq__ma__no__ch,axiom,
% 0.23/0.55 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 0.23/0.55 => ( ( Mi = Ma )
% 0.23/0.55 => ( ! [X5: vEBT_VEBT] :
% 0.23/0.55 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.55 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
% 0.23/0.55 & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % mi_eq_ma_no_ch
% 0.23/0.55 thf(fact_8336__C5_Ohyps_C_I3_J,axiom,
% 0.23/0.55 ! [X4: nat,Sx: nat] :
% 0.23/0.55 ( ( ( vEBT_vebt_succ @ summary @ X4 )
% 0.23/0.55 = ( some_nat @ Sx ) )
% 0.23/0.55 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X4 @ Sx ) ) ).
% 0.23/0.55
% 0.23/0.55 % "5.hyps"(3)
% 0.23/0.55 thf(fact_8337_atLeastAtMost__iff,axiom,
% 0.23/0.55 ! [I: extended_enat,L2: extended_enat,U: extended_enat] :
% 0.23/0.55 ( ( member_Extended_enat @ I @ ( set_or5403411693681687835d_enat @ L2 @ U ) )
% 0.23/0.55 = ( ( ord_le2932123472753598470d_enat @ L2 @ I )
% 0.23/0.55 & ( ord_le2932123472753598470d_enat @ I @ U ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastAtMost_iff
% 0.23/0.55 thf(fact_8338_atLeastAtMost__iff,axiom,
% 0.23/0.55 ! [I: set_nat,L2: set_nat,U: set_nat] :
% 0.23/0.55 ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L2 @ U ) )
% 0.23/0.55 = ( ( ord_less_eq_set_nat @ L2 @ I )
% 0.23/0.55 & ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastAtMost_iff
% 0.23/0.55 thf(fact_8339_atLeastAtMost__iff,axiom,
% 0.23/0.55 ! [I: rat,L2: rat,U: rat] :
% 0.23/0.55 ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L2 @ U ) )
% 0.23/0.55 = ( ( ord_less_eq_rat @ L2 @ I )
% 0.23/0.55 & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastAtMost_iff
% 0.23/0.55 thf(fact_8340_atLeastAtMost__iff,axiom,
% 0.23/0.55 ! [I: num,L2: num,U: num] :
% 0.23/0.55 ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L2 @ U ) )
% 0.23/0.55 = ( ( ord_less_eq_num @ L2 @ I )
% 0.23/0.55 & ( ord_less_eq_num @ I @ U ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastAtMost_iff
% 0.23/0.55 thf(fact_8341_atLeastAtMost__iff,axiom,
% 0.23/0.55 ! [I: nat,L2: nat,U: nat] :
% 0.23/0.55 ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 0.23/0.55 = ( ( ord_less_eq_nat @ L2 @ I )
% 0.23/0.55 & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastAtMost_iff
% 0.23/0.55 thf(fact_8342_atLeastAtMost__iff,axiom,
% 0.23/0.55 ! [I: int,L2: int,U: int] :
% 0.23/0.55 ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 0.23/0.55 = ( ( ord_less_eq_int @ L2 @ I )
% 0.23/0.55 & ( ord_less_eq_int @ I @ U ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastAtMost_iff
% 0.23/0.55 thf(fact_8343_atLeastAtMost__iff,axiom,
% 0.23/0.55 ! [I: real,L2: real,U: real] :
% 0.23/0.55 ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L2 @ U ) )
% 0.23/0.55 = ( ( ord_less_eq_real @ L2 @ I )
% 0.23/0.55 & ( ord_less_eq_real @ I @ U ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastAtMost_iff
% 0.23/0.55 thf(fact_8344_Icc__eq__Icc,axiom,
% 0.23/0.55 ! [L2: set_nat,H2: set_nat,L4: set_nat,H3: set_nat] :
% 0.23/0.55 ( ( ( set_or4548717258645045905et_nat @ L2 @ H2 )
% 0.23/0.55 = ( set_or4548717258645045905et_nat @ L4 @ H3 ) )
% 0.23/0.55 = ( ( ( L2 = L4 )
% 0.23/0.55 & ( H2 = H3 ) )
% 0.23/0.55 | ( ~ ( ord_less_eq_set_nat @ L2 @ H2 )
% 0.23/0.55 & ~ ( ord_less_eq_set_nat @ L4 @ H3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Icc_eq_Icc
% 0.23/0.55 thf(fact_8345_Icc__eq__Icc,axiom,
% 0.23/0.55 ! [L2: rat,H2: rat,L4: rat,H3: rat] :
% 0.23/0.55 ( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
% 0.23/0.55 = ( set_or633870826150836451st_rat @ L4 @ H3 ) )
% 0.23/0.55 = ( ( ( L2 = L4 )
% 0.23/0.55 & ( H2 = H3 ) )
% 0.23/0.55 | ( ~ ( ord_less_eq_rat @ L2 @ H2 )
% 0.23/0.55 & ~ ( ord_less_eq_rat @ L4 @ H3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Icc_eq_Icc
% 0.23/0.55 thf(fact_8346_Icc__eq__Icc,axiom,
% 0.23/0.55 ! [L2: num,H2: num,L4: num,H3: num] :
% 0.23/0.55 ( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
% 0.23/0.55 = ( set_or7049704709247886629st_num @ L4 @ H3 ) )
% 0.23/0.55 = ( ( ( L2 = L4 )
% 0.23/0.55 & ( H2 = H3 ) )
% 0.23/0.55 | ( ~ ( ord_less_eq_num @ L2 @ H2 )
% 0.23/0.55 & ~ ( ord_less_eq_num @ L4 @ H3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Icc_eq_Icc
% 0.23/0.55 thf(fact_8347_Icc__eq__Icc,axiom,
% 0.23/0.55 ! [L2: nat,H2: nat,L4: nat,H3: nat] :
% 0.23/0.55 ( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
% 0.23/0.55 = ( set_or1269000886237332187st_nat @ L4 @ H3 ) )
% 0.23/0.55 = ( ( ( L2 = L4 )
% 0.23/0.55 & ( H2 = H3 ) )
% 0.23/0.55 | ( ~ ( ord_less_eq_nat @ L2 @ H2 )
% 0.23/0.55 & ~ ( ord_less_eq_nat @ L4 @ H3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Icc_eq_Icc
% 0.23/0.55 thf(fact_8348_Icc__eq__Icc,axiom,
% 0.23/0.55 ! [L2: int,H2: int,L4: int,H3: int] :
% 0.23/0.55 ( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
% 0.23/0.55 = ( set_or1266510415728281911st_int @ L4 @ H3 ) )
% 0.23/0.55 = ( ( ( L2 = L4 )
% 0.23/0.55 & ( H2 = H3 ) )
% 0.23/0.55 | ( ~ ( ord_less_eq_int @ L2 @ H2 )
% 0.23/0.55 & ~ ( ord_less_eq_int @ L4 @ H3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Icc_eq_Icc
% 0.23/0.55 thf(fact_8349_Icc__eq__Icc,axiom,
% 0.23/0.55 ! [L2: real,H2: real,L4: real,H3: real] :
% 0.23/0.55 ( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
% 0.23/0.55 = ( set_or1222579329274155063t_real @ L4 @ H3 ) )
% 0.23/0.55 = ( ( ( L2 = L4 )
% 0.23/0.55 & ( H2 = H3 ) )
% 0.23/0.55 | ( ~ ( ord_less_eq_real @ L2 @ H2 )
% 0.23/0.55 & ~ ( ord_less_eq_real @ L4 @ H3 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % Icc_eq_Icc
% 0.23/0.55 thf(fact_8350_insert__simp__mima,axiom,
% 0.23/0.55 ! [X4: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.23/0.55 ( ( ( X4 = Mi )
% 0.23/0.55 | ( X4 = Ma ) )
% 0.23/0.55 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.23/0.55 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % insert_simp_mima
% 0.23/0.55 thf(fact_8351_Diff__cancel,axiom,
% 0.23/0.55 ! [A3: set_Extended_enat] :
% 0.23/0.55 ( ( minus_925952699566721837d_enat @ A3 @ A3 )
% 0.23/0.55 = bot_bo7653980558646680370d_enat ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_cancel
% 0.23/0.55 thf(fact_8352_Diff__cancel,axiom,
% 0.23/0.55 ! [A3: set_real] :
% 0.23/0.55 ( ( minus_minus_set_real @ A3 @ A3 )
% 0.23/0.55 = bot_bot_set_real ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_cancel
% 0.23/0.55 thf(fact_8353_Diff__cancel,axiom,
% 0.23/0.55 ! [A3: set_int] :
% 0.23/0.55 ( ( minus_minus_set_int @ A3 @ A3 )
% 0.23/0.55 = bot_bot_set_int ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_cancel
% 0.23/0.55 thf(fact_8354_Diff__cancel,axiom,
% 0.23/0.55 ! [A3: set_nat] :
% 0.23/0.55 ( ( minus_minus_set_nat @ A3 @ A3 )
% 0.23/0.55 = bot_bot_set_nat ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_cancel
% 0.23/0.55 thf(fact_8355_empty__Diff,axiom,
% 0.23/0.55 ! [A3: set_Extended_enat] :
% 0.23/0.55 ( ( minus_925952699566721837d_enat @ bot_bo7653980558646680370d_enat @ A3 )
% 0.23/0.55 = bot_bo7653980558646680370d_enat ) ).
% 0.23/0.55
% 0.23/0.55 % empty_Diff
% 0.23/0.55 thf(fact_8356_empty__Diff,axiom,
% 0.23/0.55 ! [A3: set_real] :
% 0.23/0.55 ( ( minus_minus_set_real @ bot_bot_set_real @ A3 )
% 0.23/0.55 = bot_bot_set_real ) ).
% 0.23/0.55
% 0.23/0.55 % empty_Diff
% 0.23/0.55 thf(fact_8357_empty__Diff,axiom,
% 0.23/0.55 ! [A3: set_int] :
% 0.23/0.55 ( ( minus_minus_set_int @ bot_bot_set_int @ A3 )
% 0.23/0.55 = bot_bot_set_int ) ).
% 0.23/0.55
% 0.23/0.55 % empty_Diff
% 0.23/0.55 thf(fact_8358_empty__Diff,axiom,
% 0.23/0.55 ! [A3: set_nat] :
% 0.23/0.55 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
% 0.23/0.55 = bot_bot_set_nat ) ).
% 0.23/0.55
% 0.23/0.55 % empty_Diff
% 0.23/0.55 thf(fact_8359_Diff__empty,axiom,
% 0.23/0.55 ! [A3: set_Extended_enat] :
% 0.23/0.55 ( ( minus_925952699566721837d_enat @ A3 @ bot_bo7653980558646680370d_enat )
% 0.23/0.55 = A3 ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_empty
% 0.23/0.55 thf(fact_8360_Diff__empty,axiom,
% 0.23/0.55 ! [A3: set_real] :
% 0.23/0.55 ( ( minus_minus_set_real @ A3 @ bot_bot_set_real )
% 0.23/0.55 = A3 ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_empty
% 0.23/0.55 thf(fact_8361_Diff__empty,axiom,
% 0.23/0.55 ! [A3: set_int] :
% 0.23/0.55 ( ( minus_minus_set_int @ A3 @ bot_bot_set_int )
% 0.23/0.55 = A3 ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_empty
% 0.23/0.55 thf(fact_8362_Diff__empty,axiom,
% 0.23/0.55 ! [A3: set_nat] :
% 0.23/0.55 ( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
% 0.23/0.55 = A3 ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_empty
% 0.23/0.55 thf(fact_8363_mi__ma__2__deg,axiom,
% 0.23/0.55 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.23/0.55 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.23/0.55 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.23/0.55 & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % mi_ma_2_deg
% 0.23/0.55 thf(fact_8364_succ__min,axiom,
% 0.23/0.55 ! [Deg: nat,X4: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.23/0.55 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.23/0.55 => ( ( ord_less_nat @ X4 @ Mi )
% 0.23/0.55 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.55 = ( some_nat @ Mi ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % succ_min
% 0.23/0.55 thf(fact_8365_atLeastatMost__empty__iff,axiom,
% 0.23/0.55 ! [A: extended_enat,B2: extended_enat] :
% 0.23/0.55 ( ( ( set_or5403411693681687835d_enat @ A @ B2 )
% 0.23/0.55 = bot_bo7653980558646680370d_enat )
% 0.23/0.55 = ( ~ ( ord_le2932123472753598470d_enat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff
% 0.23/0.55 thf(fact_8366_atLeastatMost__empty__iff,axiom,
% 0.23/0.55 ! [A: set_nat,B2: set_nat] :
% 0.23/0.55 ( ( ( set_or4548717258645045905et_nat @ A @ B2 )
% 0.23/0.55 = bot_bot_set_set_nat )
% 0.23/0.55 = ( ~ ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff
% 0.23/0.55 thf(fact_8367_atLeastatMost__empty__iff,axiom,
% 0.23/0.55 ! [A: rat,B2: rat] :
% 0.23/0.55 ( ( ( set_or633870826150836451st_rat @ A @ B2 )
% 0.23/0.55 = bot_bot_set_rat )
% 0.23/0.55 = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff
% 0.23/0.55 thf(fact_8368_atLeastatMost__empty__iff,axiom,
% 0.23/0.55 ! [A: num,B2: num] :
% 0.23/0.55 ( ( ( set_or7049704709247886629st_num @ A @ B2 )
% 0.23/0.55 = bot_bot_set_num )
% 0.23/0.55 = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff
% 0.23/0.55 thf(fact_8369_atLeastatMost__empty__iff,axiom,
% 0.23/0.55 ! [A: nat,B2: nat] :
% 0.23/0.55 ( ( ( set_or1269000886237332187st_nat @ A @ B2 )
% 0.23/0.55 = bot_bot_set_nat )
% 0.23/0.55 = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff
% 0.23/0.55 thf(fact_8370_atLeastatMost__empty__iff,axiom,
% 0.23/0.55 ! [A: int,B2: int] :
% 0.23/0.55 ( ( ( set_or1266510415728281911st_int @ A @ B2 )
% 0.23/0.55 = bot_bot_set_int )
% 0.23/0.55 = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff
% 0.23/0.55 thf(fact_8371_atLeastatMost__empty__iff,axiom,
% 0.23/0.55 ! [A: real,B2: real] :
% 0.23/0.55 ( ( ( set_or1222579329274155063t_real @ A @ B2 )
% 0.23/0.55 = bot_bot_set_real )
% 0.23/0.55 = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff
% 0.23/0.55 thf(fact_8372_atLeastatMost__empty__iff2,axiom,
% 0.23/0.55 ! [A: extended_enat,B2: extended_enat] :
% 0.23/0.55 ( ( bot_bo7653980558646680370d_enat
% 0.23/0.55 = ( set_or5403411693681687835d_enat @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_le2932123472753598470d_enat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff2
% 0.23/0.55 thf(fact_8373_atLeastatMost__empty__iff2,axiom,
% 0.23/0.55 ! [A: set_nat,B2: set_nat] :
% 0.23/0.55 ( ( bot_bot_set_set_nat
% 0.23/0.55 = ( set_or4548717258645045905et_nat @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff2
% 0.23/0.55 thf(fact_8374_atLeastatMost__empty__iff2,axiom,
% 0.23/0.55 ! [A: rat,B2: rat] :
% 0.23/0.55 ( ( bot_bot_set_rat
% 0.23/0.55 = ( set_or633870826150836451st_rat @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff2
% 0.23/0.55 thf(fact_8375_atLeastatMost__empty__iff2,axiom,
% 0.23/0.55 ! [A: num,B2: num] :
% 0.23/0.55 ( ( bot_bot_set_num
% 0.23/0.55 = ( set_or7049704709247886629st_num @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff2
% 0.23/0.55 thf(fact_8376_atLeastatMost__empty__iff2,axiom,
% 0.23/0.55 ! [A: nat,B2: nat] :
% 0.23/0.55 ( ( bot_bot_set_nat
% 0.23/0.55 = ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff2
% 0.23/0.55 thf(fact_8377_atLeastatMost__empty__iff2,axiom,
% 0.23/0.55 ! [A: int,B2: int] :
% 0.23/0.55 ( ( bot_bot_set_int
% 0.23/0.55 = ( set_or1266510415728281911st_int @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff2
% 0.23/0.55 thf(fact_8378_atLeastatMost__empty__iff2,axiom,
% 0.23/0.55 ! [A: real,B2: real] :
% 0.23/0.55 ( ( bot_bot_set_real
% 0.23/0.55 = ( set_or1222579329274155063t_real @ A @ B2 ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty_iff2
% 0.23/0.55 thf(fact_8379_atLeastatMost__subset__iff,axiom,
% 0.23/0.55 ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
% 0.23/0.55 ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B2 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_set_nat @ A @ B2 )
% 0.23/0.55 | ( ( ord_less_eq_set_nat @ C @ A )
% 0.23/0.55 & ( ord_less_eq_set_nat @ B2 @ D ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_subset_iff
% 0.23/0.55 thf(fact_8380_atLeastatMost__subset__iff,axiom,
% 0.23/0.55 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.23/0.55 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.55 | ( ( ord_less_eq_rat @ C @ A )
% 0.23/0.55 & ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_subset_iff
% 0.23/0.55 thf(fact_8381_atLeastatMost__subset__iff,axiom,
% 0.23/0.55 ! [A: num,B2: num,C: num,D: num] :
% 0.23/0.55 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_num @ A @ B2 )
% 0.23/0.55 | ( ( ord_less_eq_num @ C @ A )
% 0.23/0.55 & ( ord_less_eq_num @ B2 @ D ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_subset_iff
% 0.23/0.55 thf(fact_8382_atLeastatMost__subset__iff,axiom,
% 0.23/0.55 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.23/0.55 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.55 | ( ( ord_less_eq_nat @ C @ A )
% 0.23/0.55 & ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_subset_iff
% 0.23/0.55 thf(fact_8383_atLeastatMost__subset__iff,axiom,
% 0.23/0.55 ! [A: int,B2: int,C: int,D: int] :
% 0.23/0.55 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_int @ A @ B2 )
% 0.23/0.55 | ( ( ord_less_eq_int @ C @ A )
% 0.23/0.55 & ( ord_less_eq_int @ B2 @ D ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_subset_iff
% 0.23/0.55 thf(fact_8384_atLeastatMost__subset__iff,axiom,
% 0.23/0.55 ! [A: real,B2: real,C: real,D: real] :
% 0.23/0.55 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.23/0.55 = ( ~ ( ord_less_eq_real @ A @ B2 )
% 0.23/0.55 | ( ( ord_less_eq_real @ C @ A )
% 0.23/0.55 & ( ord_less_eq_real @ B2 @ D ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_subset_iff
% 0.23/0.55 thf(fact_8385_atLeastatMost__empty,axiom,
% 0.23/0.55 ! [B2: extended_enat,A: extended_enat] :
% 0.23/0.55 ( ( ord_le72135733267957522d_enat @ B2 @ A )
% 0.23/0.55 => ( ( set_or5403411693681687835d_enat @ A @ B2 )
% 0.23/0.55 = bot_bo7653980558646680370d_enat ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty
% 0.23/0.55 thf(fact_8386_atLeastatMost__empty,axiom,
% 0.23/0.55 ! [B2: rat,A: rat] :
% 0.23/0.55 ( ( ord_less_rat @ B2 @ A )
% 0.23/0.55 => ( ( set_or633870826150836451st_rat @ A @ B2 )
% 0.23/0.55 = bot_bot_set_rat ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty
% 0.23/0.55 thf(fact_8387_atLeastatMost__empty,axiom,
% 0.23/0.55 ! [B2: num,A: num] :
% 0.23/0.55 ( ( ord_less_num @ B2 @ A )
% 0.23/0.55 => ( ( set_or7049704709247886629st_num @ A @ B2 )
% 0.23/0.55 = bot_bot_set_num ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty
% 0.23/0.55 thf(fact_8388_atLeastatMost__empty,axiom,
% 0.23/0.55 ! [B2: nat,A: nat] :
% 0.23/0.55 ( ( ord_less_nat @ B2 @ A )
% 0.23/0.55 => ( ( set_or1269000886237332187st_nat @ A @ B2 )
% 0.23/0.55 = bot_bot_set_nat ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty
% 0.23/0.55 thf(fact_8389_atLeastatMost__empty,axiom,
% 0.23/0.55 ! [B2: int,A: int] :
% 0.23/0.55 ( ( ord_less_int @ B2 @ A )
% 0.23/0.55 => ( ( set_or1266510415728281911st_int @ A @ B2 )
% 0.23/0.55 = bot_bot_set_int ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty
% 0.23/0.55 thf(fact_8390_atLeastatMost__empty,axiom,
% 0.23/0.55 ! [B2: real,A: real] :
% 0.23/0.55 ( ( ord_less_real @ B2 @ A )
% 0.23/0.55 => ( ( set_or1222579329274155063t_real @ A @ B2 )
% 0.23/0.55 = bot_bot_set_real ) ) ).
% 0.23/0.55
% 0.23/0.55 % atLeastatMost_empty
% 0.23/0.55 thf(fact_8391_Diff__eq__empty__iff,axiom,
% 0.23/0.55 ! [A3: set_Extended_enat,B4: set_Extended_enat] :
% 0.23/0.55 ( ( ( minus_925952699566721837d_enat @ A3 @ B4 )
% 0.23/0.55 = bot_bo7653980558646680370d_enat )
% 0.23/0.55 = ( ord_le7203529160286727270d_enat @ A3 @ B4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_eq_empty_iff
% 0.23/0.55 thf(fact_8392_Diff__eq__empty__iff,axiom,
% 0.23/0.55 ! [A3: set_real,B4: set_real] :
% 0.23/0.55 ( ( ( minus_minus_set_real @ A3 @ B4 )
% 0.23/0.55 = bot_bot_set_real )
% 0.23/0.55 = ( ord_less_eq_set_real @ A3 @ B4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_eq_empty_iff
% 0.23/0.55 thf(fact_8393_Diff__eq__empty__iff,axiom,
% 0.23/0.55 ! [A3: set_int,B4: set_int] :
% 0.23/0.55 ( ( ( minus_minus_set_int @ A3 @ B4 )
% 0.23/0.55 = bot_bot_set_int )
% 0.23/0.55 = ( ord_less_eq_set_int @ A3 @ B4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_eq_empty_iff
% 0.23/0.55 thf(fact_8394_Diff__eq__empty__iff,axiom,
% 0.23/0.55 ! [A3: set_nat,B4: set_nat] :
% 0.23/0.55 ( ( ( minus_minus_set_nat @ A3 @ B4 )
% 0.23/0.55 = bot_bot_set_nat )
% 0.23/0.55 = ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).
% 0.23/0.55
% 0.23/0.55 % Diff_eq_empty_iff
% 0.23/0.55 thf(fact_8395__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062sc_O_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.23/0.55 ~ ! [Sc: nat] :
% 0.23/0.55 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.55 != ( some_nat @ Sc ) ) ).
% 0.23/0.55
% 0.23/0.55 % \<open>\<And>thesis. (\<And>sc. vebt_succ summary (high x (deg div 2)) = Some sc \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.23/0.55 thf(fact_8396_both__member__options__from__complete__tree__to__child,axiom,
% 0.23/0.55 ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
% 0.23/0.55 ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.23/0.55 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.55 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.55 | ( X4 = Mi )
% 0.23/0.55 | ( X4 = Ma ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % both_member_options_from_complete_tree_to_child
% 0.23/0.55 thf(fact_8397_member__inv,axiom,
% 0.23/0.55 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
% 0.23/0.55 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.55 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.23/0.55 & ( ( X4 = Mi )
% 0.23/0.55 | ( X4 = Ma )
% 0.23/0.55 | ( ( ord_less_nat @ X4 @ Ma )
% 0.23/0.55 & ( ord_less_nat @ Mi @ X4 )
% 0.23/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.55 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % member_inv
% 0.23/0.55 thf(fact_8398_both__member__options__from__chilf__to__complete__tree,axiom,
% 0.23/0.55 ! [X4: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.23/0.55 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.55 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.23/0.55 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.55 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % both_member_options_from_chilf_to_complete_tree
% 0.23/0.55 thf(fact_8399__092_060open_062Some_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Asc_J_092_060close_062,axiom,
% 0.23/0.55 ( ( some_nat @ miny )
% 0.23/0.55 = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ sc ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % \<open>Some miny = vebt_mint (treeList ! sc)\<close>
% 0.23/0.55 thf(fact_8400_greater__shift,axiom,
% 0.23/0.55 ( ord_less_nat
% 0.23/0.55 = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % greater_shift
% 0.23/0.55 thf(fact_8401_lesseq__shift,axiom,
% 0.23/0.55 ( ord_less_eq_nat
% 0.23/0.55 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.23/0.55
% 0.23/0.55 % lesseq_shift
% 0.23/0.55 thf(fact_8402_False,axiom,
% 0.23/0.55 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.55 != none_nat ) ).
% 0.23/0.55
% 0.23/0.55 % False
% 0.23/0.55 thf(fact_8403_not__psubset__empty,axiom,
% 0.23/0.55 ! [A3: set_Extended_enat] :
% 0.23/0.55 ~ ( ord_le2529575680413868914d_enat @ A3 @ bot_bo7653980558646680370d_enat ) ).
% 0.23/0.55
% 0.23/0.55 % not_psubset_empty
% 0.23/0.55 thf(fact_8404_not__psubset__empty,axiom,
% 0.23/0.55 ! [A3: set_real] :
% 0.23/0.55 ~ ( ord_less_set_real @ A3 @ bot_bot_set_real ) ).
% 0.23/0.55
% 0.23/0.55 % not_psubset_empty
% 0.23/0.55 thf(fact_8405_not__psubset__empty,axiom,
% 0.23/0.55 ! [A3: set_nat] :
% 0.23/0.55 ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).
% 0.23/0.55
% 0.23/0.55 % not_psubset_empty
% 0.23/0.55 thf(fact_8406_not__psubset__empty,axiom,
% 0.23/0.55 ! [A3: set_int] :
% 0.23/0.55 ~ ( ord_less_set_int @ A3 @ bot_bot_set_int ) ).
% 0.23/0.55
% 0.23/0.55 % not_psubset_empty
% 0.23/0.55 thf(fact_8407_bot_Onot__eq__extremum,axiom,
% 0.23/0.56 ! [A: set_Extended_enat] :
% 0.23/0.56 ( ( A != bot_bo7653980558646680370d_enat )
% 0.23/0.56 = ( ord_le2529575680413868914d_enat @ bot_bo7653980558646680370d_enat @ A ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.not_eq_extremum
% 0.23/0.56 thf(fact_8408_bot_Onot__eq__extremum,axiom,
% 0.23/0.56 ! [A: set_real] :
% 0.23/0.56 ( ( A != bot_bot_set_real )
% 0.23/0.56 = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.not_eq_extremum
% 0.23/0.56 thf(fact_8409_bot_Onot__eq__extremum,axiom,
% 0.23/0.56 ! [A: set_nat] :
% 0.23/0.56 ( ( A != bot_bot_set_nat )
% 0.23/0.56 = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.not_eq_extremum
% 0.23/0.56 thf(fact_8410_bot_Onot__eq__extremum,axiom,
% 0.23/0.56 ! [A: set_int] :
% 0.23/0.56 ( ( A != bot_bot_set_int )
% 0.23/0.56 = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.not_eq_extremum
% 0.23/0.56 thf(fact_8411_bot_Onot__eq__extremum,axiom,
% 0.23/0.56 ! [A: nat] :
% 0.23/0.56 ( ( A != bot_bot_nat )
% 0.23/0.56 = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.not_eq_extremum
% 0.23/0.56 thf(fact_8412_bot_Oextremum__strict,axiom,
% 0.23/0.56 ! [A: set_Extended_enat] :
% 0.23/0.56 ~ ( ord_le2529575680413868914d_enat @ A @ bot_bo7653980558646680370d_enat ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_strict
% 0.23/0.56 thf(fact_8413_bot_Oextremum__strict,axiom,
% 0.23/0.56 ! [A: set_real] :
% 0.23/0.56 ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_strict
% 0.23/0.56 thf(fact_8414_bot_Oextremum__strict,axiom,
% 0.23/0.56 ! [A: set_nat] :
% 0.23/0.56 ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_strict
% 0.23/0.56 thf(fact_8415_bot_Oextremum__strict,axiom,
% 0.23/0.56 ! [A: set_int] :
% 0.23/0.56 ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_strict
% 0.23/0.56 thf(fact_8416_bot_Oextremum__strict,axiom,
% 0.23/0.56 ! [A: nat] :
% 0.23/0.56 ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_strict
% 0.23/0.56 thf(fact_8417_bot_Oextremum,axiom,
% 0.23/0.56 ! [A: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ bot_bo7653980558646680370d_enat @ A ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum
% 0.23/0.56 thf(fact_8418_bot_Oextremum,axiom,
% 0.23/0.56 ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum
% 0.23/0.56 thf(fact_8419_bot_Oextremum,axiom,
% 0.23/0.56 ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum
% 0.23/0.56 thf(fact_8420_bot_Oextremum,axiom,
% 0.23/0.56 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum
% 0.23/0.56 thf(fact_8421_bot_Oextremum,axiom,
% 0.23/0.56 ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum
% 0.23/0.56 thf(fact_8422_bot_Oextremum__unique,axiom,
% 0.23/0.56 ! [A: set_Extended_enat] :
% 0.23/0.56 ( ( ord_le7203529160286727270d_enat @ A @ bot_bo7653980558646680370d_enat )
% 0.23/0.56 = ( A = bot_bo7653980558646680370d_enat ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_unique
% 0.23/0.56 thf(fact_8423_bot_Oextremum__unique,axiom,
% 0.23/0.56 ! [A: set_real] :
% 0.23/0.56 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.23/0.56 = ( A = bot_bot_set_real ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_unique
% 0.23/0.56 thf(fact_8424_bot_Oextremum__unique,axiom,
% 0.23/0.56 ! [A: set_int] :
% 0.23/0.56 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.23/0.56 = ( A = bot_bot_set_int ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_unique
% 0.23/0.56 thf(fact_8425_bot_Oextremum__unique,axiom,
% 0.23/0.56 ! [A: set_nat] :
% 0.23/0.56 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.23/0.56 = ( A = bot_bot_set_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_unique
% 0.23/0.56 thf(fact_8426_bot_Oextremum__unique,axiom,
% 0.23/0.56 ! [A: nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.23/0.56 = ( A = bot_bot_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_unique
% 0.23/0.56 thf(fact_8427_bot_Oextremum__uniqueI,axiom,
% 0.23/0.56 ! [A: set_Extended_enat] :
% 0.23/0.56 ( ( ord_le7203529160286727270d_enat @ A @ bot_bo7653980558646680370d_enat )
% 0.23/0.56 => ( A = bot_bo7653980558646680370d_enat ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_uniqueI
% 0.23/0.56 thf(fact_8428_bot_Oextremum__uniqueI,axiom,
% 0.23/0.56 ! [A: set_real] :
% 0.23/0.56 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.23/0.56 => ( A = bot_bot_set_real ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_uniqueI
% 0.23/0.56 thf(fact_8429_bot_Oextremum__uniqueI,axiom,
% 0.23/0.56 ! [A: set_int] :
% 0.23/0.56 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.23/0.56 => ( A = bot_bot_set_int ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_uniqueI
% 0.23/0.56 thf(fact_8430_bot_Oextremum__uniqueI,axiom,
% 0.23/0.56 ! [A: set_nat] :
% 0.23/0.56 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.23/0.56 => ( A = bot_bot_set_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_uniqueI
% 0.23/0.56 thf(fact_8431_bot_Oextremum__uniqueI,axiom,
% 0.23/0.56 ! [A: nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.23/0.56 => ( A = bot_bot_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % bot.extremum_uniqueI
% 0.23/0.56 thf(fact_8432_diff__shunt__var,axiom,
% 0.23/0.56 ! [X4: set_Extended_enat,Y3: set_Extended_enat] :
% 0.23/0.56 ( ( ( minus_925952699566721837d_enat @ X4 @ Y3 )
% 0.23/0.56 = bot_bo7653980558646680370d_enat )
% 0.23/0.56 = ( ord_le7203529160286727270d_enat @ X4 @ Y3 ) ) ).
% 0.23/0.56
% 0.23/0.56 % diff_shunt_var
% 0.23/0.56 thf(fact_8433_diff__shunt__var,axiom,
% 0.23/0.56 ! [X4: set_real,Y3: set_real] :
% 0.23/0.56 ( ( ( minus_minus_set_real @ X4 @ Y3 )
% 0.23/0.56 = bot_bot_set_real )
% 0.23/0.56 = ( ord_less_eq_set_real @ X4 @ Y3 ) ) ).
% 0.23/0.56
% 0.23/0.56 % diff_shunt_var
% 0.23/0.56 thf(fact_8434_diff__shunt__var,axiom,
% 0.23/0.56 ! [X4: set_int,Y3: set_int] :
% 0.23/0.56 ( ( ( minus_minus_set_int @ X4 @ Y3 )
% 0.23/0.56 = bot_bot_set_int )
% 0.23/0.56 = ( ord_less_eq_set_int @ X4 @ Y3 ) ) ).
% 0.23/0.56
% 0.23/0.56 % diff_shunt_var
% 0.23/0.56 thf(fact_8435_diff__shunt__var,axiom,
% 0.23/0.56 ! [X4: set_nat,Y3: set_nat] :
% 0.23/0.56 ( ( ( minus_minus_set_nat @ X4 @ Y3 )
% 0.23/0.56 = bot_bot_set_nat )
% 0.23/0.56 = ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% 0.23/0.56
% 0.23/0.56 % diff_shunt_var
% 0.23/0.56 thf(fact_8436_ex__nat__less,axiom,
% 0.23/0.56 ! [N: nat,P: nat > $o] :
% 0.23/0.56 ( ( ? [M6: nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M6 @ N )
% 0.23/0.56 & ( P @ M6 ) ) )
% 0.23/0.56 = ( ? [X: nat] :
% 0.23/0.56 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.23/0.56 & ( P @ X ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % ex_nat_less
% 0.23/0.56 thf(fact_8437_all__nat__less,axiom,
% 0.23/0.56 ! [N: nat,P: nat > $o] :
% 0.23/0.56 ( ( ! [M6: nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M6 @ N )
% 0.23/0.56 => ( P @ M6 ) ) )
% 0.23/0.56 = ( ! [X: nat] :
% 0.23/0.56 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.23/0.56 => ( P @ X ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % all_nat_less
% 0.23/0.56 thf(fact_8438_atLeastatMost__psubset__iff,axiom,
% 0.23/0.56 ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
% 0.23/0.56 ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B2 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
% 0.23/0.56 = ( ( ~ ( ord_less_eq_set_nat @ A @ B2 )
% 0.23/0.56 | ( ( ord_less_eq_set_nat @ C @ A )
% 0.23/0.56 & ( ord_less_eq_set_nat @ B2 @ D )
% 0.23/0.56 & ( ( ord_less_set_nat @ C @ A )
% 0.23/0.56 | ( ord_less_set_nat @ B2 @ D ) ) ) )
% 0.23/0.56 & ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % atLeastatMost_psubset_iff
% 0.23/0.56 thf(fact_8439_atLeastatMost__psubset__iff,axiom,
% 0.23/0.56 ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.23/0.56 ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.23/0.56 = ( ( ~ ( ord_less_eq_rat @ A @ B2 )
% 0.23/0.56 | ( ( ord_less_eq_rat @ C @ A )
% 0.23/0.56 & ( ord_less_eq_rat @ B2 @ D )
% 0.23/0.56 & ( ( ord_less_rat @ C @ A )
% 0.23/0.56 | ( ord_less_rat @ B2 @ D ) ) ) )
% 0.23/0.56 & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % atLeastatMost_psubset_iff
% 0.23/0.56 thf(fact_8440_atLeastatMost__psubset__iff,axiom,
% 0.23/0.56 ! [A: num,B2: num,C: num,D: num] :
% 0.23/0.56 ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.23/0.56 = ( ( ~ ( ord_less_eq_num @ A @ B2 )
% 0.23/0.56 | ( ( ord_less_eq_num @ C @ A )
% 0.23/0.56 & ( ord_less_eq_num @ B2 @ D )
% 0.23/0.56 & ( ( ord_less_num @ C @ A )
% 0.23/0.56 | ( ord_less_num @ B2 @ D ) ) ) )
% 0.23/0.56 & ( ord_less_eq_num @ C @ D ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % atLeastatMost_psubset_iff
% 0.23/0.56 thf(fact_8441_atLeastatMost__psubset__iff,axiom,
% 0.23/0.56 ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.23/0.56 ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.23/0.56 = ( ( ~ ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.56 | ( ( ord_less_eq_nat @ C @ A )
% 0.23/0.56 & ( ord_less_eq_nat @ B2 @ D )
% 0.23/0.56 & ( ( ord_less_nat @ C @ A )
% 0.23/0.56 | ( ord_less_nat @ B2 @ D ) ) ) )
% 0.23/0.56 & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % atLeastatMost_psubset_iff
% 0.23/0.56 thf(fact_8442_atLeastatMost__psubset__iff,axiom,
% 0.23/0.56 ! [A: int,B2: int,C: int,D: int] :
% 0.23/0.56 ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.23/0.56 = ( ( ~ ( ord_less_eq_int @ A @ B2 )
% 0.23/0.56 | ( ( ord_less_eq_int @ C @ A )
% 0.23/0.56 & ( ord_less_eq_int @ B2 @ D )
% 0.23/0.56 & ( ( ord_less_int @ C @ A )
% 0.23/0.56 | ( ord_less_int @ B2 @ D ) ) ) )
% 0.23/0.56 & ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % atLeastatMost_psubset_iff
% 0.23/0.56 thf(fact_8443_atLeastatMost__psubset__iff,axiom,
% 0.23/0.56 ! [A: real,B2: real,C: real,D: real] :
% 0.23/0.56 ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.23/0.56 = ( ( ~ ( ord_less_eq_real @ A @ B2 )
% 0.23/0.56 | ( ( ord_less_eq_real @ C @ A )
% 0.23/0.56 & ( ord_less_eq_real @ B2 @ D )
% 0.23/0.56 & ( ( ord_less_real @ C @ A )
% 0.23/0.56 | ( ord_less_real @ B2 @ D ) ) ) )
% 0.23/0.56 & ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % atLeastatMost_psubset_iff
% 0.23/0.56 thf(fact_8444_invar__vebt_Ointros_I4_J,axiom,
% 0.23/0.56 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.23/0.56 ( ! [X3: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.23/0.56 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.23/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.56 => ( ( M = N )
% 0.23/0.56 => ( ( Deg
% 0.23/0.56 = ( plus_plus_nat @ N @ M ) )
% 0.23/0.56 => ( ! [I3: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.56 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
% 0.23/0.56 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.23/0.56 => ( ( ( Mi = Ma )
% 0.23/0.56 => ! [X3: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.56 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 0.23/0.56 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.23/0.56 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.23/0.56 => ( ( ( Mi != Ma )
% 0.23/0.56 => ! [I3: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.23/0.56 = I3 )
% 0.23/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.23/0.56 & ! [X3: nat] :
% 0.23/0.56 ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 0.23/0.56 = I3 )
% 0.23/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 0.23/0.56 => ( ( ord_less_nat @ Mi @ X3 )
% 0.23/0.56 & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % invar_vebt.intros(4)
% 0.23/0.56 thf(fact_8445_invar__vebt_Ointros_I5_J,axiom,
% 0.23/0.56 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.23/0.56 ( ! [X3: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.23/0.56 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.23/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.56 => ( ( M
% 0.23/0.56 = ( suc @ N ) )
% 0.23/0.56 => ( ( Deg
% 0.23/0.56 = ( plus_plus_nat @ N @ M ) )
% 0.23/0.56 => ( ! [I3: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.56 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
% 0.23/0.56 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.23/0.56 => ( ( ( Mi = Ma )
% 0.23/0.56 => ! [X3: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.56 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 0.23/0.56 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.23/0.56 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.23/0.56 => ( ( ( Mi != Ma )
% 0.23/0.56 => ! [I3: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.23/0.56 = I3 )
% 0.23/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.23/0.56 & ! [X3: nat] :
% 0.23/0.56 ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 0.23/0.56 = I3 )
% 0.23/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 0.23/0.56 => ( ( ord_less_nat @ Mi @ X3 )
% 0.23/0.56 & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % invar_vebt.intros(5)
% 0.23/0.56 thf(fact_8446_mintlistlength,axiom,
% 0.23/0.56 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.23/0.56 => ( ( Mi != Ma )
% 0.23/0.56 => ( ( ord_less_nat @ Mi @ Ma )
% 0.23/0.56 & ? [M4: nat] :
% 0.23/0.56 ( ( ( some_nat @ M4 )
% 0.23/0.56 = ( vEBT_vebt_mint @ Summary ) )
% 0.23/0.56 & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % mintlistlength
% 0.23/0.56 thf(fact_8447__092_060open_062min__in__set_A_Iset__vebt_H_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_Aminy_092_060close_062,axiom,
% 0.23/0.56 vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ miny ).
% 0.23/0.56
% 0.23/0.56 % \<open>min_in_set (set_vebt' (treeList ! the (vebt_succ summary (high x (deg div 2))))) miny\<close>
% 0.23/0.56 thf(fact_8448_scmem,axiom,
% 0.23/0.56 vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ miny ).
% 0.23/0.56
% 0.23/0.56 % scmem
% 0.23/0.56 thf(fact_8449__092_060open_062invar__vebt_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_An_092_060close_062,axiom,
% 0.23/0.56 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ na ).
% 0.23/0.56
% 0.23/0.56 % \<open>invar_vebt (treeList ! the (vebt_succ summary (high x (deg div 2)))) n\<close>
% 0.23/0.56 thf(fact_8450_mint__member,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( ( vEBT_vebt_mint @ T )
% 0.23/0.56 = ( some_nat @ Maxi ) )
% 0.23/0.56 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % mint_member
% 0.23/0.56 thf(fact_8451_mint__corr__help,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat,Mini: nat,X4: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( ( vEBT_vebt_mint @ T )
% 0.23/0.56 = ( some_nat @ Mini ) )
% 0.23/0.56 => ( ( vEBT_vebt_member @ T @ X4 )
% 0.23/0.56 => ( ord_less_eq_nat @ Mini @ X4 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % mint_corr_help
% 0.23/0.56 thf(fact_8452_mint__corr__help__empty,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( ( vEBT_vebt_mint @ T )
% 0.23/0.56 = none_nat )
% 0.23/0.56 => ( ( vEBT_VEBT_set_vebt @ T )
% 0.23/0.56 = bot_bot_set_nat ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % mint_corr_help_empty
% 0.23/0.56 thf(fact_8453_mint__sound,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 )
% 0.23/0.56 => ( ( vEBT_vebt_mint @ T )
% 0.23/0.56 = ( some_nat @ X4 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % mint_sound
% 0.23/0.56 thf(fact_8454_mint__corr,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( ( vEBT_vebt_mint @ T )
% 0.23/0.56 = ( some_nat @ X4 ) )
% 0.23/0.56 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % mint_corr
% 0.23/0.56 thf(fact_8455__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062miny_O_ASome_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Asc_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.23/0.56 ~ ! [Miny: nat] :
% 0.23/0.56 ( ( some_nat @ Miny )
% 0.23/0.56 != ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ sc ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % \<open>\<And>thesis. (\<And>miny. Some miny = vebt_mint (treeList ! sc) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.23/0.56 thf(fact_8456_misiz,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat,M: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( ( some_nat @ M )
% 0.23/0.56 = ( vEBT_vebt_mint @ T ) )
% 0.23/0.56 => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % misiz
% 0.23/0.56 thf(fact_8457_bit__0__eq,axiom,
% 0.23/0.56 ( ( bit_se1146084159140164899it_int @ zero_zero_int )
% 0.23/0.56 = bot_bot_nat_o ) ).
% 0.23/0.56
% 0.23/0.56 % bit_0_eq
% 0.23/0.56 thf(fact_8458_bit__0__eq,axiom,
% 0.23/0.56 ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
% 0.23/0.56 = bot_bot_nat_o ) ).
% 0.23/0.56
% 0.23/0.56 % bit_0_eq
% 0.23/0.56 thf(fact_8459_succ__list__to__short,axiom,
% 0.23/0.56 ! [Deg: nat,Mi: nat,X4: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 0.23/0.56 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.23/0.56 => ( ( ord_less_eq_nat @ Mi @ X4 )
% 0.23/0.56 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.56 = none_nat ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % succ_list_to_short
% 0.23/0.56 thf(fact_8460_nested__mint,axiom,
% 0.23/0.56 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.23/0.56 => ( ( N
% 0.23/0.56 = ( suc @ ( suc @ Va2 ) ) )
% 0.23/0.56 => ( ~ ( ord_less_nat @ Ma @ Mi )
% 0.23/0.56 => ( ( Ma != Mi )
% 0.23/0.56 => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % nested_mint
% 0.23/0.56 thf(fact_8461__092_060open_062Some_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_092_060close_062,axiom,
% 0.23/0.56 ( ( some_nat @ miny )
% 0.23/0.56 = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % \<open>Some miny = vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2))))\<close>
% 0.23/0.56 thf(fact_8462_local_Opower__def,axiom,
% 0.23/0.56 ( vEBT_VEBT_power
% 0.23/0.56 = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % local.power_def
% 0.23/0.56 thf(fact_8463_bot__enat__def,axiom,
% 0.23/0.56 bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 0.23/0.56
% 0.23/0.56 % bot_enat_def
% 0.23/0.56 thf(fact_8464_bot__nat__def,axiom,
% 0.23/0.56 bot_bot_nat = zero_zero_nat ).
% 0.23/0.56
% 0.23/0.56 % bot_nat_def
% 0.23/0.56 thf(fact_8465_aset_I2_J,axiom,
% 0.23/0.56 ! [D4: int,A3: set_int,P: int > $o,Q: int > $o] :
% 0.23/0.56 ( ! [X3: int] :
% 0.23/0.56 ( ! [Xa2: int] :
% 0.23/0.56 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb2: int] :
% 0.23/0.56 ( ( member_int @ Xb2 @ A3 )
% 0.23/0.56 => ( X3
% 0.23/0.56 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.23/0.56 => ( ( P @ X3 )
% 0.23/0.56 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 0.23/0.56 => ( ! [X3: int] :
% 0.23/0.56 ( ! [Xa2: int] :
% 0.23/0.56 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb2: int] :
% 0.23/0.56 ( ( member_int @ Xb2 @ A3 )
% 0.23/0.56 => ( X3
% 0.23/0.56 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.23/0.56 => ( ( Q @ X3 )
% 0.23/0.56 => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ A3 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ( P @ X5 )
% 0.23/0.56 | ( Q @ X5 ) )
% 0.23/0.56 => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
% 0.23/0.56 | ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % aset(2)
% 0.23/0.56 thf(fact_8466_aset_I1_J,axiom,
% 0.23/0.56 ! [D4: int,A3: set_int,P: int > $o,Q: int > $o] :
% 0.23/0.56 ( ! [X3: int] :
% 0.23/0.56 ( ! [Xa2: int] :
% 0.23/0.56 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb2: int] :
% 0.23/0.56 ( ( member_int @ Xb2 @ A3 )
% 0.23/0.56 => ( X3
% 0.23/0.56 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.23/0.56 => ( ( P @ X3 )
% 0.23/0.56 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 0.23/0.56 => ( ! [X3: int] :
% 0.23/0.56 ( ! [Xa2: int] :
% 0.23/0.56 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb2: int] :
% 0.23/0.56 ( ( member_int @ Xb2 @ A3 )
% 0.23/0.56 => ( X3
% 0.23/0.56 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.23/0.56 => ( ( Q @ X3 )
% 0.23/0.56 => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ A3 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ( P @ X5 )
% 0.23/0.56 & ( Q @ X5 ) )
% 0.23/0.56 => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
% 0.23/0.56 & ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % aset(1)
% 0.23/0.56 thf(fact_8467_bset_I2_J,axiom,
% 0.23/0.56 ! [D4: int,B4: set_int,P: int > $o,Q: int > $o] :
% 0.23/0.56 ( ! [X3: int] :
% 0.23/0.56 ( ! [Xa2: int] :
% 0.23/0.56 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb2: int] :
% 0.23/0.56 ( ( member_int @ Xb2 @ B4 )
% 0.23/0.56 => ( X3
% 0.23/0.56 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.23/0.56 => ( ( P @ X3 )
% 0.23/0.56 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 0.23/0.56 => ( ! [X3: int] :
% 0.23/0.56 ( ! [Xa2: int] :
% 0.23/0.56 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb2: int] :
% 0.23/0.56 ( ( member_int @ Xb2 @ B4 )
% 0.23/0.56 => ( X3
% 0.23/0.56 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.23/0.56 => ( ( Q @ X3 )
% 0.23/0.56 => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ B4 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ( P @ X5 )
% 0.23/0.56 | ( Q @ X5 ) )
% 0.23/0.56 => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
% 0.23/0.56 | ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % bset(2)
% 0.23/0.56 thf(fact_8468_bset_I1_J,axiom,
% 0.23/0.56 ! [D4: int,B4: set_int,P: int > $o,Q: int > $o] :
% 0.23/0.56 ( ! [X3: int] :
% 0.23/0.56 ( ! [Xa2: int] :
% 0.23/0.56 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb2: int] :
% 0.23/0.56 ( ( member_int @ Xb2 @ B4 )
% 0.23/0.56 => ( X3
% 0.23/0.56 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.23/0.56 => ( ( P @ X3 )
% 0.23/0.56 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 0.23/0.56 => ( ! [X3: int] :
% 0.23/0.56 ( ! [Xa2: int] :
% 0.23/0.56 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb2: int] :
% 0.23/0.56 ( ( member_int @ Xb2 @ B4 )
% 0.23/0.56 => ( X3
% 0.23/0.56 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.23/0.56 => ( ( Q @ X3 )
% 0.23/0.56 => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ B4 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ( P @ X5 )
% 0.23/0.56 & ( Q @ X5 ) )
% 0.23/0.56 => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
% 0.23/0.56 & ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % bset(1)
% 0.23/0.56 thf(fact_8469_bset_I9_J,axiom,
% 0.23/0.56 ! [D: int,D4: int,B4: set_int,T: int] :
% 0.23/0.56 ( ( dvd_dvd_int @ D @ D4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ B4 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.23/0.56 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % bset(9)
% 0.23/0.56 thf(fact_8470_bset_I10_J,axiom,
% 0.23/0.56 ! [D: int,D4: int,B4: set_int,T: int] :
% 0.23/0.56 ( ( dvd_dvd_int @ D @ D4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ B4 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.23/0.56 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % bset(10)
% 0.23/0.56 thf(fact_8471_aset_I9_J,axiom,
% 0.23/0.56 ! [D: int,D4: int,A3: set_int,T: int] :
% 0.23/0.56 ( ( dvd_dvd_int @ D @ D4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ A3 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.23/0.56 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % aset(9)
% 0.23/0.56 thf(fact_8472_aset_I10_J,axiom,
% 0.23/0.56 ! [D: int,D4: int,A3: set_int,T: int] :
% 0.23/0.56 ( ( dvd_dvd_int @ D @ D4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ A3 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.23/0.56 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % aset(10)
% 0.23/0.56 thf(fact_8473_vebt__succ_Osimps_I4_J,axiom,
% 0.23/0.56 ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
% 0.23/0.56 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
% 0.23/0.56 = none_nat ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_succ.simps(4)
% 0.23/0.56 thf(fact_8474_periodic__finite__ex,axiom,
% 0.23/0.56 ! [D: int,P: int > $o] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D )
% 0.23/0.56 => ( ! [X3: int,K2: int] :
% 0.23/0.56 ( ( P @ X3 )
% 0.23/0.56 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.23/0.56 => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.23/0.56 = ( ? [X: int] :
% 0.23/0.56 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 0.23/0.56 & ( P @ X ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % periodic_finite_ex
% 0.23/0.56 thf(fact_8475_aset_I7_J,axiom,
% 0.23/0.56 ! [D4: int,A3: set_int,T: int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ A3 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ord_less_int @ T @ X5 )
% 0.23/0.56 => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % aset(7)
% 0.23/0.56 thf(fact_8476_aset_I5_J,axiom,
% 0.23/0.56 ! [D4: int,T: int,A3: set_int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ( ( member_int @ T @ A3 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ A3 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ord_less_int @ X5 @ T )
% 0.23/0.56 => ( ord_less_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % aset(5)
% 0.23/0.56 thf(fact_8477_aset_I4_J,axiom,
% 0.23/0.56 ! [D4: int,T: int,A3: set_int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ( ( member_int @ T @ A3 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ A3 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( X5 != T )
% 0.23/0.56 => ( ( plus_plus_int @ X5 @ D4 )
% 0.23/0.56 != T ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % aset(4)
% 0.23/0.56 thf(fact_8478_aset_I3_J,axiom,
% 0.23/0.56 ! [D4: int,T: int,A3: set_int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ A3 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( X5 = T )
% 0.23/0.56 => ( ( plus_plus_int @ X5 @ D4 )
% 0.23/0.56 = T ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % aset(3)
% 0.23/0.56 thf(fact_8479_bset_I7_J,axiom,
% 0.23/0.56 ! [D4: int,T: int,B4: set_int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ( ( member_int @ T @ B4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ B4 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ord_less_int @ T @ X5 )
% 0.23/0.56 => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % bset(7)
% 0.23/0.56 thf(fact_8480_bset_I5_J,axiom,
% 0.23/0.56 ! [D4: int,B4: set_int,T: int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ B4 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ord_less_int @ X5 @ T )
% 0.23/0.56 => ( ord_less_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % bset(5)
% 0.23/0.56 thf(fact_8481_bset_I4_J,axiom,
% 0.23/0.56 ! [D4: int,T: int,B4: set_int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ( ( member_int @ T @ B4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ B4 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( X5 != T )
% 0.23/0.56 => ( ( minus_minus_int @ X5 @ D4 )
% 0.23/0.56 != T ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % bset(4)
% 0.23/0.56 thf(fact_8482_bset_I3_J,axiom,
% 0.23/0.56 ! [D4: int,T: int,B4: set_int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ B4 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( X5 = T )
% 0.23/0.56 => ( ( minus_minus_int @ X5 @ D4 )
% 0.23/0.56 = T ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % bset(3)
% 0.23/0.56 thf(fact_8483_vebt__succ_Osimps_I5_J,axiom,
% 0.23/0.56 ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 0.23/0.56 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
% 0.23/0.56 = none_nat ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_succ.simps(5)
% 0.23/0.56 thf(fact_8484_aset_I8_J,axiom,
% 0.23/0.56 ! [D4: int,A3: set_int,T: int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ A3 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ord_less_eq_int @ T @ X5 )
% 0.23/0.56 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % aset(8)
% 0.23/0.56 thf(fact_8485_aset_I6_J,axiom,
% 0.23/0.56 ! [D4: int,T: int,A3: set_int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ A3 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ord_less_eq_int @ X5 @ T )
% 0.23/0.56 => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % aset(6)
% 0.23/0.56 thf(fact_8486_bset_I8_J,axiom,
% 0.23/0.56 ! [D4: int,T: int,B4: set_int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ B4 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ord_less_eq_int @ T @ X5 )
% 0.23/0.56 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % bset(8)
% 0.23/0.56 thf(fact_8487_bset_I6_J,axiom,
% 0.23/0.56 ! [D4: int,B4: set_int,T: int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ! [X5: int] :
% 0.23/0.56 ( ! [Xa3: int] :
% 0.23/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb3: int] :
% 0.23/0.56 ( ( member_int @ Xb3 @ B4 )
% 0.23/0.56 => ( X5
% 0.23/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.23/0.56 => ( ( ord_less_eq_int @ X5 @ T )
% 0.23/0.56 => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % bset(6)
% 0.23/0.56 thf(fact_8488_cpmi,axiom,
% 0.23/0.56 ! [D4: int,P: int > $o,P4: int > $o,B4: set_int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ( ? [Z3: int] :
% 0.23/0.56 ! [X3: int] :
% 0.23/0.56 ( ( ord_less_int @ X3 @ Z3 )
% 0.23/0.56 => ( ( P @ X3 )
% 0.23/0.56 = ( P4 @ X3 ) ) )
% 0.23/0.56 => ( ! [X3: int] :
% 0.23/0.56 ( ! [Xa2: int] :
% 0.23/0.56 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb2: int] :
% 0.23/0.56 ( ( member_int @ Xb2 @ B4 )
% 0.23/0.56 => ( X3
% 0.23/0.56 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.23/0.56 => ( ( P @ X3 )
% 0.23/0.56 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 0.23/0.56 => ( ! [X3: int,K2: int] :
% 0.23/0.56 ( ( P4 @ X3 )
% 0.23/0.56 = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 0.23/0.56 => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.23/0.56 = ( ? [X: int] :
% 0.23/0.56 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 & ( P4 @ X ) )
% 0.23/0.56 | ? [X: int] :
% 0.23/0.56 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 & ? [Y: int] :
% 0.23/0.56 ( ( member_int @ Y @ B4 )
% 0.23/0.56 & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % cpmi
% 0.23/0.56 thf(fact_8489_cppi,axiom,
% 0.23/0.56 ! [D4: int,P: int > $o,P4: int > $o,A3: set_int] :
% 0.23/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.23/0.56 => ( ? [Z3: int] :
% 0.23/0.56 ! [X3: int] :
% 0.23/0.56 ( ( ord_less_int @ Z3 @ X3 )
% 0.23/0.56 => ( ( P @ X3 )
% 0.23/0.56 = ( P4 @ X3 ) ) )
% 0.23/0.56 => ( ! [X3: int] :
% 0.23/0.56 ( ! [Xa2: int] :
% 0.23/0.56 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 => ! [Xb2: int] :
% 0.23/0.56 ( ( member_int @ Xb2 @ A3 )
% 0.23/0.56 => ( X3
% 0.23/0.56 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.23/0.56 => ( ( P @ X3 )
% 0.23/0.56 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 0.23/0.56 => ( ! [X3: int,K2: int] :
% 0.23/0.56 ( ( P4 @ X3 )
% 0.23/0.56 = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 0.23/0.56 => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.23/0.56 = ( ? [X: int] :
% 0.23/0.56 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 & ( P4 @ X ) )
% 0.23/0.56 | ? [X: int] :
% 0.23/0.56 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.23/0.56 & ? [Y: int] :
% 0.23/0.56 ( ( member_int @ Y @ A3 )
% 0.23/0.56 & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % cppi
% 0.23/0.56 thf(fact_8490__092_060open_062res_A_061_Athe_A_ISome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_092_060close_062,axiom,
% 0.23/0.56 ( res
% 0.23/0.56 = ( the_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % \<open>res = the (Some (2 ^ (deg div 2)) *\<^sub>o vebt_succ summary (high x (deg div 2)) +\<^sub>o vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2)))))\<close>
% 0.23/0.56 thf(fact_8491_summaxma,axiom,
% 0.23/0.56 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 0.23/0.56 => ( ( Mi != Ma )
% 0.23/0.56 => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 0.23/0.56 = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % summaxma
% 0.23/0.56 thf(fact_8492_maxt__sound,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 )
% 0.23/0.56 => ( ( vEBT_vebt_maxt @ T )
% 0.23/0.56 = ( some_nat @ X4 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % maxt_sound
% 0.23/0.56 thf(fact_8493_maxt__corr,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( ( vEBT_vebt_maxt @ T )
% 0.23/0.56 = ( some_nat @ X4 ) )
% 0.23/0.56 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % maxt_corr
% 0.23/0.56 thf(fact_8494_add__shift,axiom,
% 0.23/0.56 ! [X4: nat,Y3: nat,Z: nat] :
% 0.23/0.56 ( ( ( plus_plus_nat @ X4 @ Y3 )
% 0.23/0.56 = Z )
% 0.23/0.56 = ( ( vEBT_VEBT_add @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) )
% 0.23/0.56 = ( some_nat @ Z ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % add_shift
% 0.23/0.56 thf(fact_8495_mul__shift,axiom,
% 0.23/0.56 ! [X4: nat,Y3: nat,Z: nat] :
% 0.23/0.56 ( ( ( times_times_nat @ X4 @ Y3 )
% 0.23/0.56 = Z )
% 0.23/0.56 = ( ( vEBT_VEBT_mul @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) )
% 0.23/0.56 = ( some_nat @ Z ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % mul_shift
% 0.23/0.56 thf(fact_8496_maxbmo,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,X4: nat] :
% 0.23/0.56 ( ( ( vEBT_vebt_maxt @ T )
% 0.23/0.56 = ( some_nat @ X4 ) )
% 0.23/0.56 => ( vEBT_V8194947554948674370ptions @ T @ X4 ) ) ).
% 0.23/0.56
% 0.23/0.56 % maxbmo
% 0.23/0.56 thf(fact_8497_add__def,axiom,
% 0.23/0.56 ( vEBT_VEBT_add
% 0.23/0.56 = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % add_def
% 0.23/0.56 thf(fact_8498_mul__def,axiom,
% 0.23/0.56 ( vEBT_VEBT_mul
% 0.23/0.56 = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % mul_def
% 0.23/0.56 thf(fact_8499_maxt__member,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( ( vEBT_vebt_maxt @ T )
% 0.23/0.56 = ( some_nat @ Maxi ) )
% 0.23/0.56 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % maxt_member
% 0.23/0.56 thf(fact_8500_maxt__corr__help,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat,Maxi: nat,X4: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( ( vEBT_vebt_maxt @ T )
% 0.23/0.56 = ( some_nat @ Maxi ) )
% 0.23/0.56 => ( ( vEBT_vebt_member @ T @ X4 )
% 0.23/0.56 => ( ord_less_eq_nat @ X4 @ Maxi ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % maxt_corr_help
% 0.23/0.56 thf(fact_8501_maxt__corr__help__empty,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( ( vEBT_vebt_maxt @ T )
% 0.23/0.56 = none_nat )
% 0.23/0.56 => ( ( vEBT_VEBT_set_vebt @ T )
% 0.23/0.56 = bot_bot_set_nat ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % maxt_corr_help_empty
% 0.23/0.56 thf(fact_8502__092_060open_062vebt__succ_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ASome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_092_060close_062,axiom,
% 0.23/0.56 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.23/0.56 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % \<open>vebt_succ (Node (Some (mi, ma)) deg treeList summary) x = Some (2 ^ (deg div 2)) *\<^sub>o vebt_succ summary (high x (deg div 2)) +\<^sub>o vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2))))\<close>
% 0.23/0.56 thf(fact_8503__C2_C,axiom,
% 0.23/0.56 ( ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 = none_nat )
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.23/0.56 = none_nat ) )
% 0.23/0.56 & ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 != none_nat )
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.23/0.56 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % "2"
% 0.23/0.56 thf(fact_8504__092_060open_062vebt__member_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_A_Ithe_A_ISome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_J_092_060close_062,axiom,
% 0.23/0.56 vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ( the_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % \<open>vebt_member (Node (Some (mi, ma)) deg treeList summary) (the (Some (2 ^ (deg div 2)) *\<^sub>o vebt_succ summary (high x (deg div 2)) +\<^sub>o vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2))))))\<close>
% 0.23/0.56 thf(fact_8505__C1_C,axiom,
% 0.23/0.56 ( ( ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 != none_nat )
% 0.23/0.56 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.23/0.56 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.23/0.56 & ( ~ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 != none_nat )
% 0.23/0.56 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.23/0.56 => ( ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 = none_nat )
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.23/0.56 = none_nat ) )
% 0.23/0.56 & ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 != none_nat )
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.23/0.56 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % "1"
% 0.23/0.56 thf(fact_8506_i1,axiom,
% 0.23/0.56 ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 = none_nat )
% 0.23/0.56 | ~ ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % i1
% 0.23/0.56 thf(fact_8507_divmod__step__eq,axiom,
% 0.23/0.56 ! [L2: num,R2: nat,Q2: nat] :
% 0.23/0.56 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
% 0.23/0.56 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
% 0.23/0.56 = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L2 ) ) ) ) )
% 0.23/0.56 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
% 0.23/0.56 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
% 0.23/0.56 = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % divmod_step_eq
% 0.23/0.56 thf(fact_8508_divmod__step__eq,axiom,
% 0.23/0.56 ! [L2: num,R2: int,Q2: int] :
% 0.23/0.56 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
% 0.23/0.56 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.56 = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L2 ) ) ) ) )
% 0.23/0.56 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
% 0.23/0.56 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.56 = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % divmod_step_eq
% 0.23/0.56 thf(fact_8509_divmod__step__eq,axiom,
% 0.23/0.56 ! [L2: num,R2: code_integer,Q2: code_integer] :
% 0.23/0.56 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
% 0.23/0.56 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
% 0.23/0.56 = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
% 0.23/0.56 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
% 0.23/0.56 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
% 0.23/0.56 = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % divmod_step_eq
% 0.23/0.56 thf(fact_8510_less__shift,axiom,
% 0.23/0.56 ( ord_less_nat
% 0.23/0.56 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % less_shift
% 0.23/0.56 thf(fact_8511_fold__atLeastAtMost__nat_Ocases,axiom,
% 0.23/0.56 ! [X4: produc3368934014287244435at_num] :
% 0.23/0.56 ~ ! [F3: nat > num > num,A5: nat,B5: nat,Acc2: num] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc851828971589881931at_num @ F3 @ ( produc1195630363706982562at_num @ A5 @ ( product_Pair_nat_num @ B5 @ Acc2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % fold_atLeastAtMost_nat.cases
% 0.23/0.56 thf(fact_8512_fold__atLeastAtMost__nat_Ocases,axiom,
% 0.23/0.56 ! [X4: produc4471711990508489141at_nat] :
% 0.23/0.56 ~ ! [F3: nat > nat > nat,A5: nat,B5: nat,Acc2: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ Acc2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % fold_atLeastAtMost_nat.cases
% 0.23/0.56 thf(fact_8513_xor__num_Ocases,axiom,
% 0.23/0.56 ! [X4: product_prod_num_num] :
% 0.23/0.56 ( ( X4
% 0.23/0.56 != ( product_Pair_num_num @ one @ one ) )
% 0.23/0.56 => ( ! [N4: num] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) )
% 0.23/0.56 => ( ! [N4: num] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) )
% 0.23/0.56 => ( ! [M4: num] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
% 0.23/0.56 => ( ! [M4: num,N4: num] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) )
% 0.23/0.56 => ( ! [M4: num,N4: num] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) )
% 0.23/0.56 => ( ! [M4: num] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
% 0.23/0.56 => ( ! [M4: num,N4: num] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) )
% 0.23/0.56 => ~ ! [M4: num,N4: num] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % xor_num.cases
% 0.23/0.56 thf(fact_8514_minNullmin,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT] :
% 0.23/0.56 ( ( vEBT_VEBT_minNull @ T )
% 0.23/0.56 => ( ( vEBT_vebt_mint @ T )
% 0.23/0.56 = none_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % minNullmin
% 0.23/0.56 thf(fact_8515_minminNull,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT] :
% 0.23/0.56 ( ( ( vEBT_vebt_mint @ T )
% 0.23/0.56 = none_nat )
% 0.23/0.56 => ( vEBT_VEBT_minNull @ T ) ) ).
% 0.23/0.56
% 0.23/0.56 % minminNull
% 0.23/0.56 thf(fact_8516_vebt__member_Osimps_I4_J,axiom,
% 0.23/0.56 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
% 0.23/0.56 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X4 ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_member.simps(4)
% 0.23/0.56 thf(fact_8517_succ__less__length__list,axiom,
% 0.23/0.56 ! [Deg: nat,Mi: nat,X4: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 0.23/0.56 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.23/0.56 => ( ( ord_less_eq_nat @ Mi @ X4 )
% 0.23/0.56 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.56 = ( if_option_nat
% 0.23/0.56 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 != none_nat )
% 0.23/0.56 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.23/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 @ ( if_option_nat
% 0.23/0.56 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 = none_nat )
% 0.23/0.56 @ none_nat
% 0.23/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % succ_less_length_list
% 0.23/0.56 thf(fact_8518_not__min__Null__member,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT] :
% 0.23/0.56 ( ~ ( vEBT_VEBT_minNull @ T )
% 0.23/0.56 => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 0.23/0.56
% 0.23/0.56 % not_min_Null_member
% 0.23/0.56 thf(fact_8519_min__Null__member,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,X4: nat] :
% 0.23/0.56 ( ( vEBT_VEBT_minNull @ T )
% 0.23/0.56 => ~ ( vEBT_vebt_member @ T @ X4 ) ) ).
% 0.23/0.56
% 0.23/0.56 % min_Null_member
% 0.23/0.56 thf(fact_8520_set__vebt_H__def,axiom,
% 0.23/0.56 ( vEBT_VEBT_set_vebt
% 0.23/0.56 = ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_vebt'_def
% 0.23/0.56 thf(fact_8521_succ__greatereq__min,axiom,
% 0.23/0.56 ! [Deg: nat,Mi: nat,X4: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.23/0.56 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.23/0.56 => ( ( ord_less_eq_nat @ Mi @ X4 )
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.56 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.56 @ ( if_option_nat
% 0.23/0.56 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 != none_nat )
% 0.23/0.56 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.23/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 @ ( if_option_nat
% 0.23/0.56 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 = none_nat )
% 0.23/0.56 @ none_nat
% 0.23/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.23/0.56 @ none_nat ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % succ_greatereq_min
% 0.23/0.56 thf(fact_8522_nat__less__as__int,axiom,
% 0.23/0.56 ( ord_less_nat
% 0.23/0.56 = ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % nat_less_as_int
% 0.23/0.56 thf(fact_8523_less__set__def,axiom,
% 0.23/0.56 ( ord_le2529575680413868914d_enat
% 0.23/0.56 = ( ^ [A6: set_Extended_enat,B7: set_Extended_enat] :
% 0.23/0.56 ( ord_le8499522857272258027enat_o
% 0.23/0.56 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 )
% 0.23/0.56 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ B7 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % less_set_def
% 0.23/0.56 thf(fact_8524_less__set__def,axiom,
% 0.23/0.56 ( ord_less_set_complex
% 0.23/0.56 = ( ^ [A6: set_complex,B7: set_complex] :
% 0.23/0.56 ( ord_less_complex_o
% 0.23/0.56 @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 0.23/0.56 @ ^ [X: complex] : ( member_complex @ X @ B7 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % less_set_def
% 0.23/0.56 thf(fact_8525_less__set__def,axiom,
% 0.23/0.56 ( ord_less_set_real
% 0.23/0.56 = ( ^ [A6: set_real,B7: set_real] :
% 0.23/0.56 ( ord_less_real_o
% 0.23/0.56 @ ^ [X: real] : ( member_real @ X @ A6 )
% 0.23/0.56 @ ^ [X: real] : ( member_real @ X @ B7 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % less_set_def
% 0.23/0.56 thf(fact_8526_less__set__def,axiom,
% 0.23/0.56 ( ord_less_set_nat
% 0.23/0.56 = ( ^ [A6: set_nat,B7: set_nat] :
% 0.23/0.56 ( ord_less_nat_o
% 0.23/0.56 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 0.23/0.56 @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % less_set_def
% 0.23/0.56 thf(fact_8527_less__set__def,axiom,
% 0.23/0.56 ( ord_less_set_int
% 0.23/0.56 = ( ^ [A6: set_int,B7: set_int] :
% 0.23/0.56 ( ord_less_int_o
% 0.23/0.56 @ ^ [X: int] : ( member_int @ X @ A6 )
% 0.23/0.56 @ ^ [X: int] : ( member_int @ X @ B7 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % less_set_def
% 0.23/0.56 thf(fact_8528_strict__subset__divisors__dvd,axiom,
% 0.23/0.56 ! [A: complex,B2: complex] :
% 0.23/0.56 ( ( ord_less_set_complex
% 0.23/0.56 @ ( collect_complex
% 0.23/0.56 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
% 0.23/0.56 @ ( collect_complex
% 0.23/0.56 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B2 ) ) )
% 0.23/0.56 = ( ( dvd_dvd_complex @ A @ B2 )
% 0.23/0.56 & ~ ( dvd_dvd_complex @ B2 @ A ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % strict_subset_divisors_dvd
% 0.23/0.56 thf(fact_8529_strict__subset__divisors__dvd,axiom,
% 0.23/0.56 ! [A: real,B2: real] :
% 0.23/0.56 ( ( ord_less_set_real
% 0.23/0.56 @ ( collect_real
% 0.23/0.56 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
% 0.23/0.56 @ ( collect_real
% 0.23/0.56 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B2 ) ) )
% 0.23/0.56 = ( ( dvd_dvd_real @ A @ B2 )
% 0.23/0.56 & ~ ( dvd_dvd_real @ B2 @ A ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % strict_subset_divisors_dvd
% 0.23/0.56 thf(fact_8530_strict__subset__divisors__dvd,axiom,
% 0.23/0.56 ! [A: nat,B2: nat] :
% 0.23/0.56 ( ( ord_less_set_nat
% 0.23/0.56 @ ( collect_nat
% 0.23/0.56 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
% 0.23/0.56 @ ( collect_nat
% 0.23/0.56 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B2 ) ) )
% 0.23/0.56 = ( ( dvd_dvd_nat @ A @ B2 )
% 0.23/0.56 & ~ ( dvd_dvd_nat @ B2 @ A ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % strict_subset_divisors_dvd
% 0.23/0.56 thf(fact_8531_strict__subset__divisors__dvd,axiom,
% 0.23/0.56 ! [A: int,B2: int] :
% 0.23/0.56 ( ( ord_less_set_int
% 0.23/0.56 @ ( collect_int
% 0.23/0.56 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
% 0.23/0.56 @ ( collect_int
% 0.23/0.56 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B2 ) ) )
% 0.23/0.56 = ( ( dvd_dvd_int @ A @ B2 )
% 0.23/0.56 & ~ ( dvd_dvd_int @ B2 @ A ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % strict_subset_divisors_dvd
% 0.23/0.56 thf(fact_8532_strict__subset__divisors__dvd,axiom,
% 0.23/0.56 ! [A: code_integer,B2: code_integer] :
% 0.23/0.56 ( ( ord_le1307284697595431911nteger
% 0.23/0.56 @ ( collect_Code_integer
% 0.23/0.56 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
% 0.23/0.56 @ ( collect_Code_integer
% 0.23/0.56 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B2 ) ) )
% 0.23/0.56 = ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.23/0.56 & ~ ( dvd_dvd_Code_integer @ B2 @ A ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % strict_subset_divisors_dvd
% 0.23/0.56 thf(fact_8533_set__diff__eq,axiom,
% 0.23/0.56 ( minus_925952699566721837d_enat
% 0.23/0.56 = ( ^ [A6: set_Extended_enat,B7: set_Extended_enat] :
% 0.23/0.56 ( collec4429806609662206161d_enat
% 0.23/0.56 @ ^ [X: extended_enat] :
% 0.23/0.56 ( ( member_Extended_enat @ X @ A6 )
% 0.23/0.56 & ~ ( member_Extended_enat @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_diff_eq
% 0.23/0.56 thf(fact_8534_set__diff__eq,axiom,
% 0.23/0.56 ( minus_811609699411566653omplex
% 0.23/0.56 = ( ^ [A6: set_complex,B7: set_complex] :
% 0.23/0.56 ( collect_complex
% 0.23/0.56 @ ^ [X: complex] :
% 0.23/0.56 ( ( member_complex @ X @ A6 )
% 0.23/0.56 & ~ ( member_complex @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_diff_eq
% 0.23/0.56 thf(fact_8535_set__diff__eq,axiom,
% 0.23/0.56 ( minus_minus_set_real
% 0.23/0.56 = ( ^ [A6: set_real,B7: set_real] :
% 0.23/0.56 ( collect_real
% 0.23/0.56 @ ^ [X: real] :
% 0.23/0.56 ( ( member_real @ X @ A6 )
% 0.23/0.56 & ~ ( member_real @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_diff_eq
% 0.23/0.56 thf(fact_8536_set__diff__eq,axiom,
% 0.23/0.56 ( minus_7954133019191499631st_nat
% 0.23/0.56 = ( ^ [A6: set_list_nat,B7: set_list_nat] :
% 0.23/0.56 ( collect_list_nat
% 0.23/0.56 @ ^ [X: list_nat] :
% 0.23/0.56 ( ( member_list_nat @ X @ A6 )
% 0.23/0.56 & ~ ( member_list_nat @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_diff_eq
% 0.23/0.56 thf(fact_8537_set__diff__eq,axiom,
% 0.23/0.56 ( minus_minus_set_int
% 0.23/0.56 = ( ^ [A6: set_int,B7: set_int] :
% 0.23/0.56 ( collect_int
% 0.23/0.56 @ ^ [X: int] :
% 0.23/0.56 ( ( member_int @ X @ A6 )
% 0.23/0.56 & ~ ( member_int @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_diff_eq
% 0.23/0.56 thf(fact_8538_set__diff__eq,axiom,
% 0.23/0.56 ( minus_minus_set_nat
% 0.23/0.56 = ( ^ [A6: set_nat,B7: set_nat] :
% 0.23/0.56 ( collect_nat
% 0.23/0.56 @ ^ [X: nat] :
% 0.23/0.56 ( ( member_nat @ X @ A6 )
% 0.23/0.56 & ~ ( member_nat @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_diff_eq
% 0.23/0.56 thf(fact_8539_minus__set__def,axiom,
% 0.23/0.56 ( minus_925952699566721837d_enat
% 0.23/0.56 = ( ^ [A6: set_Extended_enat,B7: set_Extended_enat] :
% 0.23/0.56 ( collec4429806609662206161d_enat
% 0.23/0.56 @ ( minus_2020553357622893040enat_o
% 0.23/0.56 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 )
% 0.23/0.56 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % minus_set_def
% 0.23/0.56 thf(fact_8540_minus__set__def,axiom,
% 0.23/0.56 ( minus_811609699411566653omplex
% 0.23/0.56 = ( ^ [A6: set_complex,B7: set_complex] :
% 0.23/0.56 ( collect_complex
% 0.23/0.56 @ ( minus_8727706125548526216plex_o
% 0.23/0.56 @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 0.23/0.56 @ ^ [X: complex] : ( member_complex @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % minus_set_def
% 0.23/0.56 thf(fact_8541_minus__set__def,axiom,
% 0.23/0.56 ( minus_minus_set_real
% 0.23/0.56 = ( ^ [A6: set_real,B7: set_real] :
% 0.23/0.56 ( collect_real
% 0.23/0.56 @ ( minus_minus_real_o
% 0.23/0.56 @ ^ [X: real] : ( member_real @ X @ A6 )
% 0.23/0.56 @ ^ [X: real] : ( member_real @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % minus_set_def
% 0.23/0.56 thf(fact_8542_minus__set__def,axiom,
% 0.23/0.56 ( minus_7954133019191499631st_nat
% 0.23/0.56 = ( ^ [A6: set_list_nat,B7: set_list_nat] :
% 0.23/0.56 ( collect_list_nat
% 0.23/0.56 @ ( minus_1139252259498527702_nat_o
% 0.23/0.56 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
% 0.23/0.56 @ ^ [X: list_nat] : ( member_list_nat @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % minus_set_def
% 0.23/0.56 thf(fact_8543_minus__set__def,axiom,
% 0.23/0.56 ( minus_minus_set_int
% 0.23/0.56 = ( ^ [A6: set_int,B7: set_int] :
% 0.23/0.56 ( collect_int
% 0.23/0.56 @ ( minus_minus_int_o
% 0.23/0.56 @ ^ [X: int] : ( member_int @ X @ A6 )
% 0.23/0.56 @ ^ [X: int] : ( member_int @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % minus_set_def
% 0.23/0.56 thf(fact_8544_minus__set__def,axiom,
% 0.23/0.56 ( minus_minus_set_nat
% 0.23/0.56 = ( ^ [A6: set_nat,B7: set_nat] :
% 0.23/0.56 ( collect_nat
% 0.23/0.56 @ ( minus_minus_nat_o
% 0.23/0.56 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 0.23/0.56 @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % minus_set_def
% 0.23/0.56 thf(fact_8545_numeral__code_I2_J,axiom,
% 0.23/0.56 ! [N: num] :
% 0.23/0.56 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 0.23/0.56 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % numeral_code(2)
% 0.23/0.56 thf(fact_8546_numeral__code_I2_J,axiom,
% 0.23/0.56 ! [N: num] :
% 0.23/0.56 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 0.23/0.56 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % numeral_code(2)
% 0.23/0.56 thf(fact_8547_numeral__code_I2_J,axiom,
% 0.23/0.56 ! [N: num] :
% 0.23/0.56 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 0.23/0.56 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % numeral_code(2)
% 0.23/0.56 thf(fact_8548_numeral__code_I2_J,axiom,
% 0.23/0.56 ! [N: num] :
% 0.23/0.56 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.23/0.56 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % numeral_code(2)
% 0.23/0.56 thf(fact_8549_numeral__code_I2_J,axiom,
% 0.23/0.56 ! [N: num] :
% 0.23/0.56 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 0.23/0.56 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % numeral_code(2)
% 0.23/0.56 thf(fact_8550_lambda__zero,axiom,
% 0.23/0.56 ( ( ^ [H: real] : zero_zero_real )
% 0.23/0.56 = ( times_times_real @ zero_zero_real ) ) ).
% 0.23/0.56
% 0.23/0.56 % lambda_zero
% 0.23/0.56 thf(fact_8551_lambda__zero,axiom,
% 0.23/0.56 ( ( ^ [H: rat] : zero_zero_rat )
% 0.23/0.56 = ( times_times_rat @ zero_zero_rat ) ) ).
% 0.23/0.56
% 0.23/0.56 % lambda_zero
% 0.23/0.56 thf(fact_8552_lambda__zero,axiom,
% 0.23/0.56 ( ( ^ [H: nat] : zero_zero_nat )
% 0.23/0.56 = ( times_times_nat @ zero_zero_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % lambda_zero
% 0.23/0.56 thf(fact_8553_lambda__zero,axiom,
% 0.23/0.56 ( ( ^ [H: int] : zero_zero_int )
% 0.23/0.56 = ( times_times_int @ zero_zero_int ) ) ).
% 0.23/0.56
% 0.23/0.56 % lambda_zero
% 0.23/0.56 thf(fact_8554_mult__commute__abs,axiom,
% 0.23/0.56 ! [C: real] :
% 0.23/0.56 ( ( ^ [X: real] : ( times_times_real @ X @ C ) )
% 0.23/0.56 = ( times_times_real @ C ) ) ).
% 0.23/0.56
% 0.23/0.56 % mult_commute_abs
% 0.23/0.56 thf(fact_8555_mult__commute__abs,axiom,
% 0.23/0.56 ! [C: rat] :
% 0.23/0.56 ( ( ^ [X: rat] : ( times_times_rat @ X @ C ) )
% 0.23/0.56 = ( times_times_rat @ C ) ) ).
% 0.23/0.56
% 0.23/0.56 % mult_commute_abs
% 0.23/0.56 thf(fact_8556_mult__commute__abs,axiom,
% 0.23/0.56 ! [C: nat] :
% 0.23/0.56 ( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
% 0.23/0.56 = ( times_times_nat @ C ) ) ).
% 0.23/0.56
% 0.23/0.56 % mult_commute_abs
% 0.23/0.56 thf(fact_8557_mult__commute__abs,axiom,
% 0.23/0.56 ! [C: int] :
% 0.23/0.56 ( ( ^ [X: int] : ( times_times_int @ X @ C ) )
% 0.23/0.56 = ( times_times_int @ C ) ) ).
% 0.23/0.56
% 0.23/0.56 % mult_commute_abs
% 0.23/0.56 thf(fact_8558_lambda__one,axiom,
% 0.23/0.56 ( ( ^ [X: complex] : X )
% 0.23/0.56 = ( times_times_complex @ one_one_complex ) ) ).
% 0.23/0.56
% 0.23/0.56 % lambda_one
% 0.23/0.56 thf(fact_8559_lambda__one,axiom,
% 0.23/0.56 ( ( ^ [X: real] : X )
% 0.23/0.56 = ( times_times_real @ one_one_real ) ) ).
% 0.23/0.56
% 0.23/0.56 % lambda_one
% 0.23/0.56 thf(fact_8560_lambda__one,axiom,
% 0.23/0.56 ( ( ^ [X: rat] : X )
% 0.23/0.56 = ( times_times_rat @ one_one_rat ) ) ).
% 0.23/0.56
% 0.23/0.56 % lambda_one
% 0.23/0.56 thf(fact_8561_lambda__one,axiom,
% 0.23/0.56 ( ( ^ [X: nat] : X )
% 0.23/0.56 = ( times_times_nat @ one_one_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % lambda_one
% 0.23/0.56 thf(fact_8562_lambda__one,axiom,
% 0.23/0.56 ( ( ^ [X: int] : X )
% 0.23/0.56 = ( times_times_int @ one_one_int ) ) ).
% 0.23/0.56
% 0.23/0.56 % lambda_one
% 0.23/0.56 thf(fact_8563_subset__divisors__dvd,axiom,
% 0.23/0.56 ! [A: complex,B2: complex] :
% 0.23/0.56 ( ( ord_le211207098394363844omplex
% 0.23/0.56 @ ( collect_complex
% 0.23/0.56 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
% 0.23/0.56 @ ( collect_complex
% 0.23/0.56 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B2 ) ) )
% 0.23/0.56 = ( dvd_dvd_complex @ A @ B2 ) ) ).
% 0.23/0.56
% 0.23/0.56 % subset_divisors_dvd
% 0.23/0.56 thf(fact_8564_subset__divisors__dvd,axiom,
% 0.23/0.56 ! [A: real,B2: real] :
% 0.23/0.56 ( ( ord_less_eq_set_real
% 0.23/0.56 @ ( collect_real
% 0.23/0.56 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
% 0.23/0.56 @ ( collect_real
% 0.23/0.56 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B2 ) ) )
% 0.23/0.56 = ( dvd_dvd_real @ A @ B2 ) ) ).
% 0.23/0.56
% 0.23/0.56 % subset_divisors_dvd
% 0.23/0.56 thf(fact_8565_subset__divisors__dvd,axiom,
% 0.23/0.56 ! [A: int,B2: int] :
% 0.23/0.56 ( ( ord_less_eq_set_int
% 0.23/0.56 @ ( collect_int
% 0.23/0.56 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
% 0.23/0.56 @ ( collect_int
% 0.23/0.56 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B2 ) ) )
% 0.23/0.56 = ( dvd_dvd_int @ A @ B2 ) ) ).
% 0.23/0.56
% 0.23/0.56 % subset_divisors_dvd
% 0.23/0.56 thf(fact_8566_subset__divisors__dvd,axiom,
% 0.23/0.56 ! [A: code_integer,B2: code_integer] :
% 0.23/0.56 ( ( ord_le7084787975880047091nteger
% 0.23/0.56 @ ( collect_Code_integer
% 0.23/0.56 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
% 0.23/0.56 @ ( collect_Code_integer
% 0.23/0.56 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B2 ) ) )
% 0.23/0.56 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 0.23/0.56
% 0.23/0.56 % subset_divisors_dvd
% 0.23/0.56 thf(fact_8567_subset__divisors__dvd,axiom,
% 0.23/0.56 ! [A: nat,B2: nat] :
% 0.23/0.56 ( ( ord_less_eq_set_nat
% 0.23/0.56 @ ( collect_nat
% 0.23/0.56 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
% 0.23/0.56 @ ( collect_nat
% 0.23/0.56 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B2 ) ) )
% 0.23/0.56 = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 0.23/0.56
% 0.23/0.56 % subset_divisors_dvd
% 0.23/0.56 thf(fact_8568_nat__leq__as__int,axiom,
% 0.23/0.56 ( ord_less_eq_nat
% 0.23/0.56 = ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % nat_leq_as_int
% 0.23/0.56 thf(fact_8569_power__numeral__even,axiom,
% 0.23/0.56 ! [Z: complex,W: num] :
% 0.23/0.56 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.23/0.56 = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % power_numeral_even
% 0.23/0.56 thf(fact_8570_power__numeral__even,axiom,
% 0.23/0.56 ! [Z: real,W: num] :
% 0.23/0.56 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.23/0.56 = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % power_numeral_even
% 0.23/0.56 thf(fact_8571_power__numeral__even,axiom,
% 0.23/0.56 ! [Z: rat,W: num] :
% 0.23/0.56 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.23/0.56 = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % power_numeral_even
% 0.23/0.56 thf(fact_8572_power__numeral__even,axiom,
% 0.23/0.56 ! [Z: nat,W: num] :
% 0.23/0.56 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.23/0.56 = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % power_numeral_even
% 0.23/0.56 thf(fact_8573_power__numeral__even,axiom,
% 0.23/0.56 ! [Z: int,W: num] :
% 0.23/0.56 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.23/0.56 = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % power_numeral_even
% 0.23/0.56 thf(fact_8574_numeral__code_I3_J,axiom,
% 0.23/0.56 ! [N: num] :
% 0.23/0.56 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 0.23/0.56 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 0.23/0.56
% 0.23/0.56 % numeral_code(3)
% 0.23/0.56 thf(fact_8575_numeral__code_I3_J,axiom,
% 0.23/0.56 ! [N: num] :
% 0.23/0.56 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 0.23/0.56 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 0.23/0.56
% 0.23/0.56 % numeral_code(3)
% 0.23/0.56 thf(fact_8576_numeral__code_I3_J,axiom,
% 0.23/0.56 ! [N: num] :
% 0.23/0.56 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 0.23/0.56 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 0.23/0.56
% 0.23/0.56 % numeral_code(3)
% 0.23/0.56 thf(fact_8577_numeral__code_I3_J,axiom,
% 0.23/0.56 ! [N: num] :
% 0.23/0.56 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.23/0.56 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % numeral_code(3)
% 0.23/0.56 thf(fact_8578_numeral__code_I3_J,axiom,
% 0.23/0.56 ! [N: num] :
% 0.23/0.56 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 0.23/0.56 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 0.23/0.56
% 0.23/0.56 % numeral_code(3)
% 0.23/0.56 thf(fact_8579_power__numeral__odd,axiom,
% 0.23/0.56 ! [Z: complex,W: num] :
% 0.23/0.56 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.23/0.56 = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % power_numeral_odd
% 0.23/0.56 thf(fact_8580_power__numeral__odd,axiom,
% 0.23/0.56 ! [Z: real,W: num] :
% 0.23/0.56 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.23/0.56 = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % power_numeral_odd
% 0.23/0.56 thf(fact_8581_power__numeral__odd,axiom,
% 0.23/0.56 ! [Z: rat,W: num] :
% 0.23/0.56 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.23/0.56 = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % power_numeral_odd
% 0.23/0.56 thf(fact_8582_power__numeral__odd,axiom,
% 0.23/0.56 ! [Z: nat,W: num] :
% 0.23/0.56 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.23/0.56 = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % power_numeral_odd
% 0.23/0.56 thf(fact_8583_power__numeral__odd,axiom,
% 0.23/0.56 ! [Z: int,W: num] :
% 0.23/0.56 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.23/0.56 = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % power_numeral_odd
% 0.23/0.56 thf(fact_8584_nat__plus__as__int,axiom,
% 0.23/0.56 ( plus_plus_nat
% 0.23/0.56 = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % nat_plus_as_int
% 0.23/0.56 thf(fact_8585_nat__times__as__int,axiom,
% 0.23/0.56 ( times_times_nat
% 0.23/0.56 = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % nat_times_as_int
% 0.23/0.56 thf(fact_8586_nat__minus__as__int,axiom,
% 0.23/0.56 ( minus_minus_nat
% 0.23/0.56 = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % nat_minus_as_int
% 0.23/0.56 thf(fact_8587_nat__div__as__int,axiom,
% 0.23/0.56 ( divide_divide_nat
% 0.23/0.56 = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % nat_div_as_int
% 0.23/0.56 thf(fact_8588_nat__mod__as__int,axiom,
% 0.23/0.56 ( modulo_modulo_nat
% 0.23/0.56 = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % nat_mod_as_int
% 0.23/0.56 thf(fact_8589_set__conv__nth,axiom,
% 0.23/0.56 ( set_complex2
% 0.23/0.56 = ( ^ [Xs: list_complex] :
% 0.23/0.56 ( collect_complex
% 0.23/0.56 @ ^ [Uu: complex] :
% 0.23/0.56 ? [I2: nat] :
% 0.23/0.56 ( ( Uu
% 0.23/0.56 = ( nth_complex @ Xs @ I2 ) )
% 0.23/0.56 & ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_conv_nth
% 0.23/0.56 thf(fact_8590_set__conv__nth,axiom,
% 0.23/0.56 ( set_real2
% 0.23/0.56 = ( ^ [Xs: list_real] :
% 0.23/0.56 ( collect_real
% 0.23/0.56 @ ^ [Uu: real] :
% 0.23/0.56 ? [I2: nat] :
% 0.23/0.56 ( ( Uu
% 0.23/0.56 = ( nth_real @ Xs @ I2 ) )
% 0.23/0.56 & ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_conv_nth
% 0.23/0.56 thf(fact_8591_set__conv__nth,axiom,
% 0.23/0.56 ( set_list_nat2
% 0.23/0.56 = ( ^ [Xs: list_list_nat] :
% 0.23/0.56 ( collect_list_nat
% 0.23/0.56 @ ^ [Uu: list_nat] :
% 0.23/0.56 ? [I2: nat] :
% 0.23/0.56 ( ( Uu
% 0.23/0.56 = ( nth_list_nat @ Xs @ I2 ) )
% 0.23/0.56 & ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_conv_nth
% 0.23/0.56 thf(fact_8592_set__conv__nth,axiom,
% 0.23/0.56 ( set_VEBT_VEBT2
% 0.23/0.56 = ( ^ [Xs: list_VEBT_VEBT] :
% 0.23/0.56 ( collect_VEBT_VEBT
% 0.23/0.56 @ ^ [Uu: vEBT_VEBT] :
% 0.23/0.56 ? [I2: nat] :
% 0.23/0.56 ( ( Uu
% 0.23/0.56 = ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 0.23/0.56 & ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_conv_nth
% 0.23/0.56 thf(fact_8593_set__conv__nth,axiom,
% 0.23/0.56 ( set_o2
% 0.23/0.56 = ( ^ [Xs: list_o] :
% 0.23/0.56 ( collect_o
% 0.23/0.56 @ ^ [Uu: $o] :
% 0.23/0.56 ? [I2: nat] :
% 0.23/0.56 ( ( Uu
% 0.23/0.56 = ( nth_o @ Xs @ I2 ) )
% 0.23/0.56 & ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_conv_nth
% 0.23/0.56 thf(fact_8594_set__conv__nth,axiom,
% 0.23/0.56 ( set_nat2
% 0.23/0.56 = ( ^ [Xs: list_nat] :
% 0.23/0.56 ( collect_nat
% 0.23/0.56 @ ^ [Uu: nat] :
% 0.23/0.56 ? [I2: nat] :
% 0.23/0.56 ( ( Uu
% 0.23/0.56 = ( nth_nat @ Xs @ I2 ) )
% 0.23/0.56 & ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_conv_nth
% 0.23/0.56 thf(fact_8595_set__conv__nth,axiom,
% 0.23/0.56 ( set_int2
% 0.23/0.56 = ( ^ [Xs: list_int] :
% 0.23/0.56 ( collect_int
% 0.23/0.56 @ ^ [Uu: int] :
% 0.23/0.56 ? [I2: nat] :
% 0.23/0.56 ( ( Uu
% 0.23/0.56 = ( nth_int @ Xs @ I2 ) )
% 0.23/0.56 & ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_conv_nth
% 0.23/0.56 thf(fact_8596_diff__nat__eq__if,axiom,
% 0.23/0.56 ! [Z6: int,Z: int] :
% 0.23/0.56 ( ( ( ord_less_int @ Z6 @ zero_zero_int )
% 0.23/0.56 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 0.23/0.56 = ( nat2 @ Z ) ) )
% 0.23/0.56 & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
% 0.23/0.56 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 0.23/0.56 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % diff_nat_eq_if
% 0.23/0.56 thf(fact_8597_set__decode__def,axiom,
% 0.23/0.56 ( nat_set_decode
% 0.23/0.56 = ( ^ [X: nat] :
% 0.23/0.56 ( collect_nat
% 0.23/0.56 @ ^ [N2: nat] :
% 0.23/0.56 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % set_decode_def
% 0.23/0.56 thf(fact_8598_signed__take__bit__code,axiom,
% 0.23/0.56 ( bit_ri6519982836138164636nteger
% 0.23/0.56 = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( bit_se9216721137139052372nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A4 ) @ N2 ) @ ( plus_p5714425477246183910nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A4 ) @ ( bit_se7788150548672797655nteger @ ( suc @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) @ ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A4 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % signed_take_bit_code
% 0.23/0.56 thf(fact_8599_signed__take__bit__code,axiom,
% 0.23/0.56 ( bit_ri631733984087533419it_int
% 0.23/0.56 = ( ^ [N2: nat,A4: int] : ( if_int @ ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A4 ) @ N2 ) @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A4 ) @ ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ one_one_int ) ) ) @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A4 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % signed_take_bit_code
% 0.23/0.56 thf(fact_8600_pochhammer__code,axiom,
% 0.23/0.56 ( comm_s2602460028002588243omplex
% 0.23/0.56 = ( ^ [A4: complex,N2: nat] :
% 0.23/0.56 ( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
% 0.23/0.56 @ ( set_fo1517530859248394432omplex
% 0.23/0.56 @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ O ) ) )
% 0.23/0.56 @ zero_zero_nat
% 0.23/0.56 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.23/0.56 @ one_one_complex ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_code
% 0.23/0.56 thf(fact_8601_pochhammer__code,axiom,
% 0.23/0.56 ( comm_s7457072308508201937r_real
% 0.23/0.56 = ( ^ [A4: real,N2: nat] :
% 0.23/0.56 ( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
% 0.23/0.56 @ ( set_fo3111899725591712190t_real
% 0.23/0.56 @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ O ) ) )
% 0.23/0.56 @ zero_zero_nat
% 0.23/0.56 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.23/0.56 @ one_one_real ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_code
% 0.23/0.56 thf(fact_8602_pochhammer__code,axiom,
% 0.23/0.56 ( comm_s4028243227959126397er_rat
% 0.23/0.56 = ( ^ [A4: rat,N2: nat] :
% 0.23/0.56 ( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
% 0.23/0.56 @ ( set_fo1949268297981939178at_rat
% 0.23/0.56 @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ O ) ) )
% 0.23/0.56 @ zero_zero_nat
% 0.23/0.56 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.23/0.56 @ one_one_rat ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_code
% 0.23/0.56 thf(fact_8603_pochhammer__code,axiom,
% 0.23/0.56 ( comm_s4660882817536571857er_int
% 0.23/0.56 = ( ^ [A4: int,N2: nat] :
% 0.23/0.56 ( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
% 0.23/0.56 @ ( set_fo2581907887559384638at_int
% 0.23/0.56 @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ O ) ) )
% 0.23/0.56 @ zero_zero_nat
% 0.23/0.56 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.23/0.56 @ one_one_int ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_code
% 0.23/0.56 thf(fact_8604_pochhammer__code,axiom,
% 0.23/0.56 ( comm_s4663373288045622133er_nat
% 0.23/0.56 = ( ^ [A4: nat,N2: nat] :
% 0.23/0.56 ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
% 0.23/0.56 @ ( set_fo2584398358068434914at_nat
% 0.23/0.56 @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 0.23/0.56 @ zero_zero_nat
% 0.23/0.56 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.23/0.56 @ one_one_nat ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_code
% 0.23/0.56 thf(fact_8605_gbinomial__code,axiom,
% 0.23/0.56 ( gbinomial_complex
% 0.23/0.56 = ( ^ [A4: complex,K3: nat] :
% 0.23/0.56 ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
% 0.23/0.56 @ ( divide1717551699836669952omplex
% 0.23/0.56 @ ( set_fo1517530859248394432omplex
% 0.23/0.56 @ ^ [L: nat] : ( times_times_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ L ) ) )
% 0.23/0.56 @ zero_zero_nat
% 0.23/0.56 @ ( minus_minus_nat @ K3 @ one_one_nat )
% 0.23/0.56 @ one_one_complex )
% 0.23/0.56 @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % gbinomial_code
% 0.23/0.56 thf(fact_8606_gbinomial__code,axiom,
% 0.23/0.56 ( gbinomial_rat
% 0.23/0.56 = ( ^ [A4: rat,K3: nat] :
% 0.23/0.56 ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
% 0.23/0.56 @ ( divide_divide_rat
% 0.23/0.56 @ ( set_fo1949268297981939178at_rat
% 0.23/0.56 @ ^ [L: nat] : ( times_times_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ L ) ) )
% 0.23/0.56 @ zero_zero_nat
% 0.23/0.56 @ ( minus_minus_nat @ K3 @ one_one_nat )
% 0.23/0.56 @ one_one_rat )
% 0.23/0.56 @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % gbinomial_code
% 0.23/0.56 thf(fact_8607_gbinomial__code,axiom,
% 0.23/0.56 ( gbinomial_real
% 0.23/0.56 = ( ^ [A4: real,K3: nat] :
% 0.23/0.56 ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
% 0.23/0.56 @ ( divide_divide_real
% 0.23/0.56 @ ( set_fo3111899725591712190t_real
% 0.23/0.56 @ ^ [L: nat] : ( times_times_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ L ) ) )
% 0.23/0.56 @ zero_zero_nat
% 0.23/0.56 @ ( minus_minus_nat @ K3 @ one_one_nat )
% 0.23/0.56 @ one_one_real )
% 0.23/0.56 @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % gbinomial_code
% 0.23/0.56 thf(fact_8608_vebt__member_Osimps_I5_J,axiom,
% 0.23/0.56 ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
% 0.23/0.56 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.56 = ( ( X4 != Mi )
% 0.23/0.56 => ( ( X4 != Ma )
% 0.23/0.56 => ( ~ ( ord_less_nat @ X4 @ Mi )
% 0.23/0.56 & ( ~ ( ord_less_nat @ X4 @ Mi )
% 0.23/0.56 => ( ~ ( ord_less_nat @ Ma @ X4 )
% 0.23/0.56 & ( ~ ( ord_less_nat @ Ma @ X4 )
% 0.23/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.56 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_member.simps(5)
% 0.23/0.56 thf(fact_8609_vebt__succ_Osimps_I6_J,axiom,
% 0.23/0.56 ! [X4: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.23/0.56 ( ( ( ord_less_nat @ X4 @ Mi )
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.56 = ( some_nat @ Mi ) ) )
% 0.23/0.56 & ( ~ ( ord_less_nat @ X4 @ Mi )
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.56 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.56 @ ( if_option_nat
% 0.23/0.56 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 != none_nat )
% 0.23/0.56 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.23/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 @ ( if_option_nat
% 0.23/0.56 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 = none_nat )
% 0.23/0.56 @ none_nat
% 0.23/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.23/0.56 @ none_nat ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_succ.simps(6)
% 0.23/0.56 thf(fact_8610_of__int__code__if,axiom,
% 0.23/0.56 ( ring_1_of_int_real
% 0.23/0.56 = ( ^ [K3: int] :
% 0.23/0.56 ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
% 0.23/0.56 @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
% 0.23/0.56 @ ( if_real
% 0.23/0.56 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.56 = zero_zero_int )
% 0.23/0.56 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % of_int_code_if
% 0.23/0.56 thf(fact_8611_of__int__code__if,axiom,
% 0.23/0.56 ( ring_1_of_int_int
% 0.23/0.56 = ( ^ [K3: int] :
% 0.23/0.56 ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
% 0.23/0.56 @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
% 0.23/0.56 @ ( if_int
% 0.23/0.56 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.56 = zero_zero_int )
% 0.23/0.56 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % of_int_code_if
% 0.23/0.56 thf(fact_8612_of__int__code__if,axiom,
% 0.23/0.56 ( ring_17405671764205052669omplex
% 0.23/0.56 = ( ^ [K3: int] :
% 0.23/0.56 ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
% 0.23/0.56 @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
% 0.23/0.56 @ ( if_complex
% 0.23/0.56 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.56 = zero_zero_int )
% 0.23/0.56 @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % of_int_code_if
% 0.23/0.56 thf(fact_8613_of__int__code__if,axiom,
% 0.23/0.56 ( ring_18347121197199848620nteger
% 0.23/0.56 = ( ^ [K3: int] :
% 0.23/0.56 ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.23/0.56 @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
% 0.23/0.56 @ ( if_Code_integer
% 0.23/0.56 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.56 = zero_zero_int )
% 0.23/0.56 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % of_int_code_if
% 0.23/0.56 thf(fact_8614_of__int__code__if,axiom,
% 0.23/0.56 ( ring_1_of_int_rat
% 0.23/0.56 = ( ^ [K3: int] :
% 0.23/0.56 ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
% 0.23/0.56 @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
% 0.23/0.56 @ ( if_rat
% 0.23/0.56 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.56 = zero_zero_int )
% 0.23/0.56 @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % of_int_code_if
% 0.23/0.56 thf(fact_8615_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 0.23/0.56 ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
% 0.23/0.56 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X4 )
% 0.23/0.56 = ( ( X4 = Mi )
% 0.23/0.56 | ( X4 = Ma )
% 0.23/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.membermima.simps(4)
% 0.23/0.56 thf(fact_8616_monoseq__arctan__series,axiom,
% 0.23/0.56 ! [X4: real] :
% 0.23/0.56 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.56 => ( topolo6980174941875973593q_real
% 0.23/0.56 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % monoseq_arctan_series
% 0.23/0.56 thf(fact_8617_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 0.23/0.56 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
% 0.23/0.56 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X4 )
% 0.23/0.56 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.56 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.naive_member.simps(3)
% 0.23/0.56 thf(fact_8618_buildup__nothing__in__min__max,axiom,
% 0.23/0.56 ! [N: nat,X4: nat] :
% 0.23/0.56 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X4 ) ).
% 0.23/0.56
% 0.23/0.56 % buildup_nothing_in_min_max
% 0.23/0.56 thf(fact_8619_buildup__nothing__in__leaf,axiom,
% 0.23/0.56 ! [N: nat,X4: nat] :
% 0.23/0.56 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X4 ) ).
% 0.23/0.56
% 0.23/0.56 % buildup_nothing_in_leaf
% 0.23/0.56 thf(fact_8620_both__member__options__def,axiom,
% 0.23/0.56 ( vEBT_V8194947554948674370ptions
% 0.23/0.56 = ( ^ [T3: vEBT_VEBT,X: nat] :
% 0.23/0.56 ( ( vEBT_V5719532721284313246member @ T3 @ X )
% 0.23/0.56 | ( vEBT_VEBT_membermima @ T3 @ X ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % both_member_options_def
% 0.23/0.56 thf(fact_8621_member__valid__both__member__options,axiom,
% 0.23/0.56 ! [Tree: vEBT_VEBT,N: nat,X4: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ Tree @ N )
% 0.23/0.56 => ( ( vEBT_vebt_member @ Tree @ X4 )
% 0.23/0.56 => ( ( vEBT_V5719532721284313246member @ Tree @ X4 )
% 0.23/0.56 | ( vEBT_VEBT_membermima @ Tree @ X4 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % member_valid_both_member_options
% 0.23/0.56 thf(fact_8622_monoseq__realpow,axiom,
% 0.23/0.56 ! [X4: real] :
% 0.23/0.56 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.56 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.56 => ( topolo6980174941875973593q_real @ ( power_power_real @ X4 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % monoseq_realpow
% 0.23/0.56 thf(fact_8623_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.56 ( ( vEBT_VEBT_membermima @ X4 @ Xa )
% 0.23/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.23/0.56 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.23/0.56 => ~ ( ( Xa = Mi2 )
% 0.23/0.56 | ( Xa = Ma2 ) ) )
% 0.23/0.56 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [Vc2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.23/0.56 => ~ ( ( Xa = Mi2 )
% 0.23/0.56 | ( Xa = Ma2 )
% 0.23/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( 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 ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.23/0.56 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [Vd2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.23/0.56 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( 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 ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.membermima.elims(2)
% 0.23/0.56 thf(fact_8624_vebt__member_Oelims_I2_J,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.56 ( ( vEBT_vebt_member @ X4 @ Xa )
% 0.23/0.56 => ( ! [A5: $o,B5: $o] :
% 0.23/0.56 ( ( X4
% 0.23/0.56 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.56 => ~ ( ( ( Xa = zero_zero_nat )
% 0.23/0.56 => A5 )
% 0.23/0.56 & ( ( Xa != zero_zero_nat )
% 0.23/0.56 => ( ( ( Xa = one_one_nat )
% 0.23/0.56 => B5 )
% 0.23/0.56 & ( Xa = one_one_nat ) ) ) ) )
% 0.23/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [Summary2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.56 => ~ ( ( Xa != Mi2 )
% 0.23/0.56 => ( ( Xa != Ma2 )
% 0.23/0.56 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.56 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.56 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.56 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_member.elims(2)
% 0.23/0.56 thf(fact_8625_vebt__insert_Osimps_I3_J,axiom,
% 0.23/0.56 ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
% 0.23/0.56 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X4 )
% 0.23/0.56 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_insert.simps(3)
% 0.23/0.56 thf(fact_8626_pochhammer__times__pochhammer__half,axiom,
% 0.23/0.56 ! [Z: complex,N: nat] :
% 0.23/0.56 ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.23/0.56 = ( groups6464643781859351333omplex
% 0.23/0.56 @ ^ [K3: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_times_pochhammer_half
% 0.23/0.56 thf(fact_8627_pochhammer__times__pochhammer__half,axiom,
% 0.23/0.56 ! [Z: real,N: nat] :
% 0.23/0.56 ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.23/0.56 = ( groups129246275422532515t_real
% 0.23/0.56 @ ^ [K3: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_times_pochhammer_half
% 0.23/0.56 thf(fact_8628_pochhammer__times__pochhammer__half,axiom,
% 0.23/0.56 ! [Z: rat,N: nat] :
% 0.23/0.56 ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.23/0.56 = ( groups73079841787564623at_rat
% 0.23/0.56 @ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_times_pochhammer_half
% 0.23/0.56 thf(fact_8629_Leaf__0__not,axiom,
% 0.23/0.56 ! [A: $o,B2: $o] :
% 0.23/0.56 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ zero_zero_nat ) ).
% 0.23/0.56
% 0.23/0.56 % Leaf_0_not
% 0.23/0.56 thf(fact_8630_deg__1__Leafy,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT,N: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.56 => ( ( N = one_one_nat )
% 0.23/0.56 => ? [A5: $o,B5: $o] :
% 0.23/0.56 ( T
% 0.23/0.56 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % deg_1_Leafy
% 0.23/0.56 thf(fact_8631_deg__1__Leaf,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.23/0.56 => ? [A5: $o,B5: $o] :
% 0.23/0.56 ( T
% 0.23/0.56 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % deg_1_Leaf
% 0.23/0.56 thf(fact_8632_deg1Leaf,axiom,
% 0.23/0.56 ! [T: vEBT_VEBT] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.23/0.56 = ( ? [A4: $o,B3: $o] :
% 0.23/0.56 ( T
% 0.23/0.56 = ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % deg1Leaf
% 0.23/0.56 thf(fact_8633_prod_Ocl__ivl__Suc,axiom,
% 0.23/0.56 ! [N: nat,M: nat,G: nat > complex] :
% 0.23/0.56 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.23/0.56 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = one_one_complex ) )
% 0.23/0.56 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.23/0.56 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.cl_ivl_Suc
% 0.23/0.56 thf(fact_8634_prod_Ocl__ivl__Suc,axiom,
% 0.23/0.56 ! [N: nat,M: nat,G: nat > real] :
% 0.23/0.56 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.23/0.56 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = one_one_real ) )
% 0.23/0.56 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.23/0.56 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.cl_ivl_Suc
% 0.23/0.56 thf(fact_8635_prod_Ocl__ivl__Suc,axiom,
% 0.23/0.56 ! [N: nat,M: nat,G: nat > rat] :
% 0.23/0.56 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.23/0.56 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = one_one_rat ) )
% 0.23/0.56 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.23/0.56 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.cl_ivl_Suc
% 0.23/0.56 thf(fact_8636_prod_Ocl__ivl__Suc,axiom,
% 0.23/0.56 ! [N: nat,M: nat,G: nat > nat] :
% 0.23/0.56 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.23/0.56 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = one_one_nat ) )
% 0.23/0.56 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.23/0.56 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.cl_ivl_Suc
% 0.23/0.56 thf(fact_8637_prod_Ocl__ivl__Suc,axiom,
% 0.23/0.56 ! [N: nat,M: nat,G: nat > int] :
% 0.23/0.56 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.23/0.56 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = one_one_int ) )
% 0.23/0.56 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.23/0.56 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.cl_ivl_Suc
% 0.23/0.56 thf(fact_8638_mod__prod__eq,axiom,
% 0.23/0.56 ! [F: nat > nat,A: nat,A3: set_nat] :
% 0.23/0.56 ( ( modulo_modulo_nat
% 0.23/0.56 @ ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [I2: nat] : ( modulo_modulo_nat @ ( F @ I2 ) @ A )
% 0.23/0.56 @ A3 )
% 0.23/0.56 @ A )
% 0.23/0.56 = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ A ) ) ).
% 0.23/0.56
% 0.23/0.56 % mod_prod_eq
% 0.23/0.56 thf(fact_8639_mod__prod__eq,axiom,
% 0.23/0.56 ! [F: nat > int,A: int,A3: set_nat] :
% 0.23/0.56 ( ( modulo_modulo_int
% 0.23/0.56 @ ( groups705719431365010083at_int
% 0.23/0.56 @ ^ [I2: nat] : ( modulo_modulo_int @ ( F @ I2 ) @ A )
% 0.23/0.56 @ A3 )
% 0.23/0.56 @ A )
% 0.23/0.56 = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A3 ) @ A ) ) ).
% 0.23/0.56
% 0.23/0.56 % mod_prod_eq
% 0.23/0.56 thf(fact_8640_mod__prod__eq,axiom,
% 0.23/0.56 ! [F: int > int,A: int,A3: set_int] :
% 0.23/0.56 ( ( modulo_modulo_int
% 0.23/0.56 @ ( groups1705073143266064639nt_int
% 0.23/0.56 @ ^ [I2: int] : ( modulo_modulo_int @ ( F @ I2 ) @ A )
% 0.23/0.56 @ A3 )
% 0.23/0.56 @ A )
% 0.23/0.56 = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A3 ) @ A ) ) ).
% 0.23/0.56
% 0.23/0.56 % mod_prod_eq
% 0.23/0.56 thf(fact_8641_vebt__insert_Ocases,axiom,
% 0.23/0.56 ! [X4: produc9072475918466114483BT_nat] :
% 0.23/0.56 ( ! [A5: $o,B5: $o,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X3 ) )
% 0.23/0.56 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ X3 ) )
% 0.23/0.56 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X3 ) )
% 0.23/0.56 => ( ! [V3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) )
% 0.23/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_insert.cases
% 0.23/0.56 thf(fact_8642_vebt__insert_Osimps_I1_J,axiom,
% 0.23/0.56 ! [X4: nat,A: $o,B2: $o] :
% 0.23/0.56 ( ( ( X4 = zero_zero_nat )
% 0.23/0.56 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
% 0.23/0.56 = ( vEBT_Leaf @ $true @ B2 ) ) )
% 0.23/0.56 & ( ( X4 != zero_zero_nat )
% 0.23/0.56 => ( ( ( X4 = one_one_nat )
% 0.23/0.56 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
% 0.23/0.56 = ( vEBT_Leaf @ A @ $true ) ) )
% 0.23/0.56 & ( ( X4 != one_one_nat )
% 0.23/0.56 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
% 0.23/0.56 = ( vEBT_Leaf @ A @ B2 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_insert.simps(1)
% 0.23/0.56 thf(fact_8643_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.23/0.56 ! [G: nat > nat,M: nat,N: nat] :
% 0.23/0.56 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.23/0.56 = ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.shift_bounds_cl_Suc_ivl
% 0.23/0.56 thf(fact_8644_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.23/0.56 ! [G: nat > int,M: nat,N: nat] :
% 0.23/0.56 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.23/0.56 = ( groups705719431365010083at_int
% 0.23/0.56 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.shift_bounds_cl_Suc_ivl
% 0.23/0.56 thf(fact_8645_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 0.23/0.56 ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 0.23/0.56 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.23/0.56 = ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.shift_bounds_cl_nat_ivl
% 0.23/0.56 thf(fact_8646_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 0.23/0.56 ! [G: nat > int,M: nat,K: nat,N: nat] :
% 0.23/0.56 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.23/0.56 = ( groups705719431365010083at_int
% 0.23/0.56 @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.shift_bounds_cl_nat_ivl
% 0.23/0.56 thf(fact_8647_vebt__member_Ocases,axiom,
% 0.23/0.56 ! [X4: produc9072475918466114483BT_nat] :
% 0.23/0.56 ( ! [A5: $o,B5: $o,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X3 ) )
% 0.23/0.56 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) @ X3 ) )
% 0.23/0.56 => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) @ X3 ) )
% 0.23/0.56 => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
% 0.23/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_member.cases
% 0.23/0.56 thf(fact_8648_vebt__succ_Ocases,axiom,
% 0.23/0.56 ! [X4: produc9072475918466114483BT_nat] :
% 0.23/0.56 ( ! [Uu2: $o,B5: $o] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) )
% 0.23/0.56 => ( ! [Uv: $o,Uw: $o,N4: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N4 ) ) )
% 0.23/0.56 => ( ! [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Va3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) @ Va3 ) )
% 0.23/0.56 => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
% 0.23/0.56 => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
% 0.23/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_succ.cases
% 0.23/0.56 thf(fact_8649_VEBT__internal_Omembermima_Ocases,axiom,
% 0.23/0.56 ! [X4: produc9072475918466114483BT_nat] :
% 0.23/0.56 ( ! [Uu2: $o,Uv: $o,Uw: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Uw ) )
% 0.23/0.56 => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) @ Uz ) )
% 0.23/0.56 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X3 ) )
% 0.23/0.56 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ X3 ) )
% 0.23/0.56 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ X3 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.membermima.cases
% 0.23/0.56 thf(fact_8650_VEBT__internal_Onaive__member_Ocases,axiom,
% 0.23/0.56 ! [X4: produc9072475918466114483BT_nat] :
% 0.23/0.56 ( ! [A5: $o,B5: $o,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X3 ) )
% 0.23/0.56 => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) @ Ux ) )
% 0.23/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ X3 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.naive_member.cases
% 0.23/0.56 thf(fact_8651_invar__vebt_Ointros_I1_J,axiom,
% 0.23/0.56 ! [A: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ zero_zero_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % invar_vebt.intros(1)
% 0.23/0.56 thf(fact_8652_vebt__member_Osimps_I1_J,axiom,
% 0.23/0.56 ! [A: $o,B2: $o,X4: nat] :
% 0.23/0.56 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
% 0.23/0.56 = ( ( ( X4 = zero_zero_nat )
% 0.23/0.56 => A )
% 0.23/0.56 & ( ( X4 != zero_zero_nat )
% 0.23/0.56 => ( ( ( X4 = one_one_nat )
% 0.23/0.56 => B2 )
% 0.23/0.56 & ( X4 = one_one_nat ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_member.simps(1)
% 0.23/0.56 thf(fact_8653_vebt__buildup_Osimps_I2_J,axiom,
% 0.23/0.56 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 0.23/0.56 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_buildup.simps(2)
% 0.23/0.56 thf(fact_8654_vebt__succ_Osimps_I2_J,axiom,
% 0.23/0.56 ! [Uv2: $o,Uw2: $o,N: nat] :
% 0.23/0.56 ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
% 0.23/0.56 = none_nat ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_succ.simps(2)
% 0.23/0.56 thf(fact_8655_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 0.23/0.56 ! [A: $o,B2: $o,X4: nat] :
% 0.23/0.56 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
% 0.23/0.56 = ( ( ( X4 = zero_zero_nat )
% 0.23/0.56 => A )
% 0.23/0.56 & ( ( X4 != zero_zero_nat )
% 0.23/0.56 => ( ( ( X4 = one_one_nat )
% 0.23/0.56 => B2 )
% 0.23/0.56 & ( X4 = one_one_nat ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.naive_member.simps(1)
% 0.23/0.56 thf(fact_8656_prod_OatLeastAtMost__rev,axiom,
% 0.23/0.56 ! [G: nat > nat,N: nat,M: nat] :
% 0.23/0.56 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.23/0.56 = ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.atLeastAtMost_rev
% 0.23/0.56 thf(fact_8657_prod_OatLeastAtMost__rev,axiom,
% 0.23/0.56 ! [G: nat > int,N: nat,M: nat] :
% 0.23/0.56 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.23/0.56 = ( groups705719431365010083at_int
% 0.23/0.56 @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.atLeastAtMost_rev
% 0.23/0.56 thf(fact_8658_vebt__succ_Osimps_I3_J,axiom,
% 0.23/0.56 ! [Ux2: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
% 0.23/0.56 ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy @ Uz2 ) @ Va2 )
% 0.23/0.56 = none_nat ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_succ.simps(3)
% 0.23/0.56 thf(fact_8659_prod_OatLeast0__atMost__Suc,axiom,
% 0.23/0.56 ! [G: nat > real,N: nat] :
% 0.23/0.56 ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.atLeast0_atMost_Suc
% 0.23/0.56 thf(fact_8660_prod_OatLeast0__atMost__Suc,axiom,
% 0.23/0.56 ! [G: nat > rat,N: nat] :
% 0.23/0.56 ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.atLeast0_atMost_Suc
% 0.23/0.56 thf(fact_8661_prod_OatLeast0__atMost__Suc,axiom,
% 0.23/0.56 ! [G: nat > nat,N: nat] :
% 0.23/0.56 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.atLeast0_atMost_Suc
% 0.23/0.56 thf(fact_8662_prod_OatLeast0__atMost__Suc,axiom,
% 0.23/0.56 ! [G: nat > int,N: nat] :
% 0.23/0.56 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.atLeast0_atMost_Suc
% 0.23/0.56 thf(fact_8663_prod_OatLeast__Suc__atMost,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > real] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.56 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.23/0.56 = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.atLeast_Suc_atMost
% 0.23/0.56 thf(fact_8664_prod_OatLeast__Suc__atMost,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > rat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.56 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.23/0.56 = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.atLeast_Suc_atMost
% 0.23/0.56 thf(fact_8665_prod_OatLeast__Suc__atMost,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.56 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.23/0.56 = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.atLeast_Suc_atMost
% 0.23/0.56 thf(fact_8666_prod_OatLeast__Suc__atMost,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > int] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.56 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.23/0.56 = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.atLeast_Suc_atMost
% 0.23/0.56 thf(fact_8667_prod_Onat__ivl__Suc_H,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > real] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.23/0.56 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.nat_ivl_Suc'
% 0.23/0.56 thf(fact_8668_prod_Onat__ivl__Suc_H,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > rat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.23/0.56 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.nat_ivl_Suc'
% 0.23/0.56 thf(fact_8669_prod_Onat__ivl__Suc_H,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.23/0.56 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.nat_ivl_Suc'
% 0.23/0.56 thf(fact_8670_prod_Onat__ivl__Suc_H,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > int] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.23/0.56 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.nat_ivl_Suc'
% 0.23/0.56 thf(fact_8671_vebt__insert_Osimps_I4_J,axiom,
% 0.23/0.56 ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
% 0.23/0.56 ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_insert.simps(4)
% 0.23/0.56 thf(fact_8672_prod_OSuc__reindex__ivl,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > real] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.56 => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_real @ ( G @ M )
% 0.23/0.56 @ ( groups129246275422532515t_real
% 0.23/0.56 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.Suc_reindex_ivl
% 0.23/0.56 thf(fact_8673_prod_OSuc__reindex__ivl,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > rat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.56 => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_rat @ ( G @ M )
% 0.23/0.56 @ ( groups73079841787564623at_rat
% 0.23/0.56 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.Suc_reindex_ivl
% 0.23/0.56 thf(fact_8674_prod_OSuc__reindex__ivl,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.56 => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_nat @ ( G @ M )
% 0.23/0.56 @ ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.Suc_reindex_ivl
% 0.23/0.56 thf(fact_8675_prod_OSuc__reindex__ivl,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > int] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.56 => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.23/0.56 = ( times_times_int @ ( G @ M )
% 0.23/0.56 @ ( groups705719431365010083at_int
% 0.23/0.56 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.Suc_reindex_ivl
% 0.23/0.56 thf(fact_8676_fact__prod,axiom,
% 0.23/0.56 ( semiri5044797733671781792omplex
% 0.23/0.56 = ( ^ [N2: nat] :
% 0.23/0.56 ( semiri8010041392384452111omplex
% 0.23/0.56 @ ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [X: nat] : X
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % fact_prod
% 0.23/0.56 thf(fact_8677_fact__prod,axiom,
% 0.23/0.56 ( semiri773545260158071498ct_rat
% 0.23/0.56 = ( ^ [N2: nat] :
% 0.23/0.56 ( semiri681578069525770553at_rat
% 0.23/0.56 @ ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [X: nat] : X
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % fact_prod
% 0.23/0.56 thf(fact_8678_fact__prod,axiom,
% 0.23/0.56 ( semiri1406184849735516958ct_int
% 0.23/0.56 = ( ^ [N2: nat] :
% 0.23/0.56 ( semiri1314217659103216013at_int
% 0.23/0.56 @ ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [X: nat] : X
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % fact_prod
% 0.23/0.56 thf(fact_8679_fact__prod,axiom,
% 0.23/0.56 ( semiri1408675320244567234ct_nat
% 0.23/0.56 = ( ^ [N2: nat] :
% 0.23/0.56 ( semiri1316708129612266289at_nat
% 0.23/0.56 @ ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [X: nat] : X
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % fact_prod
% 0.23/0.56 thf(fact_8680_fact__prod,axiom,
% 0.23/0.56 ( semiri2265585572941072030t_real
% 0.23/0.56 = ( ^ [N2: nat] :
% 0.23/0.56 ( semiri5074537144036343181t_real
% 0.23/0.56 @ ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [X: nat] : X
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % fact_prod
% 0.23/0.56 thf(fact_8681_prod__atLeastAtMost__code,axiom,
% 0.23/0.56 ! [F: nat > complex,A: nat,B2: nat] :
% 0.23/0.56 ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.23/0.56 = ( set_fo1517530859248394432omplex
% 0.23/0.56 @ ^ [A4: nat] : ( times_times_complex @ ( F @ A4 ) )
% 0.23/0.56 @ A
% 0.23/0.56 @ B2
% 0.23/0.56 @ one_one_complex ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod_atLeastAtMost_code
% 0.23/0.56 thf(fact_8682_prod__atLeastAtMost__code,axiom,
% 0.23/0.56 ! [F: nat > real,A: nat,B2: nat] :
% 0.23/0.56 ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.23/0.56 = ( set_fo3111899725591712190t_real
% 0.23/0.56 @ ^ [A4: nat] : ( times_times_real @ ( F @ A4 ) )
% 0.23/0.56 @ A
% 0.23/0.56 @ B2
% 0.23/0.56 @ one_one_real ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod_atLeastAtMost_code
% 0.23/0.56 thf(fact_8683_prod__atLeastAtMost__code,axiom,
% 0.23/0.56 ! [F: nat > rat,A: nat,B2: nat] :
% 0.23/0.56 ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.23/0.56 = ( set_fo1949268297981939178at_rat
% 0.23/0.56 @ ^ [A4: nat] : ( times_times_rat @ ( F @ A4 ) )
% 0.23/0.56 @ A
% 0.23/0.56 @ B2
% 0.23/0.56 @ one_one_rat ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod_atLeastAtMost_code
% 0.23/0.56 thf(fact_8684_prod__atLeastAtMost__code,axiom,
% 0.23/0.56 ! [F: nat > nat,A: nat,B2: nat] :
% 0.23/0.56 ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.23/0.56 = ( set_fo2584398358068434914at_nat
% 0.23/0.56 @ ^ [A4: nat] : ( times_times_nat @ ( F @ A4 ) )
% 0.23/0.56 @ A
% 0.23/0.56 @ B2
% 0.23/0.56 @ one_one_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod_atLeastAtMost_code
% 0.23/0.56 thf(fact_8685_prod__atLeastAtMost__code,axiom,
% 0.23/0.56 ! [F: nat > int,A: nat,B2: nat] :
% 0.23/0.56 ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.23/0.56 = ( set_fo2581907887559384638at_int
% 0.23/0.56 @ ^ [A4: nat] : ( times_times_int @ ( F @ A4 ) )
% 0.23/0.56 @ A
% 0.23/0.56 @ B2
% 0.23/0.56 @ one_one_int ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod_atLeastAtMost_code
% 0.23/0.56 thf(fact_8686_prod_Oub__add__nat,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > real,P2: nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.23/0.56 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
% 0.23/0.56 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.ub_add_nat
% 0.23/0.56 thf(fact_8687_prod_Oub__add__nat,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > rat,P2: nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.23/0.56 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
% 0.23/0.56 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.ub_add_nat
% 0.23/0.56 thf(fact_8688_prod_Oub__add__nat,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > nat,P2: nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.23/0.56 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
% 0.23/0.56 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.ub_add_nat
% 0.23/0.56 thf(fact_8689_prod_Oub__add__nat,axiom,
% 0.23/0.56 ! [M: nat,N: nat,G: nat > int,P2: nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.23/0.56 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
% 0.23/0.56 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.ub_add_nat
% 0.23/0.56 thf(fact_8690_vebt__succ_Osimps_I1_J,axiom,
% 0.23/0.56 ! [B2: $o,Uu3: $o] :
% 0.23/0.56 ( ( B2
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu3 @ B2 ) @ zero_zero_nat )
% 0.23/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.23/0.56 & ( ~ B2
% 0.23/0.56 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu3 @ B2 ) @ zero_zero_nat )
% 0.23/0.56 = none_nat ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_succ.simps(1)
% 0.23/0.56 thf(fact_8691_fact__eq__fact__times,axiom,
% 0.23/0.56 ! [N: nat,M: nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.56 => ( ( semiri1408675320244567234ct_nat @ M )
% 0.23/0.56 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 0.23/0.56 @ ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [X: nat] : X
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % fact_eq_fact_times
% 0.23/0.56 thf(fact_8692_pochhammer__Suc__prod,axiom,
% 0.23/0.56 ! [A: complex,N: nat] :
% 0.23/0.56 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.23/0.56 = ( groups6464643781859351333omplex
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_Suc_prod
% 0.23/0.56 thf(fact_8693_pochhammer__Suc__prod,axiom,
% 0.23/0.56 ! [A: real,N: nat] :
% 0.23/0.56 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.23/0.56 = ( groups129246275422532515t_real
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_Suc_prod
% 0.23/0.56 thf(fact_8694_pochhammer__Suc__prod,axiom,
% 0.23/0.56 ! [A: rat,N: nat] :
% 0.23/0.56 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.23/0.56 = ( groups73079841787564623at_rat
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_Suc_prod
% 0.23/0.56 thf(fact_8695_pochhammer__Suc__prod,axiom,
% 0.23/0.56 ! [A: nat,N: nat] :
% 0.23/0.56 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.23/0.56 = ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_Suc_prod
% 0.23/0.56 thf(fact_8696_pochhammer__Suc__prod,axiom,
% 0.23/0.56 ! [A: int,N: nat] :
% 0.23/0.56 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.23/0.56 = ( groups705719431365010083at_int
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_Suc_prod
% 0.23/0.56 thf(fact_8697_pochhammer__prod__rev,axiom,
% 0.23/0.56 ( comm_s2602460028002588243omplex
% 0.23/0.56 = ( ^ [A4: complex,N2: nat] :
% 0.23/0.56 ( groups6464643781859351333omplex
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N2 @ I2 ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_prod_rev
% 0.23/0.56 thf(fact_8698_pochhammer__prod__rev,axiom,
% 0.23/0.56 ( comm_s7457072308508201937r_real
% 0.23/0.56 = ( ^ [A4: real,N2: nat] :
% 0.23/0.56 ( groups129246275422532515t_real
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I2 ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_prod_rev
% 0.23/0.56 thf(fact_8699_pochhammer__prod__rev,axiom,
% 0.23/0.56 ( comm_s4028243227959126397er_rat
% 0.23/0.56 = ( ^ [A4: rat,N2: nat] :
% 0.23/0.56 ( groups73079841787564623at_rat
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I2 ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_prod_rev
% 0.23/0.56 thf(fact_8700_pochhammer__prod__rev,axiom,
% 0.23/0.56 ( comm_s4663373288045622133er_nat
% 0.23/0.56 = ( ^ [A4: nat,N2: nat] :
% 0.23/0.56 ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I2 ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_prod_rev
% 0.23/0.56 thf(fact_8701_pochhammer__prod__rev,axiom,
% 0.23/0.56 ( comm_s4660882817536571857er_int
% 0.23/0.56 = ( ^ [A4: int,N2: nat] :
% 0.23/0.56 ( groups705719431365010083at_int
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I2 ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_prod_rev
% 0.23/0.56 thf(fact_8702_fact__div__fact,axiom,
% 0.23/0.56 ! [N: nat,M: nat] :
% 0.23/0.56 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.56 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.23/0.56 = ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [X: nat] : X
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % fact_div_fact
% 0.23/0.56 thf(fact_8703_prod_Oin__pairs,axiom,
% 0.23/0.56 ! [G: nat > real,M: nat,N: nat] :
% 0.23/0.56 ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.23/0.56 = ( groups129246275422532515t_real
% 0.23/0.56 @ ^ [I2: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.in_pairs
% 0.23/0.56 thf(fact_8704_prod_Oin__pairs,axiom,
% 0.23/0.56 ! [G: nat > rat,M: nat,N: nat] :
% 0.23/0.56 ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.23/0.56 = ( groups73079841787564623at_rat
% 0.23/0.56 @ ^ [I2: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.in_pairs
% 0.23/0.56 thf(fact_8705_prod_Oin__pairs,axiom,
% 0.23/0.56 ! [G: nat > nat,M: nat,N: nat] :
% 0.23/0.56 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.23/0.56 = ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [I2: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.in_pairs
% 0.23/0.56 thf(fact_8706_prod_Oin__pairs,axiom,
% 0.23/0.56 ! [G: nat > int,M: nat,N: nat] :
% 0.23/0.56 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.23/0.56 = ( groups705719431365010083at_int
% 0.23/0.56 @ ^ [I2: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.in_pairs
% 0.23/0.56 thf(fact_8707_pochhammer__Suc__prod__rev,axiom,
% 0.23/0.56 ! [A: complex,N: nat] :
% 0.23/0.56 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.23/0.56 = ( groups6464643781859351333omplex
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N @ I2 ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_Suc_prod_rev
% 0.23/0.56 thf(fact_8708_pochhammer__Suc__prod__rev,axiom,
% 0.23/0.56 ! [A: real,N: nat] :
% 0.23/0.56 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.23/0.56 = ( groups129246275422532515t_real
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I2 ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_Suc_prod_rev
% 0.23/0.56 thf(fact_8709_pochhammer__Suc__prod__rev,axiom,
% 0.23/0.56 ! [A: rat,N: nat] :
% 0.23/0.56 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.23/0.56 = ( groups73079841787564623at_rat
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I2 ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_Suc_prod_rev
% 0.23/0.56 thf(fact_8710_pochhammer__Suc__prod__rev,axiom,
% 0.23/0.56 ! [A: nat,N: nat] :
% 0.23/0.56 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.23/0.56 = ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I2 ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_Suc_prod_rev
% 0.23/0.56 thf(fact_8711_pochhammer__Suc__prod__rev,axiom,
% 0.23/0.56 ! [A: int,N: nat] :
% 0.23/0.56 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.23/0.56 = ( groups705719431365010083at_int
% 0.23/0.56 @ ^ [I2: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I2 ) ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % pochhammer_Suc_prod_rev
% 0.23/0.56 thf(fact_8712_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.23/0.56 ( ( ( vEBT_VEBT_membermima @ X4 @ Xa )
% 0.23/0.56 = Y3 )
% 0.23/0.56 => ( ( ? [Uu2: $o,Uv: $o] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 0.23/0.56 => Y3 )
% 0.23/0.56 => ( ( ? [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
% 0.23/0.56 => Y3 )
% 0.23/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.23/0.56 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( ~ ( ( Xa = Mi2 )
% 0.23/0.56 | ( Xa = Ma2 ) ) ) ) )
% 0.23/0.56 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [Vc2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( ~ ( ( Xa = Mi2 )
% 0.23/0.56 | ( Xa = Ma2 )
% 0.23/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( 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 ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 0.23/0.56 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [Vd2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( 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 ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.membermima.elims(1)
% 0.23/0.56 thf(fact_8713_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.56 ( ~ ( vEBT_VEBT_membermima @ X4 @ Xa )
% 0.23/0.56 => ( ! [Uu2: $o,Uv: $o] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( vEBT_Leaf @ Uu2 @ Uv ) )
% 0.23/0.56 => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
% 0.23/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.23/0.56 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.23/0.56 => ( ( Xa = Mi2 )
% 0.23/0.56 | ( Xa = Ma2 ) ) )
% 0.23/0.56 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [Vc2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.23/0.56 => ( ( Xa = Mi2 )
% 0.23/0.56 | ( Xa = Ma2 )
% 0.23/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( 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 ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.23/0.56 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [Vd2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.23/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( 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 ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.membermima.elims(3)
% 0.23/0.56 thf(fact_8714_gbinomial__Suc,axiom,
% 0.23/0.56 ! [A: complex,K: nat] :
% 0.23/0.56 ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 0.23/0.56 = ( divide1717551699836669952omplex
% 0.23/0.56 @ ( groups6464643781859351333omplex
% 0.23/0.56 @ ^ [I2: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 0.23/0.56 @ ( semiri5044797733671781792omplex @ ( suc @ K ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % gbinomial_Suc
% 0.23/0.56 thf(fact_8715_gbinomial__Suc,axiom,
% 0.23/0.56 ! [A: rat,K: nat] :
% 0.23/0.56 ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 0.23/0.56 = ( divide_divide_rat
% 0.23/0.56 @ ( groups73079841787564623at_rat
% 0.23/0.56 @ ^ [I2: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 0.23/0.56 @ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % gbinomial_Suc
% 0.23/0.56 thf(fact_8716_gbinomial__Suc,axiom,
% 0.23/0.56 ! [A: real,K: nat] :
% 0.23/0.56 ( ( gbinomial_real @ A @ ( suc @ K ) )
% 0.23/0.56 = ( divide_divide_real
% 0.23/0.56 @ ( groups129246275422532515t_real
% 0.23/0.56 @ ^ [I2: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 0.23/0.56 @ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % gbinomial_Suc
% 0.23/0.56 thf(fact_8717_gbinomial__Suc,axiom,
% 0.23/0.56 ! [A: nat,K: nat] :
% 0.23/0.56 ( ( gbinomial_nat @ A @ ( suc @ K ) )
% 0.23/0.56 = ( divide_divide_nat
% 0.23/0.56 @ ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [I2: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 0.23/0.56 @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % gbinomial_Suc
% 0.23/0.56 thf(fact_8718_gbinomial__Suc,axiom,
% 0.23/0.56 ! [A: int,K: nat] :
% 0.23/0.56 ( ( gbinomial_int @ A @ ( suc @ K ) )
% 0.23/0.56 = ( divide_divide_int
% 0.23/0.56 @ ( groups705719431365010083at_int
% 0.23/0.56 @ ^ [I2: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I2 ) )
% 0.23/0.56 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 0.23/0.56 @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % gbinomial_Suc
% 0.23/0.56 thf(fact_8719_vebt__member_Oelims_I1_J,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.23/0.56 ( ( ( vEBT_vebt_member @ X4 @ Xa )
% 0.23/0.56 = Y3 )
% 0.23/0.56 => ( ! [A5: $o,B5: $o] :
% 0.23/0.56 ( ( X4
% 0.23/0.56 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( ~ ( ( ( Xa = zero_zero_nat )
% 0.23/0.56 => A5 )
% 0.23/0.56 & ( ( Xa != zero_zero_nat )
% 0.23/0.56 => ( ( ( Xa = one_one_nat )
% 0.23/0.56 => B5 )
% 0.23/0.56 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 0.23/0.56 => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 0.23/0.56 => Y3 )
% 0.23/0.56 => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
% 0.23/0.56 => Y3 )
% 0.23/0.56 => ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.23/0.56 => Y3 )
% 0.23/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [Summary2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( ~ ( ( Xa != Mi2 )
% 0.23/0.56 => ( ( Xa != Ma2 )
% 0.23/0.56 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.56 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.56 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.56 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_member.elims(1)
% 0.23/0.56 thf(fact_8720_vebt__member_Oelims_I3_J,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.56 ( ~ ( vEBT_vebt_member @ X4 @ Xa )
% 0.23/0.56 => ( ! [A5: $o,B5: $o] :
% 0.23/0.56 ( ( X4
% 0.23/0.56 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.56 => ( ( ( Xa = zero_zero_nat )
% 0.23/0.56 => A5 )
% 0.23/0.56 & ( ( Xa != zero_zero_nat )
% 0.23/0.56 => ( ( ( Xa = one_one_nat )
% 0.23/0.56 => B5 )
% 0.23/0.56 & ( Xa = one_one_nat ) ) ) ) )
% 0.23/0.56 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 0.23/0.56 => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
% 0.23/0.56 => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.23/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [Summary2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.56 => ( ( Xa != Mi2 )
% 0.23/0.56 => ( ( Xa != Ma2 )
% 0.23/0.56 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.56 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.56 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.56 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_member.elims(3)
% 0.23/0.56 thf(fact_8721_invar__vebt_Osimps,axiom,
% 0.23/0.56 ( vEBT_invar_vebt
% 0.23/0.56 = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 0.23/0.56 ( ( ? [A4: $o,B3: $o] :
% 0.23/0.56 ( A1
% 0.23/0.56 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.23/0.56 & ( A22
% 0.23/0.56 = ( suc @ zero_zero_nat ) ) )
% 0.23/0.56 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 0.23/0.56 ( ( A1
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 0.23/0.56 & ! [X: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.23/0.56 & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 0.23/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.23/0.56 & ( A22
% 0.23/0.56 = ( plus_plus_nat @ N2 @ N2 ) )
% 0.23/0.56 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.23/0.56 & ! [X: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.23/0.56 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.56 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 0.23/0.56 ( ( A1
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 0.23/0.56 & ! [X: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.23/0.56 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 0.23/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.23/0.56 & ( A22
% 0.23/0.56 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 0.23/0.56 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.23/0.56 & ! [X: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.23/0.56 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.56 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.23/0.56 ( ( A1
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 0.23/0.56 & ! [X: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.23/0.56 & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 0.23/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.23/0.56 & ( A22
% 0.23/0.56 = ( plus_plus_nat @ N2 @ N2 ) )
% 0.23/0.56 & ! [I2: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.23/0.56 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 0.23/0.56 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 0.23/0.56 & ( ( Mi3 = Ma3 )
% 0.23/0.56 => ! [X: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.23/0.56 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.56 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.23/0.56 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.23/0.56 & ( ( Mi3 != Ma3 )
% 0.23/0.56 => ! [I2: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.23/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 0.23/0.56 = I2 )
% 0.23/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 0.23/0.56 & ! [X: nat] :
% 0.23/0.56 ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 0.23/0.56 = I2 )
% 0.23/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 0.23/0.56 => ( ( ord_less_nat @ Mi3 @ X )
% 0.23/0.56 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
% 0.23/0.56 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.23/0.56 ( ( A1
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 0.23/0.56 & ! [X: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.23/0.56 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 0.23/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.23/0.56 & ( A22
% 0.23/0.56 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 0.23/0.56 & ! [I2: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.23/0.56 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 0.23/0.56 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 0.23/0.56 & ( ( Mi3 = Ma3 )
% 0.23/0.56 => ! [X: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.23/0.56 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.56 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.23/0.56 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.23/0.56 & ( ( Mi3 != Ma3 )
% 0.23/0.56 => ! [I2: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.23/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 0.23/0.56 = I2 )
% 0.23/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 0.23/0.56 & ! [X: nat] :
% 0.23/0.56 ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 0.23/0.56 = I2 )
% 0.23/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 0.23/0.56 => ( ( ord_less_nat @ Mi3 @ X )
% 0.23/0.56 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % invar_vebt.simps
% 0.23/0.56 thf(fact_8722_invar__vebt_Ocases,axiom,
% 0.23/0.56 ! [A12: vEBT_VEBT,A23: nat] :
% 0.23/0.56 ( ( vEBT_invar_vebt @ A12 @ A23 )
% 0.23/0.56 => ( ( ? [A5: $o,B5: $o] :
% 0.23/0.56 ( A12
% 0.23/0.56 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.56 => ( A23
% 0.23/0.56 != ( suc @ zero_zero_nat ) ) )
% 0.23/0.56 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.23/0.56 ( ( A12
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.56 => ( ( A23 = Deg2 )
% 0.23/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.23/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.23/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.23/0.56 => ( ( M4 = N4 )
% 0.23/0.56 => ( ( Deg2
% 0.23/0.56 = ( plus_plus_nat @ N4 @ M4 ) )
% 0.23/0.56 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.23/0.56 => ~ ! [X5: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.23/0.56 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.23/0.56 ( ( A12
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.56 => ( ( A23 = Deg2 )
% 0.23/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.23/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.23/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.23/0.56 => ( ( M4
% 0.23/0.56 = ( suc @ N4 ) )
% 0.23/0.56 => ( ( Deg2
% 0.23/0.56 = ( plus_plus_nat @ N4 @ M4 ) )
% 0.23/0.56 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.23/0.56 => ~ ! [X5: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.23/0.56 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.23/0.56 ( ( A12
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.56 => ( ( A23 = Deg2 )
% 0.23/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.23/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.23/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.23/0.56 => ( ( M4 = N4 )
% 0.23/0.56 => ( ( Deg2
% 0.23/0.56 = ( plus_plus_nat @ N4 @ M4 ) )
% 0.23/0.56 => ( ! [I4: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.23/0.56 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.23/0.56 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.23/0.56 => ( ( ( Mi2 = Ma2 )
% 0.23/0.56 => ! [X5: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 0.23/0.56 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.23/0.56 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.56 => ~ ( ( Mi2 != Ma2 )
% 0.23/0.56 => ! [I4: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.23/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 0.23/0.56 = I4 )
% 0.23/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 0.23/0.56 & ! [X5: nat] :
% 0.23/0.56 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 0.23/0.56 = I4 )
% 0.23/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 0.23/0.56 => ( ( ord_less_nat @ Mi2 @ X5 )
% 0.23/0.56 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.23/0.56 => ~ ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.23/0.56 ( ( A12
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.56 => ( ( A23 = Deg2 )
% 0.23/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.23/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.23/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.23/0.56 => ( ( M4
% 0.23/0.56 = ( suc @ N4 ) )
% 0.23/0.56 => ( ( Deg2
% 0.23/0.56 = ( plus_plus_nat @ N4 @ M4 ) )
% 0.23/0.56 => ( ! [I4: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.23/0.56 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.23/0.56 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.23/0.56 => ( ( ( Mi2 = Ma2 )
% 0.23/0.56 => ! [X5: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 0.23/0.56 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.23/0.56 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.56 => ~ ( ( Mi2 != Ma2 )
% 0.23/0.56 => ! [I4: nat] :
% 0.23/0.56 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.23/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 0.23/0.56 = I4 )
% 0.23/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 0.23/0.56 & ! [X5: nat] :
% 0.23/0.56 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 0.23/0.56 = I4 )
% 0.23/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 0.23/0.56 => ( ( ord_less_nat @ Mi2 @ X5 )
% 0.23/0.56 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % invar_vebt.cases
% 0.23/0.56 thf(fact_8723_invar__vebt_Ointros_I2_J,axiom,
% 0.23/0.56 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.23/0.56 ( ! [X3: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.23/0.56 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.23/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.56 => ( ( M = N )
% 0.23/0.56 => ( ( Deg
% 0.23/0.56 = ( plus_plus_nat @ N @ M ) )
% 0.23/0.56 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.23/0.56 => ( ! [X3: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.56 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % invar_vebt.intros(2)
% 0.23/0.56 thf(fact_8724_invar__vebt_Ointros_I3_J,axiom,
% 0.23/0.56 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.23/0.56 ( ! [X3: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.23/0.56 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.23/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.23/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.23/0.56 => ( ( M
% 0.23/0.56 = ( suc @ N ) )
% 0.23/0.56 => ( ( Deg
% 0.23/0.56 = ( plus_plus_nat @ N @ M ) )
% 0.23/0.56 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.23/0.56 => ( ! [X3: vEBT_VEBT] :
% 0.23/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.56 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
% 0.23/0.56 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % invar_vebt.intros(3)
% 0.23/0.56 thf(fact_8725_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 0.23/0.56 ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
% 0.23/0.56 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X4 )
% 0.23/0.56 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.membermima.simps(5)
% 0.23/0.56 thf(fact_8726_vebt__succ_Oelims,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Xa: nat,Y3: option_nat] :
% 0.23/0.56 ( ( ( vEBT_vebt_succ @ X4 @ Xa )
% 0.23/0.56 = Y3 )
% 0.23/0.56 => ( ! [Uu2: $o,B5: $o] :
% 0.23/0.56 ( ( X4
% 0.23/0.56 = ( vEBT_Leaf @ Uu2 @ B5 ) )
% 0.23/0.56 => ( ( Xa = zero_zero_nat )
% 0.23/0.56 => ~ ( ( B5
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.23/0.56 & ( ~ B5
% 0.23/0.56 => ( Y3 = none_nat ) ) ) ) )
% 0.23/0.56 => ( ( ? [Uv: $o,Uw: $o] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Leaf @ Uv @ Uw ) )
% 0.23/0.56 => ( ? [N4: nat] :
% 0.23/0.56 ( Xa
% 0.23/0.56 = ( suc @ N4 ) )
% 0.23/0.56 => ( Y3 != none_nat ) ) )
% 0.23/0.56 => ( ( ? [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) )
% 0.23/0.56 => ( Y3 != none_nat ) )
% 0.23/0.56 => ( ( ? [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.23/0.56 => ( Y3 != none_nat ) )
% 0.23/0.56 => ( ( ? [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 0.23/0.56 => ( Y3 != none_nat ) )
% 0.23/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.56 ( ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.56 => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( some_nat @ Mi2 ) ) )
% 0.23/0.56 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 @ ( if_option_nat
% 0.23/0.56 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 != none_nat )
% 0.23/0.56 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.23/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.56 @ ( if_option_nat
% 0.23/0.56 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.56 = none_nat )
% 0.23/0.56 @ none_nat
% 0.23/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.23/0.56 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_succ.elims
% 0.23/0.56 thf(fact_8727_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.56 ( ~ ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 0.23/0.56 => ( ! [A5: $o,B5: $o] :
% 0.23/0.56 ( ( X4
% 0.23/0.56 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.56 => ( ( ( Xa = zero_zero_nat )
% 0.23/0.56 => A5 )
% 0.23/0.56 & ( ( Xa != zero_zero_nat )
% 0.23/0.56 => ( ( ( Xa = one_one_nat )
% 0.23/0.56 => B5 )
% 0.23/0.56 & ( Xa = one_one_nat ) ) ) ) )
% 0.23/0.56 => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
% 0.23/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [S3: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.23/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( 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 ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.naive_member.elims(3)
% 0.23/0.56 thf(fact_8728_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.56 ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 0.23/0.56 => ( ! [A5: $o,B5: $o] :
% 0.23/0.56 ( ( X4
% 0.23/0.56 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.56 => ~ ( ( ( Xa = zero_zero_nat )
% 0.23/0.56 => A5 )
% 0.23/0.56 & ( ( Xa != zero_zero_nat )
% 0.23/0.56 => ( ( ( Xa = one_one_nat )
% 0.23/0.56 => B5 )
% 0.23/0.56 & ( Xa = one_one_nat ) ) ) ) )
% 0.23/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [S3: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.23/0.56 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( 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 ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.naive_member.elims(2)
% 0.23/0.56 thf(fact_8729_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.23/0.56 ( ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 0.23/0.56 = Y3 )
% 0.23/0.56 => ( ! [A5: $o,B5: $o] :
% 0.23/0.56 ( ( X4
% 0.23/0.56 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( ~ ( ( ( Xa = zero_zero_nat )
% 0.23/0.56 => A5 )
% 0.23/0.56 & ( ( Xa != zero_zero_nat )
% 0.23/0.56 => ( ( ( Xa = one_one_nat )
% 0.23/0.56 => B5 )
% 0.23/0.56 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 0.23/0.56 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
% 0.23/0.56 => Y3 )
% 0.23/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.23/0.56 ( ? [S3: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.56 => ( 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 ) ) ) ) ) )
% 0.23/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % VEBT_internal.naive_member.elims(1)
% 0.23/0.56 thf(fact_8730_vebt__maxt_Oelims,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Y3: option_nat] :
% 0.23/0.56 ( ( ( vEBT_vebt_maxt @ X4 )
% 0.23/0.56 = Y3 )
% 0.23/0.56 => ( ! [A5: $o,B5: $o] :
% 0.23/0.56 ( ( X4
% 0.23/0.56 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.56 => ~ ( ( B5
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.23/0.56 & ( ~ B5
% 0.23/0.56 => ( ( A5
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( some_nat @ zero_zero_nat ) ) )
% 0.23/0.56 & ( ~ A5
% 0.23/0.56 => ( Y3 = none_nat ) ) ) ) ) )
% 0.23/0.56 => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 0.23/0.56 => ( Y3 != none_nat ) )
% 0.23/0.56 => ~ ! [Mi2: nat,Ma2: nat] :
% 0.23/0.56 ( ? [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
% 0.23/0.56 => ( Y3
% 0.23/0.56 != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_maxt.elims
% 0.23/0.56 thf(fact_8731_vebt__mint_Oelims,axiom,
% 0.23/0.56 ! [X4: vEBT_VEBT,Y3: option_nat] :
% 0.23/0.56 ( ( ( vEBT_vebt_mint @ X4 )
% 0.23/0.56 = Y3 )
% 0.23/0.56 => ( ! [A5: $o,B5: $o] :
% 0.23/0.56 ( ( X4
% 0.23/0.56 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.56 => ~ ( ( A5
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( some_nat @ zero_zero_nat ) ) )
% 0.23/0.56 & ( ~ A5
% 0.23/0.56 => ( ( B5
% 0.23/0.56 => ( Y3
% 0.23/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.23/0.56 & ( ~ B5
% 0.23/0.56 => ( Y3 = none_nat ) ) ) ) ) )
% 0.23/0.56 => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 0.23/0.56 => ( Y3 != none_nat ) )
% 0.23/0.56 => ~ ! [Mi2: nat] :
% 0.23/0.56 ( ? [Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.23/0.56 ( X4
% 0.23/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
% 0.23/0.56 => ( Y3
% 0.23/0.56 != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_mint.elims
% 0.23/0.56 thf(fact_8732_prod_Oempty,axiom,
% 0.23/0.56 ! [G: extended_enat > complex] :
% 0.23/0.56 ( ( groups4622424608036095791omplex @ G @ bot_bo7653980558646680370d_enat )
% 0.23/0.56 = one_one_complex ) ).
% 0.23/0.56
% 0.23/0.56 % prod.empty
% 0.23/0.56 thf(fact_8733_prod_Oempty,axiom,
% 0.23/0.56 ! [G: extended_enat > real] :
% 0.23/0.56 ( ( groups97031904164794029t_real @ G @ bot_bo7653980558646680370d_enat )
% 0.23/0.56 = one_one_real ) ).
% 0.23/0.56
% 0.23/0.56 % prod.empty
% 0.23/0.56 thf(fact_8734_prod_Oempty,axiom,
% 0.23/0.56 ! [G: extended_enat > rat] :
% 0.23/0.56 ( ( groups2245840878043517529at_rat @ G @ bot_bo7653980558646680370d_enat )
% 0.23/0.56 = one_one_rat ) ).
% 0.23/0.56
% 0.23/0.56 % prod.empty
% 0.23/0.56 thf(fact_8735_prod_Oempty,axiom,
% 0.23/0.56 ! [G: extended_enat > nat] :
% 0.23/0.56 ( ( groups2880970938130013265at_nat @ G @ bot_bo7653980558646680370d_enat )
% 0.23/0.56 = one_one_nat ) ).
% 0.23/0.56
% 0.23/0.56 % prod.empty
% 0.23/0.56 thf(fact_8736_prod_Oempty,axiom,
% 0.23/0.56 ! [G: extended_enat > int] :
% 0.23/0.56 ( ( groups2878480467620962989at_int @ G @ bot_bo7653980558646680370d_enat )
% 0.23/0.56 = one_one_int ) ).
% 0.23/0.56
% 0.23/0.56 % prod.empty
% 0.23/0.56 thf(fact_8737_prod_Oempty,axiom,
% 0.23/0.56 ! [G: real > complex] :
% 0.23/0.56 ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
% 0.23/0.56 = one_one_complex ) ).
% 0.23/0.56
% 0.23/0.56 % prod.empty
% 0.23/0.56 thf(fact_8738_prod_Oempty,axiom,
% 0.23/0.56 ! [G: real > real] :
% 0.23/0.56 ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
% 0.23/0.56 = one_one_real ) ).
% 0.23/0.56
% 0.23/0.56 % prod.empty
% 0.23/0.56 thf(fact_8739_prod_Oempty,axiom,
% 0.23/0.56 ! [G: real > rat] :
% 0.23/0.56 ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
% 0.23/0.56 = one_one_rat ) ).
% 0.23/0.56
% 0.23/0.56 % prod.empty
% 0.23/0.56 thf(fact_8740_prod_Oempty,axiom,
% 0.23/0.56 ! [G: real > nat] :
% 0.23/0.56 ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
% 0.23/0.56 = one_one_nat ) ).
% 0.23/0.56
% 0.23/0.56 % prod.empty
% 0.23/0.56 thf(fact_8741_prod_Oempty,axiom,
% 0.23/0.56 ! [G: real > int] :
% 0.23/0.56 ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
% 0.23/0.56 = one_one_int ) ).
% 0.23/0.56
% 0.23/0.56 % prod.empty
% 0.23/0.56 thf(fact_8742_vebt__maxt_Osimps_I1_J,axiom,
% 0.23/0.56 ! [B2: $o,A: $o] :
% 0.23/0.56 ( ( B2
% 0.23/0.56 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 0.23/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.23/0.56 & ( ~ B2
% 0.23/0.56 => ( ( A
% 0.23/0.56 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 0.23/0.56 = ( some_nat @ zero_zero_nat ) ) )
% 0.23/0.56 & ( ~ A
% 0.23/0.56 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 0.23/0.56 = none_nat ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % vebt_maxt.simps(1)
% 0.23/0.56 thf(fact_8743_prod_Oneutral__const,axiom,
% 0.23/0.56 ! [A3: set_nat] :
% 0.23/0.56 ( ( groups708209901874060359at_nat
% 0.23/0.56 @ ^ [Uu: nat] : one_one_nat
% 0.23/0.56 @ A3 )
% 0.23/0.56 = one_one_nat ) ).
% 0.23/0.56
% 0.23/0.56 % prod.neutral_const
% 0.23/0.56 thf(fact_8744_prod_Oneutral__const,axiom,
% 0.23/0.56 ! [A3: set_nat] :
% 0.23/0.56 ( ( groups705719431365010083at_int
% 0.23/0.56 @ ^ [Uu: nat] : one_one_int
% 0.23/0.56 @ A3 )
% 0.23/0.56 = one_one_int ) ).
% 0.23/0.56
% 0.23/0.56 % prod.neutral_const
% 0.23/0.56 thf(fact_8745_prod_Oneutral__const,axiom,
% 0.23/0.56 ! [A3: set_int] :
% 0.23/0.56 ( ( groups1705073143266064639nt_int
% 0.23/0.56 @ ^ [Uu: int] : one_one_int
% 0.23/0.56 @ A3 )
% 0.23/0.56 = one_one_int ) ).
% 0.23/0.56
% 0.23/0.56 % prod.neutral_const
% 0.23/0.56 thf(fact_8746_prod__int__eq,axiom,
% 0.23/0.56 ! [I: nat,J: nat] :
% 0.23/0.56 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 0.23/0.56 = ( groups1705073143266064639nt_int
% 0.23/0.56 @ ^ [X: int] : X
% 0.23/0.56 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod_int_eq
% 0.23/0.56 thf(fact_8747_prod__int__plus__eq,axiom,
% 0.23/0.56 ! [I: nat,J: nat] :
% 0.23/0.56 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 0.23/0.56 = ( groups1705073143266064639nt_int
% 0.23/0.56 @ ^ [X: int] : X
% 0.23/0.56 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod_int_plus_eq
% 0.23/0.56 thf(fact_8748_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.23/0.56 ! [G: extended_enat > complex,A3: set_Extended_enat] :
% 0.23/0.56 ( ( ( groups4622424608036095791omplex @ G @ A3 )
% 0.23/0.56 != one_one_complex )
% 0.23/0.56 => ~ ! [A5: extended_enat] :
% 0.23/0.56 ( ( member_Extended_enat @ A5 @ A3 )
% 0.23/0.56 => ( ( G @ A5 )
% 0.23/0.56 = one_one_complex ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.not_neutral_contains_not_neutral
% 0.23/0.56 thf(fact_8749_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.23/0.56 ! [G: complex > complex,A3: set_complex] :
% 0.23/0.56 ( ( ( groups3708469109370488835omplex @ G @ A3 )
% 0.23/0.56 != one_one_complex )
% 0.23/0.56 => ~ ! [A5: complex] :
% 0.23/0.56 ( ( member_complex @ A5 @ A3 )
% 0.23/0.56 => ( ( G @ A5 )
% 0.23/0.56 = one_one_complex ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.not_neutral_contains_not_neutral
% 0.23/0.56 thf(fact_8750_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.23/0.56 ! [G: real > complex,A3: set_real] :
% 0.23/0.56 ( ( ( groups713298508707869441omplex @ G @ A3 )
% 0.23/0.56 != one_one_complex )
% 0.23/0.56 => ~ ! [A5: real] :
% 0.23/0.56 ( ( member_real @ A5 @ A3 )
% 0.23/0.56 => ( ( G @ A5 )
% 0.23/0.56 = one_one_complex ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.not_neutral_contains_not_neutral
% 0.23/0.56 thf(fact_8751_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.23/0.56 ! [G: nat > complex,A3: set_nat] :
% 0.23/0.56 ( ( ( groups6464643781859351333omplex @ G @ A3 )
% 0.23/0.56 != one_one_complex )
% 0.23/0.56 => ~ ! [A5: nat] :
% 0.23/0.56 ( ( member_nat @ A5 @ A3 )
% 0.23/0.56 => ( ( G @ A5 )
% 0.23/0.56 = one_one_complex ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.not_neutral_contains_not_neutral
% 0.23/0.56 thf(fact_8752_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.23/0.56 ! [G: int > complex,A3: set_int] :
% 0.23/0.56 ( ( ( groups7440179247065528705omplex @ G @ A3 )
% 0.23/0.56 != one_one_complex )
% 0.23/0.56 => ~ ! [A5: int] :
% 0.23/0.56 ( ( member_int @ A5 @ A3 )
% 0.23/0.56 => ( ( G @ A5 )
% 0.23/0.56 = one_one_complex ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.not_neutral_contains_not_neutral
% 0.23/0.56 thf(fact_8753_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.23/0.56 ! [G: extended_enat > real,A3: set_Extended_enat] :
% 0.23/0.56 ( ( ( groups97031904164794029t_real @ G @ A3 )
% 0.23/0.56 != one_one_real )
% 0.23/0.56 => ~ ! [A5: extended_enat] :
% 0.23/0.56 ( ( member_Extended_enat @ A5 @ A3 )
% 0.23/0.56 => ( ( G @ A5 )
% 0.23/0.56 = one_one_real ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.not_neutral_contains_not_neutral
% 0.23/0.56 thf(fact_8754_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.23/0.56 ! [G: complex > real,A3: set_complex] :
% 0.23/0.56 ( ( ( groups766887009212190081x_real @ G @ A3 )
% 0.23/0.56 != one_one_real )
% 0.23/0.56 => ~ ! [A5: complex] :
% 0.23/0.56 ( ( member_complex @ A5 @ A3 )
% 0.23/0.56 => ( ( G @ A5 )
% 0.23/0.56 = one_one_real ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.not_neutral_contains_not_neutral
% 0.23/0.56 thf(fact_8755_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.23/0.56 ! [G: real > real,A3: set_real] :
% 0.23/0.56 ( ( ( groups1681761925125756287l_real @ G @ A3 )
% 0.23/0.56 != one_one_real )
% 0.23/0.56 => ~ ! [A5: real] :
% 0.23/0.56 ( ( member_real @ A5 @ A3 )
% 0.23/0.56 => ( ( G @ A5 )
% 0.23/0.56 = one_one_real ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.not_neutral_contains_not_neutral
% 0.23/0.56 thf(fact_8756_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.23/0.56 ! [G: nat > real,A3: set_nat] :
% 0.23/0.56 ( ( ( groups129246275422532515t_real @ G @ A3 )
% 0.23/0.56 != one_one_real )
% 0.23/0.56 => ~ ! [A5: nat] :
% 0.23/0.56 ( ( member_nat @ A5 @ A3 )
% 0.23/0.56 => ( ( G @ A5 )
% 0.23/0.56 = one_one_real ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.not_neutral_contains_not_neutral
% 0.23/0.56 thf(fact_8757_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.23/0.56 ! [G: int > real,A3: set_int] :
% 0.23/0.56 ( ( ( groups2316167850115554303t_real @ G @ A3 )
% 0.23/0.56 != one_one_real )
% 0.23/0.56 => ~ ! [A5: int] :
% 0.23/0.56 ( ( member_int @ A5 @ A3 )
% 0.23/0.56 => ( ( G @ A5 )
% 0.23/0.56 = one_one_real ) ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.not_neutral_contains_not_neutral
% 0.23/0.56 thf(fact_8758_prod_Oneutral,axiom,
% 0.23/0.56 ! [A3: set_nat,G: nat > nat] :
% 0.23/0.56 ( ! [X3: nat] :
% 0.23/0.56 ( ( member_nat @ X3 @ A3 )
% 0.23/0.56 => ( ( G @ X3 )
% 0.23/0.56 = one_one_nat ) )
% 0.23/0.56 => ( ( groups708209901874060359at_nat @ G @ A3 )
% 0.23/0.56 = one_one_nat ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.neutral
% 0.23/0.56 thf(fact_8759_prod_Oneutral,axiom,
% 0.23/0.56 ! [A3: set_nat,G: nat > int] :
% 0.23/0.56 ( ! [X3: nat] :
% 0.23/0.56 ( ( member_nat @ X3 @ A3 )
% 0.23/0.56 => ( ( G @ X3 )
% 0.23/0.56 = one_one_int ) )
% 0.23/0.56 => ( ( groups705719431365010083at_int @ G @ A3 )
% 0.23/0.56 = one_one_int ) ) ).
% 0.23/0.56
% 0.23/0.56 % prod.neutral
% 0.23/0.56 thf(fact_8760_prod_Oneutral,axiom,
% 0.23/0.56 ! [A3: set_int,G: int > int] :
% 0.23/0.57 ( ! [X3: int] :
% 0.23/0.57 ( ( member_int @ X3 @ A3 )
% 0.23/0.57 => ( ( G @ X3 )
% 0.23/0.57 = one_one_int ) )
% 0.23/0.57 => ( ( groups1705073143266064639nt_int @ G @ A3 )
% 0.23/0.57 = one_one_int ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod.neutral
% 0.23/0.57 thf(fact_8761_prod_Odistrib,axiom,
% 0.23/0.57 ! [G: nat > nat,H2: nat > nat,A3: set_nat] :
% 0.23/0.57 ( ( groups708209901874060359at_nat
% 0.23/0.57 @ ^ [X: nat] : ( times_times_nat @ ( G @ X ) @ ( H2 @ X ) )
% 0.23/0.57 @ A3 )
% 0.23/0.57 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A3 ) @ ( groups708209901874060359at_nat @ H2 @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod.distrib
% 0.23/0.57 thf(fact_8762_prod_Odistrib,axiom,
% 0.23/0.57 ! [G: nat > int,H2: nat > int,A3: set_nat] :
% 0.23/0.57 ( ( groups705719431365010083at_int
% 0.23/0.57 @ ^ [X: nat] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
% 0.23/0.57 @ A3 )
% 0.23/0.57 = ( times_times_int @ ( groups705719431365010083at_int @ G @ A3 ) @ ( groups705719431365010083at_int @ H2 @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod.distrib
% 0.23/0.57 thf(fact_8763_prod_Odistrib,axiom,
% 0.23/0.57 ! [G: int > int,H2: int > int,A3: set_int] :
% 0.23/0.57 ( ( groups1705073143266064639nt_int
% 0.23/0.57 @ ^ [X: int] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
% 0.23/0.57 @ A3 )
% 0.23/0.57 = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A3 ) @ ( groups1705073143266064639nt_int @ H2 @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod.distrib
% 0.23/0.57 thf(fact_8764_prod__power__distrib,axiom,
% 0.23/0.57 ! [F: nat > nat,A3: set_nat,N: nat] :
% 0.23/0.57 ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ N )
% 0.23/0.57 = ( groups708209901874060359at_nat
% 0.23/0.57 @ ^ [X: nat] : ( power_power_nat @ ( F @ X ) @ N )
% 0.23/0.57 @ A3 ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_power_distrib
% 0.23/0.57 thf(fact_8765_prod__power__distrib,axiom,
% 0.23/0.57 ! [F: nat > int,A3: set_nat,N: nat] :
% 0.23/0.57 ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A3 ) @ N )
% 0.23/0.57 = ( groups705719431365010083at_int
% 0.23/0.57 @ ^ [X: nat] : ( power_power_int @ ( F @ X ) @ N )
% 0.23/0.57 @ A3 ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_power_distrib
% 0.23/0.57 thf(fact_8766_prod__power__distrib,axiom,
% 0.23/0.57 ! [F: int > int,A3: set_int,N: nat] :
% 0.23/0.57 ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A3 ) @ N )
% 0.23/0.57 = ( groups1705073143266064639nt_int
% 0.23/0.57 @ ^ [X: int] : ( power_power_int @ ( F @ X ) @ N )
% 0.23/0.57 @ A3 ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_power_distrib
% 0.23/0.57 thf(fact_8767_prod__nonneg,axiom,
% 0.23/0.57 ! [A3: set_nat,F: nat > nat] :
% 0.23/0.57 ( ! [X3: nat] :
% 0.23/0.57 ( ( member_nat @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_nonneg
% 0.23/0.57 thf(fact_8768_prod__nonneg,axiom,
% 0.23/0.57 ! [A3: set_nat,F: nat > int] :
% 0.23/0.57 ( ! [X3: nat] :
% 0.23/0.57 ( ( member_nat @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_nonneg
% 0.23/0.57 thf(fact_8769_prod__nonneg,axiom,
% 0.23/0.57 ! [A3: set_int,F: int > int] :
% 0.23/0.57 ( ! [X3: int] :
% 0.23/0.57 ( ( member_int @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_nonneg
% 0.23/0.57 thf(fact_8770_prod__mono,axiom,
% 0.23/0.57 ! [A3: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
% 0.23/0.57 ( ! [I3: extended_enat] :
% 0.23/0.57 ( ( member_Extended_enat @ I3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.23/0.57 & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A3 ) @ ( groups97031904164794029t_real @ G @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_mono
% 0.23/0.57 thf(fact_8771_prod__mono,axiom,
% 0.23/0.57 ! [A3: set_complex,F: complex > real,G: complex > real] :
% 0.23/0.57 ( ! [I3: complex] :
% 0.23/0.57 ( ( member_complex @ I3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.23/0.57 & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A3 ) @ ( groups766887009212190081x_real @ G @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_mono
% 0.23/0.57 thf(fact_8772_prod__mono,axiom,
% 0.23/0.57 ! [A3: set_real,F: real > real,G: real > real] :
% 0.23/0.57 ( ! [I3: real] :
% 0.23/0.57 ( ( member_real @ I3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.23/0.57 & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A3 ) @ ( groups1681761925125756287l_real @ G @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_mono
% 0.23/0.57 thf(fact_8773_prod__mono,axiom,
% 0.23/0.57 ! [A3: set_nat,F: nat > real,G: nat > real] :
% 0.23/0.57 ( ! [I3: nat] :
% 0.23/0.57 ( ( member_nat @ I3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.23/0.57 & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A3 ) @ ( groups129246275422532515t_real @ G @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_mono
% 0.23/0.57 thf(fact_8774_prod__mono,axiom,
% 0.23/0.57 ! [A3: set_int,F: int > real,G: int > real] :
% 0.23/0.57 ( ! [I3: int] :
% 0.23/0.57 ( ( member_int @ I3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.23/0.57 & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A3 ) @ ( groups2316167850115554303t_real @ G @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_mono
% 0.23/0.57 thf(fact_8775_prod__mono,axiom,
% 0.23/0.57 ! [A3: set_Extended_enat,F: extended_enat > rat,G: extended_enat > rat] :
% 0.23/0.57 ( ! [I3: extended_enat] :
% 0.23/0.57 ( ( member_Extended_enat @ I3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.23/0.57 & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ ( groups2245840878043517529at_rat @ F @ A3 ) @ ( groups2245840878043517529at_rat @ G @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_mono
% 0.23/0.57 thf(fact_8776_prod__mono,axiom,
% 0.23/0.57 ! [A3: set_complex,F: complex > rat,G: complex > rat] :
% 0.23/0.57 ( ! [I3: complex] :
% 0.23/0.57 ( ( member_complex @ I3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.23/0.57 & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) @ ( groups225925009352817453ex_rat @ G @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_mono
% 0.23/0.57 thf(fact_8777_prod__mono,axiom,
% 0.23/0.57 ! [A3: set_real,F: real > rat,G: real > rat] :
% 0.23/0.57 ( ! [I3: real] :
% 0.23/0.57 ( ( member_real @ I3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.23/0.57 & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) @ ( groups4061424788464935467al_rat @ G @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_mono
% 0.23/0.57 thf(fact_8778_prod__mono,axiom,
% 0.23/0.57 ! [A3: set_nat,F: nat > rat,G: nat > rat] :
% 0.23/0.57 ( ! [I3: nat] :
% 0.23/0.57 ( ( member_nat @ I3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.23/0.57 & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ ( groups73079841787564623at_rat @ G @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_mono
% 0.23/0.57 thf(fact_8779_prod__mono,axiom,
% 0.23/0.57 ! [A3: set_int,F: int > rat,G: int > rat] :
% 0.23/0.57 ( ! [I3: int] :
% 0.23/0.57 ( ( member_int @ I3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.23/0.57 & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ ( groups1072433553688619179nt_rat @ G @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_mono
% 0.23/0.57 thf(fact_8780_prod__pos,axiom,
% 0.23/0.57 ! [A3: set_nat,F: nat > nat] :
% 0.23/0.57 ( ! [X3: nat] :
% 0.23/0.57 ( ( member_nat @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_pos
% 0.23/0.57 thf(fact_8781_prod__pos,axiom,
% 0.23/0.57 ! [A3: set_nat,F: nat > int] :
% 0.23/0.57 ( ! [X3: nat] :
% 0.23/0.57 ( ( member_nat @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_pos
% 0.23/0.57 thf(fact_8782_prod__pos,axiom,
% 0.23/0.57 ! [A3: set_int,F: int > int] :
% 0.23/0.57 ( ! [X3: int] :
% 0.23/0.57 ( ( member_int @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_pos
% 0.23/0.57 thf(fact_8783_prod__ge__1,axiom,
% 0.23/0.57 ! [A3: set_Extended_enat,F: extended_enat > real] :
% 0.23/0.57 ( ! [X3: extended_enat] :
% 0.23/0.57 ( ( member_Extended_enat @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ one_one_real @ ( groups97031904164794029t_real @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_ge_1
% 0.23/0.57 thf(fact_8784_prod__ge__1,axiom,
% 0.23/0.57 ! [A3: set_complex,F: complex > real] :
% 0.23/0.57 ( ! [X3: complex] :
% 0.23/0.57 ( ( member_complex @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_ge_1
% 0.23/0.57 thf(fact_8785_prod__ge__1,axiom,
% 0.23/0.57 ! [A3: set_real,F: real > real] :
% 0.23/0.57 ( ! [X3: real] :
% 0.23/0.57 ( ( member_real @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_ge_1
% 0.23/0.57 thf(fact_8786_prod__ge__1,axiom,
% 0.23/0.57 ! [A3: set_nat,F: nat > real] :
% 0.23/0.57 ( ! [X3: nat] :
% 0.23/0.57 ( ( member_nat @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_ge_1
% 0.23/0.57 thf(fact_8787_prod__ge__1,axiom,
% 0.23/0.57 ! [A3: set_int,F: int > real] :
% 0.23/0.57 ( ! [X3: int] :
% 0.23/0.57 ( ( member_int @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_ge_1
% 0.23/0.57 thf(fact_8788_prod__ge__1,axiom,
% 0.23/0.57 ! [A3: set_Extended_enat,F: extended_enat > rat] :
% 0.23/0.57 ( ! [X3: extended_enat] :
% 0.23/0.57 ( ( member_Extended_enat @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ one_one_rat @ ( groups2245840878043517529at_rat @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_ge_1
% 0.23/0.57 thf(fact_8789_prod__ge__1,axiom,
% 0.23/0.57 ! [A3: set_complex,F: complex > rat] :
% 0.23/0.57 ( ! [X3: complex] :
% 0.23/0.57 ( ( member_complex @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_ge_1
% 0.23/0.57 thf(fact_8790_prod__ge__1,axiom,
% 0.23/0.57 ! [A3: set_real,F: real > rat] :
% 0.23/0.57 ( ! [X3: real] :
% 0.23/0.57 ( ( member_real @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_ge_1
% 0.23/0.57 thf(fact_8791_prod__ge__1,axiom,
% 0.23/0.57 ! [A3: set_nat,F: nat > rat] :
% 0.23/0.57 ( ! [X3: nat] :
% 0.23/0.57 ( ( member_nat @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_ge_1
% 0.23/0.57 thf(fact_8792_prod__ge__1,axiom,
% 0.23/0.57 ! [A3: set_int,F: int > rat] :
% 0.23/0.57 ( ! [X3: int] :
% 0.23/0.57 ( ( member_int @ X3 @ A3 )
% 0.23/0.57 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_ge_1
% 0.23/0.57 thf(fact_8793_prod__le__1,axiom,
% 0.23/0.57 ! [A3: set_Extended_enat,F: extended_enat > real] :
% 0.23/0.57 ( ! [X3: extended_enat] :
% 0.23/0.57 ( ( member_Extended_enat @ X3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.23/0.57 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A3 ) @ one_one_real ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_le_1
% 0.23/0.57 thf(fact_8794_prod__le__1,axiom,
% 0.23/0.57 ! [A3: set_complex,F: complex > real] :
% 0.23/0.57 ( ! [X3: complex] :
% 0.23/0.57 ( ( member_complex @ X3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.23/0.57 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A3 ) @ one_one_real ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_le_1
% 0.23/0.57 thf(fact_8795_prod__le__1,axiom,
% 0.23/0.57 ! [A3: set_real,F: real > real] :
% 0.23/0.57 ( ! [X3: real] :
% 0.23/0.57 ( ( member_real @ X3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.23/0.57 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A3 ) @ one_one_real ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_le_1
% 0.23/0.57 thf(fact_8796_prod__le__1,axiom,
% 0.23/0.57 ! [A3: set_nat,F: nat > real] :
% 0.23/0.57 ( ! [X3: nat] :
% 0.23/0.57 ( ( member_nat @ X3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.23/0.57 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A3 ) @ one_one_real ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_le_1
% 0.23/0.57 thf(fact_8797_prod__le__1,axiom,
% 0.23/0.57 ! [A3: set_int,F: int > real] :
% 0.23/0.57 ( ! [X3: int] :
% 0.23/0.57 ( ( member_int @ X3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.23/0.57 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A3 ) @ one_one_real ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_le_1
% 0.23/0.57 thf(fact_8798_prod__le__1,axiom,
% 0.23/0.57 ! [A3: set_Extended_enat,F: extended_enat > rat] :
% 0.23/0.57 ( ! [X3: extended_enat] :
% 0.23/0.57 ( ( member_Extended_enat @ X3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 0.23/0.57 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ ( groups2245840878043517529at_rat @ F @ A3 ) @ one_one_rat ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_le_1
% 0.23/0.57 thf(fact_8799_prod__le__1,axiom,
% 0.23/0.57 ! [A3: set_complex,F: complex > rat] :
% 0.23/0.57 ( ! [X3: complex] :
% 0.23/0.57 ( ( member_complex @ X3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 0.23/0.57 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) @ one_one_rat ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_le_1
% 0.23/0.57 thf(fact_8800_prod__le__1,axiom,
% 0.23/0.57 ! [A3: set_real,F: real > rat] :
% 0.23/0.57 ( ! [X3: real] :
% 0.23/0.57 ( ( member_real @ X3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 0.23/0.57 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) @ one_one_rat ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_le_1
% 0.23/0.57 thf(fact_8801_prod__le__1,axiom,
% 0.23/0.57 ! [A3: set_nat,F: nat > rat] :
% 0.23/0.57 ( ! [X3: nat] :
% 0.23/0.57 ( ( member_nat @ X3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 0.23/0.57 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ one_one_rat ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_le_1
% 0.23/0.57 thf(fact_8802_prod__le__1,axiom,
% 0.23/0.57 ! [A3: set_int,F: int > rat] :
% 0.23/0.57 ( ! [X3: int] :
% 0.23/0.57 ( ( member_int @ X3 @ A3 )
% 0.23/0.57 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 0.23/0.57 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 0.23/0.57 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ one_one_rat ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_le_1
% 0.23/0.57 thf(fact_8803_vebt__mint_Osimps_I1_J,axiom,
% 0.23/0.57 ! [A: $o,B2: $o] :
% 0.23/0.57 ( ( A
% 0.23/0.57 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 0.23/0.57 = ( some_nat @ zero_zero_nat ) ) )
% 0.23/0.57 & ( ~ A
% 0.23/0.57 => ( ( B2
% 0.23/0.57 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 0.23/0.57 = ( some_nat @ one_one_nat ) ) )
% 0.23/0.57 & ( ~ B2
% 0.23/0.57 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 0.23/0.57 = none_nat ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % vebt_mint.simps(1)
% 0.23/0.57 thf(fact_8804_vebt__succ_Opelims,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat,Y3: option_nat] :
% 0.23/0.57 ( ( ( vEBT_vebt_succ @ X4 @ Xa )
% 0.23/0.57 = Y3 )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.57 => ( ! [Uu2: $o,B5: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ Uu2 @ B5 ) )
% 0.23/0.57 => ( ( Xa = zero_zero_nat )
% 0.23/0.57 => ( ( ( B5
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( some_nat @ one_one_nat ) ) )
% 0.23/0.57 & ( ~ B5
% 0.23/0.57 => ( Y3 = none_nat ) ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) ) ) ) )
% 0.23/0.57 => ( ! [Uv: $o,Uw: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ Uv @ Uw ) )
% 0.23/0.57 => ! [N4: nat] :
% 0.23/0.57 ( ( Xa
% 0.23/0.57 = ( suc @ N4 ) )
% 0.23/0.57 => ( ( Y3 = none_nat )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N4 ) ) ) ) ) )
% 0.23/0.57 => ( ! [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) )
% 0.23/0.57 => ( ( Y3 = none_nat )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.23/0.57 => ( ( Y3 = none_nat )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 0.23/0.57 => ( ( Y3 = none_nat )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
% 0.23/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.57 => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( some_nat @ Mi2 ) ) )
% 0.23/0.57 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 @ ( if_option_nat
% 0.23/0.57 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.57 != none_nat )
% 0.23/0.57 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.23/0.57 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.57 @ ( if_option_nat
% 0.23/0.57 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.57 = none_nat )
% 0.23/0.57 @ none_nat
% 0.23/0.57 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.23/0.57 @ none_nat ) ) ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % vebt_succ.pelims
% 0.23/0.57 thf(fact_8805_ln__series,axiom,
% 0.23/0.57 ! [X4: real] :
% 0.23/0.57 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.23/0.57 => ( ( ln_ln_real @ X4 )
% 0.23/0.57 = ( suminf_real
% 0.23/0.57 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % ln_series
% 0.23/0.57 thf(fact_8806_arctan__series,axiom,
% 0.23/0.57 ! [X4: real] :
% 0.23/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.57 => ( ( arctan @ X4 )
% 0.23/0.57 = ( suminf_real
% 0.23/0.57 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % arctan_series
% 0.23/0.57 thf(fact_8807_divmod__algorithm__code_I7_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( ( ord_less_eq_num @ M @ N )
% 0.23/0.57 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 0.23/0.57 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.23/0.57 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(7)
% 0.23/0.57 thf(fact_8808_divmod__algorithm__code_I7_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( ( ord_less_eq_num @ M @ N )
% 0.23/0.57 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 0.23/0.57 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.23/0.57 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(7)
% 0.23/0.57 thf(fact_8809_divmod__algorithm__code_I7_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( ( ord_less_eq_num @ M @ N )
% 0.23/0.57 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 0.23/0.57 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.23/0.57 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(7)
% 0.23/0.57 thf(fact_8810_divmod__algorithm__code_I2_J,axiom,
% 0.23/0.57 ! [M: num] :
% 0.23/0.57 ( ( unique5052692396658037445od_int @ M @ one )
% 0.23/0.57 = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(2)
% 0.23/0.57 thf(fact_8811_divmod__algorithm__code_I2_J,axiom,
% 0.23/0.57 ! [M: num] :
% 0.23/0.57 ( ( unique5055182867167087721od_nat @ M @ one )
% 0.23/0.57 = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(2)
% 0.23/0.57 thf(fact_8812_divmod__algorithm__code_I2_J,axiom,
% 0.23/0.57 ! [M: num] :
% 0.23/0.57 ( ( unique3479559517661332726nteger @ M @ one )
% 0.23/0.57 = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(2)
% 0.23/0.57 thf(fact_8813_powser__zero,axiom,
% 0.23/0.57 ! [F: nat > complex] :
% 0.23/0.57 ( ( suminf_complex
% 0.23/0.57 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
% 0.23/0.57 = ( F @ zero_zero_nat ) ) ).
% 0.23/0.57
% 0.23/0.57 % powser_zero
% 0.23/0.57 thf(fact_8814_powser__zero,axiom,
% 0.23/0.57 ! [F: nat > real] :
% 0.23/0.57 ( ( suminf_real
% 0.23/0.57 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
% 0.23/0.57 = ( F @ zero_zero_nat ) ) ).
% 0.23/0.57
% 0.23/0.57 % powser_zero
% 0.23/0.57 thf(fact_8815_divmod__algorithm__code_I3_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 0.23/0.57 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(3)
% 0.23/0.57 thf(fact_8816_divmod__algorithm__code_I3_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 0.23/0.57 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(3)
% 0.23/0.57 thf(fact_8817_divmod__algorithm__code_I3_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 0.23/0.57 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(3)
% 0.23/0.57 thf(fact_8818_divmod__algorithm__code_I4_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 0.23/0.57 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(4)
% 0.23/0.57 thf(fact_8819_divmod__algorithm__code_I4_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 0.23/0.57 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(4)
% 0.23/0.57 thf(fact_8820_divmod__algorithm__code_I4_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 0.23/0.57 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(4)
% 0.23/0.57 thf(fact_8821_divmod__algorithm__code_I8_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( ( ord_less_num @ M @ N )
% 0.23/0.57 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 0.23/0.57 & ( ~ ( ord_less_num @ M @ N )
% 0.23/0.57 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(8)
% 0.23/0.57 thf(fact_8822_divmod__algorithm__code_I8_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( ( ord_less_num @ M @ N )
% 0.23/0.57 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 0.23/0.57 & ( ~ ( ord_less_num @ M @ N )
% 0.23/0.57 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(8)
% 0.23/0.57 thf(fact_8823_divmod__algorithm__code_I8_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( ( ord_less_num @ M @ N )
% 0.23/0.57 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 0.23/0.57 & ( ~ ( ord_less_num @ M @ N )
% 0.23/0.57 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.57 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_algorithm_code(8)
% 0.23/0.57 thf(fact_8824_divmod__int__def,axiom,
% 0.23/0.57 ( unique5052692396658037445od_int
% 0.23/0.57 = ( ^ [M6: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_int_def
% 0.23/0.57 thf(fact_8825_divmod__def,axiom,
% 0.23/0.57 ( unique5052692396658037445od_int
% 0.23/0.57 = ( ^ [M6: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_def
% 0.23/0.57 thf(fact_8826_divmod__def,axiom,
% 0.23/0.57 ( unique5055182867167087721od_nat
% 0.23/0.57 = ( ^ [M6: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_def
% 0.23/0.57 thf(fact_8827_divmod__def,axiom,
% 0.23/0.57 ( unique3479559517661332726nteger
% 0.23/0.57 = ( ^ [M6: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_def
% 0.23/0.57 thf(fact_8828_divmod_H__nat__def,axiom,
% 0.23/0.57 ( unique5055182867167087721od_nat
% 0.23/0.57 = ( ^ [M6: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod'_nat_def
% 0.23/0.57 thf(fact_8829_divmod__divmod__step,axiom,
% 0.23/0.57 ( unique5055182867167087721od_nat
% 0.23/0.57 = ( ^ [M6: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_divmod_step
% 0.23/0.57 thf(fact_8830_divmod__divmod__step,axiom,
% 0.23/0.57 ( unique5052692396658037445od_int
% 0.23/0.57 = ( ^ [M6: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_divmod_step
% 0.23/0.57 thf(fact_8831_divmod__divmod__step,axiom,
% 0.23/0.57 ( unique3479559517661332726nteger
% 0.23/0.57 = ( ^ [M6: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M6 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M6 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_divmod_step
% 0.23/0.57 thf(fact_8832_pi__series,axiom,
% 0.23/0.57 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.23/0.57 = ( suminf_real
% 0.23/0.57 @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % pi_series
% 0.23/0.57 thf(fact_8833_vebt__member_Opelims_I3_J,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.57 ( ~ ( vEBT_vebt_member @ X4 @ Xa )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.57 => ( ! [A5: $o,B5: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
% 0.23/0.57 => ( ( ( Xa = zero_zero_nat )
% 0.23/0.57 => A5 )
% 0.23/0.57 & ( ( Xa != zero_zero_nat )
% 0.23/0.57 => ( ( ( Xa = one_one_nat )
% 0.23/0.57 => B5 )
% 0.23/0.57 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.23/0.57 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) @ Xa ) ) )
% 0.23/0.57 => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) @ Xa ) ) )
% 0.23/0.57 => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) )
% 0.23/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
% 0.23/0.57 => ( ( Xa != Mi2 )
% 0.23/0.57 => ( ( Xa != Ma2 )
% 0.23/0.57 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.57 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.57 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.57 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.57 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % vebt_member.pelims(3)
% 0.23/0.57 thf(fact_8834_vebt__member_Opelims_I1_J,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.23/0.57 ( ( ( vEBT_vebt_member @ X4 @ Xa )
% 0.23/0.57 = Y3 )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.57 => ( ! [A5: $o,B5: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( ( ( Xa = zero_zero_nat )
% 0.23/0.57 => A5 )
% 0.23/0.57 & ( ( Xa != zero_zero_nat )
% 0.23/0.57 => ( ( ( Xa = one_one_nat )
% 0.23/0.57 => B5 )
% 0.23/0.57 & ( Xa = one_one_nat ) ) ) ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 0.23/0.57 => ( ~ Y3
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
% 0.23/0.57 => ( ~ Y3
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.23/0.57 => ( ~ Y3
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
% 0.23/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( ( Xa != Mi2 )
% 0.23/0.57 => ( ( Xa != Ma2 )
% 0.23/0.57 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.57 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.57 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.57 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.57 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % vebt_member.pelims(1)
% 0.23/0.57 thf(fact_8835_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.23/0.57 ( ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 0.23/0.57 = Y3 )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.57 => ( ! [A5: $o,B5: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( ( ( Xa = zero_zero_nat )
% 0.23/0.57 => A5 )
% 0.23/0.57 & ( ( Xa != zero_zero_nat )
% 0.23/0.57 => ( ( ( Xa = one_one_nat )
% 0.23/0.57 => B5 )
% 0.23/0.57 & ( Xa = one_one_nat ) ) ) ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
% 0.23/0.57 => ( ~ Y3
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) @ Xa ) ) ) )
% 0.23/0.57 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( 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 ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % VEBT_internal.naive_member.pelims(1)
% 0.23/0.57 thf(fact_8836_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.57 ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.57 => ( ! [A5: $o,B5: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
% 0.23/0.57 => ~ ( ( ( Xa = zero_zero_nat )
% 0.23/0.57 => A5 )
% 0.23/0.57 & ( ( Xa != zero_zero_nat )
% 0.23/0.57 => ( ( ( Xa = one_one_nat )
% 0.23/0.57 => B5 )
% 0.23/0.57 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.23/0.57 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 0.23/0.57 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( 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 ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % VEBT_internal.naive_member.pelims(2)
% 0.23/0.57 thf(fact_8837_vebt__member_Opelims_I2_J,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.57 ( ( vEBT_vebt_member @ X4 @ Xa )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.57 => ( ! [A5: $o,B5: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
% 0.23/0.57 => ~ ( ( ( Xa = zero_zero_nat )
% 0.23/0.57 => A5 )
% 0.23/0.57 & ( ( Xa != zero_zero_nat )
% 0.23/0.57 => ( ( ( Xa = one_one_nat )
% 0.23/0.57 => B5 )
% 0.23/0.57 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.23/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
% 0.23/0.57 => ~ ( ( Xa != Mi2 )
% 0.23/0.57 => ( ( Xa != Ma2 )
% 0.23/0.57 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.57 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.23/0.57 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.57 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.23/0.57 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % vebt_member.pelims(2)
% 0.23/0.57 thf(fact_8838_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.57 ( ~ ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.57 => ( ! [A5: $o,B5: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
% 0.23/0.57 => ( ( ( Xa = zero_zero_nat )
% 0.23/0.57 => A5 )
% 0.23/0.57 & ( ( Xa != zero_zero_nat )
% 0.23/0.57 => ( ( ( Xa = one_one_nat )
% 0.23/0.57 => B5 )
% 0.23/0.57 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.23/0.57 => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) @ Xa ) ) )
% 0.23/0.57 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 0.23/0.57 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( 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 ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % VEBT_internal.naive_member.pelims(3)
% 0.23/0.57 thf(fact_8839_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.23/0.57 ( ( ( vEBT_VEBT_membermima @ X4 @ Xa )
% 0.23/0.57 = Y3 )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.57 => ( ! [Uu2: $o,Uv: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 0.23/0.57 => ( ~ Y3
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
% 0.23/0.57 => ( ~ Y3
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( ( Xa = Mi2 )
% 0.23/0.57 | ( Xa = Ma2 ) ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( ( Xa = Mi2 )
% 0.23/0.57 | ( Xa = Ma2 )
% 0.23/0.57 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( 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 ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) ) ) )
% 0.23/0.57 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( 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 ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % VEBT_internal.membermima.pelims(1)
% 0.23/0.57 thf(fact_8840_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.57 ( ~ ( vEBT_VEBT_membermima @ X4 @ Xa )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.57 => ( ! [Uu2: $o,Uv: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) ) )
% 0.23/0.57 => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) @ Xa ) ) )
% 0.23/0.57 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
% 0.23/0.57 => ( ( Xa = Mi2 )
% 0.23/0.57 | ( Xa = Ma2 ) ) ) )
% 0.23/0.57 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 0.23/0.57 => ( ( Xa = Mi2 )
% 0.23/0.57 | ( Xa = Ma2 )
% 0.23/0.57 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( 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 ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.23/0.57 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 0.23/0.57 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( 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 ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % VEBT_internal.membermima.pelims(3)
% 0.23/0.57 thf(fact_8841_suminf__geometric,axiom,
% 0.23/0.57 ! [C: real] :
% 0.23/0.57 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.23/0.57 => ( ( suminf_real @ ( power_power_real @ C ) )
% 0.23/0.57 = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % suminf_geometric
% 0.23/0.57 thf(fact_8842_suminf__geometric,axiom,
% 0.23/0.57 ! [C: complex] :
% 0.23/0.57 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.23/0.57 => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 0.23/0.57 = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % suminf_geometric
% 0.23/0.57 thf(fact_8843_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.57 ( ( vEBT_VEBT_membermima @ X4 @ Xa )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.57 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
% 0.23/0.57 => ~ ( ( Xa = Mi2 )
% 0.23/0.57 | ( Xa = Ma2 ) ) ) )
% 0.23/0.57 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 0.23/0.57 => ~ ( ( Xa = Mi2 )
% 0.23/0.57 | ( Xa = Ma2 )
% 0.23/0.57 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( 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 ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.23/0.57 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 0.23/0.57 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 => ( 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 ) ) ) ) ) )
% 0.23/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % VEBT_internal.membermima.pelims(2)
% 0.23/0.57 thf(fact_8844_minus__one__div__numeral,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % minus_one_div_numeral
% 0.23/0.57 thf(fact_8845_one__div__minus__numeral,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % one_div_minus_numeral
% 0.23/0.57 thf(fact_8846_dvd__numeral__simp,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.57 = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % dvd_numeral_simp
% 0.23/0.57 thf(fact_8847_dvd__numeral__simp,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.23/0.57 = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % dvd_numeral_simp
% 0.23/0.57 thf(fact_8848_dvd__numeral__simp,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 0.23/0.57 = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % dvd_numeral_simp
% 0.23/0.57 thf(fact_8849_minus__numeral__div__numeral,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % minus_numeral_div_numeral
% 0.23/0.57 thf(fact_8850_divides__aux__eq,axiom,
% 0.23/0.57 ! [Q2: nat,R2: nat] :
% 0.23/0.57 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
% 0.23/0.57 = ( R2 = zero_zero_nat ) ) ).
% 0.23/0.57
% 0.23/0.57 % divides_aux_eq
% 0.23/0.57 thf(fact_8851_divides__aux__eq,axiom,
% 0.23/0.57 ! [Q2: int,R2: int] :
% 0.23/0.57 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.57 = ( R2 = zero_zero_int ) ) ).
% 0.23/0.57
% 0.23/0.57 % divides_aux_eq
% 0.23/0.57 thf(fact_8852_Divides_Oadjust__div__eq,axiom,
% 0.23/0.57 ! [Q2: int,R2: int] :
% 0.23/0.57 ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.57 = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Divides.adjust_div_eq
% 0.23/0.57 thf(fact_8853_numeral__div__minus__numeral,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % numeral_div_minus_numeral
% 0.23/0.57 thf(fact_8854_neg__eucl__rel__int__mult__2,axiom,
% 0.23/0.57 ! [B2: int,A: int,Q2: int,R2: int] :
% 0.23/0.57 ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.23/0.57 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.57 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % neg_eucl_rel_int_mult_2
% 0.23/0.57 thf(fact_8855_summable__arctan__series,axiom,
% 0.23/0.57 ! [X4: real] :
% 0.23/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.57 => ( summable_real
% 0.23/0.57 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % summable_arctan_series
% 0.23/0.57 thf(fact_8856_pos__eucl__rel__int__mult__2,axiom,
% 0.23/0.57 ! [B2: int,A: int,Q2: int,R2: int] :
% 0.23/0.57 ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.23/0.57 => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.57 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % pos_eucl_rel_int_mult_2
% 0.23/0.57 thf(fact_8857_product__nth,axiom,
% 0.23/0.57 ! [N: nat,Xs2: list_num,Ys2: list_num] :
% 0.23/0.57 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys2 ) ) )
% 0.23/0.57 => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys2 ) @ N )
% 0.23/0.57 = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) @ ( nth_num @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % product_nth
% 0.23/0.57 thf(fact_8858_product__nth,axiom,
% 0.23/0.57 ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 0.23/0.57 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 0.23/0.57 => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys2 ) @ N )
% 0.23/0.57 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % product_nth
% 0.23/0.57 thf(fact_8859_product__nth,axiom,
% 0.23/0.57 ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_o] :
% 0.23/0.57 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) )
% 0.23/0.57 => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys2 ) @ N )
% 0.23/0.57 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % product_nth
% 0.23/0.57 thf(fact_8860_product__nth,axiom,
% 0.23/0.57 ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_nat] :
% 0.23/0.57 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) )
% 0.23/0.57 => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys2 ) @ N )
% 0.23/0.57 = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % product_nth
% 0.23/0.57 thf(fact_8861_product__nth,axiom,
% 0.23/0.57 ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_int] :
% 0.23/0.57 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) )
% 0.23/0.57 => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys2 ) @ N )
% 0.23/0.57 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % product_nth
% 0.23/0.57 thf(fact_8862_product__nth,axiom,
% 0.23/0.57 ! [N: nat,Xs2: list_o,Ys2: list_VEBT_VEBT] :
% 0.23/0.57 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 0.23/0.57 => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys2 ) @ N )
% 0.23/0.57 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % product_nth
% 0.23/0.57 thf(fact_8863_product__nth,axiom,
% 0.23/0.57 ! [N: nat,Xs2: list_o,Ys2: list_o] :
% 0.23/0.57 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) )
% 0.23/0.57 => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys2 ) @ N )
% 0.23/0.57 = ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % product_nth
% 0.23/0.57 thf(fact_8864_product__nth,axiom,
% 0.23/0.57 ! [N: nat,Xs2: list_o,Ys2: list_nat] :
% 0.23/0.57 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) )
% 0.23/0.57 => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys2 ) @ N )
% 0.23/0.57 = ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % product_nth
% 0.23/0.57 thf(fact_8865_product__nth,axiom,
% 0.23/0.57 ! [N: nat,Xs2: list_o,Ys2: list_int] :
% 0.23/0.57 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) )
% 0.23/0.57 => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys2 ) @ N )
% 0.23/0.57 = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % product_nth
% 0.23/0.57 thf(fact_8866_product__nth,axiom,
% 0.23/0.57 ! [N: nat,Xs2: list_nat,Ys2: list_num] :
% 0.23/0.57 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_num @ Ys2 ) ) )
% 0.23/0.57 => ( ( nth_Pr8326237132889035090at_num @ ( product_nat_num @ Xs2 @ Ys2 ) @ N )
% 0.23/0.57 = ( product_Pair_nat_num @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) @ ( nth_num @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % product_nth
% 0.23/0.57 thf(fact_8867_summable__iff__shift,axiom,
% 0.23/0.57 ! [F: nat > real,K: nat] :
% 0.23/0.57 ( ( summable_real
% 0.23/0.57 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.23/0.57 = ( summable_real @ F ) ) ).
% 0.23/0.57
% 0.23/0.57 % summable_iff_shift
% 0.23/0.57 thf(fact_8868_summable__cmult__iff,axiom,
% 0.23/0.57 ! [C: real,F: nat > real] :
% 0.23/0.57 ( ( summable_real
% 0.23/0.57 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 0.23/0.57 = ( ( C = zero_zero_real )
% 0.23/0.57 | ( summable_real @ F ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % summable_cmult_iff
% 0.23/0.57 thf(fact_8869_summable__divide__iff,axiom,
% 0.23/0.57 ! [F: nat > complex,C: complex] :
% 0.23/0.57 ( ( summable_complex
% 0.23/0.57 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 0.23/0.57 = ( ( C = zero_zero_complex )
% 0.23/0.57 | ( summable_complex @ F ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % summable_divide_iff
% 0.23/0.57 thf(fact_8870_summable__divide__iff,axiom,
% 0.23/0.57 ! [F: nat > real,C: real] :
% 0.23/0.57 ( ( summable_real
% 0.23/0.57 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 0.23/0.57 = ( ( C = zero_zero_real )
% 0.23/0.57 | ( summable_real @ F ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % summable_divide_iff
% 0.23/0.57 thf(fact_8871_length__product,axiom,
% 0.23/0.57 ! [Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 0.23/0.57 ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys2 ) )
% 0.23/0.57 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % length_product
% 0.23/0.57 thf(fact_8872_length__product,axiom,
% 0.23/0.57 ! [Xs2: list_VEBT_VEBT,Ys2: list_o] :
% 0.23/0.57 ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys2 ) )
% 0.23/0.57 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % length_product
% 0.23/0.57 thf(fact_8873_length__product,axiom,
% 0.23/0.57 ! [Xs2: list_VEBT_VEBT,Ys2: list_nat] :
% 0.23/0.57 ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys2 ) )
% 0.23/0.57 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % length_product
% 0.23/0.57 thf(fact_8874_length__product,axiom,
% 0.23/0.57 ! [Xs2: list_VEBT_VEBT,Ys2: list_int] :
% 0.23/0.57 ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys2 ) )
% 0.23/0.57 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % length_product
% 0.23/0.57 thf(fact_8875_length__product,axiom,
% 0.23/0.57 ! [Xs2: list_o,Ys2: list_VEBT_VEBT] :
% 0.23/0.57 ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys2 ) )
% 0.23/0.57 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % length_product
% 0.23/0.57 thf(fact_8876_length__product,axiom,
% 0.23/0.57 ! [Xs2: list_o,Ys2: list_o] :
% 0.23/0.57 ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys2 ) )
% 0.23/0.57 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % length_product
% 0.23/0.57 thf(fact_8877_length__product,axiom,
% 0.23/0.57 ! [Xs2: list_o,Ys2: list_nat] :
% 0.23/0.57 ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys2 ) )
% 0.23/0.57 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % length_product
% 0.23/0.57 thf(fact_8878_length__product,axiom,
% 0.23/0.57 ! [Xs2: list_o,Ys2: list_int] :
% 0.23/0.57 ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys2 ) )
% 0.23/0.57 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % length_product
% 0.23/0.57 thf(fact_8879_length__product,axiom,
% 0.23/0.57 ! [Xs2: list_nat,Ys2: list_VEBT_VEBT] :
% 0.23/0.57 ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys2 ) )
% 0.23/0.57 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % length_product
% 0.23/0.57 thf(fact_8880_length__product,axiom,
% 0.23/0.57 ! [Xs2: list_nat,Ys2: list_o] :
% 0.23/0.57 ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys2 ) )
% 0.23/0.57 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % length_product
% 0.23/0.57 thf(fact_8881_summable__geometric__iff,axiom,
% 0.23/0.57 ! [C: complex] :
% 0.23/0.57 ( ( summable_complex @ ( power_power_complex @ C ) )
% 0.23/0.57 = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 0.23/0.57
% 0.23/0.57 % summable_geometric_iff
% 0.23/0.57 thf(fact_8882_unique__quotient,axiom,
% 0.23/0.57 ! [A: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
% 0.23/0.57 ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.57 => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 0.23/0.57 => ( Q2 = Q5 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % unique_quotient
% 0.23/0.57 thf(fact_8883_unique__remainder,axiom,
% 0.23/0.57 ! [A: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
% 0.23/0.57 ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.57 => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 0.23/0.57 => ( R2 = R4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % unique_remainder
% 0.23/0.57 thf(fact_8884_eucl__rel__int__by0,axiom,
% 0.23/0.57 ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 0.23/0.57
% 0.23/0.57 % eucl_rel_int_by0
% 0.23/0.57 thf(fact_8885_summable__rabs__comparison__test,axiom,
% 0.23/0.57 ! [F: nat > real,G: nat > real] :
% 0.23/0.57 ( ? [N7: nat] :
% 0.23/0.57 ! [N4: nat] :
% 0.23/0.57 ( ( ord_less_eq_nat @ N7 @ N4 )
% 0.23/0.57 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.23/0.57 => ( ( summable_real @ G )
% 0.23/0.57 => ( summable_real
% 0.23/0.57 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % summable_rabs_comparison_test
% 0.23/0.57 thf(fact_8886_div__int__unique,axiom,
% 0.23/0.57 ! [K: int,L2: int,Q2: int,R2: int] :
% 0.23/0.57 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.57 => ( ( divide_divide_int @ K @ L2 )
% 0.23/0.57 = Q2 ) ) ).
% 0.23/0.57
% 0.23/0.57 % div_int_unique
% 0.23/0.57 thf(fact_8887_mod__int__unique,axiom,
% 0.23/0.57 ! [K: int,L2: int,Q2: int,R2: int] :
% 0.23/0.57 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.57 => ( ( modulo_modulo_int @ K @ L2 )
% 0.23/0.57 = R2 ) ) ).
% 0.23/0.57
% 0.23/0.57 % mod_int_unique
% 0.23/0.57 thf(fact_8888_summable__rabs,axiom,
% 0.23/0.57 ! [F: nat > real] :
% 0.23/0.57 ( ( summable_real
% 0.23/0.57 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
% 0.23/0.57 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 0.23/0.57 @ ( suminf_real
% 0.23/0.57 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % summable_rabs
% 0.23/0.57 thf(fact_8889_eucl__rel__int__dividesI,axiom,
% 0.23/0.57 ! [L2: int,K: int,Q2: int] :
% 0.23/0.57 ( ( L2 != zero_zero_int )
% 0.23/0.57 => ( ( K
% 0.23/0.57 = ( times_times_int @ Q2 @ L2 ) )
% 0.23/0.57 => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % eucl_rel_int_dividesI
% 0.23/0.57 thf(fact_8890_eucl__rel__int,axiom,
% 0.23/0.57 ! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % eucl_rel_int
% 0.23/0.57 thf(fact_8891_summable__power__series,axiom,
% 0.23/0.57 ! [F: nat > real,Z: real] :
% 0.23/0.57 ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
% 0.23/0.57 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 0.23/0.57 => ( ( ord_less_real @ Z @ one_one_real )
% 0.23/0.57 => ( summable_real
% 0.23/0.57 @ ^ [I2: nat] : ( times_times_real @ ( F @ I2 ) @ ( power_power_real @ Z @ I2 ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % summable_power_series
% 0.23/0.57 thf(fact_8892_zminus1__lemma,axiom,
% 0.23/0.57 ! [A: int,B2: int,Q2: int,R2: int] :
% 0.23/0.57 ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.57 => ( ( B2 != zero_zero_int )
% 0.23/0.57 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B2 @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B2 @ R2 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % zminus1_lemma
% 0.23/0.57 thf(fact_8893_eucl__rel__int__iff,axiom,
% 0.23/0.57 ! [K: int,L2: int,Q2: int,R2: int] :
% 0.23/0.57 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.23/0.57 = ( ( K
% 0.23/0.57 = ( plus_plus_int @ ( times_times_int @ L2 @ Q2 ) @ R2 ) )
% 0.23/0.57 & ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.23/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.23/0.57 & ( ord_less_int @ R2 @ L2 ) ) )
% 0.23/0.57 & ( ~ ( ord_less_int @ zero_zero_int @ L2 )
% 0.23/0.57 => ( ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.23/0.57 => ( ( ord_less_int @ L2 @ R2 )
% 0.23/0.57 & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 0.23/0.57 & ( ~ ( ord_less_int @ L2 @ zero_zero_int )
% 0.23/0.57 => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % eucl_rel_int_iff
% 0.23/0.57 thf(fact_8894_eucl__rel__int__remainderI,axiom,
% 0.23/0.57 ! [R2: int,L2: int,K: int,Q2: int] :
% 0.23/0.57 ( ( ( sgn_sgn_int @ R2 )
% 0.23/0.57 = ( sgn_sgn_int @ L2 ) )
% 0.23/0.57 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L2 ) )
% 0.23/0.57 => ( ( K
% 0.23/0.57 = ( plus_plus_int @ ( times_times_int @ Q2 @ L2 ) @ R2 ) )
% 0.23/0.57 => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % eucl_rel_int_remainderI
% 0.23/0.57 thf(fact_8895_eucl__rel__int_Osimps,axiom,
% 0.23/0.57 ( eucl_rel_int
% 0.23/0.57 = ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
% 0.23/0.57 ( ? [K3: int] :
% 0.23/0.57 ( ( A1 = K3 )
% 0.23/0.57 & ( A22 = zero_zero_int )
% 0.23/0.57 & ( A32
% 0.23/0.57 = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 0.23/0.57 | ? [L: int,K3: int,Q4: int] :
% 0.23/0.57 ( ( A1 = K3 )
% 0.23/0.57 & ( A22 = L )
% 0.23/0.57 & ( A32
% 0.23/0.57 = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 0.23/0.57 & ( L != zero_zero_int )
% 0.23/0.57 & ( K3
% 0.23/0.57 = ( times_times_int @ Q4 @ L ) ) )
% 0.23/0.57 | ? [R5: int,L: int,K3: int,Q4: int] :
% 0.23/0.57 ( ( A1 = K3 )
% 0.23/0.57 & ( A22 = L )
% 0.23/0.57 & ( A32
% 0.23/0.57 = ( product_Pair_int_int @ Q4 @ R5 ) )
% 0.23/0.57 & ( ( sgn_sgn_int @ R5 )
% 0.23/0.57 = ( sgn_sgn_int @ L ) )
% 0.23/0.57 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
% 0.23/0.57 & ( K3
% 0.23/0.57 = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R5 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % eucl_rel_int.simps
% 0.23/0.57 thf(fact_8896_eucl__rel__int_Ocases,axiom,
% 0.23/0.57 ! [A12: int,A23: int,A33: product_prod_int_int] :
% 0.23/0.57 ( ( eucl_rel_int @ A12 @ A23 @ A33 )
% 0.23/0.57 => ( ( ( A23 = zero_zero_int )
% 0.23/0.57 => ( A33
% 0.23/0.57 != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 0.23/0.57 => ( ! [Q3: int] :
% 0.23/0.57 ( ( A33
% 0.23/0.57 = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 0.23/0.57 => ( ( A23 != zero_zero_int )
% 0.23/0.57 => ( A12
% 0.23/0.57 != ( times_times_int @ Q3 @ A23 ) ) ) )
% 0.23/0.57 => ~ ! [R3: int,Q3: int] :
% 0.23/0.57 ( ( A33
% 0.23/0.57 = ( product_Pair_int_int @ Q3 @ R3 ) )
% 0.23/0.57 => ( ( ( sgn_sgn_int @ R3 )
% 0.23/0.57 = ( sgn_sgn_int @ A23 ) )
% 0.23/0.57 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
% 0.23/0.57 => ( A12
% 0.23/0.57 != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % eucl_rel_int.cases
% 0.23/0.57 thf(fact_8897_and__int_Oelims,axiom,
% 0.23/0.57 ! [X4: int,Xa: int,Y3: int] :
% 0.23/0.57 ( ( ( bit_se725231765392027082nd_int @ X4 @ Xa )
% 0.23/0.57 = Y3 )
% 0.23/0.57 => ( ( ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.23/0.57 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( uminus_uminus_int
% 0.23/0.57 @ ( zero_n2684676970156552555ol_int
% 0.23/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
% 0.23/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.23/0.57 & ( ~ ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.23/0.57 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( plus_plus_int
% 0.23/0.57 @ ( zero_n2684676970156552555ol_int
% 0.23/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
% 0.23/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.23/0.57 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % and_int.elims
% 0.23/0.57 thf(fact_8898_and__int_Osimps,axiom,
% 0.23/0.57 ( bit_se725231765392027082nd_int
% 0.23/0.57 = ( ^ [K3: int,L: int] :
% 0.23/0.57 ( if_int
% 0.23/0.57 @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.23/0.57 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.23/0.57 @ ( uminus_uminus_int
% 0.23/0.57 @ ( zero_n2684676970156552555ol_int
% 0.23/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.23/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 0.23/0.57 @ ( plus_plus_int
% 0.23/0.57 @ ( zero_n2684676970156552555ol_int
% 0.23/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.23/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.23/0.57 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % and_int.simps
% 0.23/0.57 thf(fact_8899_vebt__buildup_Oelims,axiom,
% 0.23/0.57 ! [X4: nat,Y3: vEBT_VEBT] :
% 0.23/0.57 ( ( ( vEBT_vebt_buildup @ X4 )
% 0.23/0.57 = Y3 )
% 0.23/0.57 => ( ( ( X4 = zero_zero_nat )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.23/0.57 => ( ( ( X4
% 0.23/0.57 = ( suc @ zero_zero_nat ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.23/0.57 => ~ ! [Va: nat] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( suc @ ( suc @ Va ) ) )
% 0.23/0.57 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( 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 ) ) ) ) ) ) )
% 0.23/0.57 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % vebt_buildup.elims
% 0.23/0.57 thf(fact_8900_sin__paired,axiom,
% 0.23/0.57 ! [X4: real] :
% 0.23/0.57 ( sums_real
% 0.23/0.57 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.23/0.57 @ ( sin_real @ X4 ) ) ).
% 0.23/0.57
% 0.23/0.57 % sin_paired
% 0.23/0.57 thf(fact_8901_atLeastAtMostPlus1__int__conv,axiom,
% 0.23/0.57 ! [M: int,N: int] :
% 0.23/0.57 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.23/0.57 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.23/0.57 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeastAtMostPlus1_int_conv
% 0.23/0.57 thf(fact_8902_simp__from__to,axiom,
% 0.23/0.57 ( set_or1266510415728281911st_int
% 0.23/0.57 = ( ^ [I2: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I2 ) @ bot_bot_set_int @ ( insert_int @ I2 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % simp_from_to
% 0.23/0.57 thf(fact_8903_power__half__series,axiom,
% 0.23/0.57 ( sums_real
% 0.23/0.57 @ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
% 0.23/0.57 @ one_one_real ) ).
% 0.23/0.57
% 0.23/0.57 % power_half_series
% 0.23/0.57 thf(fact_8904_sums__if_H,axiom,
% 0.23/0.57 ! [G: nat > real,X4: real] :
% 0.23/0.57 ( ( sums_real @ G @ X4 )
% 0.23/0.57 => ( sums_real
% 0.23/0.57 @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.57 @ X4 ) ) ).
% 0.23/0.57
% 0.23/0.57 % sums_if'
% 0.23/0.57 thf(fact_8905_sums__if,axiom,
% 0.23/0.57 ! [G: nat > real,X4: real,F: nat > real,Y3: real] :
% 0.23/0.57 ( ( sums_real @ G @ X4 )
% 0.23/0.57 => ( ( sums_real @ F @ Y3 )
% 0.23/0.57 => ( sums_real
% 0.23/0.57 @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.57 @ ( plus_plus_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % sums_if
% 0.23/0.57 thf(fact_8906_cos__paired,axiom,
% 0.23/0.57 ! [X4: real] :
% 0.23/0.57 ( sums_real
% 0.23/0.57 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.23/0.57 @ ( cos_real @ X4 ) ) ).
% 0.23/0.57
% 0.23/0.57 % cos_paired
% 0.23/0.57 thf(fact_8907_vebt__buildup_Osimps_I3_J,axiom,
% 0.23/0.57 ! [Va2: nat] :
% 0.23/0.57 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.23/0.57 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 0.23/0.57 = ( 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 ) ) ) ) ) ) )
% 0.23/0.57 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.23/0.57 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 0.23/0.57 = ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % vebt_buildup.simps(3)
% 0.23/0.57 thf(fact_8908_and__int_Opelims,axiom,
% 0.23/0.57 ! [X4: int,Xa: int,Y3: int] :
% 0.23/0.57 ( ( ( bit_se725231765392027082nd_int @ X4 @ Xa )
% 0.23/0.57 = Y3 )
% 0.23/0.57 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X4 @ Xa ) )
% 0.23/0.57 => ~ ( ( ( ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.23/0.57 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( uminus_uminus_int
% 0.23/0.57 @ ( zero_n2684676970156552555ol_int
% 0.23/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
% 0.23/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.23/0.57 & ( ~ ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.23/0.57 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( plus_plus_int
% 0.23/0.57 @ ( zero_n2684676970156552555ol_int
% 0.23/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
% 0.23/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.23/0.57 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.23/0.57 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X4 @ Xa ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % and_int.pelims
% 0.23/0.57 thf(fact_8909_and__int_Opsimps,axiom,
% 0.23/0.57 ! [K: int,L2: int] :
% 0.23/0.57 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
% 0.23/0.57 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.23/0.57 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.23/0.57 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 0.23/0.57 = ( uminus_uminus_int
% 0.23/0.57 @ ( zero_n2684676970156552555ol_int
% 0.23/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.23/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
% 0.23/0.57 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.23/0.57 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.23/0.57 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 0.23/0.57 = ( plus_plus_int
% 0.23/0.57 @ ( zero_n2684676970156552555ol_int
% 0.23/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.23/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.23/0.57 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % and_int.psimps
% 0.23/0.57 thf(fact_8910_obtain__set__succ,axiom,
% 0.23/0.57 ! [X4: nat,Z: nat,A3: set_nat,B4: set_nat] :
% 0.23/0.57 ( ( ord_less_nat @ X4 @ Z )
% 0.23/0.57 => ( ( vEBT_VEBT_max_in_set @ A3 @ Z )
% 0.23/0.57 => ( ( finite_finite_nat @ B4 )
% 0.23/0.57 => ( ( A3 = B4 )
% 0.23/0.57 => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A3 @ X4 @ X_1 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % obtain_set_succ
% 0.23/0.57 thf(fact_8911_set__vebt__finite,axiom,
% 0.23/0.57 ! [T: vEBT_VEBT,N: nat] :
% 0.23/0.57 ( ( vEBT_invar_vebt @ T @ N )
% 0.23/0.57 => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % set_vebt_finite
% 0.23/0.57 thf(fact_8912_succ__none__empty,axiom,
% 0.23/0.57 ! [Xs2: set_nat,A: nat] :
% 0.23/0.57 ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_1 )
% 0.23/0.57 => ( ( finite_finite_nat @ Xs2 )
% 0.23/0.57 => ~ ? [X5: nat] :
% 0.23/0.57 ( ( member_nat @ X5 @ Xs2 )
% 0.23/0.57 & ( ord_less_nat @ A @ X5 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % succ_none_empty
% 0.23/0.57 thf(fact_8913_finite__atLeastAtMost,axiom,
% 0.23/0.57 ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_atLeastAtMost
% 0.23/0.57 thf(fact_8914_finite__nat__set__iff__bounded,axiom,
% 0.23/0.57 ( finite_finite_nat
% 0.23/0.57 = ( ^ [N8: set_nat] :
% 0.23/0.57 ? [M6: nat] :
% 0.23/0.57 ! [X: nat] :
% 0.23/0.57 ( ( member_nat @ X @ N8 )
% 0.23/0.57 => ( ord_less_nat @ X @ M6 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_nat_set_iff_bounded
% 0.23/0.57 thf(fact_8915_bounded__nat__set__is__finite,axiom,
% 0.23/0.57 ! [N3: set_nat,N: nat] :
% 0.23/0.57 ( ! [X3: nat] :
% 0.23/0.57 ( ( member_nat @ X3 @ N3 )
% 0.23/0.57 => ( ord_less_nat @ X3 @ N ) )
% 0.23/0.57 => ( finite_finite_nat @ N3 ) ) ).
% 0.23/0.57
% 0.23/0.57 % bounded_nat_set_is_finite
% 0.23/0.57 thf(fact_8916_finite__nat__set__iff__bounded__le,axiom,
% 0.23/0.57 ( finite_finite_nat
% 0.23/0.57 = ( ^ [N8: set_nat] :
% 0.23/0.57 ? [M6: nat] :
% 0.23/0.57 ! [X: nat] :
% 0.23/0.57 ( ( member_nat @ X @ N8 )
% 0.23/0.57 => ( ord_less_eq_nat @ X @ M6 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_nat_set_iff_bounded_le
% 0.23/0.57 thf(fact_8917_finite__M__bounded__by__nat,axiom,
% 0.23/0.57 ! [P: nat > $o,I: nat] :
% 0.23/0.57 ( finite_finite_nat
% 0.23/0.57 @ ( collect_nat
% 0.23/0.57 @ ^ [K3: nat] :
% 0.23/0.57 ( ( P @ K3 )
% 0.23/0.57 & ( ord_less_nat @ K3 @ I ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_M_bounded_by_nat
% 0.23/0.57 thf(fact_8918_finite__less__ub,axiom,
% 0.23/0.57 ! [F: nat > nat,U: nat] :
% 0.23/0.57 ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
% 0.23/0.57 => ( finite_finite_nat
% 0.23/0.57 @ ( collect_nat
% 0.23/0.57 @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_less_ub
% 0.23/0.57 thf(fact_8919_atLeast0__atMost__Suc,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.23/0.57 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeast0_atMost_Suc
% 0.23/0.57 thf(fact_8920_Icc__eq__insert__lb__nat,axiom,
% 0.23/0.57 ! [M: nat,N: nat] :
% 0.23/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.57 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 0.23/0.57 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Icc_eq_insert_lb_nat
% 0.23/0.57 thf(fact_8921_atLeastAtMostSuc__conv,axiom,
% 0.23/0.57 ! [M: nat,N: nat] :
% 0.23/0.57 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.23/0.57 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 0.23/0.57 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeastAtMostSuc_conv
% 0.23/0.57 thf(fact_8922_atLeastAtMost__insertL,axiom,
% 0.23/0.57 ! [M: nat,N: nat] :
% 0.23/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.57 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.23/0.57 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeastAtMost_insertL
% 0.23/0.57 thf(fact_8923_finite__divisors__nat,axiom,
% 0.23/0.57 ! [M: nat] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.57 => ( finite_finite_nat
% 0.23/0.57 @ ( collect_nat
% 0.23/0.57 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_divisors_nat
% 0.23/0.57 thf(fact_8924_subset__eq__atLeast0__atMost__finite,axiom,
% 0.23/0.57 ! [N3: set_nat,N: nat] :
% 0.23/0.57 ( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.23/0.57 => ( finite_finite_nat @ N3 ) ) ).
% 0.23/0.57
% 0.23/0.57 % subset_eq_atLeast0_atMost_finite
% 0.23/0.57 thf(fact_8925_set__decode__plus__power__2,axiom,
% 0.23/0.57 ! [N: nat,Z: nat] :
% 0.23/0.57 ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 0.23/0.57 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 0.23/0.57 = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % set_decode_plus_power_2
% 0.23/0.57 thf(fact_8926_and__int_Opinduct,axiom,
% 0.23/0.57 ! [A0: int,A12: int,P: int > int > $o] :
% 0.23/0.57 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.23/0.57 => ( ! [K2: int,L3: int] :
% 0.23/0.57 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L3 ) )
% 0.23/0.57 => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.23/0.57 & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.23/0.57 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.23/0.57 => ( P @ K2 @ L3 ) ) )
% 0.23/0.57 => ( P @ A0 @ A12 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % and_int.pinduct
% 0.23/0.57 thf(fact_8927_finite__Collect__le__nat,axiom,
% 0.23/0.57 ! [K: nat] :
% 0.23/0.57 ( finite_finite_nat
% 0.23/0.57 @ ( collect_nat
% 0.23/0.57 @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_Collect_le_nat
% 0.23/0.57 thf(fact_8928_finite__Collect__less__nat,axiom,
% 0.23/0.57 ! [K: nat] :
% 0.23/0.57 ( finite_finite_nat
% 0.23/0.57 @ ( collect_nat
% 0.23/0.57 @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_Collect_less_nat
% 0.23/0.57 thf(fact_8929_finite__atLeastAtMost__int,axiom,
% 0.23/0.57 ! [L2: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L2 @ U ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_atLeastAtMost_int
% 0.23/0.57 thf(fact_8930_finite__interval__int4,axiom,
% 0.23/0.57 ! [A: int,B2: int] :
% 0.23/0.57 ( finite_finite_int
% 0.23/0.57 @ ( collect_int
% 0.23/0.57 @ ^ [I2: int] :
% 0.23/0.57 ( ( ord_less_int @ A @ I2 )
% 0.23/0.57 & ( ord_less_int @ I2 @ B2 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_interval_int4
% 0.23/0.57 thf(fact_8931_finite__interval__int3,axiom,
% 0.23/0.57 ! [A: int,B2: int] :
% 0.23/0.57 ( finite_finite_int
% 0.23/0.57 @ ( collect_int
% 0.23/0.57 @ ^ [I2: int] :
% 0.23/0.57 ( ( ord_less_int @ A @ I2 )
% 0.23/0.57 & ( ord_less_eq_int @ I2 @ B2 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_interval_int3
% 0.23/0.57 thf(fact_8932_finite__interval__int2,axiom,
% 0.23/0.57 ! [A: int,B2: int] :
% 0.23/0.57 ( finite_finite_int
% 0.23/0.57 @ ( collect_int
% 0.23/0.57 @ ^ [I2: int] :
% 0.23/0.57 ( ( ord_less_eq_int @ A @ I2 )
% 0.23/0.57 & ( ord_less_int @ I2 @ B2 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_interval_int2
% 0.23/0.57 thf(fact_8933_finite__nth__roots,axiom,
% 0.23/0.57 ! [N: nat,C: complex] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( finite3207457112153483333omplex
% 0.23/0.57 @ ( collect_complex
% 0.23/0.57 @ ^ [Z5: complex] :
% 0.23/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.23/0.57 = C ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_nth_roots
% 0.23/0.57 thf(fact_8934_set__encode__insert,axiom,
% 0.23/0.57 ! [A3: set_nat,N: nat] :
% 0.23/0.57 ( ( finite_finite_nat @ A3 )
% 0.23/0.57 => ( ~ ( member_nat @ N @ A3 )
% 0.23/0.57 => ( ( nat_set_encode @ ( insert_nat @ N @ A3 ) )
% 0.23/0.57 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % set_encode_insert
% 0.23/0.57 thf(fact_8935_upto_Opinduct,axiom,
% 0.23/0.57 ! [A0: int,A12: int,P: int > int > $o] :
% 0.23/0.57 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.23/0.57 => ( ! [I3: int,J2: int] :
% 0.23/0.57 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
% 0.23/0.57 => ( ( ( ord_less_eq_int @ I3 @ J2 )
% 0.23/0.57 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
% 0.23/0.57 => ( P @ I3 @ J2 ) ) )
% 0.23/0.57 => ( P @ A0 @ A12 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % upto.pinduct
% 0.23/0.57 thf(fact_8936_set__encode__def,axiom,
% 0.23/0.57 ( nat_set_encode
% 0.23/0.57 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % set_encode_def
% 0.23/0.57 thf(fact_8937_mask__eq__sum__exp__nat,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 0.23/0.57 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.57 @ ( collect_nat
% 0.23/0.57 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % mask_eq_sum_exp_nat
% 0.23/0.57 thf(fact_8938_gauss__sum__nat,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [X: nat] : X
% 0.23/0.57 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.23/0.57 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % gauss_sum_nat
% 0.23/0.57 thf(fact_8939_arith__series__nat,axiom,
% 0.23/0.57 ! [A: nat,D: nat,N: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I2 @ D ) )
% 0.23/0.57 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.23/0.57 = ( 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 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % arith_series_nat
% 0.23/0.57 thf(fact_8940_Sum__Icc__nat,axiom,
% 0.23/0.57 ! [M: nat,N: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [X: nat] : X
% 0.23/0.57 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.23/0.57 = ( 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 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Sum_Icc_nat
% 0.23/0.57 thf(fact_8941_even__set__encode__iff,axiom,
% 0.23/0.57 ! [A3: set_nat] :
% 0.23/0.57 ( ( finite_finite_nat @ A3 )
% 0.23/0.57 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A3 ) )
% 0.23/0.57 = ( ~ ( member_nat @ zero_zero_nat @ A3 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % even_set_encode_iff
% 0.23/0.57 thf(fact_8942_Maclaurin__minus__cos__expansion,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 0.23/0.57 => ? [T2: real] :
% 0.23/0.57 ( ( ord_less_real @ X4 @ T2 )
% 0.23/0.57 & ( ord_less_real @ T2 @ zero_zero_real )
% 0.23/0.57 & ( ( cos_real @ X4 )
% 0.23/0.57 = ( plus_plus_real
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.57 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Maclaurin_minus_cos_expansion
% 0.23/0.57 thf(fact_8943_Maclaurin__cos__expansion2,axiom,
% 0.23/0.57 ! [X4: real,N: nat] :
% 0.23/0.57 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ? [T2: real] :
% 0.23/0.57 ( ( ord_less_real @ zero_zero_real @ T2 )
% 0.23/0.57 & ( ord_less_real @ T2 @ X4 )
% 0.23/0.57 & ( ( cos_real @ X4 )
% 0.23/0.57 = ( plus_plus_real
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.57 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Maclaurin_cos_expansion2
% 0.23/0.57 thf(fact_8944_Maclaurin__sin__expansion3,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ? [T2: real] :
% 0.23/0.57 ( ( ord_less_real @ zero_zero_real @ T2 )
% 0.23/0.57 & ( ord_less_real @ T2 @ X4 )
% 0.23/0.57 & ( ( sin_real @ X4 )
% 0.23/0.57 = ( plus_plus_real
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.57 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Maclaurin_sin_expansion3
% 0.23/0.57 thf(fact_8945_Maclaurin__sin__expansion4,axiom,
% 0.23/0.57 ! [X4: real,N: nat] :
% 0.23/0.57 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ? [T2: real] :
% 0.23/0.57 ( ( ord_less_real @ zero_zero_real @ T2 )
% 0.23/0.57 & ( ord_less_eq_real @ T2 @ X4 )
% 0.23/0.57 & ( ( sin_real @ X4 )
% 0.23/0.57 = ( plus_plus_real
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.57 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Maclaurin_sin_expansion4
% 0.23/0.57 thf(fact_8946_finite__lessThan,axiom,
% 0.23/0.57 ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_lessThan
% 0.23/0.57 thf(fact_8947_lessThan__0,axiom,
% 0.23/0.57 ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 0.23/0.57 = bot_bot_set_nat ) ).
% 0.23/0.57
% 0.23/0.57 % lessThan_0
% 0.23/0.57 thf(fact_8948_sumr__cos__zero__one,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.23/0.57 = one_one_real ) ).
% 0.23/0.57
% 0.23/0.57 % sumr_cos_zero_one
% 0.23/0.57 thf(fact_8949_lessThan__Suc,axiom,
% 0.23/0.57 ! [K: nat] :
% 0.23/0.57 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.23/0.57 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % lessThan_Suc
% 0.23/0.57 thf(fact_8950_lessThan__empty__iff,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( ( set_ord_lessThan_nat @ N )
% 0.23/0.57 = bot_bot_set_nat )
% 0.23/0.57 = ( N = zero_zero_nat ) ) ).
% 0.23/0.57
% 0.23/0.57 % lessThan_empty_iff
% 0.23/0.57 thf(fact_8951_lessThan__nat__numeral,axiom,
% 0.23/0.57 ! [K: num] :
% 0.23/0.57 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 0.23/0.57 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % lessThan_nat_numeral
% 0.23/0.57 thf(fact_8952_sum__nth__roots,axiom,
% 0.23/0.57 ! [N: nat,C: complex] :
% 0.23/0.57 ( ( ord_less_nat @ one_one_nat @ N )
% 0.23/0.57 => ( ( groups7754918857620584856omplex
% 0.23/0.57 @ ^ [X: complex] : X
% 0.23/0.57 @ ( collect_complex
% 0.23/0.57 @ ^ [Z5: complex] :
% 0.23/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.23/0.57 = C ) ) )
% 0.23/0.57 = zero_zero_complex ) ) ).
% 0.23/0.57
% 0.23/0.57 % sum_nth_roots
% 0.23/0.57 thf(fact_8953_sum__roots__unity,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( ord_less_nat @ one_one_nat @ N )
% 0.23/0.57 => ( ( groups7754918857620584856omplex
% 0.23/0.57 @ ^ [X: complex] : X
% 0.23/0.57 @ ( collect_complex
% 0.23/0.57 @ ^ [Z5: complex] :
% 0.23/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.23/0.57 = one_one_complex ) ) )
% 0.23/0.57 = zero_zero_complex ) ) ).
% 0.23/0.57
% 0.23/0.57 % sum_roots_unity
% 0.23/0.57 thf(fact_8954_Maclaurin__lemma,axiom,
% 0.23/0.57 ! [H2: real,F: real > real,J: nat > real,N: nat] :
% 0.23/0.57 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.23/0.57 => ? [B8: real] :
% 0.23/0.57 ( ( F @ H2 )
% 0.23/0.57 = ( plus_plus_real
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.57 @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Maclaurin_lemma
% 0.23/0.57 thf(fact_8955_sum__split__even__odd,axiom,
% 0.23/0.57 ! [F: nat > real,G: nat > real,N: nat] :
% 0.23/0.57 ( ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [I2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ ( F @ I2 ) @ ( G @ I2 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.57 = ( plus_plus_real
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [I2: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ one_one_nat ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % sum_split_even_odd
% 0.23/0.57 thf(fact_8956_Maclaurin__exp__le,axiom,
% 0.23/0.57 ! [X4: real,N: nat] :
% 0.23/0.57 ? [T2: real] :
% 0.23/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 0.23/0.57 & ( ( exp_real @ X4 )
% 0.23/0.57 = ( plus_plus_real
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X4 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.57 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Maclaurin_exp_le
% 0.23/0.57 thf(fact_8957_Maclaurin__sin__bound,axiom,
% 0.23/0.57 ! [X4: real,N: nat] :
% 0.23/0.57 ( ord_less_eq_real
% 0.23/0.57 @ ( abs_abs_real
% 0.23/0.57 @ ( minus_minus_real @ ( sin_real @ X4 )
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 0.23/0.57 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X4 ) @ N ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Maclaurin_sin_bound
% 0.23/0.57 thf(fact_8958_Sum__Icc__int,axiom,
% 0.23/0.57 ! [M: int,N: int] :
% 0.23/0.57 ( ( ord_less_eq_int @ M @ N )
% 0.23/0.57 => ( ( groups4538972089207619220nt_int
% 0.23/0.57 @ ^ [X: int] : X
% 0.23/0.57 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 0.23/0.57 = ( 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 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Sum_Icc_int
% 0.23/0.57 thf(fact_8959_sum__pos__lt__pair,axiom,
% 0.23/0.57 ! [F: nat > real,K: nat] :
% 0.23/0.57 ( ( summable_real @ F )
% 0.23/0.57 => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
% 0.23/0.57 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % sum_pos_lt_pair
% 0.23/0.57 thf(fact_8960_Maclaurin__exp__lt,axiom,
% 0.23/0.57 ! [X4: real,N: nat] :
% 0.23/0.57 ( ( X4 != zero_zero_real )
% 0.23/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ? [T2: real] :
% 0.23/0.57 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T2 ) )
% 0.23/0.57 & ( ord_less_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 0.23/0.57 & ( ( exp_real @ X4 )
% 0.23/0.57 = ( plus_plus_real
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X4 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.57 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Maclaurin_exp_lt
% 0.23/0.57 thf(fact_8961_Maclaurin__sin__expansion,axiom,
% 0.23/0.57 ! [X4: real,N: nat] :
% 0.23/0.57 ? [T2: real] :
% 0.23/0.57 ( ( sin_real @ X4 )
% 0.23/0.57 = ( plus_plus_real
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.57 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Maclaurin_sin_expansion
% 0.23/0.57 thf(fact_8962_Maclaurin__sin__expansion2,axiom,
% 0.23/0.57 ! [X4: real,N: nat] :
% 0.23/0.57 ? [T2: real] :
% 0.23/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 0.23/0.57 & ( ( sin_real @ X4 )
% 0.23/0.57 = ( plus_plus_real
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.57 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Maclaurin_sin_expansion2
% 0.23/0.57 thf(fact_8963_Maclaurin__cos__expansion,axiom,
% 0.23/0.57 ! [X4: real,N: nat] :
% 0.23/0.57 ? [T2: real] :
% 0.23/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 0.23/0.57 & ( ( cos_real @ X4 )
% 0.23/0.57 = ( plus_plus_real
% 0.23/0.57 @ ( groups6591440286371151544t_real
% 0.23/0.57 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.57 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Maclaurin_cos_expansion
% 0.23/0.57 thf(fact_8964_bij__betw__roots__unity,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( bij_betw_nat_complex
% 0.23/0.57 @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.23/0.57 @ ( set_ord_lessThan_nat @ N )
% 0.23/0.57 @ ( collect_complex
% 0.23/0.57 @ ^ [Z5: complex] :
% 0.23/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.23/0.57 = one_one_complex ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % bij_betw_roots_unity
% 0.23/0.57 thf(fact_8965_finite__atMost,axiom,
% 0.23/0.57 ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_atMost
% 0.23/0.57 thf(fact_8966_atMost__0,axiom,
% 0.23/0.57 ( ( set_ord_atMost_nat @ zero_zero_nat )
% 0.23/0.57 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 0.23/0.57
% 0.23/0.57 % atMost_0
% 0.23/0.57 thf(fact_8967_atMost__atLeast0,axiom,
% 0.23/0.57 ( set_ord_atMost_nat
% 0.23/0.57 = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 0.23/0.57
% 0.23/0.57 % atMost_atLeast0
% 0.23/0.57 thf(fact_8968_lessThan__Suc__atMost,axiom,
% 0.23/0.57 ! [K: nat] :
% 0.23/0.57 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.23/0.57 = ( set_ord_atMost_nat @ K ) ) ).
% 0.23/0.57
% 0.23/0.57 % lessThan_Suc_atMost
% 0.23/0.57 thf(fact_8969_atMost__Suc,axiom,
% 0.23/0.57 ! [K: nat] :
% 0.23/0.57 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 0.23/0.57 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % atMost_Suc
% 0.23/0.57 thf(fact_8970_atMost__nat__numeral,axiom,
% 0.23/0.57 ! [K: num] :
% 0.23/0.57 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 0.23/0.57 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % atMost_nat_numeral
% 0.23/0.57 thf(fact_8971_sum__choose__upper,axiom,
% 0.23/0.57 ! [M: nat,N: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [K3: nat] : ( binomial @ K3 @ M )
% 0.23/0.57 @ ( set_ord_atMost_nat @ N ) )
% 0.23/0.57 = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % sum_choose_upper
% 0.23/0.57 thf(fact_8972_sum__choose__lower,axiom,
% 0.23/0.57 ! [R2: nat,N: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
% 0.23/0.57 @ ( set_ord_atMost_nat @ N ) )
% 0.23/0.57 = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% 0.23/0.57
% 0.23/0.57 % sum_choose_lower
% 0.23/0.57 thf(fact_8973_choose__rising__sum_I2_J,axiom,
% 0.23/0.57 ! [N: nat,M: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 0.23/0.57 @ ( set_ord_atMost_nat @ M ) )
% 0.23/0.57 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 0.23/0.57
% 0.23/0.57 % choose_rising_sum(2)
% 0.23/0.57 thf(fact_8974_choose__rising__sum_I1_J,axiom,
% 0.23/0.57 ! [N: nat,M: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 0.23/0.57 @ ( set_ord_atMost_nat @ M ) )
% 0.23/0.57 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % choose_rising_sum(1)
% 0.23/0.57 thf(fact_8975_atLeast1__atMost__eq__remove0,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.23/0.57 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeast1_atMost_eq_remove0
% 0.23/0.57 thf(fact_8976_sum__choose__diagonal,axiom,
% 0.23/0.57 ! [M: nat,N: nat] :
% 0.23/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.57 => ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 0.23/0.57 @ ( set_ord_atMost_nat @ M ) )
% 0.23/0.57 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % sum_choose_diagonal
% 0.23/0.57 thf(fact_8977_vandermonde,axiom,
% 0.23/0.57 ! [M: nat,N: nat,R2: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
% 0.23/0.57 @ ( set_ord_atMost_nat @ R2 ) )
% 0.23/0.57 = ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).
% 0.23/0.57
% 0.23/0.57 % vandermonde
% 0.23/0.57 thf(fact_8978_choose__row__sum,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 0.23/0.57 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.23/0.57
% 0.23/0.57 % choose_row_sum
% 0.23/0.57 thf(fact_8979_binomial,axiom,
% 0.23/0.57 ! [A: nat,B2: nat,N: nat] :
% 0.23/0.57 ( ( power_power_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
% 0.23/0.57 = ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N @ K3 ) ) )
% 0.23/0.57 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % binomial
% 0.23/0.57 thf(fact_8980_polynomial__product__nat,axiom,
% 0.23/0.57 ! [M: nat,A: nat > nat,N: nat,B2: nat > nat,X4: nat] :
% 0.23/0.57 ( ! [I3: nat] :
% 0.23/0.57 ( ( ord_less_nat @ M @ I3 )
% 0.23/0.57 => ( ( A @ I3 )
% 0.23/0.57 = zero_zero_nat ) )
% 0.23/0.57 => ( ! [J2: nat] :
% 0.23/0.57 ( ( ord_less_nat @ N @ J2 )
% 0.23/0.57 => ( ( B2 @ J2 )
% 0.23/0.57 = zero_zero_nat ) )
% 0.23/0.57 => ( ( times_times_nat
% 0.23/0.57 @ ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( power_power_nat @ X4 @ I2 ) )
% 0.23/0.57 @ ( set_ord_atMost_nat @ M ) )
% 0.23/0.57 @ ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [J3: nat] : ( times_times_nat @ ( B2 @ J3 ) @ ( power_power_nat @ X4 @ J3 ) )
% 0.23/0.57 @ ( set_ord_atMost_nat @ N ) ) )
% 0.23/0.57 = ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [R5: nat] :
% 0.23/0.57 ( times_times_nat
% 0.23/0.57 @ ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 0.23/0.57 @ ( set_ord_atMost_nat @ R5 ) )
% 0.23/0.57 @ ( power_power_nat @ X4 @ R5 ) )
% 0.23/0.57 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % polynomial_product_nat
% 0.23/0.57 thf(fact_8981_choose__square__sum,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.57 @ ( set_ord_atMost_nat @ N ) )
% 0.23/0.57 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 0.23/0.57
% 0.23/0.57 % choose_square_sum
% 0.23/0.57 thf(fact_8982_binomial__r__part__sum,axiom,
% 0.23/0.57 ! [M: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.23/0.57 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % binomial_r_part_sum
% 0.23/0.57 thf(fact_8983_choose__linear__sum,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [I2: nat] : ( times_times_nat @ I2 @ ( binomial @ N @ I2 ) )
% 0.23/0.57 @ ( set_ord_atMost_nat @ N ) )
% 0.23/0.57 = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % choose_linear_sum
% 0.23/0.57 thf(fact_8984_of__nat__id,axiom,
% 0.23/0.57 ( semiri1316708129612266289at_nat
% 0.23/0.57 = ( ^ [N2: nat] : N2 ) ) ).
% 0.23/0.57
% 0.23/0.57 % of_nat_id
% 0.23/0.57 thf(fact_8985_real__scaleR__def,axiom,
% 0.23/0.57 real_V1485227260804924795R_real = times_times_real ).
% 0.23/0.57
% 0.23/0.57 % real_scaleR_def
% 0.23/0.57 thf(fact_8986_complex__scaleR,axiom,
% 0.23/0.57 ! [R2: real,A: real,B2: real] :
% 0.23/0.57 ( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B2 ) )
% 0.23/0.57 = ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % complex_scaleR
% 0.23/0.57 thf(fact_8987_bij__betw__nth__root__unity,axiom,
% 0.23/0.57 ! [C: complex,N: nat] :
% 0.23/0.57 ( ( C != zero_zero_complex )
% 0.23/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 0.23/0.57 @ ( collect_complex
% 0.23/0.57 @ ^ [Z5: complex] :
% 0.23/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.23/0.57 = one_one_complex ) )
% 0.23/0.57 @ ( collect_complex
% 0.23/0.57 @ ^ [Z5: complex] :
% 0.23/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.23/0.57 = C ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % bij_betw_nth_root_unity
% 0.23/0.57 thf(fact_8988_infinite__int__iff__unbounded,axiom,
% 0.23/0.57 ! [S2: set_int] :
% 0.23/0.57 ( ( ~ ( finite_finite_int @ S2 ) )
% 0.23/0.57 = ( ! [M6: int] :
% 0.23/0.57 ? [N2: int] :
% 0.23/0.57 ( ( ord_less_int @ M6 @ ( abs_abs_int @ N2 ) )
% 0.23/0.57 & ( member_int @ N2 @ S2 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % infinite_int_iff_unbounded
% 0.23/0.57 thf(fact_8989_Arg__def,axiom,
% 0.23/0.57 ( arg
% 0.23/0.57 = ( ^ [Z5: complex] :
% 0.23/0.57 ( if_real @ ( Z5 = zero_zero_complex ) @ zero_zero_real
% 0.23/0.57 @ ( fChoice_real
% 0.23/0.57 @ ^ [A4: real] :
% 0.23/0.57 ( ( ( sgn_sgn_complex @ Z5 )
% 0.23/0.57 = ( cis @ A4 ) )
% 0.23/0.57 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
% 0.23/0.57 & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Arg_def
% 0.23/0.57 thf(fact_8990_real__root__zero,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( root @ N @ zero_zero_real )
% 0.23/0.57 = zero_zero_real ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_zero
% 0.23/0.57 thf(fact_8991_real__root__Suc__0,axiom,
% 0.23/0.57 ! [X4: real] :
% 0.23/0.57 ( ( root @ ( suc @ zero_zero_nat ) @ X4 )
% 0.23/0.57 = X4 ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_Suc_0
% 0.23/0.57 thf(fact_8992_real__root__eq__iff,axiom,
% 0.23/0.57 ! [N: nat,X4: real,Y3: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ( root @ N @ X4 )
% 0.23/0.57 = ( root @ N @ Y3 ) )
% 0.23/0.57 = ( X4 = Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_eq_iff
% 0.23/0.57 thf(fact_8993_root__0,axiom,
% 0.23/0.57 ! [X4: real] :
% 0.23/0.57 ( ( root @ zero_zero_nat @ X4 )
% 0.23/0.57 = zero_zero_real ) ).
% 0.23/0.57
% 0.23/0.57 % root_0
% 0.23/0.57 thf(fact_8994_real__root__eq__0__iff,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ( root @ N @ X4 )
% 0.23/0.57 = zero_zero_real )
% 0.23/0.57 = ( X4 = zero_zero_real ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_eq_0_iff
% 0.23/0.57 thf(fact_8995_real__root__less__iff,axiom,
% 0.23/0.57 ! [N: nat,X4: real,Y3: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) )
% 0.23/0.57 = ( ord_less_real @ X4 @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_less_iff
% 0.23/0.57 thf(fact_8996_real__root__le__iff,axiom,
% 0.23/0.57 ! [N: nat,X4: real,Y3: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_eq_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) )
% 0.23/0.57 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_le_iff
% 0.23/0.57 thf(fact_8997_real__root__one,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( root @ N @ one_one_real )
% 0.23/0.57 = one_one_real ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_one
% 0.23/0.57 thf(fact_8998_real__root__eq__1__iff,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ( root @ N @ X4 )
% 0.23/0.57 = one_one_real )
% 0.23/0.57 = ( X4 = one_one_real ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_eq_1_iff
% 0.23/0.57 thf(fact_8999_real__root__gt__0__iff,axiom,
% 0.23/0.57 ! [N: nat,Y3: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 0.23/0.57 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_gt_0_iff
% 0.23/0.57 thf(fact_9000_real__root__lt__0__iff,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ ( root @ N @ X4 ) @ zero_zero_real )
% 0.23/0.57 = ( ord_less_real @ X4 @ zero_zero_real ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_lt_0_iff
% 0.23/0.57 thf(fact_9001_real__root__le__0__iff,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_eq_real @ ( root @ N @ X4 ) @ zero_zero_real )
% 0.23/0.57 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_le_0_iff
% 0.23/0.57 thf(fact_9002_real__root__ge__0__iff,axiom,
% 0.23/0.57 ! [N: nat,Y3: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 0.23/0.57 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_ge_0_iff
% 0.23/0.57 thf(fact_9003_real__root__gt__1__iff,axiom,
% 0.23/0.57 ! [N: nat,Y3: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
% 0.23/0.57 = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_gt_1_iff
% 0.23/0.57 thf(fact_9004_real__root__lt__1__iff,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ ( root @ N @ X4 ) @ one_one_real )
% 0.23/0.57 = ( ord_less_real @ X4 @ one_one_real ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_lt_1_iff
% 0.23/0.57 thf(fact_9005_real__root__ge__1__iff,axiom,
% 0.23/0.57 ! [N: nat,Y3: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
% 0.23/0.57 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_ge_1_iff
% 0.23/0.57 thf(fact_9006_real__root__le__1__iff,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_eq_real @ ( root @ N @ X4 ) @ one_one_real )
% 0.23/0.57 = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_le_1_iff
% 0.23/0.57 thf(fact_9007_real__root__pow__pos2,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ( ( power_power_real @ ( root @ N @ X4 ) @ N )
% 0.23/0.57 = X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_pow_pos2
% 0.23/0.57 thf(fact_9008_real__root__minus,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( root @ N @ ( uminus_uminus_real @ X4 ) )
% 0.23/0.57 = ( uminus_uminus_real @ ( root @ N @ X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_minus
% 0.23/0.57 thf(fact_9009_real__root__inverse,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( root @ N @ ( inverse_inverse_real @ X4 ) )
% 0.23/0.57 = ( inverse_inverse_real @ ( root @ N @ X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_inverse
% 0.23/0.57 thf(fact_9010_real__root__commute,axiom,
% 0.23/0.57 ! [M: nat,N: nat,X4: real] :
% 0.23/0.57 ( ( root @ M @ ( root @ N @ X4 ) )
% 0.23/0.57 = ( root @ N @ ( root @ M @ X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_commute
% 0.23/0.57 thf(fact_9011_real__root__mult,axiom,
% 0.23/0.57 ! [N: nat,X4: real,Y3: real] :
% 0.23/0.57 ( ( root @ N @ ( times_times_real @ X4 @ Y3 ) )
% 0.23/0.57 = ( times_times_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_mult
% 0.23/0.57 thf(fact_9012_real__root__mult__exp,axiom,
% 0.23/0.57 ! [M: nat,N: nat,X4: real] :
% 0.23/0.57 ( ( root @ ( times_times_nat @ M @ N ) @ X4 )
% 0.23/0.57 = ( root @ M @ ( root @ N @ X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_mult_exp
% 0.23/0.57 thf(fact_9013_real__root__divide,axiom,
% 0.23/0.57 ! [N: nat,X4: real,Y3: real] :
% 0.23/0.57 ( ( root @ N @ ( divide_divide_real @ X4 @ Y3 ) )
% 0.23/0.57 = ( divide_divide_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_divide
% 0.23/0.57 thf(fact_9014_real__root__pos__pos__le,axiom,
% 0.23/0.57 ! [X4: real,N: nat] :
% 0.23/0.57 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_pos_pos_le
% 0.23/0.57 thf(fact_9015_real__root__less__mono,axiom,
% 0.23/0.57 ! [N: nat,X4: real,Y3: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.57 => ( ord_less_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_less_mono
% 0.23/0.57 thf(fact_9016_real__root__le__mono,axiom,
% 0.23/0.57 ! [N: nat,X4: real,Y3: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.23/0.57 => ( ord_less_eq_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_le_mono
% 0.23/0.57 thf(fact_9017_real__root__power,axiom,
% 0.23/0.57 ! [N: nat,X4: real,K: nat] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( root @ N @ ( power_power_real @ X4 @ K ) )
% 0.23/0.57 = ( power_power_real @ ( root @ N @ X4 ) @ K ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_power
% 0.23/0.57 thf(fact_9018_real__root__abs,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( root @ N @ ( abs_abs_real @ X4 ) )
% 0.23/0.57 = ( abs_abs_real @ ( root @ N @ X4 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_abs
% 0.23/0.57 thf(fact_9019_sgn__root,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( sgn_sgn_real @ ( root @ N @ X4 ) )
% 0.23/0.57 = ( sgn_sgn_real @ X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % sgn_root
% 0.23/0.57 thf(fact_9020_real__root__gt__zero,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X4 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_gt_zero
% 0.23/0.57 thf(fact_9021_real__root__strict__decreasing,axiom,
% 0.23/0.57 ! [N: nat,N3: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_nat @ N @ N3 )
% 0.23/0.57 => ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.57 => ( ord_less_real @ ( root @ N3 @ X4 ) @ ( root @ N @ X4 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_strict_decreasing
% 0.23/0.57 thf(fact_9022_sqrt__def,axiom,
% 0.23/0.57 ( sqrt
% 0.23/0.57 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % sqrt_def
% 0.23/0.57 thf(fact_9023_root__abs__power,axiom,
% 0.23/0.57 ! [N: nat,Y3: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
% 0.23/0.57 = ( abs_abs_real @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % root_abs_power
% 0.23/0.57 thf(fact_9024_real__root__pos__pos,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X4 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_pos_pos
% 0.23/0.57 thf(fact_9025_real__root__strict__increasing,axiom,
% 0.23/0.57 ! [N: nat,N3: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_nat @ N @ N3 )
% 0.23/0.57 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.23/0.57 => ( ord_less_real @ ( root @ N @ X4 ) @ ( root @ N3 @ X4 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_strict_increasing
% 0.23/0.57 thf(fact_9026_real__root__decreasing,axiom,
% 0.23/0.57 ! [N: nat,N3: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ one_one_real @ X4 )
% 0.23/0.57 => ( ord_less_eq_real @ ( root @ N3 @ X4 ) @ ( root @ N @ X4 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_decreasing
% 0.23/0.57 thf(fact_9027_real__root__pow__pos,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ( ( power_power_real @ ( root @ N @ X4 ) @ N )
% 0.23/0.57 = X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_pow_pos
% 0.23/0.57 thf(fact_9028_real__root__pos__unique,axiom,
% 0.23/0.57 ! [N: nat,Y3: real,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.23/0.57 => ( ( ( power_power_real @ Y3 @ N )
% 0.23/0.57 = X4 )
% 0.23/0.57 => ( ( root @ N @ X4 )
% 0.23/0.57 = Y3 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_pos_unique
% 0.23/0.57 thf(fact_9029_real__root__power__cancel,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ( ( root @ N @ ( power_power_real @ X4 @ N ) )
% 0.23/0.57 = X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_power_cancel
% 0.23/0.57 thf(fact_9030_odd__real__root__pow,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.57 => ( ( power_power_real @ ( root @ N @ X4 ) @ N )
% 0.23/0.57 = X4 ) ) ).
% 0.23/0.57
% 0.23/0.57 % odd_real_root_pow
% 0.23/0.57 thf(fact_9031_odd__real__root__unique,axiom,
% 0.23/0.57 ! [N: nat,Y3: real,X4: real] :
% 0.23/0.57 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.57 => ( ( ( power_power_real @ Y3 @ N )
% 0.23/0.57 = X4 )
% 0.23/0.57 => ( ( root @ N @ X4 )
% 0.23/0.57 = Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % odd_real_root_unique
% 0.23/0.57 thf(fact_9032_odd__real__root__power__cancel,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.57 => ( ( root @ N @ ( power_power_real @ X4 @ N ) )
% 0.23/0.57 = X4 ) ) ).
% 0.23/0.57
% 0.23/0.57 % odd_real_root_power_cancel
% 0.23/0.57 thf(fact_9033_real__root__increasing,axiom,
% 0.23/0.57 ! [N: nat,N3: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.23/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.57 => ( ord_less_eq_real @ ( root @ N @ X4 ) @ ( root @ N3 @ X4 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % real_root_increasing
% 0.23/0.57 thf(fact_9034_sgn__power__root,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X4 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X4 ) ) @ N ) )
% 0.23/0.57 = X4 ) ) ).
% 0.23/0.57
% 0.23/0.57 % sgn_power_root
% 0.23/0.57 thf(fact_9035_root__sgn__power,axiom,
% 0.23/0.57 ! [N: nat,Y3: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
% 0.23/0.57 = Y3 ) ) ).
% 0.23/0.57
% 0.23/0.57 % root_sgn_power
% 0.23/0.57 thf(fact_9036_ln__root,axiom,
% 0.23/0.57 ! [N: nat,B2: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.23/0.57 => ( ( ln_ln_real @ ( root @ N @ B2 ) )
% 0.23/0.57 = ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % ln_root
% 0.23/0.57 thf(fact_9037_log__root,axiom,
% 0.23/0.57 ! [N: nat,A: real,B2: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.23/0.57 => ( ( log @ B2 @ ( root @ N @ A ) )
% 0.23/0.57 = ( divide_divide_real @ ( log @ B2 @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % log_root
% 0.23/0.57 thf(fact_9038_log__base__root,axiom,
% 0.23/0.57 ! [N: nat,B2: real,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.23/0.57 => ( ( log @ ( root @ N @ B2 ) @ X4 )
% 0.23/0.57 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X4 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % log_base_root
% 0.23/0.57 thf(fact_9039_split__root,axiom,
% 0.23/0.57 ! [P: real > $o,N: nat,X4: real] :
% 0.23/0.57 ( ( P @ ( root @ N @ X4 ) )
% 0.23/0.57 = ( ( ( N = zero_zero_nat )
% 0.23/0.57 => ( P @ zero_zero_real ) )
% 0.23/0.57 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ! [Y: real] :
% 0.23/0.57 ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 0.23/0.57 = X4 )
% 0.23/0.57 => ( P @ Y ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % split_root
% 0.23/0.57 thf(fact_9040_infinite__nat__iff__unbounded,axiom,
% 0.23/0.57 ! [S2: set_nat] :
% 0.23/0.57 ( ( ~ ( finite_finite_nat @ S2 ) )
% 0.23/0.57 = ( ! [M6: nat] :
% 0.23/0.57 ? [N2: nat] :
% 0.23/0.57 ( ( ord_less_nat @ M6 @ N2 )
% 0.23/0.57 & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % infinite_nat_iff_unbounded
% 0.23/0.57 thf(fact_9041_unbounded__k__infinite,axiom,
% 0.23/0.57 ! [K: nat,S2: set_nat] :
% 0.23/0.57 ( ! [M4: nat] :
% 0.23/0.57 ( ( ord_less_nat @ K @ M4 )
% 0.23/0.57 => ? [N6: nat] :
% 0.23/0.57 ( ( ord_less_nat @ M4 @ N6 )
% 0.23/0.57 & ( member_nat @ N6 @ S2 ) ) )
% 0.23/0.57 => ~ ( finite_finite_nat @ S2 ) ) ).
% 0.23/0.57
% 0.23/0.57 % unbounded_k_infinite
% 0.23/0.57 thf(fact_9042_infinite__nat__iff__unbounded__le,axiom,
% 0.23/0.57 ! [S2: set_nat] :
% 0.23/0.57 ( ( ~ ( finite_finite_nat @ S2 ) )
% 0.23/0.57 = ( ! [M6: nat] :
% 0.23/0.57 ? [N2: nat] :
% 0.23/0.57 ( ( ord_less_eq_nat @ M6 @ N2 )
% 0.23/0.57 & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % infinite_nat_iff_unbounded_le
% 0.23/0.57 thf(fact_9043_root__powr__inverse,axiom,
% 0.23/0.57 ! [N: nat,X4: real] :
% 0.23/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.57 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.57 => ( ( root @ N @ X4 )
% 0.23/0.57 = ( powr_real @ X4 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % root_powr_inverse
% 0.23/0.57 thf(fact_9044_divmod__BitM__2__eq,axiom,
% 0.23/0.57 ! [M: num] :
% 0.23/0.57 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 0.23/0.57 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_BitM_2_eq
% 0.23/0.57 thf(fact_9045_insert__simp__excp,axiom,
% 0.23/0.57 ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X4: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.23/0.57 ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.57 => ( ( ord_less_nat @ X4 @ Mi )
% 0.23/0.57 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.23/0.57 => ( ( X4 != Ma )
% 0.23/0.57 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X4 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % insert_simp_excp
% 0.23/0.57 thf(fact_9046_insert__simp__norm,axiom,
% 0.23/0.57 ! [X4: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.23/0.57 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.57 => ( ( ord_less_nat @ Mi @ X4 )
% 0.23/0.57 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.23/0.57 => ( ( X4 != Ma )
% 0.23/0.57 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X4 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % insert_simp_norm
% 0.23/0.57 thf(fact_9047_max__Suc__Suc,axiom,
% 0.23/0.57 ! [M: nat,N: nat] :
% 0.23/0.57 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.23/0.57 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % max_Suc_Suc
% 0.23/0.57 thf(fact_9048_max__0R,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( ord_max_nat @ N @ zero_zero_nat )
% 0.23/0.57 = N ) ).
% 0.23/0.57
% 0.23/0.57 % max_0R
% 0.23/0.57 thf(fact_9049_max__0L,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( ord_max_nat @ zero_zero_nat @ N )
% 0.23/0.57 = N ) ).
% 0.23/0.57
% 0.23/0.57 % max_0L
% 0.23/0.57 thf(fact_9050_max__nat_Oright__neutral,axiom,
% 0.23/0.57 ! [A: nat] :
% 0.23/0.57 ( ( ord_max_nat @ A @ zero_zero_nat )
% 0.23/0.57 = A ) ).
% 0.23/0.57
% 0.23/0.57 % max_nat.right_neutral
% 0.23/0.57 thf(fact_9051_max__nat_Oneutr__eq__iff,axiom,
% 0.23/0.57 ! [A: nat,B2: nat] :
% 0.23/0.57 ( ( zero_zero_nat
% 0.23/0.57 = ( ord_max_nat @ A @ B2 ) )
% 0.23/0.57 = ( ( A = zero_zero_nat )
% 0.23/0.57 & ( B2 = zero_zero_nat ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % max_nat.neutr_eq_iff
% 0.23/0.57 thf(fact_9052_max__nat_Oleft__neutral,axiom,
% 0.23/0.57 ! [A: nat] :
% 0.23/0.57 ( ( ord_max_nat @ zero_zero_nat @ A )
% 0.23/0.57 = A ) ).
% 0.23/0.57
% 0.23/0.57 % max_nat.left_neutral
% 0.23/0.57 thf(fact_9053_max__nat_Oeq__neutr__iff,axiom,
% 0.23/0.57 ! [A: nat,B2: nat] :
% 0.23/0.57 ( ( ( ord_max_nat @ A @ B2 )
% 0.23/0.57 = zero_zero_nat )
% 0.23/0.57 = ( ( A = zero_zero_nat )
% 0.23/0.57 & ( B2 = zero_zero_nat ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % max_nat.eq_neutr_iff
% 0.23/0.57 thf(fact_9054_max__numeral__Suc,axiom,
% 0.23/0.57 ! [K: num,N: nat] :
% 0.23/0.57 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.23/0.57 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % max_numeral_Suc
% 0.23/0.57 thf(fact_9055_max__Suc__numeral,axiom,
% 0.23/0.57 ! [N: nat,K: num] :
% 0.23/0.57 ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.23/0.57 = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % max_Suc_numeral
% 0.23/0.57 thf(fact_9056_pred__numeral__simps_I2_J,axiom,
% 0.23/0.57 ! [K: num] :
% 0.23/0.57 ( ( pred_numeral @ ( bit0 @ K ) )
% 0.23/0.57 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % pred_numeral_simps(2)
% 0.23/0.57 thf(fact_9057_nat__add__max__right,axiom,
% 0.23/0.57 ! [M: nat,N: nat,Q2: nat] :
% 0.23/0.57 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 0.23/0.57 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % nat_add_max_right
% 0.23/0.57 thf(fact_9058_nat__add__max__left,axiom,
% 0.23/0.57 ! [M: nat,N: nat,Q2: nat] :
% 0.23/0.57 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 0.23/0.57 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % nat_add_max_left
% 0.23/0.57 thf(fact_9059_nat__mult__max__right,axiom,
% 0.23/0.57 ! [M: nat,N: nat,Q2: nat] :
% 0.23/0.57 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 0.23/0.57 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % nat_mult_max_right
% 0.23/0.57 thf(fact_9060_nat__mult__max__left,axiom,
% 0.23/0.57 ! [M: nat,N: nat,Q2: nat] :
% 0.23/0.57 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 0.23/0.57 = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % nat_mult_max_left
% 0.23/0.57 thf(fact_9061_semiring__norm_I26_J,axiom,
% 0.23/0.57 ( ( bitM @ one )
% 0.23/0.57 = one ) ).
% 0.23/0.57
% 0.23/0.57 % semiring_norm(26)
% 0.23/0.57 thf(fact_9062_nat__minus__add__max,axiom,
% 0.23/0.57 ! [N: nat,M: nat] :
% 0.23/0.57 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 0.23/0.57 = ( ord_max_nat @ N @ M ) ) ).
% 0.23/0.57
% 0.23/0.57 % nat_minus_add_max
% 0.23/0.57 thf(fact_9063_semiring__norm_I27_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( bitM @ ( bit0 @ N ) )
% 0.23/0.57 = ( bit1 @ ( bitM @ N ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % semiring_norm(27)
% 0.23/0.57 thf(fact_9064_semiring__norm_I28_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( bitM @ ( bit1 @ N ) )
% 0.23/0.57 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % semiring_norm(28)
% 0.23/0.57 thf(fact_9065_inc__BitM__eq,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( inc @ ( bitM @ N ) )
% 0.23/0.57 = ( bit0 @ N ) ) ).
% 0.23/0.57
% 0.23/0.57 % inc_BitM_eq
% 0.23/0.57 thf(fact_9066_BitM__inc__eq,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( bitM @ ( inc @ N ) )
% 0.23/0.57 = ( bit1 @ N ) ) ).
% 0.23/0.57
% 0.23/0.57 % BitM_inc_eq
% 0.23/0.57 thf(fact_9067_eval__nat__numeral_I2_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.23/0.57 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % eval_nat_numeral(2)
% 0.23/0.57 thf(fact_9068_one__plus__BitM,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 0.23/0.57 = ( bit0 @ N ) ) ).
% 0.23/0.57
% 0.23/0.57 % one_plus_BitM
% 0.23/0.57 thf(fact_9069_BitM__plus__one,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 0.23/0.57 = ( bit0 @ N ) ) ).
% 0.23/0.57
% 0.23/0.57 % BitM_plus_one
% 0.23/0.57 thf(fact_9070_vebt__insert_Osimps_I5_J,axiom,
% 0.23/0.57 ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
% 0.23/0.57 ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
% 0.23/0.57 = ( if_VEBT_VEBT
% 0.23/0.57 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.23/0.57 & ~ ( ( X4 = Mi )
% 0.23/0.57 | ( X4 = Ma ) ) )
% 0.23/0.57 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ X4 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 0.23/0.57 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % vebt_insert.simps(5)
% 0.23/0.57 thf(fact_9071_vebt__insert_Oelims,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
% 0.23/0.57 ( ( ( vEBT_vebt_insert @ X4 @ Xa )
% 0.23/0.57 = Y3 )
% 0.23/0.57 => ( ! [A5: $o,B5: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.57 => ~ ( ( ( Xa = zero_zero_nat )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( vEBT_Leaf @ $true @ B5 ) ) )
% 0.23/0.57 & ( ( Xa != zero_zero_nat )
% 0.23/0.57 => ( ( ( Xa = one_one_nat )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ $true ) ) )
% 0.23/0.57 & ( ( Xa != one_one_nat )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) ) )
% 0.23/0.57 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) ) )
% 0.23/0.57 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
% 0.23/0.57 => ( ! [V3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) ) )
% 0.23/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( if_VEBT_VEBT
% 0.23/0.57 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 & ~ ( ( Xa = Mi2 )
% 0.23/0.57 | ( Xa = Ma2 ) ) )
% 0.23/0.57 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 0.23/0.57 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % vebt_insert.elims
% 0.23/0.57 thf(fact_9072_vebt__insert_Opelims,axiom,
% 0.23/0.57 ! [X4: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
% 0.23/0.57 ( ( ( vEBT_vebt_insert @ X4 @ Xa )
% 0.23/0.57 = Y3 )
% 0.23/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.57 => ( ! [A5: $o,B5: $o] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.57 => ( ( ( ( Xa = zero_zero_nat )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( vEBT_Leaf @ $true @ B5 ) ) )
% 0.23/0.57 & ( ( Xa != zero_zero_nat )
% 0.23/0.57 => ( ( ( Xa = one_one_nat )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ $true ) ) )
% 0.23/0.57 & ( ( Xa != one_one_nat )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
% 0.23/0.57 => ( ! [V3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
% 0.23/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( if_VEBT_VEBT
% 0.23/0.57 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.23/0.57 & ~ ( ( Xa = Mi2 )
% 0.23/0.57 | ( Xa = Ma2 ) ) )
% 0.23/0.57 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 0.23/0.57 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) )
% 0.23/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % vebt_insert.pelims
% 0.23/0.57 thf(fact_9073_vebt__buildup_Opelims,axiom,
% 0.23/0.57 ! [X4: nat,Y3: vEBT_VEBT] :
% 0.23/0.57 ( ( ( vEBT_vebt_buildup @ X4 )
% 0.23/0.57 = Y3 )
% 0.23/0.57 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X4 )
% 0.23/0.57 => ( ( ( X4 = zero_zero_nat )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( vEBT_Leaf @ $false @ $false ) )
% 0.23/0.57 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 0.23/0.57 => ( ( ( X4
% 0.23/0.57 = ( suc @ zero_zero_nat ) )
% 0.23/0.57 => ( ( Y3
% 0.23/0.57 = ( vEBT_Leaf @ $false @ $false ) )
% 0.23/0.57 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.23/0.57 => ~ ! [Va: nat] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( suc @ ( suc @ Va ) ) )
% 0.23/0.57 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( 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 ) ) ) ) ) ) )
% 0.23/0.57 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 = ( 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 ) ) ) ) ) ) ) ) )
% 0.23/0.57 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % vebt_buildup.pelims
% 0.23/0.57 thf(fact_9074_max__enat__simps_I2_J,axiom,
% 0.23/0.57 ! [Q2: extended_enat] :
% 0.23/0.57 ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 0.23/0.57 = Q2 ) ).
% 0.23/0.57
% 0.23/0.57 % max_enat_simps(2)
% 0.23/0.57 thf(fact_9075_max__enat__simps_I3_J,axiom,
% 0.23/0.57 ! [Q2: extended_enat] :
% 0.23/0.57 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 0.23/0.57 = Q2 ) ).
% 0.23/0.57
% 0.23/0.57 % max_enat_simps(3)
% 0.23/0.57 thf(fact_9076_divmod__step__nat__def,axiom,
% 0.23/0.57 ( unique5026877609467782581ep_nat
% 0.23/0.57 = ( ^ [L: num] :
% 0.23/0.57 ( produc2626176000494625587at_nat
% 0.23/0.57 @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ 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 @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_step_nat_def
% 0.23/0.57 thf(fact_9077_divmod__step__int__def,axiom,
% 0.23/0.57 ( unique5024387138958732305ep_int
% 0.23/0.57 = ( ^ [L: num] :
% 0.23/0.57 ( produc4245557441103728435nt_int
% 0.23/0.57 @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ 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 @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_step_int_def
% 0.23/0.57 thf(fact_9078_divmod__nat__if,axiom,
% 0.23/0.57 ( divmod_nat
% 0.23/0.57 = ( ^ [M6: nat,N2: nat] :
% 0.23/0.57 ( if_Pro6206227464963214023at_nat
% 0.23/0.57 @ ( ( N2 = zero_zero_nat )
% 0.23/0.57 | ( ord_less_nat @ M6 @ N2 ) )
% 0.23/0.57 @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
% 0.23/0.57 @ ( produc2626176000494625587at_nat
% 0.23/0.57 @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
% 0.23/0.57 @ ( divmod_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_nat_if
% 0.23/0.57 thf(fact_9079_arctan__def,axiom,
% 0.23/0.57 ( arctan
% 0.23/0.57 = ( ^ [Y: real] :
% 0.23/0.57 ( the_real
% 0.23/0.57 @ ^ [X: real] :
% 0.23/0.57 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.23/0.57 & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.57 & ( ( tan_real @ X )
% 0.23/0.57 = Y ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % arctan_def
% 0.23/0.57 thf(fact_9080_ln__neg__is__const,axiom,
% 0.23/0.57 ! [X4: real] :
% 0.23/0.57 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 0.23/0.57 => ( ( ln_ln_real @ X4 )
% 0.23/0.57 = ( the_real
% 0.23/0.57 @ ^ [X: real] : $false ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % ln_neg_is_const
% 0.23/0.57 thf(fact_9081_arccos__def,axiom,
% 0.23/0.57 ( arccos
% 0.23/0.57 = ( ^ [Y: real] :
% 0.23/0.57 ( the_real
% 0.23/0.57 @ ^ [X: real] :
% 0.23/0.57 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.23/0.57 & ( ord_less_eq_real @ X @ pi )
% 0.23/0.57 & ( ( cos_real @ X )
% 0.23/0.57 = Y ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % arccos_def
% 0.23/0.57 thf(fact_9082_divmod__nat__def,axiom,
% 0.23/0.57 ( divmod_nat
% 0.23/0.57 = ( ^ [M6: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N2 ) @ ( modulo_modulo_nat @ M6 @ N2 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_nat_def
% 0.23/0.57 thf(fact_9083_pi__half,axiom,
% 0.23/0.57 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.23/0.57 = ( the_real
% 0.23/0.57 @ ^ [X: real] :
% 0.23/0.57 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.23/0.57 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.23/0.57 & ( ( cos_real @ X )
% 0.23/0.57 = zero_zero_real ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % pi_half
% 0.23/0.57 thf(fact_9084_pi__def,axiom,
% 0.23/0.57 ( pi
% 0.23/0.57 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.23/0.57 @ ( the_real
% 0.23/0.57 @ ^ [X: real] :
% 0.23/0.57 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.23/0.57 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.23/0.57 & ( ( cos_real @ X )
% 0.23/0.57 = zero_zero_real ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % pi_def
% 0.23/0.57 thf(fact_9085_arcsin__def,axiom,
% 0.23/0.57 ( arcsin
% 0.23/0.57 = ( ^ [Y: real] :
% 0.23/0.57 ( the_real
% 0.23/0.57 @ ^ [X: real] :
% 0.23/0.57 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.23/0.57 & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.57 & ( ( sin_real @ X )
% 0.23/0.57 = Y ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % arcsin_def
% 0.23/0.57 thf(fact_9086_or__int__unfold,axiom,
% 0.23/0.57 ( bit_se1409905431419307370or_int
% 0.23/0.57 = ( ^ [K3: int,L: int] :
% 0.23/0.57 ( if_int
% 0.23/0.57 @ ( ( K3
% 0.23/0.57 = ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.57 | ( L
% 0.23/0.57 = ( uminus_uminus_int @ one_one_int ) ) )
% 0.23/0.57 @ ( uminus_uminus_int @ one_one_int )
% 0.23/0.57 @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_int_unfold
% 0.23/0.57 thf(fact_9087_horner__sum__of__bool__2__less,axiom,
% 0.23/0.57 ! [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 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % horner_sum_of_bool_2_less
% 0.23/0.57 thf(fact_9088_or__nonnegative__int__iff,axiom,
% 0.23/0.57 ! [K: int,L2: int] :
% 0.23/0.57 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
% 0.23/0.57 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.23/0.57 & ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_nonnegative_int_iff
% 0.23/0.57 thf(fact_9089_or__negative__int__iff,axiom,
% 0.23/0.57 ! [K: int,L2: int] :
% 0.23/0.57 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ zero_zero_int )
% 0.23/0.57 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.23/0.57 | ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_negative_int_iff
% 0.23/0.57 thf(fact_9090_or__minus__numerals_I2_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_minus_numerals(2)
% 0.23/0.57 thf(fact_9091_or__minus__numerals_I6_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.23/0.57 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_minus_numerals(6)
% 0.23/0.57 thf(fact_9092_or__minus__minus__numerals,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.57 = ( 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 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_minus_minus_numerals
% 0.23/0.57 thf(fact_9093_and__minus__minus__numerals,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.57 = ( 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 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % and_minus_minus_numerals
% 0.23/0.57 thf(fact_9094_bit__or__int__iff,axiom,
% 0.23/0.57 ! [K: int,L2: int,N: nat] :
% 0.23/0.57 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N )
% 0.23/0.57 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.23/0.57 | ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % bit_or_int_iff
% 0.23/0.57 thf(fact_9095_OR__lower,axiom,
% 0.23/0.57 ! [X4: int,Y3: int] :
% 0.23/0.57 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.23/0.57 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % OR_lower
% 0.23/0.57 thf(fact_9096_or__greater__eq,axiom,
% 0.23/0.57 ! [L2: int,K: int] :
% 0.23/0.57 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 0.23/0.57 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L2 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_greater_eq
% 0.23/0.57 thf(fact_9097_plus__and__or,axiom,
% 0.23/0.57 ! [X4: int,Y3: int] :
% 0.23/0.57 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ ( bit_se1409905431419307370or_int @ X4 @ Y3 ) )
% 0.23/0.57 = ( plus_plus_int @ X4 @ Y3 ) ) ).
% 0.23/0.57
% 0.23/0.57 % plus_and_or
% 0.23/0.57 thf(fact_9098_or__int__def,axiom,
% 0.23/0.57 ( bit_se1409905431419307370or_int
% 0.23/0.57 = ( ^ [K3: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_int_def
% 0.23/0.57 thf(fact_9099_or__not__numerals_I1_J,axiom,
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.23/0.57 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_numerals(1)
% 0.23/0.57 thf(fact_9100_xor__int__def,axiom,
% 0.23/0.57 ( bit_se6526347334894502574or_int
% 0.23/0.57 = ( ^ [K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % xor_int_def
% 0.23/0.57 thf(fact_9101_concat__bit__def,axiom,
% 0.23/0.57 ( bit_concat_bit
% 0.23/0.57 = ( ^ [N2: nat,K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % concat_bit_def
% 0.23/0.57 thf(fact_9102_set__bit__int__def,axiom,
% 0.23/0.57 ( bit_se7879613467334960850it_int
% 0.23/0.57 = ( ^ [N2: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % set_bit_int_def
% 0.23/0.57 thf(fact_9103_or__not__numerals_I4_J,axiom,
% 0.23/0.57 ! [M: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.23/0.57 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_numerals(4)
% 0.23/0.57 thf(fact_9104_or__not__numerals_I2_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.23/0.57 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_numerals(2)
% 0.23/0.57 thf(fact_9105_or__not__numerals_I3_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.23/0.57 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_numerals(3)
% 0.23/0.57 thf(fact_9106_or__not__numerals_I7_J,axiom,
% 0.23/0.57 ! [M: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.23/0.57 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_numerals(7)
% 0.23/0.57 thf(fact_9107_or__not__numerals_I6_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.23/0.57 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_numerals(6)
% 0.23/0.57 thf(fact_9108_OR__upper,axiom,
% 0.23/0.57 ! [X4: int,N: nat,Y3: int] :
% 0.23/0.57 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.23/0.57 => ( ( ord_less_int @ X4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.57 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.23/0.57 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X4 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % OR_upper
% 0.23/0.57 thf(fact_9109_or__not__numerals_I5_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.23/0.57 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_numerals(5)
% 0.23/0.57 thf(fact_9110_or__not__numerals_I9_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.23/0.57 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_numerals(9)
% 0.23/0.57 thf(fact_9111_or__not__numerals_I8_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.23/0.57 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_numerals(8)
% 0.23/0.57 thf(fact_9112_or__int__rec,axiom,
% 0.23/0.57 ( bit_se1409905431419307370or_int
% 0.23/0.57 = ( ^ [K3: int,L: int] :
% 0.23/0.57 ( plus_plus_int
% 0.23/0.57 @ ( zero_n2684676970156552555ol_int
% 0.23/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.23/0.57 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.23/0.57 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_int_rec
% 0.23/0.57 thf(fact_9113_or__minus__numerals_I5_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.23/0.57 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_minus_numerals(5)
% 0.23/0.57 thf(fact_9114_or__minus__numerals_I1_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_minus_numerals(1)
% 0.23/0.57 thf(fact_9115_or__nat__numerals_I2_J,axiom,
% 0.23/0.57 ! [Y3: num] :
% 0.23/0.57 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.23/0.57 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_nat_numerals(2)
% 0.23/0.57 thf(fact_9116_or__nat__numerals_I4_J,axiom,
% 0.23/0.57 ! [X4: num] :
% 0.23/0.57 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 0.23/0.57 = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_nat_numerals(4)
% 0.23/0.57 thf(fact_9117_or__nat__numerals_I1_J,axiom,
% 0.23/0.57 ! [Y3: num] :
% 0.23/0.57 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.23/0.57 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_nat_numerals(1)
% 0.23/0.57 thf(fact_9118_or__nat__numerals_I3_J,axiom,
% 0.23/0.57 ! [X4: num] :
% 0.23/0.57 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 0.23/0.57 = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_nat_numerals(3)
% 0.23/0.57 thf(fact_9119_or__minus__numerals_I4_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_minus_numerals(4)
% 0.23/0.57 thf(fact_9120_or__minus__numerals_I8_J,axiom,
% 0.23/0.57 ! [N: num,M: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_minus_numerals(8)
% 0.23/0.57 thf(fact_9121_or__minus__numerals_I7_J,axiom,
% 0.23/0.57 ! [N: num,M: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_minus_numerals(7)
% 0.23/0.57 thf(fact_9122_or__minus__numerals_I3_J,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_minus_numerals(3)
% 0.23/0.57 thf(fact_9123_or__not__num__neg_Osimps_I1_J,axiom,
% 0.23/0.57 ( ( bit_or_not_num_neg @ one @ one )
% 0.23/0.57 = one ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_num_neg.simps(1)
% 0.23/0.57 thf(fact_9124_set__bit__nat__def,axiom,
% 0.23/0.57 ( bit_se7882103937844011126it_nat
% 0.23/0.57 = ( ^ [M6: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % set_bit_nat_def
% 0.23/0.57 thf(fact_9125_or__not__num__neg_Osimps_I4_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 0.23/0.57 = ( bit0 @ one ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_num_neg.simps(4)
% 0.23/0.57 thf(fact_9126_or__not__num__neg_Osimps_I6_J,axiom,
% 0.23/0.57 ! [N: num,M: num] :
% 0.23/0.57 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 0.23/0.57 = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_num_neg.simps(6)
% 0.23/0.57 thf(fact_9127_or__not__num__neg_Osimps_I3_J,axiom,
% 0.23/0.57 ! [M: num] :
% 0.23/0.57 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 0.23/0.57 = ( bit1 @ M ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_num_neg.simps(3)
% 0.23/0.57 thf(fact_9128_or__not__num__neg_Osimps_I7_J,axiom,
% 0.23/0.57 ! [N: num] :
% 0.23/0.57 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 0.23/0.57 = one ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_num_neg.simps(7)
% 0.23/0.57 thf(fact_9129_or__not__num__neg_Osimps_I5_J,axiom,
% 0.23/0.57 ! [N: num,M: num] :
% 0.23/0.57 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 0.23/0.57 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_num_neg.simps(5)
% 0.23/0.57 thf(fact_9130_or__not__num__neg_Osimps_I9_J,axiom,
% 0.23/0.57 ! [N: num,M: num] :
% 0.23/0.57 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 0.23/0.57 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_num_neg.simps(9)
% 0.23/0.57 thf(fact_9131_or__nat__def,axiom,
% 0.23/0.57 ( bit_se1412395901928357646or_nat
% 0.23/0.57 = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_nat_def
% 0.23/0.57 thf(fact_9132_or__not__num__neg_Osimps_I2_J,axiom,
% 0.23/0.57 ! [M: num] :
% 0.23/0.57 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 0.23/0.57 = ( bit1 @ M ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_num_neg.simps(2)
% 0.23/0.57 thf(fact_9133_or__not__num__neg_Osimps_I8_J,axiom,
% 0.23/0.57 ! [N: num,M: num] :
% 0.23/0.57 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 0.23/0.57 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_num_neg.simps(8)
% 0.23/0.57 thf(fact_9134_or__not__num__neg_Oelims,axiom,
% 0.23/0.57 ! [X4: num,Xa: num,Y3: num] :
% 0.23/0.57 ( ( ( bit_or_not_num_neg @ X4 @ Xa )
% 0.23/0.57 = Y3 )
% 0.23/0.57 => ( ( ( X4 = one )
% 0.23/0.57 => ( ( Xa = one )
% 0.23/0.57 => ( Y3 != one ) ) )
% 0.23/0.57 => ( ( ( X4 = one )
% 0.23/0.57 => ! [M4: num] :
% 0.23/0.57 ( ( Xa
% 0.23/0.57 = ( bit0 @ M4 ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( bit1 @ M4 ) ) ) )
% 0.23/0.57 => ( ( ( X4 = one )
% 0.23/0.57 => ! [M4: num] :
% 0.23/0.57 ( ( Xa
% 0.23/0.57 = ( bit1 @ M4 ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( bit1 @ M4 ) ) ) )
% 0.23/0.57 => ( ( ? [N4: num] :
% 0.23/0.57 ( X4
% 0.23/0.57 = ( bit0 @ N4 ) )
% 0.23/0.57 => ( ( Xa = one )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( bit0 @ one ) ) ) )
% 0.23/0.57 => ( ! [N4: num] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( bit0 @ N4 ) )
% 0.23/0.57 => ! [M4: num] :
% 0.23/0.57 ( ( Xa
% 0.23/0.57 = ( bit0 @ M4 ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 0.23/0.57 => ( ! [N4: num] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( bit0 @ N4 ) )
% 0.23/0.57 => ! [M4: num] :
% 0.23/0.57 ( ( Xa
% 0.23/0.57 = ( bit1 @ M4 ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( bit0 @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 0.23/0.57 => ( ( ? [N4: num] :
% 0.23/0.57 ( X4
% 0.23/0.57 = ( bit1 @ N4 ) )
% 0.23/0.57 => ( ( Xa = one )
% 0.23/0.57 => ( Y3 != one ) ) )
% 0.23/0.57 => ( ! [N4: num] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( bit1 @ N4 ) )
% 0.23/0.57 => ! [M4: num] :
% 0.23/0.57 ( ( Xa
% 0.23/0.57 = ( bit0 @ M4 ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 0.23/0.57 => ~ ! [N4: num] :
% 0.23/0.57 ( ( X4
% 0.23/0.57 = ( bit1 @ N4 ) )
% 0.23/0.57 => ! [M4: num] :
% 0.23/0.57 ( ( Xa
% 0.23/0.57 = ( bit1 @ M4 ) )
% 0.23/0.57 => ( Y3
% 0.23/0.57 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_not_num_neg.elims
% 0.23/0.57 thf(fact_9135_numeral__or__not__num__eq,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % numeral_or_not_num_eq
% 0.23/0.57 thf(fact_9136_int__numeral__not__or__num__neg,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % int_numeral_not_or_num_neg
% 0.23/0.57 thf(fact_9137_int__numeral__or__not__num__neg,axiom,
% 0.23/0.57 ! [M: num,N: num] :
% 0.23/0.57 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.57 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % int_numeral_or_not_num_neg
% 0.23/0.57 thf(fact_9138_or__Suc__0__eq,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.23/0.57 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_Suc_0_eq
% 0.23/0.57 thf(fact_9139_Suc__0__or__eq,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.23/0.57 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Suc_0_or_eq
% 0.23/0.57 thf(fact_9140_or__nat__rec,axiom,
% 0.23/0.57 ( bit_se1412395901928357646or_nat
% 0.23/0.57 = ( ^ [M6: nat,N2: nat] :
% 0.23/0.57 ( plus_plus_nat
% 0.23/0.57 @ ( zero_n2687167440665602831ol_nat
% 0.23/0.57 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 0.23/0.57 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.23/0.57 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_nat_rec
% 0.23/0.57 thf(fact_9141_or__nat__unfold,axiom,
% 0.23/0.57 ( bit_se1412395901928357646or_nat
% 0.23/0.57 = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % or_nat_unfold
% 0.23/0.57 thf(fact_9142_Sum__Ico__nat,axiom,
% 0.23/0.57 ! [M: nat,N: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [X: nat] : X
% 0.23/0.57 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.23/0.57 = ( 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 ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Sum_Ico_nat
% 0.23/0.57 thf(fact_9143_sum__power2,axiom,
% 0.23/0.57 ! [K: nat] :
% 0.23/0.57 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 0.23/0.57 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 0.23/0.57
% 0.23/0.57 % sum_power2
% 0.23/0.57 thf(fact_9144_finite__atLeastLessThan,axiom,
% 0.23/0.57 ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_atLeastLessThan
% 0.23/0.57 thf(fact_9145_atLeastLessThan__singleton,axiom,
% 0.23/0.57 ! [M: nat] :
% 0.23/0.57 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 0.23/0.57 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeastLessThan_singleton
% 0.23/0.57 thf(fact_9146_all__nat__less__eq,axiom,
% 0.23/0.57 ! [N: nat,P: nat > $o] :
% 0.23/0.57 ( ( ! [M6: nat] :
% 0.23/0.57 ( ( ord_less_nat @ M6 @ N )
% 0.23/0.57 => ( P @ M6 ) ) )
% 0.23/0.57 = ( ! [X: nat] :
% 0.23/0.57 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.23/0.57 => ( P @ X ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % all_nat_less_eq
% 0.23/0.57 thf(fact_9147_ex__nat__less__eq,axiom,
% 0.23/0.57 ! [N: nat,P: nat > $o] :
% 0.23/0.57 ( ( ? [M6: nat] :
% 0.23/0.57 ( ( ord_less_nat @ M6 @ N )
% 0.23/0.57 & ( P @ M6 ) ) )
% 0.23/0.57 = ( ? [X: nat] :
% 0.23/0.57 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.23/0.57 & ( P @ X ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % ex_nat_less_eq
% 0.23/0.57 thf(fact_9148_atLeastLessThanSuc__atLeastAtMost,axiom,
% 0.23/0.57 ! [L2: nat,U: nat] :
% 0.23/0.57 ( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U ) )
% 0.23/0.57 = ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeastLessThanSuc_atLeastAtMost
% 0.23/0.57 thf(fact_9149_lessThan__atLeast0,axiom,
% 0.23/0.57 ( set_ord_lessThan_nat
% 0.23/0.57 = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 0.23/0.57
% 0.23/0.57 % lessThan_atLeast0
% 0.23/0.57 thf(fact_9150_atLeastLessThan0,axiom,
% 0.23/0.57 ! [M: nat] :
% 0.23/0.57 ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 0.23/0.57 = bot_bot_set_nat ) ).
% 0.23/0.57
% 0.23/0.57 % atLeastLessThan0
% 0.23/0.57 thf(fact_9151_atLeast0__lessThan__Suc,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.23/0.57 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeast0_lessThan_Suc
% 0.23/0.57 thf(fact_9152_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.23/0.57 ! [N3: set_nat,N: nat] :
% 0.23/0.57 ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.23/0.57 => ( finite_finite_nat @ N3 ) ) ).
% 0.23/0.57
% 0.23/0.57 % subset_eq_atLeast0_lessThan_finite
% 0.23/0.57 thf(fact_9153_atLeastLessThanSuc,axiom,
% 0.23/0.57 ! [M: nat,N: nat] :
% 0.23/0.57 ( ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.57 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.23/0.57 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 0.23/0.57 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.23/0.57 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.23/0.57 = bot_bot_set_nat ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeastLessThanSuc
% 0.23/0.57 thf(fact_9154_prod__Suc__fact,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.23/0.57 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_Suc_fact
% 0.23/0.57 thf(fact_9155_prod__Suc__Suc__fact,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.23/0.57 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.23/0.57
% 0.23/0.57 % prod_Suc_Suc_fact
% 0.23/0.57 thf(fact_9156_atLeastLessThan__nat__numeral,axiom,
% 0.23/0.57 ! [M: nat,K: num] :
% 0.23/0.57 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.23/0.57 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.23/0.57 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.23/0.57 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.23/0.57 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.23/0.57 = bot_bot_set_nat ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeastLessThan_nat_numeral
% 0.23/0.57 thf(fact_9157_atLeast1__lessThan__eq__remove0,axiom,
% 0.23/0.57 ! [N: nat] :
% 0.23/0.57 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.23/0.57 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeast1_lessThan_eq_remove0
% 0.23/0.57 thf(fact_9158_Chebyshev__sum__upper__nat,axiom,
% 0.23/0.57 ! [N: nat,A: nat > nat,B2: nat > nat] :
% 0.23/0.57 ( ! [I3: nat,J2: nat] :
% 0.23/0.57 ( ( ord_less_eq_nat @ I3 @ J2 )
% 0.23/0.57 => ( ( ord_less_nat @ J2 @ N )
% 0.23/0.57 => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
% 0.23/0.57 => ( ! [I3: nat,J2: nat] :
% 0.23/0.57 ( ( ord_less_eq_nat @ I3 @ J2 )
% 0.23/0.57 => ( ( ord_less_nat @ J2 @ N )
% 0.23/0.57 => ( ord_less_eq_nat @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
% 0.23/0.57 => ( ord_less_eq_nat
% 0.23/0.57 @ ( times_times_nat @ N
% 0.23/0.57 @ ( groups3542108847815614940at_nat
% 0.23/0.57 @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( B2 @ I2 ) )
% 0.23/0.57 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 0.23/0.57 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Chebyshev_sum_upper_nat
% 0.23/0.57 thf(fact_9159_finite__atLeastLessThan__int,axiom,
% 0.23/0.57 ! [L2: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L2 @ U ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_atLeastLessThan_int
% 0.23/0.57 thf(fact_9160_finite__atLeastZeroLessThan__int,axiom,
% 0.23/0.57 ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 0.23/0.57
% 0.23/0.57 % finite_atLeastZeroLessThan_int
% 0.23/0.57 thf(fact_9161_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 0.23/0.57 ! [L2: int,U: int] :
% 0.23/0.57 ( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
% 0.23/0.57 = ( set_or1266510415728281911st_int @ L2 @ U ) ) ).
% 0.23/0.57
% 0.23/0.57 % atLeastLessThanPlusOne_atLeastAtMost_int
% 0.23/0.57 thf(fact_9162_valid__eq,axiom,
% 0.23/0.57 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 0.23/0.57
% 0.23/0.57 % valid_eq
% 0.23/0.57 thf(fact_9163_valid__eq1,axiom,
% 0.23/0.57 ! [T: vEBT_VEBT,D: nat] :
% 0.23/0.57 ( ( vEBT_invar_vebt @ T @ D )
% 0.23/0.57 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 0.23/0.57
% 0.23/0.57 % valid_eq1
% 0.23/0.57 thf(fact_9164_valid__eq2,axiom,
% 0.23/0.57 ! [T: vEBT_VEBT,D: nat] :
% 0.23/0.57 ( ( vEBT_VEBT_valid @ T @ D )
% 0.23/0.57 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 0.23/0.57
% 0.23/0.57 % valid_eq2
% 0.23/0.57 thf(fact_9165_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 0.23/0.57 ! [Uu3: $o,Uv2: $o,D: nat] :
% 0.23/0.57 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ D )
% 0.23/0.57 = ( D = one_one_nat ) ) ).
% 0.23/0.57
% 0.23/0.57 % VEBT_internal.valid'.simps(1)
% 0.23/0.57 thf(fact_9166_divmod__step__integer__def,axiom,
% 0.23/0.57 ( unique4921790084139445826nteger
% 0.23/0.57 = ( ^ [L: num] :
% 0.23/0.57 ( produc6916734918728496179nteger
% 0.23/0.57 @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_step_integer_def
% 0.23/0.57 thf(fact_9167_csqrt_Osimps_I1_J,axiom,
% 0.23/0.57 ! [Z: complex] :
% 0.23/0.57 ( ( re @ ( csqrt @ Z ) )
% 0.23/0.57 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % csqrt.simps(1)
% 0.23/0.57 thf(fact_9168_complex__Re__numeral,axiom,
% 0.23/0.57 ! [V: num] :
% 0.23/0.57 ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 0.23/0.57 = ( numeral_numeral_real @ V ) ) ).
% 0.23/0.57
% 0.23/0.57 % complex_Re_numeral
% 0.23/0.57 thf(fact_9169_Re__divide__of__nat,axiom,
% 0.23/0.57 ! [Z: complex,N: nat] :
% 0.23/0.57 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.23/0.57 = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Re_divide_of_nat
% 0.23/0.57 thf(fact_9170_Re__divide__of__real,axiom,
% 0.23/0.57 ! [Z: complex,R2: real] :
% 0.23/0.57 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 0.23/0.57 = ( divide_divide_real @ ( re @ Z ) @ R2 ) ) ).
% 0.23/0.57
% 0.23/0.57 % Re_divide_of_real
% 0.23/0.57 thf(fact_9171_Re__sgn,axiom,
% 0.23/0.57 ! [Z: complex] :
% 0.23/0.57 ( ( re @ ( sgn_sgn_complex @ Z ) )
% 0.23/0.57 = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Re_sgn
% 0.23/0.57 thf(fact_9172_Re__divide__numeral,axiom,
% 0.23/0.57 ! [Z: complex,W: num] :
% 0.23/0.57 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.23/0.57 = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % Re_divide_numeral
% 0.23/0.57 thf(fact_9173_plus__integer__code_I1_J,axiom,
% 0.23/0.57 ! [K: code_integer] :
% 0.23/0.57 ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
% 0.23/0.57 = K ) ).
% 0.23/0.57
% 0.23/0.57 % plus_integer_code(1)
% 0.23/0.57 thf(fact_9174_plus__integer__code_I2_J,axiom,
% 0.23/0.57 ! [L2: code_integer] :
% 0.23/0.57 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.23/0.57 = L2 ) ).
% 0.23/0.57
% 0.23/0.57 % plus_integer_code(2)
% 0.23/0.57 thf(fact_9175_times__integer__code_I1_J,axiom,
% 0.23/0.57 ! [K: code_integer] :
% 0.23/0.57 ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 0.23/0.57 = zero_z3403309356797280102nteger ) ).
% 0.23/0.57
% 0.23/0.57 % times_integer_code(1)
% 0.23/0.57 thf(fact_9176_times__integer__code_I2_J,axiom,
% 0.23/0.57 ! [L2: code_integer] :
% 0.23/0.57 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.23/0.57 = zero_z3403309356797280102nteger ) ).
% 0.23/0.57
% 0.23/0.57 % times_integer_code(2)
% 0.23/0.57 thf(fact_9177_divmod__integer_H__def,axiom,
% 0.23/0.57 ( unique3479559517661332726nteger
% 0.23/0.57 = ( ^ [M6: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % divmod_integer'_def
% 0.23/0.57 thf(fact_9178_sgn__integer__code,axiom,
% 0.23/0.57 ( sgn_sgn_Code_integer
% 0.23/0.57 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % sgn_integer_code
% 0.23/0.57 thf(fact_9179_minus__integer__code_I2_J,axiom,
% 0.23/0.57 ! [L2: code_integer] :
% 0.23/0.57 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.23/0.57 = ( uminus1351360451143612070nteger @ L2 ) ) ).
% 0.23/0.57
% 0.23/0.57 % minus_integer_code(2)
% 0.23/0.57 thf(fact_9180_minus__integer__code_I1_J,axiom,
% 0.23/0.57 ! [K: code_integer] :
% 0.23/0.57 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 0.23/0.57 = K ) ).
% 0.23/0.57
% 0.23/0.57 % minus_integer_code(1)
% 0.23/0.57 thf(fact_9181_complex__Re__le__cmod,axiom,
% 0.23/0.57 ! [X4: complex] : ( ord_less_eq_real @ ( re @ X4 ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).
% 0.23/0.57
% 0.23/0.57 % complex_Re_le_cmod
% 0.23/0.57 thf(fact_9182_one__complex_Osimps_I1_J,axiom,
% 0.23/0.57 ( ( re @ one_one_complex )
% 0.23/0.57 = one_one_real ) ).
% 0.23/0.57
% 0.23/0.57 % one_complex.simps(1)
% 0.23/0.57 thf(fact_9183_plus__complex_Osimps_I1_J,axiom,
% 0.23/0.57 ! [X4: complex,Y3: complex] :
% 0.23/0.57 ( ( re @ ( plus_plus_complex @ X4 @ Y3 ) )
% 0.23/0.57 = ( plus_plus_real @ ( re @ X4 ) @ ( re @ Y3 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % plus_complex.simps(1)
% 0.23/0.57 thf(fact_9184_scaleR__complex_Osimps_I1_J,axiom,
% 0.23/0.57 ! [R2: real,X4: complex] :
% 0.23/0.57 ( ( re @ ( real_V2046097035970521341omplex @ R2 @ X4 ) )
% 0.23/0.57 = ( times_times_real @ R2 @ ( re @ X4 ) ) ) ).
% 0.23/0.57
% 0.23/0.57 % scaleR_complex.simps(1)
% 0.23/0.57 thf(fact_9185_minus__complex_Osimps_I1_J,axiom,
% 0.23/0.57 ! [X4: complex,Y3: complex] :
% 0.23/0.57 ( ( re @ ( minus_minus_complex @ X4 @ Y3 ) )
% 0.23/0.57 = ( minus_minus_real @ ( re @ X4 ) @ ( re @ Y3 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % minus_complex.simps(1)
% 0.23/0.58 thf(fact_9186_abs__Re__le__cmod,axiom,
% 0.23/0.58 ! [X4: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).
% 0.23/0.58
% 0.23/0.58 % abs_Re_le_cmod
% 0.23/0.58 thf(fact_9187_Re__csqrt,axiom,
% 0.23/0.58 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_csqrt
% 0.23/0.58 thf(fact_9188_one__natural_Orsp,axiom,
% 0.23/0.58 one_one_nat = one_one_nat ).
% 0.23/0.58
% 0.23/0.58 % one_natural.rsp
% 0.23/0.58 thf(fact_9189_cmod__plus__Re__le__0__iff,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 0.23/0.58 = ( ( re @ Z )
% 0.23/0.58 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % cmod_plus_Re_le_0_iff
% 0.23/0.58 thf(fact_9190_cos__n__Re__cis__pow__n,axiom,
% 0.23/0.58 ! [N: nat,A: real] :
% 0.23/0.58 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.23/0.58 = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % cos_n_Re_cis_pow_n
% 0.23/0.58 thf(fact_9191_csqrt_Ocode,axiom,
% 0.23/0.58 ( csqrt
% 0.23/0.58 = ( ^ [Z5: complex] :
% 0.23/0.58 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.58 @ ( times_times_real
% 0.23/0.58 @ ( if_real
% 0.23/0.58 @ ( ( im @ Z5 )
% 0.23/0.58 = zero_zero_real )
% 0.23/0.58 @ one_one_real
% 0.23/0.58 @ ( sgn_sgn_real @ ( im @ Z5 ) ) )
% 0.23/0.58 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % csqrt.code
% 0.23/0.58 thf(fact_9192_csqrt_Osimps_I2_J,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( im @ ( csqrt @ Z ) )
% 0.23/0.58 = ( times_times_real
% 0.23/0.58 @ ( if_real
% 0.23/0.58 @ ( ( im @ Z )
% 0.23/0.58 = zero_zero_real )
% 0.23/0.58 @ one_one_real
% 0.23/0.58 @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 0.23/0.58 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % csqrt.simps(2)
% 0.23/0.58 thf(fact_9193_integer__of__int__code,axiom,
% 0.23/0.58 ( code_integer_of_int
% 0.23/0.58 = ( ^ [K3: int] :
% 0.23/0.58 ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 0.23/0.58 @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.23/0.58 @ ( if_Code_integer
% 0.23/0.58 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.23/0.58 = zero_zero_int )
% 0.23/0.58 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % integer_of_int_code
% 0.23/0.58 thf(fact_9194_Im__i__times,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
% 0.23/0.58 = ( re @ Z ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_i_times
% 0.23/0.58 thf(fact_9195_Im__divide__of__real,axiom,
% 0.23/0.58 ! [Z: complex,R2: real] :
% 0.23/0.58 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 0.23/0.58 = ( divide_divide_real @ ( im @ Z ) @ R2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_divide_of_real
% 0.23/0.58 thf(fact_9196_Im__sgn,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( im @ ( sgn_sgn_complex @ Z ) )
% 0.23/0.58 = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_sgn
% 0.23/0.58 thf(fact_9197_Re__power__real,axiom,
% 0.23/0.58 ! [X4: complex,N: nat] :
% 0.23/0.58 ( ( ( im @ X4 )
% 0.23/0.58 = zero_zero_real )
% 0.23/0.58 => ( ( re @ ( power_power_complex @ X4 @ N ) )
% 0.23/0.58 = ( power_power_real @ ( re @ X4 ) @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_power_real
% 0.23/0.58 thf(fact_9198_Re__i__times,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
% 0.23/0.58 = ( uminus_uminus_real @ ( im @ Z ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_i_times
% 0.23/0.58 thf(fact_9199_Im__divide__numeral,axiom,
% 0.23/0.58 ! [Z: complex,W: num] :
% 0.23/0.58 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.23/0.58 = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_divide_numeral
% 0.23/0.58 thf(fact_9200_Im__divide__of__nat,axiom,
% 0.23/0.58 ! [Z: complex,N: nat] :
% 0.23/0.58 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.23/0.58 = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_divide_of_nat
% 0.23/0.58 thf(fact_9201_csqrt__of__real__nonneg,axiom,
% 0.23/0.58 ! [X4: complex] :
% 0.23/0.58 ( ( ( im @ X4 )
% 0.23/0.58 = zero_zero_real )
% 0.23/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X4 ) )
% 0.23/0.58 => ( ( csqrt @ X4 )
% 0.23/0.58 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X4 ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % csqrt_of_real_nonneg
% 0.23/0.58 thf(fact_9202_csqrt__minus,axiom,
% 0.23/0.58 ! [X4: complex] :
% 0.23/0.58 ( ( ( ord_less_real @ ( im @ X4 ) @ zero_zero_real )
% 0.23/0.58 | ( ( ( im @ X4 )
% 0.23/0.58 = zero_zero_real )
% 0.23/0.58 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X4 ) ) ) )
% 0.23/0.58 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X4 ) )
% 0.23/0.58 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X4 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % csqrt_minus
% 0.23/0.58 thf(fact_9203_csqrt__of__real__nonpos,axiom,
% 0.23/0.58 ! [X4: complex] :
% 0.23/0.58 ( ( ( im @ X4 )
% 0.23/0.58 = zero_zero_real )
% 0.23/0.58 => ( ( ord_less_eq_real @ ( re @ X4 ) @ zero_zero_real )
% 0.23/0.58 => ( ( csqrt @ X4 )
% 0.23/0.58 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X4 ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % csqrt_of_real_nonpos
% 0.23/0.58 thf(fact_9204_divide__integer_Oabs__eq,axiom,
% 0.23/0.58 ! [Xa: int,X4: int] :
% 0.23/0.58 ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 0.23/0.58 = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X4 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % divide_integer.abs_eq
% 0.23/0.58 thf(fact_9205_modulo__integer_Oabs__eq,axiom,
% 0.23/0.58 ! [Xa: int,X4: int] :
% 0.23/0.58 ( ( modulo364778990260209775nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 0.23/0.58 = ( code_integer_of_int @ ( modulo_modulo_int @ Xa @ X4 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % modulo_integer.abs_eq
% 0.23/0.58 thf(fact_9206_less__integer__code_I1_J,axiom,
% 0.23/0.58 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 0.23/0.58
% 0.23/0.58 % less_integer_code(1)
% 0.23/0.58 thf(fact_9207_less__integer_Oabs__eq,axiom,
% 0.23/0.58 ! [Xa: int,X4: int] :
% 0.23/0.58 ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 0.23/0.58 = ( ord_less_int @ Xa @ X4 ) ) ).
% 0.23/0.58
% 0.23/0.58 % less_integer.abs_eq
% 0.23/0.58 thf(fact_9208_abs__integer__code,axiom,
% 0.23/0.58 ( abs_abs_Code_integer
% 0.23/0.58 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % abs_integer_code
% 0.23/0.58 thf(fact_9209_imaginary__unit_Osimps_I2_J,axiom,
% 0.23/0.58 ( ( im @ imaginary_unit )
% 0.23/0.58 = one_one_real ) ).
% 0.23/0.58
% 0.23/0.58 % imaginary_unit.simps(2)
% 0.23/0.58 thf(fact_9210_plus__integer_Oabs__eq,axiom,
% 0.23/0.58 ! [Xa: int,X4: int] :
% 0.23/0.58 ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 0.23/0.58 = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X4 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % plus_integer.abs_eq
% 0.23/0.58 thf(fact_9211_times__integer_Oabs__eq,axiom,
% 0.23/0.58 ! [Xa: int,X4: int] :
% 0.23/0.58 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 0.23/0.58 = ( code_integer_of_int @ ( times_times_int @ Xa @ X4 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % times_integer.abs_eq
% 0.23/0.58 thf(fact_9212_minus__integer_Oabs__eq,axiom,
% 0.23/0.58 ! [Xa: int,X4: int] :
% 0.23/0.58 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 0.23/0.58 = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X4 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % minus_integer.abs_eq
% 0.23/0.58 thf(fact_9213_plus__complex_Osimps_I2_J,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( im @ ( plus_plus_complex @ X4 @ Y3 ) )
% 0.23/0.58 = ( plus_plus_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % plus_complex.simps(2)
% 0.23/0.58 thf(fact_9214_scaleR__complex_Osimps_I2_J,axiom,
% 0.23/0.58 ! [R2: real,X4: complex] :
% 0.23/0.58 ( ( im @ ( real_V2046097035970521341omplex @ R2 @ X4 ) )
% 0.23/0.58 = ( times_times_real @ R2 @ ( im @ X4 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % scaleR_complex.simps(2)
% 0.23/0.58 thf(fact_9215_minus__complex_Osimps_I2_J,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( im @ ( minus_minus_complex @ X4 @ Y3 ) )
% 0.23/0.58 = ( minus_minus_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % minus_complex.simps(2)
% 0.23/0.58 thf(fact_9216_abs__Im__le__cmod,axiom,
% 0.23/0.58 ! [X4: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).
% 0.23/0.58
% 0.23/0.58 % abs_Im_le_cmod
% 0.23/0.58 thf(fact_9217_times__complex_Osimps_I2_J,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( im @ ( times_times_complex @ X4 @ Y3 ) )
% 0.23/0.58 = ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y3 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % times_complex.simps(2)
% 0.23/0.58 thf(fact_9218_cmod__Im__le__iff,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( ( re @ X4 )
% 0.23/0.58 = ( re @ Y3 ) )
% 0.23/0.58 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 0.23/0.58 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X4 ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % cmod_Im_le_iff
% 0.23/0.58 thf(fact_9219_cmod__Re__le__iff,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( ( im @ X4 )
% 0.23/0.58 = ( im @ Y3 ) )
% 0.23/0.58 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 0.23/0.58 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X4 ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % cmod_Re_le_iff
% 0.23/0.58 thf(fact_9220_times__complex_Osimps_I1_J,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( re @ ( times_times_complex @ X4 @ Y3 ) )
% 0.23/0.58 = ( minus_minus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % times_complex.simps(1)
% 0.23/0.58 thf(fact_9221_plus__complex_Ocode,axiom,
% 0.23/0.58 ( plus_plus_complex
% 0.23/0.58 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) @ ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % plus_complex.code
% 0.23/0.58 thf(fact_9222_scaleR__complex_Ocode,axiom,
% 0.23/0.58 ( real_V2046097035970521341omplex
% 0.23/0.58 = ( ^ [R5: real,X: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X ) ) @ ( times_times_real @ R5 @ ( im @ X ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % scaleR_complex.code
% 0.23/0.58 thf(fact_9223_minus__complex_Ocode,axiom,
% 0.23/0.58 ( minus_minus_complex
% 0.23/0.58 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) @ ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % minus_complex.code
% 0.23/0.58 thf(fact_9224_csqrt__principal,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 0.23/0.58 | ( ( ( re @ ( csqrt @ Z ) )
% 0.23/0.58 = zero_zero_real )
% 0.23/0.58 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % csqrt_principal
% 0.23/0.58 thf(fact_9225_cmod__le,axiom,
% 0.23/0.58 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % cmod_le
% 0.23/0.58 thf(fact_9226_sin__n__Im__cis__pow__n,axiom,
% 0.23/0.58 ! [N: nat,A: real] :
% 0.23/0.58 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.23/0.58 = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sin_n_Im_cis_pow_n
% 0.23/0.58 thf(fact_9227_Re__exp,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( re @ ( exp_complex @ Z ) )
% 0.23/0.58 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_exp
% 0.23/0.58 thf(fact_9228_Im__exp,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( im @ ( exp_complex @ Z ) )
% 0.23/0.58 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_exp
% 0.23/0.58 thf(fact_9229_complex__eq,axiom,
% 0.23/0.58 ! [A: complex] :
% 0.23/0.58 ( A
% 0.23/0.58 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_eq
% 0.23/0.58 thf(fact_9230_times__complex_Ocode,axiom,
% 0.23/0.58 ( times_times_complex
% 0.23/0.58 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % times_complex.code
% 0.23/0.58 thf(fact_9231_exp__eq__polar,axiom,
% 0.23/0.58 ( exp_complex
% 0.23/0.58 = ( ^ [Z5: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z5 ) ) ) @ ( cis @ ( im @ Z5 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % exp_eq_polar
% 0.23/0.58 thf(fact_9232_cmod__power2,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.58 = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % cmod_power2
% 0.23/0.58 thf(fact_9233_Im__power2,axiom,
% 0.23/0.58 ! [X4: complex] :
% 0.23/0.58 ( ( im @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.58 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X4 ) ) @ ( im @ X4 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_power2
% 0.23/0.58 thf(fact_9234_Re__power2,axiom,
% 0.23/0.58 ! [X4: complex] :
% 0.23/0.58 ( ( re @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.58 = ( minus_minus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_power2
% 0.23/0.58 thf(fact_9235_complex__eq__0,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( Z = zero_zero_complex )
% 0.23/0.58 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.58 = zero_zero_real ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_eq_0
% 0.23/0.58 thf(fact_9236_norm__complex__def,axiom,
% 0.23/0.58 ( real_V1022390504157884413omplex
% 0.23/0.58 = ( ^ [Z5: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % norm_complex_def
% 0.23/0.58 thf(fact_9237_inverse__complex_Osimps_I1_J,axiom,
% 0.23/0.58 ! [X4: complex] :
% 0.23/0.58 ( ( re @ ( invers8013647133539491842omplex @ X4 ) )
% 0.23/0.58 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % inverse_complex.simps(1)
% 0.23/0.58 thf(fact_9238_complex__neq__0,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( Z != zero_zero_complex )
% 0.23/0.58 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_neq_0
% 0.23/0.58 thf(fact_9239_Re__divide,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( re @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
% 0.23/0.58 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_divide
% 0.23/0.58 thf(fact_9240_csqrt__unique,axiom,
% 0.23/0.58 ! [W: complex,Z: complex] :
% 0.23/0.58 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.23/0.58 = Z )
% 0.23/0.58 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 0.23/0.58 | ( ( ( re @ W )
% 0.23/0.58 = zero_zero_real )
% 0.23/0.58 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 0.23/0.58 => ( ( csqrt @ Z )
% 0.23/0.58 = W ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % csqrt_unique
% 0.23/0.58 thf(fact_9241_csqrt__square,axiom,
% 0.23/0.58 ! [B2: complex] :
% 0.23/0.58 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B2 ) )
% 0.23/0.58 | ( ( ( re @ B2 )
% 0.23/0.58 = zero_zero_real )
% 0.23/0.58 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B2 ) ) ) )
% 0.23/0.58 => ( ( csqrt @ ( power_power_complex @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.58 = B2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % csqrt_square
% 0.23/0.58 thf(fact_9242_inverse__complex_Osimps_I2_J,axiom,
% 0.23/0.58 ! [X4: complex] :
% 0.23/0.58 ( ( im @ ( invers8013647133539491842omplex @ X4 ) )
% 0.23/0.58 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % inverse_complex.simps(2)
% 0.23/0.58 thf(fact_9243_Im__divide,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( im @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
% 0.23/0.58 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_divide
% 0.23/0.58 thf(fact_9244_complex__abs__le__norm,axiom,
% 0.23/0.58 ! [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 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_abs_le_norm
% 0.23/0.58 thf(fact_9245_complex__unit__circle,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( Z != zero_zero_complex )
% 0.23/0.58 => ( ( 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 ) ) ) )
% 0.23/0.58 = one_one_real ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_unit_circle
% 0.23/0.58 thf(fact_9246_inverse__complex_Ocode,axiom,
% 0.23/0.58 ( invers8013647133539491842omplex
% 0.23/0.58 = ( ^ [X: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % inverse_complex.code
% 0.23/0.58 thf(fact_9247_Complex__divide,axiom,
% 0.23/0.58 ( divide1717551699836669952omplex
% 0.23/0.58 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Complex_divide
% 0.23/0.58 thf(fact_9248_Im__Reals__divide,axiom,
% 0.23/0.58 ! [R2: complex,Z: complex] :
% 0.23/0.58 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.23/0.58 => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 0.23/0.58 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_Reals_divide
% 0.23/0.58 thf(fact_9249_Re__Reals__divide,axiom,
% 0.23/0.58 ! [R2: complex,Z: complex] :
% 0.23/0.58 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.23/0.58 => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 0.23/0.58 = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_Reals_divide
% 0.23/0.58 thf(fact_9250_Re__divide__Reals,axiom,
% 0.23/0.58 ! [R2: complex,Z: complex] :
% 0.23/0.58 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.23/0.58 => ( ( re @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 0.23/0.58 = ( divide_divide_real @ ( re @ Z ) @ ( re @ R2 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_divide_Reals
% 0.23/0.58 thf(fact_9251_real__eq__imaginary__iff,axiom,
% 0.23/0.58 ! [Y3: complex,X4: complex] :
% 0.23/0.58 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 0.23/0.58 => ( ( member_complex @ X4 @ real_V2521375963428798218omplex )
% 0.23/0.58 => ( ( X4
% 0.23/0.58 = ( times_times_complex @ imaginary_unit @ Y3 ) )
% 0.23/0.58 = ( ( X4 = zero_zero_complex )
% 0.23/0.58 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % real_eq_imaginary_iff
% 0.23/0.58 thf(fact_9252_imaginary__eq__real__iff,axiom,
% 0.23/0.58 ! [Y3: complex,X4: complex] :
% 0.23/0.58 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 0.23/0.58 => ( ( member_complex @ X4 @ real_V2521375963428798218omplex )
% 0.23/0.58 => ( ( ( times_times_complex @ imaginary_unit @ Y3 )
% 0.23/0.58 = X4 )
% 0.23/0.58 = ( ( X4 = zero_zero_complex )
% 0.23/0.58 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % imaginary_eq_real_iff
% 0.23/0.58 thf(fact_9253_Im__divide__Reals,axiom,
% 0.23/0.58 ! [R2: complex,Z: complex] :
% 0.23/0.58 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.23/0.58 => ( ( im @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 0.23/0.58 = ( divide_divide_real @ ( im @ Z ) @ ( re @ R2 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_divide_Reals
% 0.23/0.58 thf(fact_9254_complex__diff__cnj,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 0.23/0.58 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_diff_cnj
% 0.23/0.58 thf(fact_9255_complex__mult__cnj,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 0.23/0.58 = ( 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 ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_mult_cnj
% 0.23/0.58 thf(fact_9256_integer__of__num_I3_J,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 0.23/0.58 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 0.23/0.58
% 0.23/0.58 % integer_of_num(3)
% 0.23/0.58 thf(fact_9257_complex__cnj__mult,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( cnj @ ( times_times_complex @ X4 @ Y3 ) )
% 0.23/0.58 = ( times_times_complex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_cnj_mult
% 0.23/0.58 thf(fact_9258_complex__cnj__divide,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( cnj @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
% 0.23/0.58 = ( divide1717551699836669952omplex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_cnj_divide
% 0.23/0.58 thf(fact_9259_complex__cnj__add,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( cnj @ ( plus_plus_complex @ X4 @ Y3 ) )
% 0.23/0.58 = ( plus_plus_complex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_cnj_add
% 0.23/0.58 thf(fact_9260_complex__cnj__diff,axiom,
% 0.23/0.58 ! [X4: complex,Y3: complex] :
% 0.23/0.58 ( ( cnj @ ( minus_minus_complex @ X4 @ Y3 ) )
% 0.23/0.58 = ( minus_minus_complex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_cnj_diff
% 0.23/0.58 thf(fact_9261_complex__In__mult__cnj__zero,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.23/0.58 = zero_zero_real ) ).
% 0.23/0.58
% 0.23/0.58 % complex_In_mult_cnj_zero
% 0.23/0.58 thf(fact_9262_Re__complex__div__eq__0,axiom,
% 0.23/0.58 ! [A: complex,B2: complex] :
% 0.23/0.58 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.23/0.58 = zero_zero_real )
% 0.23/0.58 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
% 0.23/0.58 = zero_zero_real ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_complex_div_eq_0
% 0.23/0.58 thf(fact_9263_Im__complex__div__eq__0,axiom,
% 0.23/0.58 ! [A: complex,B2: complex] :
% 0.23/0.58 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.23/0.58 = zero_zero_real )
% 0.23/0.58 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
% 0.23/0.58 = zero_zero_real ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_complex_div_eq_0
% 0.23/0.58 thf(fact_9264_complex__mod__sqrt__Re__mult__cnj,axiom,
% 0.23/0.58 ( real_V1022390504157884413omplex
% 0.23/0.58 = ( ^ [Z5: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z5 @ ( cnj @ Z5 ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_mod_sqrt_Re_mult_cnj
% 0.23/0.58 thf(fact_9265_integer__of__num__triv_I1_J,axiom,
% 0.23/0.58 ( ( code_integer_of_num @ one )
% 0.23/0.58 = one_one_Code_integer ) ).
% 0.23/0.58
% 0.23/0.58 % integer_of_num_triv(1)
% 0.23/0.58 thf(fact_9266_Re__complex__div__lt__0,axiom,
% 0.23/0.58 ! [A: complex,B2: complex] :
% 0.23/0.58 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 0.23/0.58 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_complex_div_lt_0
% 0.23/0.58 thf(fact_9267_Re__complex__div__gt__0,axiom,
% 0.23/0.58 ! [A: complex,B2: complex] :
% 0.23/0.58 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.23/0.58 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_complex_div_gt_0
% 0.23/0.58 thf(fact_9268_Re__complex__div__ge__0,axiom,
% 0.23/0.58 ! [A: complex,B2: complex] :
% 0.23/0.58 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.23/0.58 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_complex_div_ge_0
% 0.23/0.58 thf(fact_9269_Re__complex__div__le__0,axiom,
% 0.23/0.58 ! [A: complex,B2: complex] :
% 0.23/0.58 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 0.23/0.58 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 0.23/0.58
% 0.23/0.58 % Re_complex_div_le_0
% 0.23/0.58 thf(fact_9270_Im__complex__div__lt__0,axiom,
% 0.23/0.58 ! [A: complex,B2: complex] :
% 0.23/0.58 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 0.23/0.58 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_complex_div_lt_0
% 0.23/0.58 thf(fact_9271_Im__complex__div__gt__0,axiom,
% 0.23/0.58 ! [A: complex,B2: complex] :
% 0.23/0.58 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.23/0.58 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_complex_div_gt_0
% 0.23/0.58 thf(fact_9272_Im__complex__div__ge__0,axiom,
% 0.23/0.58 ! [A: complex,B2: complex] :
% 0.23/0.58 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.23/0.58 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_complex_div_ge_0
% 0.23/0.58 thf(fact_9273_Im__complex__div__le__0,axiom,
% 0.23/0.58 ! [A: complex,B2: complex] :
% 0.23/0.58 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 0.23/0.58 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 0.23/0.58
% 0.23/0.58 % Im_complex_div_le_0
% 0.23/0.58 thf(fact_9274_integer__of__num_I2_J,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 0.23/0.58 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % integer_of_num(2)
% 0.23/0.58 thf(fact_9275_complex__mod__mult__cnj,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.23/0.58 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_mod_mult_cnj
% 0.23/0.58 thf(fact_9276_complex__div__gt__0,axiom,
% 0.23/0.58 ! [A: complex,B2: complex] :
% 0.23/0.58 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.23/0.58 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) )
% 0.23/0.58 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.23/0.58 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_div_gt_0
% 0.23/0.58 thf(fact_9277_integer__of__num__triv_I2_J,axiom,
% 0.23/0.58 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 0.23/0.58 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % integer_of_num_triv(2)
% 0.23/0.58 thf(fact_9278_complex__norm__square,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.58 = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_norm_square
% 0.23/0.58 thf(fact_9279_complex__add__cnj,axiom,
% 0.23/0.58 ! [Z: complex] :
% 0.23/0.58 ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 0.23/0.58 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_add_cnj
% 0.23/0.58 thf(fact_9280_complex__div__cnj,axiom,
% 0.23/0.58 ( divide1717551699836669952omplex
% 0.23/0.58 = ( ^ [A4: complex,B3: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A4 @ ( cnj @ B3 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % complex_div_cnj
% 0.23/0.58 thf(fact_9281_cnj__add__mult__eq__Re,axiom,
% 0.23/0.58 ! [Z: complex,W: complex] :
% 0.23/0.58 ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 0.23/0.58 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % cnj_add_mult_eq_Re
% 0.23/0.58 thf(fact_9282_bit__cut__integer__def,axiom,
% 0.23/0.58 ( code_bit_cut_integer
% 0.23/0.58 = ( ^ [K3: code_integer] :
% 0.23/0.58 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.23/0.58 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % bit_cut_integer_def
% 0.23/0.58 thf(fact_9283_divmod__integer__def,axiom,
% 0.23/0.58 ( code_divmod_integer
% 0.23/0.58 = ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L ) @ ( modulo364778990260209775nteger @ K3 @ L ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % divmod_integer_def
% 0.23/0.58 thf(fact_9284_card__lessThan,axiom,
% 0.23/0.58 ! [U: nat] :
% 0.23/0.58 ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
% 0.23/0.58 = U ) ).
% 0.23/0.58
% 0.23/0.58 % card_lessThan
% 0.23/0.58 thf(fact_9285_card__Collect__less__nat,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( finite_card_nat
% 0.23/0.58 @ ( collect_nat
% 0.23/0.58 @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
% 0.23/0.58 = N ) ).
% 0.23/0.58
% 0.23/0.58 % card_Collect_less_nat
% 0.23/0.58 thf(fact_9286_card__atMost,axiom,
% 0.23/0.58 ! [U: nat] :
% 0.23/0.58 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 0.23/0.58 = ( suc @ U ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_atMost
% 0.23/0.58 thf(fact_9287_card__atLeastLessThan,axiom,
% 0.23/0.58 ! [L2: nat,U: nat] :
% 0.23/0.58 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
% 0.23/0.58 = ( minus_minus_nat @ U @ L2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_atLeastLessThan
% 0.23/0.58 thf(fact_9288_card__Collect__le__nat,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( finite_card_nat
% 0.23/0.58 @ ( collect_nat
% 0.23/0.58 @ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
% 0.23/0.58 = ( suc @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_Collect_le_nat
% 0.23/0.58 thf(fact_9289_card__atLeastAtMost,axiom,
% 0.23/0.58 ! [L2: nat,U: nat] :
% 0.23/0.58 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 0.23/0.58 = ( minus_minus_nat @ ( suc @ U ) @ L2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_atLeastAtMost
% 0.23/0.58 thf(fact_9290_card__atLeastLessThan__int,axiom,
% 0.23/0.58 ! [L2: int,U: int] :
% 0.23/0.58 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L2 @ U ) )
% 0.23/0.58 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_atLeastLessThan_int
% 0.23/0.58 thf(fact_9291_card__atLeastAtMost__int,axiom,
% 0.23/0.58 ! [L2: int,U: int] :
% 0.23/0.58 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 0.23/0.58 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L2 ) @ one_one_int ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_atLeastAtMost_int
% 0.23/0.58 thf(fact_9292_card__less__Suc2,axiom,
% 0.23/0.58 ! [M7: set_nat,I: nat] :
% 0.23/0.58 ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 0.23/0.58 => ( ( finite_card_nat
% 0.23/0.58 @ ( collect_nat
% 0.23/0.58 @ ^ [K3: nat] :
% 0.23/0.58 ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 0.23/0.58 & ( ord_less_nat @ K3 @ I ) ) ) )
% 0.23/0.58 = ( finite_card_nat
% 0.23/0.58 @ ( collect_nat
% 0.23/0.58 @ ^ [K3: nat] :
% 0.23/0.58 ( ( member_nat @ K3 @ M7 )
% 0.23/0.58 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_less_Suc2
% 0.23/0.58 thf(fact_9293_card__less__Suc,axiom,
% 0.23/0.58 ! [M7: set_nat,I: nat] :
% 0.23/0.58 ( ( member_nat @ zero_zero_nat @ M7 )
% 0.23/0.58 => ( ( suc
% 0.23/0.58 @ ( finite_card_nat
% 0.23/0.58 @ ( collect_nat
% 0.23/0.58 @ ^ [K3: nat] :
% 0.23/0.58 ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 0.23/0.58 & ( ord_less_nat @ K3 @ I ) ) ) ) )
% 0.23/0.58 = ( finite_card_nat
% 0.23/0.58 @ ( collect_nat
% 0.23/0.58 @ ^ [K3: nat] :
% 0.23/0.58 ( ( member_nat @ K3 @ M7 )
% 0.23/0.58 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_less_Suc
% 0.23/0.58 thf(fact_9294_card__less,axiom,
% 0.23/0.58 ! [M7: set_nat,I: nat] :
% 0.23/0.58 ( ( member_nat @ zero_zero_nat @ M7 )
% 0.23/0.58 => ( ( finite_card_nat
% 0.23/0.58 @ ( collect_nat
% 0.23/0.58 @ ^ [K3: nat] :
% 0.23/0.58 ( ( member_nat @ K3 @ M7 )
% 0.23/0.58 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
% 0.23/0.58 != zero_zero_nat ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_less
% 0.23/0.58 thf(fact_9295_card__atLeastZeroLessThan__int,axiom,
% 0.23/0.58 ! [U: int] :
% 0.23/0.58 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
% 0.23/0.58 = ( nat2 @ U ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_atLeastZeroLessThan_int
% 0.23/0.58 thf(fact_9296_subset__card__intvl__is__intvl,axiom,
% 0.23/0.58 ! [A3: set_nat,K: nat] :
% 0.23/0.58 ( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) )
% 0.23/0.58 => ( A3
% 0.23/0.58 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % subset_card_intvl_is_intvl
% 0.23/0.58 thf(fact_9297_subset__eq__atLeast0__lessThan__card,axiom,
% 0.23/0.58 ! [N3: set_nat,N: nat] :
% 0.23/0.58 ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.23/0.58 => ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % subset_eq_atLeast0_lessThan_card
% 0.23/0.58 thf(fact_9298_card__sum__le__nat__sum,axiom,
% 0.23/0.58 ! [S2: set_nat] :
% 0.23/0.58 ( ord_less_eq_nat
% 0.23/0.58 @ ( groups3542108847815614940at_nat
% 0.23/0.58 @ ^ [X: nat] : X
% 0.23/0.58 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 0.23/0.58 @ ( groups3542108847815614940at_nat
% 0.23/0.58 @ ^ [X: nat] : X
% 0.23/0.58 @ S2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_sum_le_nat_sum
% 0.23/0.58 thf(fact_9299_card__nth__roots,axiom,
% 0.23/0.58 ! [C: complex,N: nat] :
% 0.23/0.58 ( ( C != zero_zero_complex )
% 0.23/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.58 => ( ( finite_card_complex
% 0.23/0.58 @ ( collect_complex
% 0.23/0.58 @ ^ [Z5: complex] :
% 0.23/0.58 ( ( power_power_complex @ Z5 @ N )
% 0.23/0.58 = C ) ) )
% 0.23/0.58 = N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_nth_roots
% 0.23/0.58 thf(fact_9300_card__roots__unity__eq,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.58 => ( ( finite_card_complex
% 0.23/0.58 @ ( collect_complex
% 0.23/0.58 @ ^ [Z5: complex] :
% 0.23/0.58 ( ( power_power_complex @ Z5 @ N )
% 0.23/0.58 = one_one_complex ) ) )
% 0.23/0.58 = N ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_roots_unity_eq
% 0.23/0.58 thf(fact_9301_bit__cut__integer__code,axiom,
% 0.23/0.58 ( code_bit_cut_integer
% 0.23/0.58 = ( ^ [K3: code_integer] :
% 0.23/0.58 ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 0.23/0.58 @ ( produc9125791028180074456eger_o
% 0.23/0.58 @ ^ [R5: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
% 0.23/0.58 @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % bit_cut_integer_code
% 0.23/0.58 thf(fact_9302_divmod__abs__def,axiom,
% 0.23/0.58 ( code_divmod_abs
% 0.23/0.58 = ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % divmod_abs_def
% 0.23/0.58 thf(fact_9303_divmod__integer__code,axiom,
% 0.23/0.58 ( code_divmod_integer
% 0.23/0.58 = ( ^ [K3: code_integer,L: code_integer] :
% 0.23/0.58 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.23/0.58 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
% 0.23/0.58 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L )
% 0.23/0.58 @ ( produc6916734918728496179nteger
% 0.23/0.58 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S4 ) ) )
% 0.23/0.58 @ ( code_divmod_abs @ K3 @ L ) ) )
% 0.23/0.58 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 0.23/0.58 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 0.23/0.58 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L )
% 0.23/0.58 @ ( produc6916734918728496179nteger
% 0.23/0.58 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S4 ) ) )
% 0.23/0.58 @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % divmod_integer_code
% 0.23/0.58 thf(fact_9304_vebt__maxt_Opelims,axiom,
% 0.23/0.58 ! [X4: vEBT_VEBT,Y3: option_nat] :
% 0.23/0.58 ( ( ( vEBT_vebt_maxt @ X4 )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X4 )
% 0.23/0.58 => ( ! [A5: $o,B5: $o] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.58 => ( ( ( B5
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( some_nat @ one_one_nat ) ) )
% 0.23/0.58 & ( ~ B5
% 0.23/0.58 => ( ( A5
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( some_nat @ zero_zero_nat ) ) )
% 0.23/0.58 & ( ~ A5
% 0.23/0.58 => ( Y3 = none_nat ) ) ) ) )
% 0.23/0.58 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
% 0.23/0.58 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 0.23/0.58 => ( ( Y3 = none_nat )
% 0.23/0.58 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) ) ) )
% 0.23/0.58 => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
% 0.23/0.58 => ( ( Y3
% 0.23/0.58 = ( some_nat @ Ma2 ) )
% 0.23/0.58 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % vebt_maxt.pelims
% 0.23/0.58 thf(fact_9305_vebt__mint_Opelims,axiom,
% 0.23/0.58 ! [X4: vEBT_VEBT,Y3: option_nat] :
% 0.23/0.58 ( ( ( vEBT_vebt_mint @ X4 )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X4 )
% 0.23/0.58 => ( ! [A5: $o,B5: $o] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.58 => ( ( ( A5
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( some_nat @ zero_zero_nat ) ) )
% 0.23/0.58 & ( ~ A5
% 0.23/0.58 => ( ( B5
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( some_nat @ one_one_nat ) ) )
% 0.23/0.58 & ( ~ B5
% 0.23/0.58 => ( Y3 = none_nat ) ) ) ) )
% 0.23/0.58 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
% 0.23/0.58 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 0.23/0.58 => ( ( Y3 = none_nat )
% 0.23/0.58 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) ) ) )
% 0.23/0.58 => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
% 0.23/0.58 => ( ( Y3
% 0.23/0.58 = ( some_nat @ Mi2 ) )
% 0.23/0.58 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % vebt_mint.pelims
% 0.23/0.58 thf(fact_9306_prod__decode__aux_Osimps,axiom,
% 0.23/0.58 ( nat_prod_decode_aux
% 0.23/0.58 = ( ^ [K3: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K3 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K3 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M6 @ ( suc @ K3 ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % prod_decode_aux.simps
% 0.23/0.58 thf(fact_9307_nat_Odisc__eq__case_I1_J,axiom,
% 0.23/0.58 ! [Nat: nat] :
% 0.23/0.58 ( ( Nat = zero_zero_nat )
% 0.23/0.58 = ( case_nat_o @ $true
% 0.23/0.58 @ ^ [Uu: nat] : $false
% 0.23/0.58 @ Nat ) ) ).
% 0.23/0.58
% 0.23/0.58 % nat.disc_eq_case(1)
% 0.23/0.58 thf(fact_9308_nat_Odisc__eq__case_I2_J,axiom,
% 0.23/0.58 ! [Nat: nat] :
% 0.23/0.58 ( ( Nat != zero_zero_nat )
% 0.23/0.58 = ( case_nat_o @ $false
% 0.23/0.58 @ ^ [Uu: nat] : $true
% 0.23/0.58 @ Nat ) ) ).
% 0.23/0.58
% 0.23/0.58 % nat.disc_eq_case(2)
% 0.23/0.58 thf(fact_9309_less__eq__nat_Osimps_I2_J,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.23/0.58 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % less_eq_nat.simps(2)
% 0.23/0.58 thf(fact_9310_max__Suc2,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 0.23/0.58 = ( case_nat_nat @ ( suc @ N )
% 0.23/0.58 @ ^ [M2: nat] : ( suc @ ( ord_max_nat @ M2 @ N ) )
% 0.23/0.58 @ M ) ) ).
% 0.23/0.58
% 0.23/0.58 % max_Suc2
% 0.23/0.58 thf(fact_9311_max__Suc1,axiom,
% 0.23/0.58 ! [N: nat,M: nat] :
% 0.23/0.58 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 0.23/0.58 = ( case_nat_nat @ ( suc @ N )
% 0.23/0.58 @ ^ [M2: nat] : ( suc @ ( ord_max_nat @ N @ M2 ) )
% 0.23/0.58 @ M ) ) ).
% 0.23/0.58
% 0.23/0.58 % max_Suc1
% 0.23/0.58 thf(fact_9312_diff__Suc,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.23/0.58 = ( case_nat_nat @ zero_zero_nat
% 0.23/0.58 @ ^ [K3: nat] : K3
% 0.23/0.58 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % diff_Suc
% 0.23/0.58 thf(fact_9313_prod__decode__aux_Oelims,axiom,
% 0.23/0.58 ! [X4: nat,Xa: nat,Y3: product_prod_nat_nat] :
% 0.23/0.58 ( ( ( nat_prod_decode_aux @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( ( ord_less_eq_nat @ Xa @ X4 )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X4 @ Xa ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_eq_nat @ Xa @ X4 )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( nat_prod_decode_aux @ ( suc @ X4 ) @ ( minus_minus_nat @ Xa @ ( suc @ X4 ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % prod_decode_aux.elims
% 0.23/0.58 thf(fact_9314_pred__def,axiom,
% 0.23/0.58 ( pred
% 0.23/0.58 = ( case_nat_nat @ zero_zero_nat
% 0.23/0.58 @ ^ [X23: nat] : X23 ) ) ).
% 0.23/0.58
% 0.23/0.58 % pred_def
% 0.23/0.58 thf(fact_9315_floor__real__def,axiom,
% 0.23/0.58 ( archim6058952711729229775r_real
% 0.23/0.58 = ( ^ [X: real] :
% 0.23/0.58 ( the_int
% 0.23/0.58 @ ^ [Z5: int] :
% 0.23/0.58 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X )
% 0.23/0.58 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % floor_real_def
% 0.23/0.58 thf(fact_9316_floor__rat__def,axiom,
% 0.23/0.58 ( archim3151403230148437115or_rat
% 0.23/0.58 = ( ^ [X: rat] :
% 0.23/0.58 ( the_int
% 0.23/0.58 @ ^ [Z5: int] :
% 0.23/0.58 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X )
% 0.23/0.58 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % floor_rat_def
% 0.23/0.58 thf(fact_9317_drop__bit__numeral__minus__bit1,axiom,
% 0.23/0.58 ! [L2: num,K: num] :
% 0.23/0.58 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.23/0.58 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_numeral_minus_bit1
% 0.23/0.58 thf(fact_9318_Suc__0__mod__numeral,axiom,
% 0.23/0.58 ! [K: num] :
% 0.23/0.58 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.23/0.58 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Suc_0_mod_numeral
% 0.23/0.58 thf(fact_9319_drop__bit__nonnegative__int__iff,axiom,
% 0.23/0.58 ! [N: nat,K: int] :
% 0.23/0.58 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 0.23/0.58 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_nonnegative_int_iff
% 0.23/0.58 thf(fact_9320_drop__bit__negative__int__iff,axiom,
% 0.23/0.58 ! [N: nat,K: int] :
% 0.23/0.58 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 0.23/0.58 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_negative_int_iff
% 0.23/0.58 thf(fact_9321_drop__bit__minus__one,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.23/0.58 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_minus_one
% 0.23/0.58 thf(fact_9322_snd__divmod__nat,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
% 0.23/0.58 = ( modulo_modulo_nat @ M @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % snd_divmod_nat
% 0.23/0.58 thf(fact_9323_drop__bit__Suc__minus__bit0,axiom,
% 0.23/0.58 ! [N: nat,K: num] :
% 0.23/0.58 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.23/0.58 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_Suc_minus_bit0
% 0.23/0.58 thf(fact_9324_drop__bit__numeral__minus__bit0,axiom,
% 0.23/0.58 ! [L2: num,K: num] :
% 0.23/0.58 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.23/0.58 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_numeral_minus_bit0
% 0.23/0.58 thf(fact_9325_drop__bit__Suc__minus__bit1,axiom,
% 0.23/0.58 ! [N: nat,K: num] :
% 0.23/0.58 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.23/0.58 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_Suc_minus_bit1
% 0.23/0.58 thf(fact_9326_abs__rat__def,axiom,
% 0.23/0.58 ( abs_abs_rat
% 0.23/0.58 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % abs_rat_def
% 0.23/0.58 thf(fact_9327_sgn__rat__def,axiom,
% 0.23/0.58 ( sgn_sgn_rat
% 0.23/0.58 = ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sgn_rat_def
% 0.23/0.58 thf(fact_9328_obtain__pos__sum,axiom,
% 0.23/0.58 ! [R2: rat] :
% 0.23/0.58 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.23/0.58 => ~ ! [S3: rat] :
% 0.23/0.58 ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 0.23/0.58 => ! [T2: rat] :
% 0.23/0.58 ( ( ord_less_rat @ zero_zero_rat @ T2 )
% 0.23/0.58 => ( R2
% 0.23/0.58 != ( plus_plus_rat @ S3 @ T2 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % obtain_pos_sum
% 0.23/0.58 thf(fact_9329_less__eq__rat__def,axiom,
% 0.23/0.58 ( ord_less_eq_rat
% 0.23/0.58 = ( ^ [X: rat,Y: rat] :
% 0.23/0.58 ( ( ord_less_rat @ X @ Y )
% 0.23/0.58 | ( X = Y ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % less_eq_rat_def
% 0.23/0.58 thf(fact_9330_drop__bit__push__bit__int,axiom,
% 0.23/0.58 ! [M: nat,N: nat,K: int] :
% 0.23/0.58 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.23/0.58 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_push_bit_int
% 0.23/0.58 thf(fact_9331_drop__bit__int__def,axiom,
% 0.23/0.58 ( bit_se8568078237143864401it_int
% 0.23/0.58 = ( ^ [N2: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_int_def
% 0.23/0.58 thf(fact_9332_rat__inverse__code,axiom,
% 0.23/0.58 ! [P2: rat] :
% 0.23/0.58 ( ( quotient_of @ ( inverse_inverse_rat @ P2 ) )
% 0.23/0.58 = ( produc4245557441103728435nt_int
% 0.23/0.58 @ ^ [A4: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A4 ) @ B3 ) @ ( abs_abs_int @ A4 ) ) )
% 0.23/0.58 @ ( quotient_of @ P2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % rat_inverse_code
% 0.23/0.58 thf(fact_9333_normalize__negative,axiom,
% 0.23/0.58 ! [Q2: int,P2: int] :
% 0.23/0.58 ( ( ord_less_int @ Q2 @ zero_zero_int )
% 0.23/0.58 => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 0.23/0.58 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P2 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % normalize_negative
% 0.23/0.58 thf(fact_9334_Suc__0__div__numeral,axiom,
% 0.23/0.58 ! [K: num] :
% 0.23/0.58 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.23/0.58 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Suc_0_div_numeral
% 0.23/0.58 thf(fact_9335_snd__divmod__integer,axiom,
% 0.23/0.58 ! [K: code_integer,L2: code_integer] :
% 0.23/0.58 ( ( produc6174133586879617921nteger @ ( code_divmod_integer @ K @ L2 ) )
% 0.23/0.58 = ( modulo364778990260209775nteger @ K @ L2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % snd_divmod_integer
% 0.23/0.58 thf(fact_9336_snd__divmod__abs,axiom,
% 0.23/0.58 ! [K: code_integer,L2: code_integer] :
% 0.23/0.58 ( ( produc6174133586879617921nteger @ ( code_divmod_abs @ K @ L2 ) )
% 0.23/0.58 = ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % snd_divmod_abs
% 0.23/0.58 thf(fact_9337_drop__bit__of__Suc__0,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.23/0.58 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_of_Suc_0
% 0.23/0.58 thf(fact_9338_fst__divmod__nat,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 0.23/0.58 = ( divide_divide_nat @ M @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % fst_divmod_nat
% 0.23/0.58 thf(fact_9339_quotient__of__denom__pos_H,axiom,
% 0.23/0.58 ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % quotient_of_denom_pos'
% 0.23/0.58 thf(fact_9340_diff__rat__def,axiom,
% 0.23/0.58 ( minus_minus_rat
% 0.23/0.58 = ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % diff_rat_def
% 0.23/0.58 thf(fact_9341_divide__rat__def,axiom,
% 0.23/0.58 ( divide_divide_rat
% 0.23/0.58 = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % divide_rat_def
% 0.23/0.58 thf(fact_9342_rat__divide__code,axiom,
% 0.23/0.58 ! [P2: rat,Q2: rat] :
% 0.23/0.58 ( ( quotient_of @ ( divide_divide_rat @ P2 @ Q2 ) )
% 0.23/0.58 = ( produc4245557441103728435nt_int
% 0.23/0.58 @ ^ [A4: int,C2: int] :
% 0.23/0.58 ( produc4245557441103728435nt_int
% 0.23/0.58 @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) ) )
% 0.23/0.58 @ ( quotient_of @ Q2 ) )
% 0.23/0.58 @ ( quotient_of @ P2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % rat_divide_code
% 0.23/0.58 thf(fact_9343_rat__times__code,axiom,
% 0.23/0.58 ! [P2: rat,Q2: rat] :
% 0.23/0.58 ( ( quotient_of @ ( times_times_rat @ P2 @ Q2 ) )
% 0.23/0.58 = ( produc4245557441103728435nt_int
% 0.23/0.58 @ ^ [A4: int,C2: int] :
% 0.23/0.58 ( produc4245557441103728435nt_int
% 0.23/0.58 @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ B3 ) @ ( times_times_int @ C2 @ D2 ) ) )
% 0.23/0.58 @ ( quotient_of @ Q2 ) )
% 0.23/0.58 @ ( quotient_of @ P2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % rat_times_code
% 0.23/0.58 thf(fact_9344_drop__bit__nat__eq,axiom,
% 0.23/0.58 ! [N: nat,K: int] :
% 0.23/0.58 ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 0.23/0.58 = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_nat_eq
% 0.23/0.58 thf(fact_9345_quotient__of__div,axiom,
% 0.23/0.58 ! [R2: rat,N: int,D: int] :
% 0.23/0.58 ( ( ( quotient_of @ R2 )
% 0.23/0.58 = ( product_Pair_int_int @ N @ D ) )
% 0.23/0.58 => ( R2
% 0.23/0.58 = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % quotient_of_div
% 0.23/0.58 thf(fact_9346_rat__plus__code,axiom,
% 0.23/0.58 ! [P2: rat,Q2: rat] :
% 0.23/0.58 ( ( quotient_of @ ( plus_plus_rat @ P2 @ Q2 ) )
% 0.23/0.58 = ( produc4245557441103728435nt_int
% 0.23/0.58 @ ^ [A4: int,C2: int] :
% 0.23/0.58 ( produc4245557441103728435nt_int
% 0.23/0.58 @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B3 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
% 0.23/0.58 @ ( quotient_of @ Q2 ) )
% 0.23/0.58 @ ( quotient_of @ P2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % rat_plus_code
% 0.23/0.58 thf(fact_9347_rat__minus__code,axiom,
% 0.23/0.58 ! [P2: rat,Q2: rat] :
% 0.23/0.58 ( ( quotient_of @ ( minus_minus_rat @ P2 @ Q2 ) )
% 0.23/0.58 = ( produc4245557441103728435nt_int
% 0.23/0.58 @ ^ [A4: int,C2: int] :
% 0.23/0.58 ( produc4245557441103728435nt_int
% 0.23/0.58 @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B3 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
% 0.23/0.58 @ ( quotient_of @ Q2 ) )
% 0.23/0.58 @ ( quotient_of @ P2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % rat_minus_code
% 0.23/0.58 thf(fact_9348_quotient__of__denom__pos,axiom,
% 0.23/0.58 ! [R2: rat,P2: int,Q2: int] :
% 0.23/0.58 ( ( ( quotient_of @ R2 )
% 0.23/0.58 = ( product_Pair_int_int @ P2 @ Q2 ) )
% 0.23/0.58 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % quotient_of_denom_pos
% 0.23/0.58 thf(fact_9349_normalize__denom__pos,axiom,
% 0.23/0.58 ! [R2: product_prod_int_int,P2: int,Q2: int] :
% 0.23/0.58 ( ( ( normalize @ R2 )
% 0.23/0.58 = ( product_Pair_int_int @ P2 @ Q2 ) )
% 0.23/0.58 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % normalize_denom_pos
% 0.23/0.58 thf(fact_9350_normalize__crossproduct,axiom,
% 0.23/0.58 ! [Q2: int,S: int,P2: int,R2: int] :
% 0.23/0.58 ( ( Q2 != zero_zero_int )
% 0.23/0.58 => ( ( S != zero_zero_int )
% 0.23/0.58 => ( ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 0.23/0.58 = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
% 0.23/0.58 => ( ( times_times_int @ P2 @ S )
% 0.23/0.58 = ( times_times_int @ R2 @ Q2 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % normalize_crossproduct
% 0.23/0.58 thf(fact_9351_drop__bit__nat__def,axiom,
% 0.23/0.58 ( bit_se8570568707652914677it_nat
% 0.23/0.58 = ( ^ [N2: nat,M6: nat] : ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_bit_nat_def
% 0.23/0.58 thf(fact_9352_one__mod__minus__numeral,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % one_mod_minus_numeral
% 0.23/0.58 thf(fact_9353_minus__one__mod__numeral,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.58 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % minus_one_mod_numeral
% 0.23/0.58 thf(fact_9354_numeral__mod__minus__numeral,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % numeral_mod_minus_numeral
% 0.23/0.58 thf(fact_9355_fst__divmod__integer,axiom,
% 0.23/0.58 ! [K: code_integer,L2: code_integer] :
% 0.23/0.58 ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L2 ) )
% 0.23/0.58 = ( divide6298287555418463151nteger @ K @ L2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % fst_divmod_integer
% 0.23/0.58 thf(fact_9356_fst__divmod__abs,axiom,
% 0.23/0.58 ! [K: code_integer,L2: code_integer] :
% 0.23/0.58 ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L2 ) )
% 0.23/0.58 = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % fst_divmod_abs
% 0.23/0.58 thf(fact_9357_minus__numeral__mod__numeral,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.58 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % minus_numeral_mod_numeral
% 0.23/0.58 thf(fact_9358_Divides_Oadjust__mod__def,axiom,
% 0.23/0.58 ( adjust_mod
% 0.23/0.58 = ( ^ [L: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L @ R5 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Divides.adjust_mod_def
% 0.23/0.58 thf(fact_9359_bezw_Osimps,axiom,
% 0.23/0.58 ( bezw
% 0.23/0.58 = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % bezw.simps
% 0.23/0.58 thf(fact_9360_bezw_Oelims,axiom,
% 0.23/0.58 ! [X4: nat,Xa: nat,Y3: product_prod_int_int] :
% 0.23/0.58 ( ( ( bezw @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( ( Xa = zero_zero_nat )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.23/0.58 & ( ( Xa != zero_zero_nat )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Xa ) ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % bezw.elims
% 0.23/0.58 thf(fact_9361_bezw__non__0,axiom,
% 0.23/0.58 ! [Y3: nat,X4: nat] :
% 0.23/0.58 ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 0.23/0.58 => ( ( bezw @ X4 @ Y3 )
% 0.23/0.58 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % bezw_non_0
% 0.23/0.58 thf(fact_9362_bezw_Opelims,axiom,
% 0.23/0.58 ! [X4: nat,Xa: nat,Y3: product_prod_int_int] :
% 0.23/0.58 ( ( ( bezw @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
% 0.23/0.58 => ~ ( ( ( ( Xa = zero_zero_nat )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.23/0.58 & ( ( Xa != zero_zero_nat )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Xa ) ) ) ) ) ) ) )
% 0.23/0.58 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % bezw.pelims
% 0.23/0.58 thf(fact_9363_normalize__def,axiom,
% 0.23/0.58 ( normalize
% 0.23/0.58 = ( ^ [P3: product_prod_int_int] :
% 0.23/0.58 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P3 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) )
% 0.23/0.58 @ ( if_Pro3027730157355071871nt_int
% 0.23/0.58 @ ( ( product_snd_int_int @ P3 )
% 0.23/0.58 = zero_zero_int )
% 0.23/0.58 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.23/0.58 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % normalize_def
% 0.23/0.58 thf(fact_9364_Frct__code__post_I5_J,axiom,
% 0.23/0.58 ! [K: num] :
% 0.23/0.58 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 0.23/0.58 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Frct_code_post(5)
% 0.23/0.58 thf(fact_9365_gcd__pos__int,axiom,
% 0.23/0.58 ! [M: int,N: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 0.23/0.58 = ( ( M != zero_zero_int )
% 0.23/0.58 | ( N != zero_zero_int ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_pos_int
% 0.23/0.58 thf(fact_9366_gcd__red__int,axiom,
% 0.23/0.58 ( gcd_gcd_int
% 0.23/0.58 = ( ^ [X: int,Y: int] : ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_red_int
% 0.23/0.58 thf(fact_9367_bezout__int,axiom,
% 0.23/0.58 ! [X4: int,Y3: int] :
% 0.23/0.58 ? [U3: int,V3: int] :
% 0.23/0.58 ( ( plus_plus_int @ ( times_times_int @ U3 @ X4 ) @ ( times_times_int @ V3 @ Y3 ) )
% 0.23/0.58 = ( gcd_gcd_int @ X4 @ Y3 ) ) ).
% 0.23/0.58
% 0.23/0.58 % bezout_int
% 0.23/0.58 thf(fact_9368_gcd__mult__distrib__int,axiom,
% 0.23/0.58 ! [K: int,M: int,N: int] :
% 0.23/0.58 ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 0.23/0.58 = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_mult_distrib_int
% 0.23/0.58 thf(fact_9369_gcd__le1__int,axiom,
% 0.23/0.58 ! [A: int,B2: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ A )
% 0.23/0.58 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ A ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_le1_int
% 0.23/0.58 thf(fact_9370_gcd__le2__int,axiom,
% 0.23/0.58 ! [B2: int,A: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.58 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ B2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_le2_int
% 0.23/0.58 thf(fact_9371_gcd__non__0__int,axiom,
% 0.23/0.58 ! [Y3: int,X4: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ Y3 )
% 0.23/0.58 => ( ( gcd_gcd_int @ X4 @ Y3 )
% 0.23/0.58 = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_non_0_int
% 0.23/0.58 thf(fact_9372_gcd__code__int,axiom,
% 0.23/0.58 ( gcd_gcd_int
% 0.23/0.58 = ( ^ [K3: int,L: int] : ( abs_abs_int @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_code_int
% 0.23/0.58 thf(fact_9373_Frct__code__post_I6_J,axiom,
% 0.23/0.58 ! [K: num,L2: num] :
% 0.23/0.58 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
% 0.23/0.58 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Frct_code_post(6)
% 0.23/0.58 thf(fact_9374_prod__decode__aux_Opelims,axiom,
% 0.23/0.58 ! [X4: nat,Xa: nat,Y3: product_prod_nat_nat] :
% 0.23/0.58 ( ( ( nat_prod_decode_aux @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
% 0.23/0.58 => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X4 )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X4 @ Xa ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_eq_nat @ Xa @ X4 )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( nat_prod_decode_aux @ ( suc @ X4 ) @ ( minus_minus_nat @ Xa @ ( suc @ X4 ) ) ) ) ) )
% 0.23/0.58 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % prod_decode_aux.pelims
% 0.23/0.58 thf(fact_9375_finite__enumerate,axiom,
% 0.23/0.58 ! [S2: set_nat] :
% 0.23/0.58 ( ( finite_finite_nat @ S2 )
% 0.23/0.58 => ? [R3: nat > nat] :
% 0.23/0.58 ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
% 0.23/0.58 & ! [N6: nat] :
% 0.23/0.58 ( ( ord_less_nat @ N6 @ ( finite_card_nat @ S2 ) )
% 0.23/0.58 => ( member_nat @ ( R3 @ N6 ) @ S2 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % finite_enumerate
% 0.23/0.58 thf(fact_9376_gcd__1__nat,axiom,
% 0.23/0.58 ! [M: nat] :
% 0.23/0.58 ( ( gcd_gcd_nat @ M @ one_one_nat )
% 0.23/0.58 = one_one_nat ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_1_nat
% 0.23/0.58 thf(fact_9377_gcd__Suc__0,axiom,
% 0.23/0.58 ! [M: nat] :
% 0.23/0.58 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.23/0.58 = ( suc @ zero_zero_nat ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_Suc_0
% 0.23/0.58 thf(fact_9378_gcd__pos__nat,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 0.23/0.58 = ( ( M != zero_zero_nat )
% 0.23/0.58 | ( N != zero_zero_nat ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_pos_nat
% 0.23/0.58 thf(fact_9379_gcd__red__nat,axiom,
% 0.23/0.58 ( gcd_gcd_nat
% 0.23/0.58 = ( ^ [X: nat,Y: nat] : ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_red_nat
% 0.23/0.58 thf(fact_9380_gcd__mult__distrib__nat,axiom,
% 0.23/0.58 ! [K: nat,M: nat,N: nat] :
% 0.23/0.58 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 0.23/0.58 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_mult_distrib_nat
% 0.23/0.58 thf(fact_9381_gcd__le2__nat,axiom,
% 0.23/0.58 ! [B2: nat,A: nat] :
% 0.23/0.58 ( ( B2 != zero_zero_nat )
% 0.23/0.58 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ B2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_le2_nat
% 0.23/0.58 thf(fact_9382_gcd__le1__nat,axiom,
% 0.23/0.58 ! [A: nat,B2: nat] :
% 0.23/0.58 ( ( A != zero_zero_nat )
% 0.23/0.58 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ A ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_le1_nat
% 0.23/0.58 thf(fact_9383_gcd__diff2__nat,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.58 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 0.23/0.58 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_diff2_nat
% 0.23/0.58 thf(fact_9384_gcd__diff1__nat,axiom,
% 0.23/0.58 ! [N: nat,M: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ N @ M )
% 0.23/0.58 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 0.23/0.58 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_diff1_nat
% 0.23/0.58 thf(fact_9385_gcd__non__0__nat,axiom,
% 0.23/0.58 ! [Y3: nat,X4: nat] :
% 0.23/0.58 ( ( Y3 != zero_zero_nat )
% 0.23/0.58 => ( ( gcd_gcd_nat @ X4 @ Y3 )
% 0.23/0.58 = ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_non_0_nat
% 0.23/0.58 thf(fact_9386_gcd__nat_Osimps,axiom,
% 0.23/0.58 ( gcd_gcd_nat
% 0.23/0.58 = ( ^ [X: nat,Y: nat] : ( if_nat @ ( Y = zero_zero_nat ) @ X @ ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_nat.simps
% 0.23/0.58 thf(fact_9387_gcd__nat_Oelims,axiom,
% 0.23/0.58 ! [X4: nat,Xa: nat,Y3: nat] :
% 0.23/0.58 ( ( ( gcd_gcd_nat @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( ( Xa = zero_zero_nat )
% 0.23/0.58 => ( Y3 = X4 ) )
% 0.23/0.58 & ( ( Xa != zero_zero_nat )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_nat.elims
% 0.23/0.58 thf(fact_9388_bezout__nat,axiom,
% 0.23/0.58 ! [A: nat,B2: nat] :
% 0.23/0.58 ( ( A != zero_zero_nat )
% 0.23/0.58 => ? [X3: nat,Y4: nat] :
% 0.23/0.58 ( ( times_times_nat @ A @ X3 )
% 0.23/0.58 = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % bezout_nat
% 0.23/0.58 thf(fact_9389_bezout__gcd__nat_H,axiom,
% 0.23/0.58 ! [B2: nat,A: nat] :
% 0.23/0.58 ? [X3: nat,Y4: nat] :
% 0.23/0.58 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( times_times_nat @ A @ X3 ) )
% 0.23/0.58 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B2 @ Y4 ) )
% 0.23/0.58 = ( gcd_gcd_nat @ A @ B2 ) ) )
% 0.23/0.58 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B2 @ X3 ) )
% 0.23/0.58 & ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.23/0.58 = ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % bezout_gcd_nat'
% 0.23/0.58 thf(fact_9390_gcd__code__integer,axiom,
% 0.23/0.58 ( gcd_gcd_Code_integer
% 0.23/0.58 = ( ^ [K3: code_integer,L: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_code_integer
% 0.23/0.58 thf(fact_9391_bezw__aux,axiom,
% 0.23/0.58 ! [X4: nat,Y3: nat] :
% 0.23/0.58 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X4 @ Y3 ) )
% 0.23/0.58 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X4 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X4 ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X4 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % bezw_aux
% 0.23/0.58 thf(fact_9392_nat__descend__induct,axiom,
% 0.23/0.58 ! [N: nat,P: nat > $o,M: nat] :
% 0.23/0.58 ( ! [K2: nat] :
% 0.23/0.58 ( ( ord_less_nat @ N @ K2 )
% 0.23/0.58 => ( P @ K2 ) )
% 0.23/0.58 => ( ! [K2: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ K2 @ N )
% 0.23/0.58 => ( ! [I4: nat] :
% 0.23/0.58 ( ( ord_less_nat @ K2 @ I4 )
% 0.23/0.58 => ( P @ I4 ) )
% 0.23/0.58 => ( P @ K2 ) ) )
% 0.23/0.58 => ( P @ M ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % nat_descend_induct
% 0.23/0.58 thf(fact_9393_gcd__nat_Opelims,axiom,
% 0.23/0.58 ! [X4: nat,Xa: nat,Y3: nat] :
% 0.23/0.58 ( ( ( gcd_gcd_nat @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
% 0.23/0.58 => ~ ( ( ( ( Xa = zero_zero_nat )
% 0.23/0.58 => ( Y3 = X4 ) )
% 0.23/0.58 & ( ( Xa != zero_zero_nat )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) ) )
% 0.23/0.58 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_nat.pelims
% 0.23/0.58 thf(fact_9394_divmod__integer__eq__cases,axiom,
% 0.23/0.58 ( code_divmod_integer
% 0.23/0.58 = ( ^ [K3: code_integer,L: code_integer] :
% 0.23/0.58 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.23/0.58 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 0.23/0.58 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
% 0.23/0.58 @ ( if_Pro6119634080678213985nteger
% 0.23/0.58 @ ( ( sgn_sgn_Code_integer @ K3 )
% 0.23/0.58 = ( sgn_sgn_Code_integer @ L ) )
% 0.23/0.58 @ ( code_divmod_abs @ K3 @ L )
% 0.23/0.58 @ ( produc6916734918728496179nteger
% 0.23/0.58 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S4 ) ) )
% 0.23/0.58 @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % divmod_integer_eq_cases
% 0.23/0.58 thf(fact_9395_card__greaterThanLessThan__int,axiom,
% 0.23/0.58 ! [L2: int,U: int] :
% 0.23/0.58 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U ) )
% 0.23/0.58 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_greaterThanLessThan_int
% 0.23/0.58 thf(fact_9396_xor__minus__numerals_I2_J,axiom,
% 0.23/0.58 ! [K: int,N: num] :
% 0.23/0.58 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % xor_minus_numerals(2)
% 0.23/0.58 thf(fact_9397_xor__minus__numerals_I1_J,axiom,
% 0.23/0.58 ! [N: num,K: int] :
% 0.23/0.58 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 0.23/0.58 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % xor_minus_numerals(1)
% 0.23/0.58 thf(fact_9398_finite__greaterThanLessThan__int,axiom,
% 0.23/0.58 ! [L2: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L2 @ U ) ) ).
% 0.23/0.58
% 0.23/0.58 % finite_greaterThanLessThan_int
% 0.23/0.58 thf(fact_9399_card_Ocomp__fun__commute__on,axiom,
% 0.23/0.58 ( ( comp_nat_nat_nat @ suc @ suc )
% 0.23/0.58 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 0.23/0.58
% 0.23/0.58 % card.comp_fun_commute_on
% 0.23/0.58 thf(fact_9400_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 0.23/0.58 ! [L2: int,U: int] :
% 0.23/0.58 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 0.23/0.58 = ( set_or5832277885323065728an_int @ L2 @ U ) ) ).
% 0.23/0.58
% 0.23/0.58 % atLeastPlusOneLessThan_greaterThanLessThan_int
% 0.23/0.58 thf(fact_9401_sub__BitM__One__eq,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 0.23/0.58 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sub_BitM_One_eq
% 0.23/0.58 thf(fact_9402_finite__greaterThanLessThan,axiom,
% 0.23/0.58 ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).
% 0.23/0.58
% 0.23/0.58 % finite_greaterThanLessThan
% 0.23/0.58 thf(fact_9403_card__greaterThanLessThan,axiom,
% 0.23/0.58 ! [L2: nat,U: nat] :
% 0.23/0.58 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
% 0.23/0.58 = ( minus_minus_nat @ U @ ( suc @ L2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_greaterThanLessThan
% 0.23/0.58 thf(fact_9404_atLeastSucLessThan__greaterThanLessThan,axiom,
% 0.23/0.58 ! [L2: nat,U: nat] :
% 0.23/0.58 ( ( set_or4665077453230672383an_nat @ ( suc @ L2 ) @ U )
% 0.23/0.58 = ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).
% 0.23/0.58
% 0.23/0.58 % atLeastSucLessThan_greaterThanLessThan
% 0.23/0.58 thf(fact_9405_tanh__real__bounds,axiom,
% 0.23/0.58 ! [X4: real] : ( member_real @ ( tanh_real @ X4 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 0.23/0.58
% 0.23/0.58 % tanh_real_bounds
% 0.23/0.58 thf(fact_9406_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.23/0.58 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 0.23/0.58 @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 0.23/0.58 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).
% 0.23/0.58
% 0.23/0.58 % max_nat.semilattice_neutr_order_axioms
% 0.23/0.58 thf(fact_9407_Suc__funpow,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( compow_nat_nat @ N @ suc )
% 0.23/0.58 = ( plus_plus_nat @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % Suc_funpow
% 0.23/0.58 thf(fact_9408_int__of__integer__code,axiom,
% 0.23/0.58 ( code_int_of_integer
% 0.23/0.58 = ( ^ [K3: code_integer] :
% 0.23/0.58 ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 0.23/0.58 @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.23/0.58 @ ( produc1553301316500091796er_int
% 0.23/0.58 @ ^ [L: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
% 0.23/0.58 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % int_of_integer_code
% 0.23/0.58 thf(fact_9409_plus__integer_Orep__eq,axiom,
% 0.23/0.58 ! [X4: code_integer,Xa: code_integer] :
% 0.23/0.58 ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X4 @ Xa ) )
% 0.23/0.58 = ( plus_plus_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % plus_integer.rep_eq
% 0.23/0.58 thf(fact_9410_times__integer_Orep__eq,axiom,
% 0.23/0.58 ! [X4: code_integer,Xa: code_integer] :
% 0.23/0.58 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X4 @ Xa ) )
% 0.23/0.58 = ( times_times_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % times_integer.rep_eq
% 0.23/0.58 thf(fact_9411_minus__integer_Orep__eq,axiom,
% 0.23/0.58 ! [X4: code_integer,Xa: code_integer] :
% 0.23/0.58 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X4 @ Xa ) )
% 0.23/0.58 = ( minus_minus_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % minus_integer.rep_eq
% 0.23/0.58 thf(fact_9412_divide__integer_Orep__eq,axiom,
% 0.23/0.58 ! [X4: code_integer,Xa: code_integer] :
% 0.23/0.58 ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X4 @ Xa ) )
% 0.23/0.58 = ( divide_divide_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % divide_integer.rep_eq
% 0.23/0.58 thf(fact_9413_modulo__integer_Orep__eq,axiom,
% 0.23/0.58 ! [X4: code_integer,Xa: code_integer] :
% 0.23/0.58 ( ( code_int_of_integer @ ( modulo364778990260209775nteger @ X4 @ Xa ) )
% 0.23/0.58 = ( modulo_modulo_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % modulo_integer.rep_eq
% 0.23/0.58 thf(fact_9414_less__integer_Orep__eq,axiom,
% 0.23/0.58 ( ord_le6747313008572928689nteger
% 0.23/0.58 = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % less_integer.rep_eq
% 0.23/0.58 thf(fact_9415_integer__less__iff,axiom,
% 0.23/0.58 ( ord_le6747313008572928689nteger
% 0.23/0.58 = ( ^ [K3: code_integer,L: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % integer_less_iff
% 0.23/0.58 thf(fact_9416_times__int_Oabs__eq,axiom,
% 0.23/0.58 ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
% 0.23/0.58 ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
% 0.23/0.58 = ( abs_Integ
% 0.23/0.58 @ ( produc27273713700761075at_nat
% 0.23/0.58 @ ^ [X: nat,Y: nat] :
% 0.23/0.58 ( produc2626176000494625587at_nat
% 0.23/0.58 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) )
% 0.23/0.58 @ Xa
% 0.23/0.58 @ X4 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % times_int.abs_eq
% 0.23/0.58 thf(fact_9417_Gcd__remove0__nat,axiom,
% 0.23/0.58 ! [M7: set_nat] :
% 0.23/0.58 ( ( finite_finite_nat @ M7 )
% 0.23/0.58 => ( ( gcd_Gcd_nat @ M7 )
% 0.23/0.58 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Gcd_remove0_nat
% 0.23/0.58 thf(fact_9418_Gcd__nat__eq__one,axiom,
% 0.23/0.58 ! [N3: set_nat] :
% 0.23/0.58 ( ( member_nat @ one_one_nat @ N3 )
% 0.23/0.58 => ( ( gcd_Gcd_nat @ N3 )
% 0.23/0.58 = one_one_nat ) ) ).
% 0.23/0.58
% 0.23/0.58 % Gcd_nat_eq_one
% 0.23/0.58 thf(fact_9419_nat_Oabs__eq,axiom,
% 0.23/0.58 ! [X4: product_prod_nat_nat] :
% 0.23/0.58 ( ( nat2 @ ( abs_Integ @ X4 ) )
% 0.23/0.58 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X4 ) ) ).
% 0.23/0.58
% 0.23/0.58 % nat.abs_eq
% 0.23/0.58 thf(fact_9420_one__int__def,axiom,
% 0.23/0.58 ( one_one_int
% 0.23/0.58 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % one_int_def
% 0.23/0.58 thf(fact_9421_less__int_Oabs__eq,axiom,
% 0.23/0.58 ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
% 0.23/0.58 ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
% 0.23/0.58 = ( produc8739625826339149834_nat_o
% 0.23/0.58 @ ^ [X: nat,Y: nat] :
% 0.23/0.58 ( produc6081775807080527818_nat_o
% 0.23/0.58 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) )
% 0.23/0.58 @ Xa
% 0.23/0.58 @ X4 ) ) ).
% 0.23/0.58
% 0.23/0.58 % less_int.abs_eq
% 0.23/0.58 thf(fact_9422_less__eq__int_Oabs__eq,axiom,
% 0.23/0.58 ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
% 0.23/0.58 ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
% 0.23/0.58 = ( produc8739625826339149834_nat_o
% 0.23/0.58 @ ^ [X: nat,Y: nat] :
% 0.23/0.58 ( produc6081775807080527818_nat_o
% 0.23/0.58 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) )
% 0.23/0.58 @ Xa
% 0.23/0.58 @ X4 ) ) ).
% 0.23/0.58
% 0.23/0.58 % less_eq_int.abs_eq
% 0.23/0.58 thf(fact_9423_plus__int_Oabs__eq,axiom,
% 0.23/0.58 ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
% 0.23/0.58 ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
% 0.23/0.58 = ( abs_Integ
% 0.23/0.58 @ ( produc27273713700761075at_nat
% 0.23/0.58 @ ^ [X: nat,Y: nat] :
% 0.23/0.58 ( produc2626176000494625587at_nat
% 0.23/0.58 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
% 0.23/0.58 @ Xa
% 0.23/0.58 @ X4 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % plus_int.abs_eq
% 0.23/0.58 thf(fact_9424_minus__int_Oabs__eq,axiom,
% 0.23/0.58 ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
% 0.23/0.58 ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
% 0.23/0.58 = ( abs_Integ
% 0.23/0.58 @ ( produc27273713700761075at_nat
% 0.23/0.58 @ ^ [X: nat,Y: nat] :
% 0.23/0.58 ( produc2626176000494625587at_nat
% 0.23/0.58 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) )
% 0.23/0.58 @ Xa
% 0.23/0.58 @ X4 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % minus_int.abs_eq
% 0.23/0.58 thf(fact_9425_num__of__nat_Osimps_I2_J,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.58 => ( ( num_of_nat @ ( suc @ N ) )
% 0.23/0.58 = ( inc @ ( num_of_nat @ N ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.58 => ( ( num_of_nat @ ( suc @ N ) )
% 0.23/0.58 = one ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % num_of_nat.simps(2)
% 0.23/0.58 thf(fact_9426_num__of__nat__numeral__eq,axiom,
% 0.23/0.58 ! [Q2: num] :
% 0.23/0.58 ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 0.23/0.58 = Q2 ) ).
% 0.23/0.58
% 0.23/0.58 % num_of_nat_numeral_eq
% 0.23/0.58 thf(fact_9427_num__of__nat_Osimps_I1_J,axiom,
% 0.23/0.58 ( ( num_of_nat @ zero_zero_nat )
% 0.23/0.58 = one ) ).
% 0.23/0.58
% 0.23/0.58 % num_of_nat.simps(1)
% 0.23/0.58 thf(fact_9428_numeral__num__of__nat,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.58 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 0.23/0.58 = N ) ) ).
% 0.23/0.58
% 0.23/0.58 % numeral_num_of_nat
% 0.23/0.58 thf(fact_9429_num__of__nat__One,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 0.23/0.58 => ( ( num_of_nat @ N )
% 0.23/0.58 = one ) ) ).
% 0.23/0.58
% 0.23/0.58 % num_of_nat_One
% 0.23/0.58 thf(fact_9430_num__of__nat__double,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.58 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 0.23/0.58 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % num_of_nat_double
% 0.23/0.58 thf(fact_9431_num__of__nat__plus__distrib,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.58 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 0.23/0.58 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % num_of_nat_plus_distrib
% 0.23/0.58 thf(fact_9432_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.23/0.58 ! [N: nat,J: nat,I: nat] :
% 0.23/0.58 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 0.23/0.58 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 0.23/0.58 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % nth_sorted_list_of_set_greaterThanLessThan
% 0.23/0.58 thf(fact_9433_less__eq__int_Orep__eq,axiom,
% 0.23/0.58 ( ord_less_eq_int
% 0.23/0.58 = ( ^ [X: int,Xa4: int] :
% 0.23/0.58 ( produc8739625826339149834_nat_o
% 0.23/0.58 @ ^ [Y: nat,Z5: nat] :
% 0.23/0.58 ( produc6081775807080527818_nat_o
% 0.23/0.58 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U4 @ Z5 ) ) )
% 0.23/0.58 @ ( rep_Integ @ X )
% 0.23/0.58 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % less_eq_int.rep_eq
% 0.23/0.58 thf(fact_9434_less__int_Orep__eq,axiom,
% 0.23/0.58 ( ord_less_int
% 0.23/0.58 = ( ^ [X: int,Xa4: int] :
% 0.23/0.58 ( produc8739625826339149834_nat_o
% 0.23/0.58 @ ^ [Y: nat,Z5: nat] :
% 0.23/0.58 ( produc6081775807080527818_nat_o
% 0.23/0.58 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U4 @ Z5 ) ) )
% 0.23/0.58 @ ( rep_Integ @ X )
% 0.23/0.58 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % less_int.rep_eq
% 0.23/0.58 thf(fact_9435_nat_Orep__eq,axiom,
% 0.23/0.58 ( nat2
% 0.23/0.58 = ( ^ [X: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % nat.rep_eq
% 0.23/0.58 thf(fact_9436_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.23/0.58 ! [N: nat,J: nat,I: nat] :
% 0.23/0.58 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 0.23/0.58 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 0.23/0.58 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % nth_sorted_list_of_set_greaterThanAtMost
% 0.23/0.58 thf(fact_9437_prod__encode__def,axiom,
% 0.23/0.58 ( nat_prod_encode
% 0.23/0.58 = ( produc6842872674320459806at_nat
% 0.23/0.58 @ ^ [M6: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M6 @ N2 ) ) @ M6 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % prod_encode_def
% 0.23/0.58 thf(fact_9438_pow_Osimps_I3_J,axiom,
% 0.23/0.58 ! [X4: num,Y3: num] :
% 0.23/0.58 ( ( pow @ X4 @ ( bit1 @ Y3 ) )
% 0.23/0.58 = ( times_times_num @ ( sqr @ ( pow @ X4 @ Y3 ) ) @ X4 ) ) ).
% 0.23/0.58
% 0.23/0.58 % pow.simps(3)
% 0.23/0.58 thf(fact_9439_finite__greaterThanAtMost,axiom,
% 0.23/0.58 ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).
% 0.23/0.58
% 0.23/0.58 % finite_greaterThanAtMost
% 0.23/0.58 thf(fact_9440_card__greaterThanAtMost,axiom,
% 0.23/0.58 ! [L2: nat,U: nat] :
% 0.23/0.58 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
% 0.23/0.58 = ( minus_minus_nat @ U @ L2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_greaterThanAtMost
% 0.23/0.58 thf(fact_9441_sqr_Osimps_I2_J,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( sqr @ ( bit0 @ N ) )
% 0.23/0.58 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sqr.simps(2)
% 0.23/0.58 thf(fact_9442_sqr_Osimps_I1_J,axiom,
% 0.23/0.58 ( ( sqr @ one )
% 0.23/0.58 = one ) ).
% 0.23/0.58
% 0.23/0.58 % sqr.simps(1)
% 0.23/0.58 thf(fact_9443_atLeastSucAtMost__greaterThanAtMost,axiom,
% 0.23/0.58 ! [L2: nat,U: nat] :
% 0.23/0.58 ( ( set_or1269000886237332187st_nat @ ( suc @ L2 ) @ U )
% 0.23/0.58 = ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).
% 0.23/0.58
% 0.23/0.58 % atLeastSucAtMost_greaterThanAtMost
% 0.23/0.58 thf(fact_9444_sqr__conv__mult,axiom,
% 0.23/0.58 ( sqr
% 0.23/0.58 = ( ^ [X: num] : ( times_times_num @ X @ X ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sqr_conv_mult
% 0.23/0.58 thf(fact_9445_le__prod__encode__1,axiom,
% 0.23/0.58 ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % le_prod_encode_1
% 0.23/0.58 thf(fact_9446_le__prod__encode__2,axiom,
% 0.23/0.58 ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % le_prod_encode_2
% 0.23/0.58 thf(fact_9447_pow_Osimps_I2_J,axiom,
% 0.23/0.58 ! [X4: num,Y3: num] :
% 0.23/0.58 ( ( pow @ X4 @ ( bit0 @ Y3 ) )
% 0.23/0.58 = ( sqr @ ( pow @ X4 @ Y3 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % pow.simps(2)
% 0.23/0.58 thf(fact_9448_sqr_Osimps_I3_J,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( sqr @ ( bit1 @ N ) )
% 0.23/0.58 = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sqr.simps(3)
% 0.23/0.58 thf(fact_9449_prod__encode__prod__decode__aux,axiom,
% 0.23/0.58 ! [K: nat,M: nat] :
% 0.23/0.58 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 0.23/0.58 = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 0.23/0.58
% 0.23/0.58 % prod_encode_prod_decode_aux
% 0.23/0.58 thf(fact_9450_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 0.23/0.58 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.23/0.58 ( ( ( vEBT_VEBT_valid @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( ? [Uu2: $o,Uv: $o] :
% 0.23/0.58 ( X4
% 0.23/0.58 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( Xa != one_one_nat ) ) )
% 0.23/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( ~ ( ( Deg2 = Xa )
% 0.23/0.58 & ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.23/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 & ( case_o184042715313410164at_nat
% 0.23/0.58 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.23/0.58 & ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 @ ( produc6081775807080527818_nat_o
% 0.23/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.23/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 & ! [I2: nat] :
% 0.23/0.58 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 0.23/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.23/0.58 & ( ( Mi3 = Ma3 )
% 0.23/0.58 => ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 & ( ( Mi3 != Ma3 )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.23/0.58 & ! [X: nat] :
% 0.23/0.58 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.23/0.58 => ( ( ord_less_nat @ Mi3 @ X )
% 0.23/0.58 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.23/0.58 @ Mima ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % VEBT_internal.valid'.elims(1)
% 0.23/0.58 thf(fact_9451_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 0.23/0.58 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.58 ( ( vEBT_VEBT_valid @ X4 @ Xa )
% 0.23/0.58 => ( ( ? [Uu2: $o,Uv: $o] :
% 0.23/0.58 ( X4
% 0.23/0.58 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 0.23/0.58 => ( Xa != one_one_nat ) )
% 0.23/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.58 => ~ ( ( Deg2 = Xa )
% 0.23/0.58 & ! [X5: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.23/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 & ( case_o184042715313410164at_nat
% 0.23/0.58 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.23/0.58 & ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 @ ( produc6081775807080527818_nat_o
% 0.23/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.23/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 & ! [I2: nat] :
% 0.23/0.58 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 0.23/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.23/0.58 & ( ( Mi3 = Ma3 )
% 0.23/0.58 => ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 & ( ( Mi3 != Ma3 )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.23/0.58 & ! [X: nat] :
% 0.23/0.58 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.23/0.58 => ( ( ord_less_nat @ Mi3 @ X )
% 0.23/0.58 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.23/0.58 @ Mima ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % VEBT_internal.valid'.elims(2)
% 0.23/0.58 thf(fact_9452_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 0.23/0.58 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.58 ( ~ ( vEBT_VEBT_valid @ X4 @ Xa )
% 0.23/0.58 => ( ( ? [Uu2: $o,Uv: $o] :
% 0.23/0.58 ( X4
% 0.23/0.58 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 0.23/0.58 => ( Xa = one_one_nat ) )
% 0.23/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.58 => ( ( Deg2 = Xa )
% 0.23/0.58 & ! [X3: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.23/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 & ( case_o184042715313410164at_nat
% 0.23/0.58 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.23/0.58 & ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 @ ( produc6081775807080527818_nat_o
% 0.23/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.23/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 & ! [I2: nat] :
% 0.23/0.58 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 0.23/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.23/0.58 & ( ( Mi3 = Ma3 )
% 0.23/0.58 => ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 & ( ( Mi3 != Ma3 )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.23/0.58 & ! [X: nat] :
% 0.23/0.58 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.23/0.58 => ( ( ord_less_nat @ Mi3 @ X )
% 0.23/0.58 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.23/0.58 @ Mima ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % VEBT_internal.valid'.elims(3)
% 0.23/0.58 thf(fact_9453_finite__greaterThanAtMost__int,axiom,
% 0.23/0.58 ! [L2: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L2 @ U ) ) ).
% 0.23/0.58
% 0.23/0.58 % finite_greaterThanAtMost_int
% 0.23/0.58 thf(fact_9454_card__greaterThanAtMost__int,axiom,
% 0.23/0.58 ! [L2: int,U: int] :
% 0.23/0.58 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L2 @ U ) )
% 0.23/0.58 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_greaterThanAtMost_int
% 0.23/0.58 thf(fact_9455_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 0.23/0.58 ! [L2: int,U: int] :
% 0.23/0.58 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 0.23/0.58 = ( set_or6656581121297822940st_int @ L2 @ U ) ) ).
% 0.23/0.58
% 0.23/0.58 % atLeastPlusOneAtMost_greaterThanAtMost_int
% 0.23/0.58 thf(fact_9456_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 0.23/0.58 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
% 0.23/0.58 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
% 0.23/0.58 = ( ( Deg = Deg3 )
% 0.23/0.58 & ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.58 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.23/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 & ( case_o184042715313410164at_nat
% 0.23/0.58 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
% 0.23/0.58 & ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 @ ( produc6081775807080527818_nat_o
% 0.23/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.23/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.23/0.58 & ! [I2: nat] :
% 0.23/0.58 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X6 ) )
% 0.23/0.58 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 0.23/0.58 & ( ( Mi3 = Ma3 )
% 0.23/0.58 => ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 & ( ( Mi3 != Ma3 )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.23/0.58 & ! [X: nat] :
% 0.23/0.58 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 0.23/0.58 => ( ( ord_less_nat @ Mi3 @ X )
% 0.23/0.58 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.23/0.58 @ Mima2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % VEBT_internal.valid'.simps(2)
% 0.23/0.58 thf(fact_9457_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 0.23/0.58 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.23/0.58 ( ( ( vEBT_VEBT_valid @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.58 => ( ! [Uu2: $o,Uv: $o] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 0.23/0.58 => ( ( Y3
% 0.23/0.58 = ( Xa = one_one_nat ) )
% 0.23/0.58 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) ) ) )
% 0.23/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.58 => ( ( Y3
% 0.23/0.58 = ( ( Deg2 = Xa )
% 0.23/0.58 & ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.23/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 & ( case_o184042715313410164at_nat
% 0.23/0.58 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.23/0.58 & ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 @ ( produc6081775807080527818_nat_o
% 0.23/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.23/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 & ! [I2: nat] :
% 0.23/0.58 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 0.23/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.23/0.58 & ( ( Mi3 = Ma3 )
% 0.23/0.58 => ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 & ( ( Mi3 != Ma3 )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.23/0.58 & ! [X: nat] :
% 0.23/0.58 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.23/0.58 => ( ( ord_less_nat @ Mi3 @ X )
% 0.23/0.58 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.23/0.58 @ Mima ) ) )
% 0.23/0.58 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % VEBT_internal.valid'.pelims(1)
% 0.23/0.58 thf(fact_9458_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 0.23/0.58 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.58 ( ( vEBT_VEBT_valid @ X4 @ Xa )
% 0.23/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.58 => ( ! [Uu2: $o,Uv: $o] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 0.23/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) )
% 0.23/0.58 => ( Xa != one_one_nat ) ) )
% 0.23/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 0.23/0.58 => ~ ( ( Deg2 = Xa )
% 0.23/0.58 & ! [X5: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.23/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 & ( case_o184042715313410164at_nat
% 0.23/0.58 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.23/0.58 & ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 @ ( produc6081775807080527818_nat_o
% 0.23/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.23/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 & ! [I2: nat] :
% 0.23/0.58 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 0.23/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.23/0.58 & ( ( Mi3 = Ma3 )
% 0.23/0.58 => ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 & ( ( Mi3 != Ma3 )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.23/0.58 & ! [X: nat] :
% 0.23/0.58 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.23/0.58 => ( ( ord_less_nat @ Mi3 @ X )
% 0.23/0.58 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.23/0.58 @ Mima ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % VEBT_internal.valid'.pelims(2)
% 0.23/0.58 thf(fact_9459_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 0.23/0.58 ! [X4: vEBT_VEBT,Xa: nat] :
% 0.23/0.58 ( ~ ( vEBT_VEBT_valid @ X4 @ Xa )
% 0.23/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 0.23/0.58 => ( ! [Uu2: $o,Uv: $o] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 0.23/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) )
% 0.23/0.58 => ( Xa = one_one_nat ) ) )
% 0.23/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 0.23/0.58 => ( ( Deg2 = Xa )
% 0.23/0.58 & ! [X3: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.23/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 & ( case_o184042715313410164at_nat
% 0.23/0.58 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.23/0.58 & ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 @ ( produc6081775807080527818_nat_o
% 0.23/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.23/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 & ! [I2: nat] :
% 0.23/0.58 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.58 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 0.23/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.23/0.58 & ( ( Mi3 = Ma3 )
% 0.23/0.58 => ! [X: vEBT_VEBT] :
% 0.23/0.58 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.23/0.58 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 0.23/0.58 & ( ( Mi3 != Ma3 )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.23/0.58 & ! [X: nat] :
% 0.23/0.58 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.23/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.23/0.58 => ( ( ord_less_nat @ Mi3 @ X )
% 0.23/0.58 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.23/0.58 @ Mima ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % VEBT_internal.valid'.pelims(3)
% 0.23/0.58 thf(fact_9460_GreatestI__nat,axiom,
% 0.23/0.58 ! [P: nat > $o,K: nat,B2: nat] :
% 0.23/0.58 ( ( P @ K )
% 0.23/0.58 => ( ! [Y4: nat] :
% 0.23/0.58 ( ( P @ Y4 )
% 0.23/0.58 => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 0.23/0.58 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % GreatestI_nat
% 0.23/0.58 thf(fact_9461_Greatest__le__nat,axiom,
% 0.23/0.58 ! [P: nat > $o,K: nat,B2: nat] :
% 0.23/0.58 ( ( P @ K )
% 0.23/0.58 => ( ! [Y4: nat] :
% 0.23/0.58 ( ( P @ Y4 )
% 0.23/0.58 => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 0.23/0.58 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Greatest_le_nat
% 0.23/0.58 thf(fact_9462_GreatestI__ex__nat,axiom,
% 0.23/0.58 ! [P: nat > $o,B2: nat] :
% 0.23/0.58 ( ? [X_12: nat] : ( P @ X_12 )
% 0.23/0.58 => ( ! [Y4: nat] :
% 0.23/0.58 ( ( P @ Y4 )
% 0.23/0.58 => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 0.23/0.58 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % GreatestI_ex_nat
% 0.23/0.58 thf(fact_9463_rat__floor__lemma,axiom,
% 0.23/0.58 ! [A: int,B2: int] :
% 0.23/0.58 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B2 ) ) @ ( fract @ A @ B2 ) )
% 0.23/0.58 & ( ord_less_rat @ ( fract @ A @ B2 ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % rat_floor_lemma
% 0.23/0.58 thf(fact_9464_image__minus__const__atLeastLessThan__nat,axiom,
% 0.23/0.58 ! [C: nat,Y3: nat,X4: nat] :
% 0.23/0.58 ( ( ( ord_less_nat @ C @ Y3 )
% 0.23/0.58 => ( ( image_nat_nat
% 0.23/0.58 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 0.23/0.58 @ ( set_or4665077453230672383an_nat @ X4 @ Y3 ) )
% 0.23/0.58 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X4 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_nat @ C @ Y3 )
% 0.23/0.58 => ( ( ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.58 => ( ( image_nat_nat
% 0.23/0.58 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 0.23/0.58 @ ( set_or4665077453230672383an_nat @ X4 @ Y3 ) )
% 0.23/0.58 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.23/0.58 & ( ~ ( ord_less_nat @ X4 @ Y3 )
% 0.23/0.58 => ( ( image_nat_nat
% 0.23/0.58 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 0.23/0.58 @ ( set_or4665077453230672383an_nat @ X4 @ Y3 ) )
% 0.23/0.58 = bot_bot_set_nat ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % image_minus_const_atLeastLessThan_nat
% 0.23/0.58 thf(fact_9465_take__bit__numeral__minus__numeral__int,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 = ( case_option_int_num @ zero_zero_int
% 0.23/0.58 @ ^ [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 ) ) )
% 0.23/0.58 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % take_bit_numeral_minus_numeral_int
% 0.23/0.58 thf(fact_9466_take__bit__num__simps_I1_J,axiom,
% 0.23/0.58 ! [M: num] :
% 0.23/0.58 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 0.23/0.58 = none_num ) ).
% 0.23/0.58
% 0.23/0.58 % take_bit_num_simps(1)
% 0.23/0.58 thf(fact_9467_bij__betw__Suc,axiom,
% 0.23/0.58 ! [M7: set_nat,N3: set_nat] :
% 0.23/0.58 ( ( bij_betw_nat_nat @ suc @ M7 @ N3 )
% 0.23/0.58 = ( ( image_nat_nat @ suc @ M7 )
% 0.23/0.58 = N3 ) ) ).
% 0.23/0.58
% 0.23/0.58 % bij_betw_Suc
% 0.23/0.58 thf(fact_9468_take__bit__num__simps_I2_J,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 0.23/0.58 = ( some_num @ one ) ) ).
% 0.23/0.58
% 0.23/0.58 % take_bit_num_simps(2)
% 0.23/0.58 thf(fact_9469_take__bit__num__simps_I5_J,axiom,
% 0.23/0.58 ! [R2: num] :
% 0.23/0.58 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
% 0.23/0.58 = ( some_num @ one ) ) ).
% 0.23/0.58
% 0.23/0.58 % take_bit_num_simps(5)
% 0.23/0.58 thf(fact_9470_image__Suc__atLeastAtMost,axiom,
% 0.23/0.58 ! [I: nat,J: nat] :
% 0.23/0.58 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 0.23/0.58 = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % image_Suc_atLeastAtMost
% 0.23/0.58 thf(fact_9471_image__Suc__atLeastLessThan,axiom,
% 0.23/0.58 ! [I: nat,J: nat] :
% 0.23/0.58 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 0.23/0.58 = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % image_Suc_atLeastLessThan
% 0.23/0.58 thf(fact_9472_mult__rat,axiom,
% 0.23/0.58 ! [A: int,B2: int,C: int,D: int] :
% 0.23/0.58 ( ( times_times_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.23/0.58 = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % mult_rat
% 0.23/0.58 thf(fact_9473_divide__rat,axiom,
% 0.23/0.58 ! [A: int,B2: int,C: int,D: int] :
% 0.23/0.58 ( ( divide_divide_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.23/0.58 = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % divide_rat
% 0.23/0.58 thf(fact_9474_floor__Fract,axiom,
% 0.23/0.58 ! [A: int,B2: int] :
% 0.23/0.58 ( ( archim3151403230148437115or_rat @ ( fract @ A @ B2 ) )
% 0.23/0.58 = ( divide_divide_int @ A @ B2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % floor_Fract
% 0.23/0.58 thf(fact_9475_less__rat,axiom,
% 0.23/0.58 ! [B2: int,D: int,A: int,C: int] :
% 0.23/0.58 ( ( B2 != zero_zero_int )
% 0.23/0.58 => ( ( D != zero_zero_int )
% 0.23/0.58 => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.23/0.58 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % less_rat
% 0.23/0.58 thf(fact_9476_add__rat,axiom,
% 0.23/0.58 ! [B2: int,D: int,A: int,C: int] :
% 0.23/0.58 ( ( B2 != zero_zero_int )
% 0.23/0.58 => ( ( D != zero_zero_int )
% 0.23/0.58 => ( ( plus_plus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.23/0.58 = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % add_rat
% 0.23/0.58 thf(fact_9477_le__rat,axiom,
% 0.23/0.58 ! [B2: int,D: int,A: int,C: int] :
% 0.23/0.58 ( ( B2 != zero_zero_int )
% 0.23/0.58 => ( ( D != zero_zero_int )
% 0.23/0.58 => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.23/0.58 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % le_rat
% 0.23/0.58 thf(fact_9478_diff__rat,axiom,
% 0.23/0.58 ! [B2: int,D: int,A: int,C: int] :
% 0.23/0.58 ( ( B2 != zero_zero_int )
% 0.23/0.58 => ( ( D != zero_zero_int )
% 0.23/0.58 => ( ( minus_minus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.23/0.58 = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % diff_rat
% 0.23/0.58 thf(fact_9479_sgn__rat,axiom,
% 0.23/0.58 ! [A: int,B2: int] :
% 0.23/0.58 ( ( sgn_sgn_rat @ ( fract @ A @ B2 ) )
% 0.23/0.58 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B2 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sgn_rat
% 0.23/0.58 thf(fact_9480_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( bit_take_bit_num @ N @ one )
% 0.23/0.58 = ( case_nat_option_num @ none_num
% 0.23/0.58 @ ^ [N2: nat] : ( some_num @ one )
% 0.23/0.58 @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % Code_Abstract_Nat.take_bit_num_code(1)
% 0.23/0.58 thf(fact_9481_zero__notin__Suc__image,axiom,
% 0.23/0.58 ! [A3: set_nat] :
% 0.23/0.58 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).
% 0.23/0.58
% 0.23/0.58 % zero_notin_Suc_image
% 0.23/0.58 thf(fact_9482_Rat__induct__pos,axiom,
% 0.23/0.58 ! [P: rat > $o,Q2: rat] :
% 0.23/0.58 ( ! [A5: int,B5: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ B5 )
% 0.23/0.58 => ( P @ ( fract @ A5 @ B5 ) ) )
% 0.23/0.58 => ( P @ Q2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rat_induct_pos
% 0.23/0.58 thf(fact_9483_eq__rat_I1_J,axiom,
% 0.23/0.58 ! [B2: int,D: int,A: int,C: int] :
% 0.23/0.58 ( ( B2 != zero_zero_int )
% 0.23/0.58 => ( ( D != zero_zero_int )
% 0.23/0.58 => ( ( ( fract @ A @ B2 )
% 0.23/0.58 = ( fract @ C @ D ) )
% 0.23/0.58 = ( ( times_times_int @ A @ D )
% 0.23/0.58 = ( times_times_int @ C @ B2 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % eq_rat(1)
% 0.23/0.58 thf(fact_9484_mult__rat__cancel,axiom,
% 0.23/0.58 ! [C: int,A: int,B2: int] :
% 0.23/0.58 ( ( C != zero_zero_int )
% 0.23/0.58 => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.23/0.58 = ( fract @ A @ B2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % mult_rat_cancel
% 0.23/0.58 thf(fact_9485_Fract__coprime,axiom,
% 0.23/0.58 ! [A: int,B2: int] :
% 0.23/0.58 ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B2 ) ) @ ( divide_divide_int @ B2 @ ( gcd_gcd_int @ A @ B2 ) ) )
% 0.23/0.58 = ( fract @ A @ B2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % Fract_coprime
% 0.23/0.58 thf(fact_9486_Fract__of__int__quotient,axiom,
% 0.23/0.58 ( fract
% 0.23/0.58 = ( ^ [K3: int,L: int] : ( divide_divide_rat @ ( ring_1_of_int_rat @ K3 ) @ ( ring_1_of_int_rat @ L ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Fract_of_int_quotient
% 0.23/0.58 thf(fact_9487_image__Suc__lessThan,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.58 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % image_Suc_lessThan
% 0.23/0.58 thf(fact_9488_image__Suc__atMost,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 0.23/0.58 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % image_Suc_atMost
% 0.23/0.58 thf(fact_9489_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.23/0.58 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % atLeast0_atMost_Suc_eq_insert_0
% 0.23/0.58 thf(fact_9490_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.23/0.58 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % atLeast0_lessThan_Suc_eq_insert_0
% 0.23/0.58 thf(fact_9491_lessThan__Suc__eq__insert__0,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 0.23/0.58 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % lessThan_Suc_eq_insert_0
% 0.23/0.58 thf(fact_9492_atMost__Suc__eq__insert__0,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 0.23/0.58 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % atMost_Suc_eq_insert_0
% 0.23/0.58 thf(fact_9493_Fract__less__zero__iff,axiom,
% 0.23/0.58 ! [B2: int,A: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.58 => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
% 0.23/0.58 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Fract_less_zero_iff
% 0.23/0.58 thf(fact_9494_zero__less__Fract__iff,axiom,
% 0.23/0.58 ! [B2: int,A: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.58 => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
% 0.23/0.58 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % zero_less_Fract_iff
% 0.23/0.58 thf(fact_9495_Fract__less__one__iff,axiom,
% 0.23/0.58 ! [B2: int,A: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.58 => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ one_one_rat )
% 0.23/0.58 = ( ord_less_int @ A @ B2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Fract_less_one_iff
% 0.23/0.58 thf(fact_9496_one__less__Fract__iff,axiom,
% 0.23/0.58 ! [B2: int,A: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.58 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B2 ) )
% 0.23/0.58 = ( ord_less_int @ B2 @ A ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % one_less_Fract_iff
% 0.23/0.58 thf(fact_9497_Fract__add__one,axiom,
% 0.23/0.58 ! [N: int,M: int] :
% 0.23/0.58 ( ( N != zero_zero_int )
% 0.23/0.58 => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 0.23/0.58 = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Fract_add_one
% 0.23/0.58 thf(fact_9498_Fract__le__zero__iff,axiom,
% 0.23/0.58 ! [B2: int,A: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.58 => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
% 0.23/0.58 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Fract_le_zero_iff
% 0.23/0.58 thf(fact_9499_zero__le__Fract__iff,axiom,
% 0.23/0.58 ! [B2: int,A: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
% 0.23/0.58 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % zero_le_Fract_iff
% 0.23/0.58 thf(fact_9500_one__le__Fract__iff,axiom,
% 0.23/0.58 ! [B2: int,A: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.58 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B2 ) )
% 0.23/0.58 = ( ord_less_eq_int @ B2 @ A ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % one_le_Fract_iff
% 0.23/0.58 thf(fact_9501_Fract__le__one__iff,axiom,
% 0.23/0.58 ! [B2: int,A: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.23/0.58 => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ one_one_rat )
% 0.23/0.58 = ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Fract_le_one_iff
% 0.23/0.58 thf(fact_9502_take__bit__num__def,axiom,
% 0.23/0.58 ( bit_take_bit_num
% 0.23/0.58 = ( ^ [N2: nat,M6: num] :
% 0.23/0.58 ( if_option_num
% 0.23/0.58 @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M6 ) )
% 0.23/0.58 = zero_zero_nat )
% 0.23/0.58 @ none_num
% 0.23/0.58 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % take_bit_num_def
% 0.23/0.58 thf(fact_9503_and__minus__numerals_I3_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.23/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_minus_numerals(3)
% 0.23/0.58 thf(fact_9504_and__minus__numerals_I7_J,axiom,
% 0.23/0.58 ! [N: num,M: num] :
% 0.23/0.58 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.23/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_minus_numerals(7)
% 0.23/0.58 thf(fact_9505_and__minus__numerals_I4_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.23/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_minus_numerals(4)
% 0.23/0.58 thf(fact_9506_take__bit__num__simps_I4_J,axiom,
% 0.23/0.58 ! [N: nat,M: num] :
% 0.23/0.58 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 0.23/0.58 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % take_bit_num_simps(4)
% 0.23/0.58 thf(fact_9507_take__bit__num__simps_I3_J,axiom,
% 0.23/0.58 ! [N: nat,M: num] :
% 0.23/0.58 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 0.23/0.58 = ( case_o6005452278849405969um_num @ none_num
% 0.23/0.58 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.23/0.58 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % take_bit_num_simps(3)
% 0.23/0.58 thf(fact_9508_take__bit__num__simps_I7_J,axiom,
% 0.23/0.58 ! [R2: num,M: num] :
% 0.23/0.58 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
% 0.23/0.58 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % take_bit_num_simps(7)
% 0.23/0.58 thf(fact_9509_take__bit__num__simps_I6_J,axiom,
% 0.23/0.58 ! [R2: num,M: num] :
% 0.23/0.58 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 0.23/0.58 = ( case_o6005452278849405969um_num @ none_num
% 0.23/0.58 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.23/0.58 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % take_bit_num_simps(6)
% 0.23/0.58 thf(fact_9510_and__minus__numerals_I8_J,axiom,
% 0.23/0.58 ! [N: num,M: num] :
% 0.23/0.58 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.23/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_minus_numerals(8)
% 0.23/0.58 thf(fact_9511_and__not__num_Osimps_I8_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.23/0.58 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.23/0.58 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.23/0.58 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num.simps(8)
% 0.23/0.58 thf(fact_9512_and__not__num_Osimps_I1_J,axiom,
% 0.23/0.58 ( ( bit_and_not_num @ one @ one )
% 0.23/0.58 = none_num ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num.simps(1)
% 0.23/0.58 thf(fact_9513_and__not__num_Osimps_I2_J,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 0.23/0.58 = ( some_num @ one ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num.simps(2)
% 0.23/0.58 thf(fact_9514_and__not__num_Osimps_I4_J,axiom,
% 0.23/0.58 ! [M: num] :
% 0.23/0.58 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 0.23/0.58 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num.simps(4)
% 0.23/0.58 thf(fact_9515_and__not__num_Osimps_I3_J,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 0.23/0.58 = none_num ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num.simps(3)
% 0.23/0.58 thf(fact_9516_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 0.23/0.58 ! [N: nat,M: num] :
% 0.23/0.58 ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 0.23/0.58 = ( case_nat_option_num @ none_num
% 0.23/0.58 @ ^ [N2: nat] :
% 0.23/0.58 ( case_o6005452278849405969um_num @ none_num
% 0.23/0.58 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.23/0.58 @ ( bit_take_bit_num @ N2 @ M ) )
% 0.23/0.58 @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % Code_Abstract_Nat.take_bit_num_code(2)
% 0.23/0.58 thf(fact_9517_image__int__atLeastAtMost,axiom,
% 0.23/0.58 ! [A: nat,B2: nat] :
% 0.23/0.58 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.23/0.58 = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % image_int_atLeastAtMost
% 0.23/0.58 thf(fact_9518_image__int__atLeastLessThan,axiom,
% 0.23/0.58 ! [A: nat,B2: nat] :
% 0.23/0.58 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B2 ) )
% 0.23/0.58 = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % image_int_atLeastLessThan
% 0.23/0.58 thf(fact_9519_and__not__num_Osimps_I7_J,axiom,
% 0.23/0.58 ! [M: num] :
% 0.23/0.58 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 0.23/0.58 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num.simps(7)
% 0.23/0.58 thf(fact_9520_and__not__num__eq__Some__iff,axiom,
% 0.23/0.58 ! [M: num,N: num,Q2: num] :
% 0.23/0.58 ( ( ( bit_and_not_num @ M @ N )
% 0.23/0.58 = ( some_num @ Q2 ) )
% 0.23/0.58 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 = ( numeral_numeral_int @ Q2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num_eq_Some_iff
% 0.23/0.58 thf(fact_9521_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 0.23/0.58 ! [N: nat,M: num] :
% 0.23/0.58 ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 0.23/0.58 = ( case_nat_option_num @ none_num
% 0.23/0.58 @ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
% 0.23/0.58 @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % Code_Abstract_Nat.take_bit_num_code(3)
% 0.23/0.58 thf(fact_9522_image__add__int__atLeastLessThan,axiom,
% 0.23/0.58 ! [L2: int,U: int] :
% 0.23/0.58 ( ( image_int_int
% 0.23/0.58 @ ^ [X: int] : ( plus_plus_int @ X @ L2 )
% 0.23/0.58 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
% 0.23/0.58 = ( set_or4662586982721622107an_int @ L2 @ U ) ) ).
% 0.23/0.58
% 0.23/0.58 % image_add_int_atLeastLessThan
% 0.23/0.58 thf(fact_9523_and__not__num__eq__None__iff,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( ( bit_and_not_num @ M @ N )
% 0.23/0.58 = none_num )
% 0.23/0.58 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 = zero_zero_int ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num_eq_None_iff
% 0.23/0.58 thf(fact_9524_image__atLeastZeroLessThan__int,axiom,
% 0.23/0.58 ! [U: int] :
% 0.23/0.58 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.23/0.58 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 0.23/0.58 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % image_atLeastZeroLessThan_int
% 0.23/0.58 thf(fact_9525_int__numeral__and__not__num,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % int_numeral_and_not_num
% 0.23/0.58 thf(fact_9526_int__numeral__not__and__num,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % int_numeral_not_and_num
% 0.23/0.58 thf(fact_9527_positive__rat,axiom,
% 0.23/0.58 ! [A: int,B2: int] :
% 0.23/0.58 ( ( positive @ ( fract @ A @ B2 ) )
% 0.23/0.58 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % positive_rat
% 0.23/0.58 thf(fact_9528_Rat_Opositive__add,axiom,
% 0.23/0.58 ! [X4: rat,Y3: rat] :
% 0.23/0.58 ( ( positive @ X4 )
% 0.23/0.58 => ( ( positive @ Y3 )
% 0.23/0.58 => ( positive @ ( plus_plus_rat @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rat.positive_add
% 0.23/0.58 thf(fact_9529_Rat_Opositive__mult,axiom,
% 0.23/0.58 ! [X4: rat,Y3: rat] :
% 0.23/0.58 ( ( positive @ X4 )
% 0.23/0.58 => ( ( positive @ Y3 )
% 0.23/0.58 => ( positive @ ( times_times_rat @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rat.positive_mult
% 0.23/0.58 thf(fact_9530_less__rat__def,axiom,
% 0.23/0.58 ( ord_less_rat
% 0.23/0.58 = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % less_rat_def
% 0.23/0.58 thf(fact_9531_Rat_Opositive_Orep__eq,axiom,
% 0.23/0.58 ( positive
% 0.23/0.58 = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rat.positive.rep_eq
% 0.23/0.58 thf(fact_9532_and__not__num_Oelims,axiom,
% 0.23/0.58 ! [X4: num,Xa: num,Y3: option_num] :
% 0.23/0.58 ( ( ( bit_and_not_num @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( ( X4 = one )
% 0.23/0.58 => ( ( Xa = one )
% 0.23/0.58 => ( Y3 != none_num ) ) )
% 0.23/0.58 => ( ( ( X4 = one )
% 0.23/0.58 => ( ? [N4: num] :
% 0.23/0.58 ( Xa
% 0.23/0.58 = ( bit0 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ one ) ) ) )
% 0.23/0.58 => ( ( ( X4 = one )
% 0.23/0.58 => ( ? [N4: num] :
% 0.23/0.58 ( Xa
% 0.23/0.58 = ( bit1 @ N4 ) )
% 0.23/0.58 => ( Y3 != none_num ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit0 @ M4 ) )
% 0.23/0.58 => ( ( Xa = one )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit0 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit0 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit0 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit1 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit1 @ M4 ) )
% 0.23/0.58 => ( ( Xa = one )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit1 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit0 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.23/0.58 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.23/0.58 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 0.23/0.58 => ~ ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit1 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit1 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num.elims
% 0.23/0.58 thf(fact_9533_Bit__Operations_Otake__bit__num__code,axiom,
% 0.23/0.58 ( bit_take_bit_num
% 0.23/0.58 = ( ^ [N2: nat,M6: num] :
% 0.23/0.58 ( produc478579273971653890on_num
% 0.23/0.58 @ ^ [A4: nat,X: num] :
% 0.23/0.58 ( case_nat_option_num @ none_num
% 0.23/0.58 @ ^ [O: nat] :
% 0.23/0.58 ( case_num_option_num @ ( some_num @ one )
% 0.23/0.58 @ ^ [P3: num] :
% 0.23/0.58 ( case_o6005452278849405969um_num @ none_num
% 0.23/0.58 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.23/0.58 @ ( bit_take_bit_num @ O @ P3 ) )
% 0.23/0.58 @ ^ [P3: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P3 ) ) )
% 0.23/0.58 @ X )
% 0.23/0.58 @ A4 )
% 0.23/0.58 @ ( product_Pair_nat_num @ N2 @ M6 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Bit_Operations.take_bit_num_code
% 0.23/0.58 thf(fact_9534_and__not__num_Osimps_I5_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.23/0.58 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num.simps(5)
% 0.23/0.58 thf(fact_9535_and__not__num_Osimps_I9_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.58 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num.simps(9)
% 0.23/0.58 thf(fact_9536_and__not__num_Osimps_I6_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.58 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_not_num.simps(6)
% 0.23/0.58 thf(fact_9537_and__num_Oelims,axiom,
% 0.23/0.58 ! [X4: num,Xa: num,Y3: option_num] :
% 0.23/0.58 ( ( ( bit_un7362597486090784418nd_num @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( ( X4 = one )
% 0.23/0.58 => ( ( Xa = one )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ one ) ) ) )
% 0.23/0.58 => ( ( ( X4 = one )
% 0.23/0.58 => ( ? [N4: num] :
% 0.23/0.58 ( Xa
% 0.23/0.58 = ( bit0 @ N4 ) )
% 0.23/0.58 => ( Y3 != none_num ) ) )
% 0.23/0.58 => ( ( ( X4 = one )
% 0.23/0.58 => ( ? [N4: num] :
% 0.23/0.58 ( Xa
% 0.23/0.58 = ( bit1 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ one ) ) ) )
% 0.23/0.58 => ( ( ? [M4: num] :
% 0.23/0.58 ( X4
% 0.23/0.58 = ( bit0 @ M4 ) )
% 0.23/0.58 => ( ( Xa = one )
% 0.23/0.58 => ( Y3 != none_num ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit0 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit0 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit0 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit1 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 0.23/0.58 => ( ( ? [M4: num] :
% 0.23/0.58 ( X4
% 0.23/0.58 = ( bit1 @ M4 ) )
% 0.23/0.58 => ( ( Xa = one )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ one ) ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit1 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit0 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 0.23/0.58 => ~ ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit1 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit1 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.23/0.58 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.23/0.58 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_num.elims
% 0.23/0.58 thf(fact_9538_xor__num_Oelims,axiom,
% 0.23/0.58 ! [X4: num,Xa: num,Y3: option_num] :
% 0.23/0.58 ( ( ( bit_un2480387367778600638or_num @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( ( X4 = one )
% 0.23/0.58 => ( ( Xa = one )
% 0.23/0.58 => ( Y3 != none_num ) ) )
% 0.23/0.58 => ( ( ( X4 = one )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit0 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ ( bit1 @ N4 ) ) ) ) )
% 0.23/0.58 => ( ( ( X4 = one )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit1 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ ( bit0 @ N4 ) ) ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit0 @ M4 ) )
% 0.23/0.58 => ( ( Xa = one )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ ( bit1 @ M4 ) ) ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit0 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit0 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit0 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit1 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit1 @ M4 ) )
% 0.23/0.58 => ( ( Xa = one )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.23/0.58 => ( ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit1 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit0 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) )
% 0.23/0.58 => ~ ! [M4: num] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( bit1 @ M4 ) )
% 0.23/0.58 => ! [N4: num] :
% 0.23/0.58 ( ( Xa
% 0.23/0.58 = ( bit1 @ N4 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % xor_num.elims
% 0.23/0.58 thf(fact_9539_xor__num_Osimps_I8_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.23/0.58 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % xor_num.simps(8)
% 0.23/0.58 thf(fact_9540_and__num_Osimps_I1_J,axiom,
% 0.23/0.58 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 0.23/0.58 = ( some_num @ one ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_num.simps(1)
% 0.23/0.58 thf(fact_9541_xor__num_Osimps_I1_J,axiom,
% 0.23/0.58 ( ( bit_un2480387367778600638or_num @ one @ one )
% 0.23/0.58 = none_num ) ).
% 0.23/0.58
% 0.23/0.58 % xor_num.simps(1)
% 0.23/0.58 thf(fact_9542_and__num_Osimps_I5_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.23/0.58 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_num.simps(5)
% 0.23/0.58 thf(fact_9543_xor__num_Osimps_I5_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.23/0.58 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % xor_num.simps(5)
% 0.23/0.58 thf(fact_9544_and__num_Osimps_I7_J,axiom,
% 0.23/0.58 ! [M: num] :
% 0.23/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 0.23/0.58 = ( some_num @ one ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_num.simps(7)
% 0.23/0.58 thf(fact_9545_and__num_Osimps_I3_J,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 0.23/0.58 = ( some_num @ one ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_num.simps(3)
% 0.23/0.58 thf(fact_9546_and__num_Osimps_I2_J,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 0.23/0.58 = none_num ) ).
% 0.23/0.58
% 0.23/0.58 % and_num.simps(2)
% 0.23/0.58 thf(fact_9547_and__num_Osimps_I4_J,axiom,
% 0.23/0.58 ! [M: num] :
% 0.23/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 0.23/0.58 = none_num ) ).
% 0.23/0.58
% 0.23/0.58 % and_num.simps(4)
% 0.23/0.58 thf(fact_9548_xor__num_Osimps_I9_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.58 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % xor_num.simps(9)
% 0.23/0.58 thf(fact_9549_and__num_Osimps_I6_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.58 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_num.simps(6)
% 0.23/0.58 thf(fact_9550_and__num_Osimps_I8_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.23/0.58 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_num.simps(8)
% 0.23/0.58 thf(fact_9551_xor__num_Osimps_I7_J,axiom,
% 0.23/0.58 ! [M: num] :
% 0.23/0.58 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 0.23/0.58 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % xor_num.simps(7)
% 0.23/0.58 thf(fact_9552_xor__num_Osimps_I4_J,axiom,
% 0.23/0.58 ! [M: num] :
% 0.23/0.58 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 0.23/0.58 = ( some_num @ ( bit1 @ M ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % xor_num.simps(4)
% 0.23/0.58 thf(fact_9553_xor__num_Osimps_I3_J,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 0.23/0.58 = ( some_num @ ( bit0 @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % xor_num.simps(3)
% 0.23/0.58 thf(fact_9554_xor__num_Osimps_I2_J,axiom,
% 0.23/0.58 ! [N: num] :
% 0.23/0.58 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 0.23/0.58 = ( some_num @ ( bit1 @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % xor_num.simps(2)
% 0.23/0.58 thf(fact_9555_and__num_Osimps_I9_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.23/0.58 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.23/0.58 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.23/0.58 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % and_num.simps(9)
% 0.23/0.58 thf(fact_9556_xor__num_Osimps_I6_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.23/0.58 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % xor_num.simps(6)
% 0.23/0.58 thf(fact_9557_and__num__dict,axiom,
% 0.23/0.58 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 0.23/0.58
% 0.23/0.58 % and_num_dict
% 0.23/0.58 thf(fact_9558_xor__num__dict,axiom,
% 0.23/0.58 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 0.23/0.58
% 0.23/0.58 % xor_num_dict
% 0.23/0.58 thf(fact_9559_num__of__integer__code,axiom,
% 0.23/0.58 ( code_num_of_integer
% 0.23/0.58 = ( ^ [K3: code_integer] :
% 0.23/0.58 ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
% 0.23/0.58 @ ( produc7336495610019696514er_num
% 0.23/0.58 @ ^ [L: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
% 0.23/0.58 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % num_of_integer_code
% 0.23/0.58 thf(fact_9560_min__Suc__Suc,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.23/0.58 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % min_Suc_Suc
% 0.23/0.58 thf(fact_9561_min__0R,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( ord_min_nat @ N @ zero_zero_nat )
% 0.23/0.58 = zero_zero_nat ) ).
% 0.23/0.58
% 0.23/0.58 % min_0R
% 0.23/0.58 thf(fact_9562_min__0L,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( ord_min_nat @ zero_zero_nat @ N )
% 0.23/0.58 = zero_zero_nat ) ).
% 0.23/0.58
% 0.23/0.58 % min_0L
% 0.23/0.58 thf(fact_9563_min__numeral__Suc,axiom,
% 0.23/0.58 ! [K: num,N: nat] :
% 0.23/0.58 ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.23/0.58 = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % min_numeral_Suc
% 0.23/0.58 thf(fact_9564_min__Suc__numeral,axiom,
% 0.23/0.58 ! [N: nat,K: num] :
% 0.23/0.58 ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.23/0.58 = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % min_Suc_numeral
% 0.23/0.58 thf(fact_9565_nat__mult__min__right,axiom,
% 0.23/0.58 ! [M: nat,N: nat,Q2: nat] :
% 0.23/0.58 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
% 0.23/0.58 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % nat_mult_min_right
% 0.23/0.58 thf(fact_9566_nat__mult__min__left,axiom,
% 0.23/0.58 ! [M: nat,N: nat,Q2: nat] :
% 0.23/0.58 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
% 0.23/0.58 = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % nat_mult_min_left
% 0.23/0.58 thf(fact_9567_min__diff,axiom,
% 0.23/0.58 ! [M: nat,I: nat,N: nat] :
% 0.23/0.58 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 0.23/0.58 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 0.23/0.58
% 0.23/0.58 % min_diff
% 0.23/0.58 thf(fact_9568_concat__bit__assoc__sym,axiom,
% 0.23/0.58 ! [M: nat,N: nat,K: int,L2: int,R2: int] :
% 0.23/0.58 ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L2 ) @ R2 )
% 0.23/0.58 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L2 @ R2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % concat_bit_assoc_sym
% 0.23/0.58 thf(fact_9569_take__bit__concat__bit__eq,axiom,
% 0.23/0.58 ! [M: nat,N: nat,K: int,L2: int] :
% 0.23/0.58 ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L2 ) )
% 0.23/0.58 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % take_bit_concat_bit_eq
% 0.23/0.58 thf(fact_9570_min__Suc1,axiom,
% 0.23/0.58 ! [N: nat,M: nat] :
% 0.23/0.58 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 0.23/0.58 = ( case_nat_nat @ zero_zero_nat
% 0.23/0.58 @ ^ [M2: nat] : ( suc @ ( ord_min_nat @ N @ M2 ) )
% 0.23/0.58 @ M ) ) ).
% 0.23/0.58
% 0.23/0.58 % min_Suc1
% 0.23/0.58 thf(fact_9571_min__Suc2,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 0.23/0.58 = ( case_nat_nat @ zero_zero_nat
% 0.23/0.58 @ ^ [M2: nat] : ( suc @ ( ord_min_nat @ M2 @ N ) )
% 0.23/0.58 @ M ) ) ).
% 0.23/0.58
% 0.23/0.58 % min_Suc2
% 0.23/0.58 thf(fact_9572_card__le__Suc__Max,axiom,
% 0.23/0.58 ! [S2: set_nat] :
% 0.23/0.58 ( ( finite_finite_nat @ S2 )
% 0.23/0.58 => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_le_Suc_Max
% 0.23/0.58 thf(fact_9573_divide__nat__def,axiom,
% 0.23/0.58 ( divide_divide_nat
% 0.23/0.58 = ( ^ [M6: nat,N2: nat] :
% 0.23/0.58 ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 0.23/0.58 @ ( lattic8265883725875713057ax_nat
% 0.23/0.58 @ ( collect_nat
% 0.23/0.58 @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M6 ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % divide_nat_def
% 0.23/0.58 thf(fact_9574_gcd__is__Max__divisors__nat,axiom,
% 0.23/0.58 ! [N: nat,M: nat] :
% 0.23/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.58 => ( ( gcd_gcd_nat @ M @ N )
% 0.23/0.58 = ( lattic8265883725875713057ax_nat
% 0.23/0.58 @ ( collect_nat
% 0.23/0.58 @ ^ [D2: nat] :
% 0.23/0.58 ( ( dvd_dvd_nat @ D2 @ M )
% 0.23/0.58 & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % gcd_is_Max_divisors_nat
% 0.23/0.58 thf(fact_9575_min__enat__simps_I2_J,axiom,
% 0.23/0.58 ! [Q2: extended_enat] :
% 0.23/0.58 ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 0.23/0.58 = zero_z5237406670263579293d_enat ) ).
% 0.23/0.58
% 0.23/0.58 % min_enat_simps(2)
% 0.23/0.58 thf(fact_9576_min__enat__simps_I3_J,axiom,
% 0.23/0.58 ! [Q2: extended_enat] :
% 0.23/0.58 ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 0.23/0.58 = zero_z5237406670263579293d_enat ) ).
% 0.23/0.58
% 0.23/0.58 % min_enat_simps(3)
% 0.23/0.58 thf(fact_9577_sorted__list__of__set__greaterThanAtMost,axiom,
% 0.23/0.58 ! [I: nat,J: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 0.23/0.58 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 0.23/0.58 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sorted_list_of_set_greaterThanAtMost
% 0.23/0.58 thf(fact_9578_sorted__list__of__set__greaterThanLessThan,axiom,
% 0.23/0.58 ! [I: nat,J: nat] :
% 0.23/0.58 ( ( ord_less_nat @ ( suc @ I ) @ J )
% 0.23/0.58 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 0.23/0.58 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sorted_list_of_set_greaterThanLessThan
% 0.23/0.58 thf(fact_9579_upto__aux__rec,axiom,
% 0.23/0.58 ( upto_aux
% 0.23/0.58 = ( ^ [I2: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_aux_rec
% 0.23/0.58 thf(fact_9580_list__encode_Osimps_I2_J,axiom,
% 0.23/0.58 ! [X4: nat,Xs2: list_nat] :
% 0.23/0.58 ( ( nat_list_encode @ ( cons_nat @ X4 @ Xs2 ) )
% 0.23/0.58 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % list_encode.simps(2)
% 0.23/0.58 thf(fact_9581_list__encode_Oelims,axiom,
% 0.23/0.58 ! [X4: list_nat,Y3: nat] :
% 0.23/0.58 ( ( ( nat_list_encode @ X4 )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( ( X4 = nil_nat )
% 0.23/0.58 => ( Y3 != zero_zero_nat ) )
% 0.23/0.58 => ~ ! [X3: nat,Xs3: list_nat] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( cons_nat @ X3 @ Xs3 ) )
% 0.23/0.58 => ( Y3
% 0.23/0.58 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % list_encode.elims
% 0.23/0.58 thf(fact_9582_quotient__of__def,axiom,
% 0.23/0.58 ( quotient_of
% 0.23/0.58 = ( ^ [X: rat] :
% 0.23/0.58 ( the_Pr4378521158711661632nt_int
% 0.23/0.58 @ ^ [Pair: product_prod_int_int] :
% 0.23/0.58 ( ( X
% 0.23/0.58 = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 0.23/0.58 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 0.23/0.58 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % quotient_of_def
% 0.23/0.58 thf(fact_9583_coprime__abs__right__iff,axiom,
% 0.23/0.58 ! [K: int,L2: int] :
% 0.23/0.58 ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L2 ) )
% 0.23/0.58 = ( algebr932160517623751201me_int @ K @ L2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_abs_right_iff
% 0.23/0.58 thf(fact_9584_coprime__abs__left__iff,axiom,
% 0.23/0.58 ! [K: int,L2: int] :
% 0.23/0.58 ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L2 )
% 0.23/0.58 = ( algebr932160517623751201me_int @ K @ L2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_abs_left_iff
% 0.23/0.58 thf(fact_9585_normalize__stable,axiom,
% 0.23/0.58 ! [Q2: int,P2: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ Q2 )
% 0.23/0.58 => ( ( algebr932160517623751201me_int @ P2 @ Q2 )
% 0.23/0.58 => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 0.23/0.58 = ( product_Pair_int_int @ P2 @ Q2 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % normalize_stable
% 0.23/0.58 thf(fact_9586_coprime__crossproduct__int,axiom,
% 0.23/0.58 ! [A: int,D: int,B2: int,C: int] :
% 0.23/0.58 ( ( algebr932160517623751201me_int @ A @ D )
% 0.23/0.58 => ( ( algebr932160517623751201me_int @ B2 @ C )
% 0.23/0.58 => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
% 0.23/0.58 = ( times_times_int @ ( abs_abs_int @ B2 ) @ ( abs_abs_int @ D ) ) )
% 0.23/0.58 = ( ( ( abs_abs_int @ A )
% 0.23/0.58 = ( abs_abs_int @ B2 ) )
% 0.23/0.58 & ( ( abs_abs_int @ C )
% 0.23/0.58 = ( abs_abs_int @ D ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_crossproduct_int
% 0.23/0.58 thf(fact_9587_Rat__cases,axiom,
% 0.23/0.58 ! [Q2: rat] :
% 0.23/0.58 ~ ! [A5: int,B5: int] :
% 0.23/0.58 ( ( Q2
% 0.23/0.58 = ( fract @ A5 @ B5 ) )
% 0.23/0.58 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 0.23/0.58 => ~ ( algebr932160517623751201me_int @ A5 @ B5 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rat_cases
% 0.23/0.58 thf(fact_9588_Rat__induct,axiom,
% 0.23/0.58 ! [P: rat > $o,Q2: rat] :
% 0.23/0.58 ( ! [A5: int,B5: int] :
% 0.23/0.58 ( ( ord_less_int @ zero_zero_int @ B5 )
% 0.23/0.58 => ( ( algebr932160517623751201me_int @ A5 @ B5 )
% 0.23/0.58 => ( P @ ( fract @ A5 @ B5 ) ) ) )
% 0.23/0.58 => ( P @ Q2 ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rat_induct
% 0.23/0.58 thf(fact_9589_coprime__common__divisor__int,axiom,
% 0.23/0.58 ! [A: int,B2: int,X4: int] :
% 0.23/0.58 ( ( algebr932160517623751201me_int @ A @ B2 )
% 0.23/0.58 => ( ( dvd_dvd_int @ X4 @ A )
% 0.23/0.58 => ( ( dvd_dvd_int @ X4 @ B2 )
% 0.23/0.58 => ( ( abs_abs_int @ X4 )
% 0.23/0.58 = one_one_int ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_common_divisor_int
% 0.23/0.58 thf(fact_9590_Rat__cases__nonzero,axiom,
% 0.23/0.58 ! [Q2: rat] :
% 0.23/0.58 ( ! [A5: int,B5: int] :
% 0.23/0.58 ( ( Q2
% 0.23/0.58 = ( fract @ A5 @ B5 ) )
% 0.23/0.58 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 0.23/0.58 => ( ( A5 != zero_zero_int )
% 0.23/0.58 => ~ ( algebr932160517623751201me_int @ A5 @ B5 ) ) ) )
% 0.23/0.58 => ( Q2 = zero_zero_rat ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rat_cases_nonzero
% 0.23/0.58 thf(fact_9591_card__length__sum__list__rec,axiom,
% 0.23/0.58 ! [M: nat,N3: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.23/0.58 => ( ( finite_card_list_nat
% 0.23/0.58 @ ( collect_list_nat
% 0.23/0.58 @ ^ [L: list_nat] :
% 0.23/0.58 ( ( ( size_size_list_nat @ L )
% 0.23/0.58 = M )
% 0.23/0.58 & ( ( groups4561878855575611511st_nat @ L )
% 0.23/0.58 = N3 ) ) ) )
% 0.23/0.58 = ( plus_plus_nat
% 0.23/0.58 @ ( finite_card_list_nat
% 0.23/0.58 @ ( collect_list_nat
% 0.23/0.58 @ ^ [L: list_nat] :
% 0.23/0.58 ( ( ( size_size_list_nat @ L )
% 0.23/0.58 = ( minus_minus_nat @ M @ one_one_nat ) )
% 0.23/0.58 & ( ( groups4561878855575611511st_nat @ L )
% 0.23/0.58 = N3 ) ) ) )
% 0.23/0.58 @ ( finite_card_list_nat
% 0.23/0.58 @ ( collect_list_nat
% 0.23/0.58 @ ^ [L: list_nat] :
% 0.23/0.58 ( ( ( size_size_list_nat @ L )
% 0.23/0.58 = M )
% 0.23/0.58 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L ) @ one_one_nat )
% 0.23/0.58 = N3 ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_length_sum_list_rec
% 0.23/0.58 thf(fact_9592_card__length__sum__list,axiom,
% 0.23/0.58 ! [M: nat,N3: nat] :
% 0.23/0.58 ( ( finite_card_list_nat
% 0.23/0.58 @ ( collect_list_nat
% 0.23/0.58 @ ^ [L: list_nat] :
% 0.23/0.58 ( ( ( size_size_list_nat @ L )
% 0.23/0.58 = M )
% 0.23/0.58 & ( ( groups4561878855575611511st_nat @ L )
% 0.23/0.58 = N3 ) ) ) )
% 0.23/0.58 = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M ) @ one_one_nat ) @ N3 ) ) ).
% 0.23/0.58
% 0.23/0.58 % card_length_sum_list
% 0.23/0.58 thf(fact_9593_quotient__of__unique,axiom,
% 0.23/0.58 ! [R2: rat] :
% 0.23/0.58 ? [X3: product_prod_int_int] :
% 0.23/0.58 ( ( R2
% 0.23/0.58 = ( fract @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 0.23/0.58 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X3 ) )
% 0.23/0.58 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) )
% 0.23/0.58 & ! [Y5: product_prod_int_int] :
% 0.23/0.58 ( ( ( R2
% 0.23/0.58 = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 0.23/0.58 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
% 0.23/0.58 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 0.23/0.58 => ( Y5 = X3 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % quotient_of_unique
% 0.23/0.58 thf(fact_9594_sorted__list__of__set__atMost__Suc,axiom,
% 0.23/0.58 ! [K: nat] :
% 0.23/0.58 ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 0.23/0.58 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sorted_list_of_set_atMost_Suc
% 0.23/0.58 thf(fact_9595_sorted__list__of__set__lessThan__Suc,axiom,
% 0.23/0.58 ! [K: nat] :
% 0.23/0.58 ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 0.23/0.58 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sorted_list_of_set_lessThan_Suc
% 0.23/0.58 thf(fact_9596_coprime__int__iff,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.23/0.58 = ( algebr934650988132801477me_nat @ M @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_int_iff
% 0.23/0.58 thf(fact_9597_coprime__nat__abs__right__iff,axiom,
% 0.23/0.58 ! [N: nat,K: int] :
% 0.23/0.58 ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.23/0.58 = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_nat_abs_right_iff
% 0.23/0.58 thf(fact_9598_coprime__nat__abs__left__iff,axiom,
% 0.23/0.58 ! [K: int,N: nat] :
% 0.23/0.58 ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.23/0.58 = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_nat_abs_left_iff
% 0.23/0.58 thf(fact_9599_coprime__common__divisor__nat,axiom,
% 0.23/0.58 ! [A: nat,B2: nat,X4: nat] :
% 0.23/0.58 ( ( algebr934650988132801477me_nat @ A @ B2 )
% 0.23/0.58 => ( ( dvd_dvd_nat @ X4 @ A )
% 0.23/0.58 => ( ( dvd_dvd_nat @ X4 @ B2 )
% 0.23/0.58 => ( X4 = one_one_nat ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_common_divisor_nat
% 0.23/0.58 thf(fact_9600_coprime__Suc__0__right,axiom,
% 0.23/0.58 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_Suc_0_right
% 0.23/0.58 thf(fact_9601_coprime__Suc__0__left,axiom,
% 0.23/0.58 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_Suc_0_left
% 0.23/0.58 thf(fact_9602_coprime__Suc__left__nat,axiom,
% 0.23/0.58 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_Suc_left_nat
% 0.23/0.58 thf(fact_9603_coprime__Suc__right__nat,axiom,
% 0.23/0.58 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_Suc_right_nat
% 0.23/0.58 thf(fact_9604_coprime__crossproduct__nat,axiom,
% 0.23/0.58 ! [A: nat,D: nat,B2: nat,C: nat] :
% 0.23/0.58 ( ( algebr934650988132801477me_nat @ A @ D )
% 0.23/0.58 => ( ( algebr934650988132801477me_nat @ B2 @ C )
% 0.23/0.58 => ( ( ( times_times_nat @ A @ C )
% 0.23/0.58 = ( times_times_nat @ B2 @ D ) )
% 0.23/0.58 = ( ( A = B2 )
% 0.23/0.58 & ( C = D ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_crossproduct_nat
% 0.23/0.58 thf(fact_9605_coprime__diff__one__right__nat,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.58 => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_diff_one_right_nat
% 0.23/0.58 thf(fact_9606_coprime__diff__one__left__nat,axiom,
% 0.23/0.58 ! [N: nat] :
% 0.23/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.58 => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 0.23/0.58
% 0.23/0.58 % coprime_diff_one_left_nat
% 0.23/0.58 thf(fact_9607_upto_Opsimps,axiom,
% 0.23/0.58 ! [I: int,J: int] :
% 0.23/0.58 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 0.23/0.58 => ( ( ( ord_less_eq_int @ I @ J )
% 0.23/0.58 => ( ( upto @ I @ J )
% 0.23/0.58 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_eq_int @ I @ J )
% 0.23/0.58 => ( ( upto @ I @ J )
% 0.23/0.58 = nil_int ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto.psimps
% 0.23/0.58 thf(fact_9608_upto__Nil,axiom,
% 0.23/0.58 ! [I: int,J: int] :
% 0.23/0.58 ( ( ( upto @ I @ J )
% 0.23/0.58 = nil_int )
% 0.23/0.58 = ( ord_less_int @ J @ I ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_Nil
% 0.23/0.58 thf(fact_9609_upto__Nil2,axiom,
% 0.23/0.58 ! [I: int,J: int] :
% 0.23/0.58 ( ( nil_int
% 0.23/0.58 = ( upto @ I @ J ) )
% 0.23/0.58 = ( ord_less_int @ J @ I ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_Nil2
% 0.23/0.58 thf(fact_9610_upto__empty,axiom,
% 0.23/0.58 ! [J: int,I: int] :
% 0.23/0.58 ( ( ord_less_int @ J @ I )
% 0.23/0.58 => ( ( upto @ I @ J )
% 0.23/0.58 = nil_int ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_empty
% 0.23/0.58 thf(fact_9611_nth__upto,axiom,
% 0.23/0.58 ! [I: int,K: nat,J: int] :
% 0.23/0.58 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 0.23/0.58 => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 0.23/0.58 = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % nth_upto
% 0.23/0.58 thf(fact_9612_length__upto,axiom,
% 0.23/0.58 ! [I: int,J: int] :
% 0.23/0.58 ( ( size_size_list_int @ ( upto @ I @ J ) )
% 0.23/0.58 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % length_upto
% 0.23/0.58 thf(fact_9613_upto__rec__numeral_I1_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.58 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.58 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.58 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.58 = nil_int ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_rec_numeral(1)
% 0.23/0.58 thf(fact_9614_upto__rec__numeral_I2_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 = nil_int ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_rec_numeral(2)
% 0.23/0.58 thf(fact_9615_upto__rec__numeral_I3_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.58 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.58 = ( 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 ) ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.58 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.23/0.58 = nil_int ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_rec_numeral(3)
% 0.23/0.58 thf(fact_9616_upto__rec__numeral_I4_J,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 = ( 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 ) ) ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.23/0.58 = nil_int ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_rec_numeral(4)
% 0.23/0.58 thf(fact_9617_upto__split2,axiom,
% 0.23/0.58 ! [I: int,J: int,K: int] :
% 0.23/0.58 ( ( ord_less_eq_int @ I @ J )
% 0.23/0.58 => ( ( ord_less_eq_int @ J @ K )
% 0.23/0.58 => ( ( upto @ I @ K )
% 0.23/0.58 = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_split2
% 0.23/0.58 thf(fact_9618_upto__split1,axiom,
% 0.23/0.58 ! [I: int,J: int,K: int] :
% 0.23/0.58 ( ( ord_less_eq_int @ I @ J )
% 0.23/0.58 => ( ( ord_less_eq_int @ J @ K )
% 0.23/0.58 => ( ( upto @ I @ K )
% 0.23/0.58 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_split1
% 0.23/0.58 thf(fact_9619_atLeastLessThan__upto,axiom,
% 0.23/0.58 ( set_or4662586982721622107an_int
% 0.23/0.58 = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % atLeastLessThan_upto
% 0.23/0.58 thf(fact_9620_greaterThanAtMost__upto,axiom,
% 0.23/0.58 ( set_or6656581121297822940st_int
% 0.23/0.58 = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % greaterThanAtMost_upto
% 0.23/0.58 thf(fact_9621_upto__rec1,axiom,
% 0.23/0.58 ! [I: int,J: int] :
% 0.23/0.58 ( ( ord_less_eq_int @ I @ J )
% 0.23/0.58 => ( ( upto @ I @ J )
% 0.23/0.58 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_rec1
% 0.23/0.58 thf(fact_9622_upto_Oelims,axiom,
% 0.23/0.58 ! [X4: int,Xa: int,Y3: list_int] :
% 0.23/0.58 ( ( ( upto @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( ( ord_less_eq_int @ X4 @ Xa )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_eq_int @ X4 @ Xa )
% 0.23/0.58 => ( Y3 = nil_int ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto.elims
% 0.23/0.58 thf(fact_9623_upto_Osimps,axiom,
% 0.23/0.58 ( upto
% 0.23/0.58 = ( ^ [I2: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I2 @ J3 ) @ ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto.simps
% 0.23/0.58 thf(fact_9624_upto__rec2,axiom,
% 0.23/0.58 ! [I: int,J: int] :
% 0.23/0.58 ( ( ord_less_eq_int @ I @ J )
% 0.23/0.58 => ( ( upto @ I @ J )
% 0.23/0.58 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_rec2
% 0.23/0.58 thf(fact_9625_greaterThanLessThan__upto,axiom,
% 0.23/0.58 ( set_or5832277885323065728an_int
% 0.23/0.58 = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % greaterThanLessThan_upto
% 0.23/0.58 thf(fact_9626_upto__split3,axiom,
% 0.23/0.58 ! [I: int,J: int,K: int] :
% 0.23/0.58 ( ( ord_less_eq_int @ I @ J )
% 0.23/0.58 => ( ( ord_less_eq_int @ J @ K )
% 0.23/0.58 => ( ( upto @ I @ K )
% 0.23/0.58 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto_split3
% 0.23/0.58 thf(fact_9627_upto_Opelims,axiom,
% 0.23/0.58 ! [X4: int,Xa: int,Y3: list_int] :
% 0.23/0.58 ( ( ( upto @ X4 @ Xa )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) )
% 0.23/0.58 => ~ ( ( ( ( ord_less_eq_int @ X4 @ Xa )
% 0.23/0.58 => ( Y3
% 0.23/0.58 = ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_eq_int @ X4 @ Xa )
% 0.23/0.58 => ( Y3 = nil_int ) ) )
% 0.23/0.58 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upto.pelims
% 0.23/0.58 thf(fact_9628_list__encode_Opelims,axiom,
% 0.23/0.58 ! [X4: list_nat,Y3: nat] :
% 0.23/0.58 ( ( ( nat_list_encode @ X4 )
% 0.23/0.58 = Y3 )
% 0.23/0.58 => ( ( accp_list_nat @ nat_list_encode_rel @ X4 )
% 0.23/0.58 => ( ( ( X4 = nil_nat )
% 0.23/0.58 => ( ( Y3 = zero_zero_nat )
% 0.23/0.58 => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 0.23/0.58 => ~ ! [X3: nat,Xs3: list_nat] :
% 0.23/0.58 ( ( X4
% 0.23/0.58 = ( cons_nat @ X3 @ Xs3 ) )
% 0.23/0.58 => ( ( Y3
% 0.23/0.58 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) )
% 0.23/0.58 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % list_encode.pelims
% 0.23/0.58 thf(fact_9629_Rats__abs__iff,axiom,
% 0.23/0.58 ! [X4: real] :
% 0.23/0.58 ( ( member_real @ ( abs_abs_real @ X4 ) @ field_5140801741446780682s_real )
% 0.23/0.58 = ( member_real @ X4 @ field_5140801741446780682s_real ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rats_abs_iff
% 0.23/0.58 thf(fact_9630_Rats__no__top__le,axiom,
% 0.23/0.58 ! [X4: real] :
% 0.23/0.58 ? [X3: real] :
% 0.23/0.58 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 0.23/0.58 & ( ord_less_eq_real @ X4 @ X3 ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rats_no_top_le
% 0.23/0.58 thf(fact_9631_Rats__no__bot__less,axiom,
% 0.23/0.58 ! [X4: real] :
% 0.23/0.58 ? [X3: real] :
% 0.23/0.58 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 0.23/0.58 & ( ord_less_real @ X3 @ X4 ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rats_no_bot_less
% 0.23/0.58 thf(fact_9632_Rats__dense__in__real,axiom,
% 0.23/0.58 ! [X4: real,Y3: real] :
% 0.23/0.58 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.58 => ? [X3: real] :
% 0.23/0.58 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 0.23/0.58 & ( ord_less_real @ X4 @ X3 )
% 0.23/0.58 & ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rats_dense_in_real
% 0.23/0.58 thf(fact_9633_Rats__abs__nat__div__natE,axiom,
% 0.23/0.58 ! [X4: real] :
% 0.23/0.58 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 0.23/0.58 => ~ ! [M4: nat,N4: nat] :
% 0.23/0.58 ( ( N4 != zero_zero_nat )
% 0.23/0.58 => ( ( ( abs_abs_real @ X4 )
% 0.23/0.58 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N4 ) ) )
% 0.23/0.58 => ~ ( algebr934650988132801477me_nat @ M4 @ N4 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % Rats_abs_nat_div_natE
% 0.23/0.58 thf(fact_9634_upt__rec__numeral,axiom,
% 0.23/0.58 ! [M: num,N: num] :
% 0.23/0.58 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.23/0.58 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.23/0.58 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 0.23/0.58 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.23/0.58 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.23/0.58 = nil_nat ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upt_rec_numeral
% 0.23/0.58 thf(fact_9635_hd__upt,axiom,
% 0.23/0.58 ! [I: nat,J: nat] :
% 0.23/0.58 ( ( ord_less_nat @ I @ J )
% 0.23/0.58 => ( ( hd_nat @ ( upt @ I @ J ) )
% 0.23/0.58 = I ) ) ).
% 0.23/0.58
% 0.23/0.58 % hd_upt
% 0.23/0.58 thf(fact_9636_drop__upt,axiom,
% 0.23/0.58 ! [M: nat,I: nat,J: nat] :
% 0.23/0.58 ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 0.23/0.58 = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 0.23/0.58
% 0.23/0.58 % drop_upt
% 0.23/0.58 thf(fact_9637_take__upt,axiom,
% 0.23/0.58 ! [I: nat,M: nat,N: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 0.23/0.58 => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 0.23/0.58 = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % take_upt
% 0.23/0.58 thf(fact_9638_upt__conv__Nil,axiom,
% 0.23/0.58 ! [J: nat,I: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ J @ I )
% 0.23/0.58 => ( ( upt @ I @ J )
% 0.23/0.58 = nil_nat ) ) ).
% 0.23/0.58
% 0.23/0.58 % upt_conv_Nil
% 0.23/0.58 thf(fact_9639_length__upt,axiom,
% 0.23/0.58 ! [I: nat,J: nat] :
% 0.23/0.58 ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 0.23/0.58 = ( minus_minus_nat @ J @ I ) ) ).
% 0.23/0.58
% 0.23/0.58 % length_upt
% 0.23/0.58 thf(fact_9640_upt__eq__Nil__conv,axiom,
% 0.23/0.58 ! [I: nat,J: nat] :
% 0.23/0.58 ( ( ( upt @ I @ J )
% 0.23/0.58 = nil_nat )
% 0.23/0.58 = ( ( J = zero_zero_nat )
% 0.23/0.58 | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upt_eq_Nil_conv
% 0.23/0.58 thf(fact_9641_nth__upt,axiom,
% 0.23/0.58 ! [I: nat,K: nat,J: nat] :
% 0.23/0.58 ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 0.23/0.58 => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 0.23/0.58 = ( plus_plus_nat @ I @ K ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % nth_upt
% 0.23/0.58 thf(fact_9642_sum__list__upt,axiom,
% 0.23/0.58 ! [M: nat,N: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ M @ N )
% 0.23/0.58 => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 0.23/0.58 = ( groups3542108847815614940at_nat
% 0.23/0.58 @ ^ [X: nat] : X
% 0.23/0.58 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % sum_list_upt
% 0.23/0.58 thf(fact_9643_atLeastAtMost__upt,axiom,
% 0.23/0.58 ( set_or1269000886237332187st_nat
% 0.23/0.58 = ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M6 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % atLeastAtMost_upt
% 0.23/0.58 thf(fact_9644_upt__conv__Cons__Cons,axiom,
% 0.23/0.58 ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
% 0.23/0.58 ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
% 0.23/0.58 = ( upt @ M @ Q2 ) )
% 0.23/0.58 = ( ( cons_nat @ N @ Ns )
% 0.23/0.58 = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upt_conv_Cons_Cons
% 0.23/0.58 thf(fact_9645_greaterThanAtMost__upt,axiom,
% 0.23/0.58 ( set_or6659071591806873216st_nat
% 0.23/0.58 = ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M6 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % greaterThanAtMost_upt
% 0.23/0.58 thf(fact_9646_greaterThanLessThan__upt,axiom,
% 0.23/0.58 ( set_or5834768355832116004an_nat
% 0.23/0.58 = ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M6 ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % greaterThanLessThan_upt
% 0.23/0.58 thf(fact_9647_atMost__upto,axiom,
% 0.23/0.58 ( set_ord_atMost_nat
% 0.23/0.58 = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % atMost_upto
% 0.23/0.58 thf(fact_9648_upt__conv__Cons,axiom,
% 0.23/0.58 ! [I: nat,J: nat] :
% 0.23/0.58 ( ( ord_less_nat @ I @ J )
% 0.23/0.58 => ( ( upt @ I @ J )
% 0.23/0.58 = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upt_conv_Cons
% 0.23/0.58 thf(fact_9649_upt__add__eq__append,axiom,
% 0.23/0.58 ! [I: nat,J: nat,K: nat] :
% 0.23/0.58 ( ( ord_less_eq_nat @ I @ J )
% 0.23/0.58 => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 0.23/0.58 = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upt_add_eq_append
% 0.23/0.58 thf(fact_9650_upt__eq__Cons__conv,axiom,
% 0.23/0.58 ! [I: nat,J: nat,X4: nat,Xs2: list_nat] :
% 0.23/0.58 ( ( ( upt @ I @ J )
% 0.23/0.58 = ( cons_nat @ X4 @ Xs2 ) )
% 0.23/0.58 = ( ( ord_less_nat @ I @ J )
% 0.23/0.58 & ( I = X4 )
% 0.23/0.58 & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 0.23/0.58 = Xs2 ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upt_eq_Cons_conv
% 0.23/0.58 thf(fact_9651_upt__rec,axiom,
% 0.23/0.58 ( upt
% 0.23/0.58 = ( ^ [I2: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J3 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 0.23/0.58
% 0.23/0.58 % upt_rec
% 0.23/0.58 thf(fact_9652_upt__Suc__append,axiom,
% 0.23/0.58 ! [I: nat,J: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ I @ J )
% 0.23/0.59 => ( ( upt @ I @ ( suc @ J ) )
% 0.23/0.59 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % upt_Suc_append
% 0.23/0.59 thf(fact_9653_upt__Suc,axiom,
% 0.23/0.59 ! [I: nat,J: nat] :
% 0.23/0.59 ( ( ( ord_less_eq_nat @ I @ J )
% 0.23/0.59 => ( ( upt @ I @ ( suc @ J ) )
% 0.23/0.59 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 0.23/0.59 & ( ~ ( ord_less_eq_nat @ I @ J )
% 0.23/0.59 => ( ( upt @ I @ ( suc @ J ) )
% 0.23/0.59 = nil_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % upt_Suc
% 0.23/0.59 thf(fact_9654_mono__Suc,axiom,
% 0.23/0.59 order_mono_nat_nat @ suc ).
% 0.23/0.59
% 0.23/0.59 % mono_Suc
% 0.23/0.59 thf(fact_9655_mono__times__nat,axiom,
% 0.23/0.59 ! [N: nat] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % mono_times_nat
% 0.23/0.59 thf(fact_9656_mono__ge2__power__minus__self,axiom,
% 0.23/0.59 ! [K: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.23/0.59 => ( order_mono_nat_nat
% 0.23/0.59 @ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M6 ) @ M6 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % mono_ge2_power_minus_self
% 0.23/0.59 thf(fact_9657_map__Suc__upt,axiom,
% 0.23/0.59 ! [M: nat,N: nat] :
% 0.23/0.59 ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
% 0.23/0.59 = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % map_Suc_upt
% 0.23/0.59 thf(fact_9658_sorted__upt,axiom,
% 0.23/0.59 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 0.23/0.59
% 0.23/0.59 % sorted_upt
% 0.23/0.59 thf(fact_9659_sorted__wrt__upt,axiom,
% 0.23/0.59 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 0.23/0.59
% 0.23/0.59 % sorted_wrt_upt
% 0.23/0.59 thf(fact_9660_map__add__upt,axiom,
% 0.23/0.59 ! [N: nat,M: nat] :
% 0.23/0.59 ( ( map_nat_nat
% 0.23/0.59 @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N )
% 0.23/0.59 @ ( upt @ zero_zero_nat @ M ) )
% 0.23/0.59 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % map_add_upt
% 0.23/0.59 thf(fact_9661_map__decr__upt,axiom,
% 0.23/0.59 ! [M: nat,N: nat] :
% 0.23/0.59 ( ( map_nat_nat
% 0.23/0.59 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.23/0.59 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.23/0.59 = ( upt @ M @ N ) ) ).
% 0.23/0.59
% 0.23/0.59 % map_decr_upt
% 0.23/0.59 thf(fact_9662_sorted__wrt__less__idx,axiom,
% 0.23/0.59 ! [Ns: list_nat,I: nat] :
% 0.23/0.59 ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 0.23/0.59 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 0.23/0.59 => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % sorted_wrt_less_idx
% 0.23/0.59 thf(fact_9663_sorted__wrt__upto,axiom,
% 0.23/0.59 ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 0.23/0.59
% 0.23/0.59 % sorted_wrt_upto
% 0.23/0.59 thf(fact_9664_Rats__eq__int__div__int,axiom,
% 0.23/0.59 ( field_5140801741446780682s_real
% 0.23/0.59 = ( collect_real
% 0.23/0.59 @ ^ [Uu: real] :
% 0.23/0.59 ? [I2: int,J3: int] :
% 0.23/0.59 ( ( Uu
% 0.23/0.59 = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 0.23/0.59 & ( J3 != zero_zero_int ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Rats_eq_int_div_int
% 0.23/0.59 thf(fact_9665_Rats__eq__int__div__nat,axiom,
% 0.23/0.59 ( field_5140801741446780682s_real
% 0.23/0.59 = ( collect_real
% 0.23/0.59 @ ^ [Uu: real] :
% 0.23/0.59 ? [I2: int,N2: nat] :
% 0.23/0.59 ( ( Uu
% 0.23/0.59 = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.23/0.59 & ( N2 != zero_zero_nat ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Rats_eq_int_div_nat
% 0.23/0.59 thf(fact_9666_range__mod,axiom,
% 0.23/0.59 ! [N: nat] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( image_nat_nat
% 0.23/0.59 @ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N )
% 0.23/0.59 @ top_top_set_nat )
% 0.23/0.59 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % range_mod
% 0.23/0.59 thf(fact_9667_UNIV__nat__eq,axiom,
% 0.23/0.59 ( top_top_set_nat
% 0.23/0.59 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % UNIV_nat_eq
% 0.23/0.59 thf(fact_9668_card__UNIV__unit,axiom,
% 0.23/0.59 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 0.23/0.59 = one_one_nat ) ).
% 0.23/0.59
% 0.23/0.59 % card_UNIV_unit
% 0.23/0.59 thf(fact_9669_range__mult,axiom,
% 0.23/0.59 ! [A: real] :
% 0.23/0.59 ( ( ( A = zero_zero_real )
% 0.23/0.59 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.23/0.59 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 0.23/0.59 & ( ( A != zero_zero_real )
% 0.23/0.59 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.23/0.59 = top_top_set_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % range_mult
% 0.23/0.59 thf(fact_9670_card__UNIV__bool,axiom,
% 0.23/0.59 ( ( finite_card_o @ top_top_set_o )
% 0.23/0.59 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % card_UNIV_bool
% 0.23/0.59 thf(fact_9671_root__def,axiom,
% 0.23/0.59 ( root
% 0.23/0.59 = ( ^ [N2: nat,X: real] :
% 0.23/0.59 ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 0.23/0.59 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 0.23/0.59 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 0.23/0.59 @ X ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % root_def
% 0.23/0.59 thf(fact_9672_card__UNIV__char,axiom,
% 0.23/0.59 ( ( finite_card_char @ top_top_set_char )
% 0.23/0.59 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % card_UNIV_char
% 0.23/0.59 thf(fact_9673_UNIV__char__of__nat,axiom,
% 0.23/0.59 ( top_top_set_char
% 0.23/0.59 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % UNIV_char_of_nat
% 0.23/0.59 thf(fact_9674_nat__of__char__less__256,axiom,
% 0.23/0.59 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % nat_of_char_less_256
% 0.23/0.59 thf(fact_9675_range__nat__of__char,axiom,
% 0.23/0.59 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 0.23/0.59 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % range_nat_of_char
% 0.23/0.59 thf(fact_9676_integer__of__char__code,axiom,
% 0.23/0.59 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 0.23/0.59 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 0.23/0.59 = ( 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 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % integer_of_char_code
% 0.23/0.59 thf(fact_9677_String_Ochar__of__ascii__of,axiom,
% 0.23/0.59 ! [C: char] :
% 0.23/0.59 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 0.23/0.59 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % String.char_of_ascii_of
% 0.23/0.59 thf(fact_9678_DERIV__even__real__root,axiom,
% 0.23/0.59 ! [N: nat,X4: real] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_even_real_root
% 0.23/0.59 thf(fact_9679_DERIV__real__root__generic,axiom,
% 0.23/0.59 ! [N: nat,X4: real,D4: real] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( X4 != zero_zero_real )
% 0.23/0.59 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.59 => ( D4
% 0.23/0.59 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 0.23/0.59 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 0.23/0.59 => ( D4
% 0.23/0.59 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 0.23/0.59 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.59 => ( D4
% 0.23/0.59 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_real_root_generic
% 0.23/0.59 thf(fact_9680_DERIV__neg__dec__right,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,X4: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.23/0.59 => ? [D3: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.23/0.59 & ! [H4: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.23/0.59 => ( ( ord_less_real @ H4 @ D3 )
% 0.23/0.59 => ( ord_less_real @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_neg_dec_right
% 0.23/0.59 thf(fact_9681_DERIV__pos__inc__right,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,X4: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.23/0.59 => ? [D3: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.23/0.59 & ! [H4: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.23/0.59 => ( ( ord_less_real @ H4 @ D3 )
% 0.23/0.59 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_pos_inc_right
% 0.23/0.59 thf(fact_9682_DERIV__pos__imp__increasing,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_eq_real @ X3 @ B2 )
% 0.23/0.59 => ? [Y5: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.23/0.59 => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_pos_imp_increasing
% 0.23/0.59 thf(fact_9683_DERIV__neg__imp__decreasing,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_eq_real @ X3 @ B2 )
% 0.23/0.59 => ? [Y5: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.23/0.59 => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_neg_imp_decreasing
% 0.23/0.59 thf(fact_9684_DERIV__nonpos__imp__nonincreasing,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_eq_real @ X3 @ B2 )
% 0.23/0.59 => ? [Y5: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 0.23/0.59 => ( ord_less_eq_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_nonpos_imp_nonincreasing
% 0.23/0.59 thf(fact_9685_DERIV__nonneg__imp__nondecreasing,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_eq_real @ X3 @ B2 )
% 0.23/0.59 => ? [Y5: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 0.23/0.59 => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_nonneg_imp_nondecreasing
% 0.23/0.59 thf(fact_9686_deriv__nonneg__imp__mono,axiom,
% 0.23/0.59 ! [A: real,B2: real,G: real > real,G2: real > real] :
% 0.23/0.59 ( ! [X3: real] :
% 0.23/0.59 ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 0.23/0.59 => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
% 0.23/0.59 => ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.59 => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % deriv_nonneg_imp_mono
% 0.23/0.59 thf(fact_9687_DERIV__isconst__all,axiom,
% 0.23/0.59 ! [F: real > real,X4: real,Y3: real] :
% 0.23/0.59 ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 => ( ( F @ X4 )
% 0.23/0.59 = ( F @ Y3 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_isconst_all
% 0.23/0.59 thf(fact_9688_DERIV__mirror,axiom,
% 0.23/0.59 ! [F: real > real,Y3: real,X4: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X4 ) @ top_top_set_real ) )
% 0.23/0.59 = ( has_fi5821293074295781190e_real
% 0.23/0.59 @ ^ [X: real] : ( F @ ( uminus_uminus_real @ X ) )
% 0.23/0.59 @ ( uminus_uminus_real @ Y3 )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_mirror
% 0.23/0.59 thf(fact_9689_DERIV__const__ratio__const2,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real,K: real] :
% 0.23/0.59 ( ( A != B2 )
% 0.23/0.59 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ ( minus_minus_real @ B2 @ A ) )
% 0.23/0.59 = K ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_const_ratio_const2
% 0.23/0.59 thf(fact_9690_DERIV__const__ratio__const,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real,K: real] :
% 0.23/0.59 ( ( A != B2 )
% 0.23/0.59 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 0.23/0.59 = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ K ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_const_ratio_const
% 0.23/0.59 thf(fact_9691_DERIV__pos__inc__left,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,X4: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.23/0.59 => ? [D3: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.23/0.59 & ! [H4: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.23/0.59 => ( ( ord_less_real @ H4 @ D3 )
% 0.23/0.59 => ( ord_less_real @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_pos_inc_left
% 0.23/0.59 thf(fact_9692_DERIV__neg__dec__left,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,X4: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.23/0.59 => ? [D3: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.23/0.59 & ! [H4: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.23/0.59 => ( ( ord_less_real @ H4 @ D3 )
% 0.23/0.59 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_neg_dec_left
% 0.23/0.59 thf(fact_9693_DERIV__isconst3,axiom,
% 0.23/0.59 ! [A: real,B2: real,X4: real,Y3: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.23/0.59 => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.23/0.59 => ( ( F @ X4 )
% 0.23/0.59 = ( F @ Y3 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_isconst3
% 0.23/0.59 thf(fact_9694_has__real__derivative__pos__inc__left,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,X4: real,S2: set_real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.23/0.59 => ? [D3: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.23/0.59 & ! [H4: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.23/0.59 => ( ( member_real @ ( minus_minus_real @ X4 @ H4 ) @ S2 )
% 0.23/0.59 => ( ( ord_less_real @ H4 @ D3 )
% 0.23/0.59 => ( ord_less_real @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % has_real_derivative_pos_inc_left
% 0.23/0.59 thf(fact_9695_has__real__derivative__neg__dec__left,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,X4: real,S2: set_real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
% 0.23/0.59 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.23/0.59 => ? [D3: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.23/0.59 & ! [H4: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.23/0.59 => ( ( member_real @ ( minus_minus_real @ X4 @ H4 ) @ S2 )
% 0.23/0.59 => ( ( ord_less_real @ H4 @ D3 )
% 0.23/0.59 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % has_real_derivative_neg_dec_left
% 0.23/0.59 thf(fact_9696_has__real__derivative__pos__inc__right,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,X4: real,S2: set_real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.23/0.59 => ? [D3: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.23/0.59 & ! [H4: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.23/0.59 => ( ( member_real @ ( plus_plus_real @ X4 @ H4 ) @ S2 )
% 0.23/0.59 => ( ( ord_less_real @ H4 @ D3 )
% 0.23/0.59 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % has_real_derivative_pos_inc_right
% 0.23/0.59 thf(fact_9697_has__real__derivative__neg__dec__right,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,X4: real,S2: set_real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
% 0.23/0.59 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.23/0.59 => ? [D3: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.23/0.59 & ! [H4: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.23/0.59 => ( ( member_real @ ( plus_plus_real @ X4 @ H4 ) @ S2 )
% 0.23/0.59 => ( ( ord_less_real @ H4 @ D3 )
% 0.23/0.59 => ( ord_less_real @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % has_real_derivative_neg_dec_right
% 0.23/0.59 thf(fact_9698_MVT2,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real,F4: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_eq_real @ X3 @ B2 )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.23/0.59 => ? [Z2: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ Z2 )
% 0.23/0.59 & ( ord_less_real @ Z2 @ B2 )
% 0.23/0.59 & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 0.23/0.59 = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ ( F4 @ Z2 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % MVT2
% 0.23/0.59 thf(fact_9699_DERIV__local__const,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,X4: real,D: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.23/0.59 => ( ! [Y4: real] :
% 0.23/0.59 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) @ D )
% 0.23/0.59 => ( ( F @ X4 )
% 0.23/0.59 = ( F @ Y4 ) ) )
% 0.23/0.59 => ( L2 = zero_zero_real ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_local_const
% 0.23/0.59 thf(fact_9700_DERIV__ln,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_ln
% 0.23/0.59 thf(fact_9701_DERIV__const__average,axiom,
% 0.23/0.59 ! [A: real,B2: real,V: real > real,K: real] :
% 0.23/0.59 ( ( A != B2 )
% 0.23/0.59 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.23/0.59 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_const_average
% 0.23/0.59 thf(fact_9702_DERIV__local__max,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,X4: real,D: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.23/0.59 => ( ! [Y4: real] :
% 0.23/0.59 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) @ D )
% 0.23/0.59 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 0.23/0.59 => ( L2 = zero_zero_real ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_local_max
% 0.23/0.59 thf(fact_9703_DERIV__local__min,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,X4: real,D: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.23/0.59 => ( ! [Y4: real] :
% 0.23/0.59 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) @ D )
% 0.23/0.59 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.23/0.59 => ( L2 = zero_zero_real ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_local_min
% 0.23/0.59 thf(fact_9704_DERIV__ln__divide,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_ln_divide
% 0.23/0.59 thf(fact_9705_DERIV__pow,axiom,
% 0.23/0.59 ! [N: nat,X4: real,S: set_real] :
% 0.23/0.59 ( has_fi5821293074295781190e_real
% 0.23/0.59 @ ^ [X: real] : ( power_power_real @ X @ N )
% 0.23/0.59 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X4 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ S ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_pow
% 0.23/0.59 thf(fact_9706_DERIV__fun__pow,axiom,
% 0.23/0.59 ! [G: real > real,M: real,X4: real,N: nat] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real
% 0.23/0.59 @ ^ [X: real] : ( power_power_real @ ( G @ X ) @ N )
% 0.23/0.59 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X4 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_fun_pow
% 0.23/0.59 thf(fact_9707_has__real__derivative__powr,axiom,
% 0.23/0.59 ! [Z: real,R2: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.23/0.59 => ( has_fi5821293074295781190e_real
% 0.23/0.59 @ ^ [Z5: real] : ( powr_real @ Z5 @ R2 )
% 0.23/0.59 @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % has_real_derivative_powr
% 0.23/0.59 thf(fact_9708_DERIV__log,axiom,
% 0.23/0.59 ! [X4: real,B2: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( log @ B2 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B2 ) @ X4 ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_log
% 0.23/0.59 thf(fact_9709_DERIV__fun__powr,axiom,
% 0.23/0.59 ! [G: real > real,M: real,X4: real,R2: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ ( G @ X4 ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real
% 0.23/0.59 @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R2 )
% 0.23/0.59 @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X4 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_fun_powr
% 0.23/0.59 thf(fact_9710_DERIV__powr,axiom,
% 0.23/0.59 ! [G: real > real,M: real,X4: real,F: real > real,R2: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ ( G @ X4 ) )
% 0.23/0.59 => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real
% 0.23/0.59 @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
% 0.23/0.59 @ ( times_times_real @ ( powr_real @ ( G @ X4 ) @ ( F @ X4 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X4 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X4 ) ) @ ( G @ X4 ) ) ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_powr
% 0.23/0.59 thf(fact_9711_DERIV__real__sqrt,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_real_sqrt
% 0.23/0.59 thf(fact_9712_DERIV__series_H,axiom,
% 0.23/0.59 ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B2: real,L5: nat > real] :
% 0.23/0.59 ( ! [N4: nat] :
% 0.23/0.59 ( has_fi5821293074295781190e_real
% 0.23/0.59 @ ^ [X: real] : ( F @ X @ N4 )
% 0.23/0.59 @ ( F4 @ X0 @ N4 )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.23/0.59 => ( summable_real @ ( F @ X3 ) ) )
% 0.23/0.59 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.23/0.59 => ( ( summable_real @ ( F4 @ X0 ) )
% 0.23/0.59 => ( ( summable_real @ L5 )
% 0.23/0.59 => ( ! [N4: nat,X3: real,Y4: real] :
% 0.23/0.59 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.23/0.59 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.23/0.59 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N4 ) @ ( F @ Y4 @ N4 ) ) ) @ ( times_times_real @ ( L5 @ N4 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) ) ) ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real
% 0.23/0.59 @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
% 0.23/0.59 @ ( suminf_real @ ( F4 @ X0 ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_series'
% 0.23/0.59 thf(fact_9713_DERIV__arctan,axiom,
% 0.23/0.59 ! [X4: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_arctan
% 0.23/0.59 thf(fact_9714_arsinh__real__has__field__derivative,axiom,
% 0.23/0.59 ! [X4: real,A3: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A3 ) ) ).
% 0.23/0.59
% 0.23/0.59 % arsinh_real_has_field_derivative
% 0.23/0.59 thf(fact_9715_DERIV__real__sqrt__generic,axiom,
% 0.23/0.59 ! [X4: real,D4: real] :
% 0.23/0.59 ( ( X4 != zero_zero_real )
% 0.23/0.59 => ( ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.59 => ( D4
% 0.23/0.59 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.23/0.59 => ( ( ( ord_less_real @ X4 @ zero_zero_real )
% 0.23/0.59 => ( D4
% 0.23/0.59 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_real_sqrt_generic
% 0.23/0.59 thf(fact_9716_arcosh__real__has__field__derivative,axiom,
% 0.23/0.59 ! [X4: real,A3: set_real] :
% 0.23/0.59 ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A3 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % arcosh_real_has_field_derivative
% 0.23/0.59 thf(fact_9717_artanh__real__has__field__derivative,axiom,
% 0.23/0.59 ! [X4: real,A3: set_real] :
% 0.23/0.59 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A3 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % artanh_real_has_field_derivative
% 0.23/0.59 thf(fact_9718_DERIV__power__series_H,axiom,
% 0.23/0.59 ! [R: real,F: nat > real,X0: real] :
% 0.23/0.59 ( ! [X3: real] :
% 0.23/0.59 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.23/0.59 => ( summable_real
% 0.23/0.59 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X3 @ N2 ) ) ) )
% 0.23/0.59 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ R )
% 0.23/0.59 => ( has_fi5821293074295781190e_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( suminf_real
% 0.23/0.59 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) )
% 0.23/0.59 @ ( suminf_real
% 0.23/0.59 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_power_series'
% 0.23/0.59 thf(fact_9719_DERIV__real__root,axiom,
% 0.23/0.59 ! [N: nat,X4: real] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_real_root
% 0.23/0.59 thf(fact_9720_DERIV__arccos,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_arccos
% 0.23/0.59 thf(fact_9721_DERIV__arcsin,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_arcsin
% 0.23/0.59 thf(fact_9722_Maclaurin__all__le__objl,axiom,
% 0.23/0.59 ! [Diff: nat > real > real,F: real > real,X4: real,N: nat] :
% 0.23/0.59 ( ( ( ( Diff @ zero_zero_nat )
% 0.23/0.59 = F )
% 0.23/0.59 & ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.23/0.59 => ? [T2: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 0.23/0.59 & ( ( F @ X4 )
% 0.23/0.59 = ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.59 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Maclaurin_all_le_objl
% 0.23/0.59 thf(fact_9723_Maclaurin__all__le,axiom,
% 0.23/0.59 ! [Diff: nat > real > real,F: real > real,X4: real,N: nat] :
% 0.23/0.59 ( ( ( Diff @ zero_zero_nat )
% 0.23/0.59 = F )
% 0.23/0.59 => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 => ? [T2: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 0.23/0.59 & ( ( F @ X4 )
% 0.23/0.59 = ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.59 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Maclaurin_all_le
% 0.23/0.59 thf(fact_9724_DERIV__odd__real__root,axiom,
% 0.23/0.59 ! [N: nat,X4: real] :
% 0.23/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.59 => ( ( X4 != zero_zero_real )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_odd_real_root
% 0.23/0.59 thf(fact_9725_Maclaurin,axiom,
% 0.23/0.59 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.23/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( ( Diff @ zero_zero_nat )
% 0.23/0.59 = F )
% 0.23/0.59 => ( ! [M4: nat,T2: real] :
% 0.23/0.59 ( ( ( ord_less_nat @ M4 @ N )
% 0.23/0.59 & ( ord_less_eq_real @ zero_zero_real @ T2 )
% 0.23/0.59 & ( ord_less_eq_real @ T2 @ H2 ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 0.23/0.59 => ? [T2: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ T2 )
% 0.23/0.59 & ( ord_less_real @ T2 @ H2 )
% 0.23/0.59 & ( ( F @ H2 )
% 0.23/0.59 = ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.59 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Maclaurin
% 0.23/0.59 thf(fact_9726_Maclaurin2,axiom,
% 0.23/0.59 ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.23/0.59 => ( ( ( Diff @ zero_zero_nat )
% 0.23/0.59 = F )
% 0.23/0.59 => ( ! [M4: nat,T2: real] :
% 0.23/0.59 ( ( ( ord_less_nat @ M4 @ N )
% 0.23/0.59 & ( ord_less_eq_real @ zero_zero_real @ T2 )
% 0.23/0.59 & ( ord_less_eq_real @ T2 @ H2 ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 0.23/0.59 => ? [T2: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ T2 )
% 0.23/0.59 & ( ord_less_eq_real @ T2 @ H2 )
% 0.23/0.59 & ( ( F @ H2 )
% 0.23/0.59 = ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.59 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Maclaurin2
% 0.23/0.59 thf(fact_9727_Maclaurin__minus,axiom,
% 0.23/0.59 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ H2 @ zero_zero_real )
% 0.23/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( ( Diff @ zero_zero_nat )
% 0.23/0.59 = F )
% 0.23/0.59 => ( ! [M4: nat,T2: real] :
% 0.23/0.59 ( ( ( ord_less_nat @ M4 @ N )
% 0.23/0.59 & ( ord_less_eq_real @ H2 @ T2 )
% 0.23/0.59 & ( ord_less_eq_real @ T2 @ zero_zero_real ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 0.23/0.59 => ? [T2: real] :
% 0.23/0.59 ( ( ord_less_real @ H2 @ T2 )
% 0.23/0.59 & ( ord_less_real @ T2 @ zero_zero_real )
% 0.23/0.59 & ( ( F @ H2 )
% 0.23/0.59 = ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.59 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Maclaurin_minus
% 0.23/0.59 thf(fact_9728_Maclaurin__all__lt,axiom,
% 0.23/0.59 ! [Diff: nat > real > real,F: real > real,N: nat,X4: real] :
% 0.23/0.59 ( ( ( Diff @ zero_zero_nat )
% 0.23/0.59 = F )
% 0.23/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( X4 != zero_zero_real )
% 0.23/0.59 => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 => ? [T2: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T2 ) )
% 0.23/0.59 & ( ord_less_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 0.23/0.59 & ( ( F @ X4 )
% 0.23/0.59 = ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.59 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Maclaurin_all_lt
% 0.23/0.59 thf(fact_9729_Maclaurin__bi__le,axiom,
% 0.23/0.59 ! [Diff: nat > real > real,F: real > real,N: nat,X4: real] :
% 0.23/0.59 ( ( ( Diff @ zero_zero_nat )
% 0.23/0.59 = F )
% 0.23/0.59 => ( ! [M4: nat,T2: real] :
% 0.23/0.59 ( ( ( ord_less_nat @ M4 @ N )
% 0.23/0.59 & ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 0.23/0.59 => ? [T2: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 0.23/0.59 & ( ( F @ X4 )
% 0.23/0.59 = ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.59 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Maclaurin_bi_le
% 0.23/0.59 thf(fact_9730_Taylor,axiom,
% 0.23/0.59 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real,X4: real] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( ( Diff @ zero_zero_nat )
% 0.23/0.59 = F )
% 0.23/0.59 => ( ! [M4: nat,T2: real] :
% 0.23/0.59 ( ( ( ord_less_nat @ M4 @ N )
% 0.23/0.59 & ( ord_less_eq_real @ A @ T2 )
% 0.23/0.59 & ( ord_less_eq_real @ T2 @ B2 ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 0.23/0.59 => ( ( ord_less_eq_real @ A @ C )
% 0.23/0.59 => ( ( ord_less_eq_real @ C @ B2 )
% 0.23/0.59 => ( ( ord_less_eq_real @ A @ X4 )
% 0.23/0.59 => ( ( ord_less_eq_real @ X4 @ B2 )
% 0.23/0.59 => ( ( X4 != C )
% 0.23/0.59 => ? [T2: real] :
% 0.23/0.59 ( ( ( ord_less_real @ X4 @ C )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ T2 )
% 0.23/0.59 & ( ord_less_real @ T2 @ C ) ) )
% 0.23/0.59 & ( ~ ( ord_less_real @ X4 @ C )
% 0.23/0.59 => ( ( ord_less_real @ C @ T2 )
% 0.23/0.59 & ( ord_less_real @ T2 @ X4 ) ) )
% 0.23/0.59 & ( ( F @ X4 )
% 0.23/0.59 = ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ C ) @ M6 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.59 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Taylor
% 0.23/0.59 thf(fact_9731_Taylor__up,axiom,
% 0.23/0.59 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( ( Diff @ zero_zero_nat )
% 0.23/0.59 = F )
% 0.23/0.59 => ( ! [M4: nat,T2: real] :
% 0.23/0.59 ( ( ( ord_less_nat @ M4 @ N )
% 0.23/0.59 & ( ord_less_eq_real @ A @ T2 )
% 0.23/0.59 & ( ord_less_eq_real @ T2 @ B2 ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 0.23/0.59 => ( ( ord_less_eq_real @ A @ C )
% 0.23/0.59 => ( ( ord_less_real @ C @ B2 )
% 0.23/0.59 => ? [T2: real] :
% 0.23/0.59 ( ( ord_less_real @ C @ T2 )
% 0.23/0.59 & ( ord_less_real @ T2 @ B2 )
% 0.23/0.59 & ( ( F @ B2 )
% 0.23/0.59 = ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ M6 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.59 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Taylor_up
% 0.23/0.59 thf(fact_9732_Taylor__down,axiom,
% 0.23/0.59 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( ( Diff @ zero_zero_nat )
% 0.23/0.59 = F )
% 0.23/0.59 => ( ! [M4: nat,T2: real] :
% 0.23/0.59 ( ( ( ord_less_nat @ M4 @ N )
% 0.23/0.59 & ( ord_less_eq_real @ A @ T2 )
% 0.23/0.59 & ( ord_less_eq_real @ T2 @ B2 ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 0.23/0.59 => ( ( ord_less_real @ A @ C )
% 0.23/0.59 => ( ( ord_less_eq_real @ C @ B2 )
% 0.23/0.59 => ? [T2: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ T2 )
% 0.23/0.59 & ( ord_less_real @ T2 @ C )
% 0.23/0.59 & ( ( F @ A )
% 0.23/0.59 = ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M6 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ N ) )
% 0.23/0.59 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Taylor_down
% 0.23/0.59 thf(fact_9733_Maclaurin__lemma2,axiom,
% 0.23/0.59 ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B4: real] :
% 0.23/0.59 ( ! [M4: nat,T2: real] :
% 0.23/0.59 ( ( ( ord_less_nat @ M4 @ N )
% 0.23/0.59 & ( ord_less_eq_real @ zero_zero_real @ T2 )
% 0.23/0.59 & ( ord_less_eq_real @ T2 @ H2 ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 0.23/0.59 => ( ( N
% 0.23/0.59 = ( suc @ K ) )
% 0.23/0.59 => ! [M5: nat,T4: real] :
% 0.23/0.59 ( ( ( ord_less_nat @ M5 @ N )
% 0.23/0.59 & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.23/0.59 & ( ord_less_eq_real @ T4 @ H2 ) )
% 0.23/0.59 => ( has_fi5821293074295781190e_real
% 0.23/0.59 @ ^ [U4: real] :
% 0.23/0.59 ( minus_minus_real @ ( Diff @ M5 @ U4 )
% 0.23/0.59 @ ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M5 @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ U4 @ P3 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M5 ) ) )
% 0.23/0.59 @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ U4 @ ( minus_minus_nat @ N @ M5 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M5 ) ) ) ) ) )
% 0.23/0.59 @ ( minus_minus_real @ ( Diff @ ( suc @ M5 ) @ T4 )
% 0.23/0.59 @ ( plus_plus_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M5 ) @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ T4 @ P3 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) )
% 0.23/0.59 @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) ) ) ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Maclaurin_lemma2
% 0.23/0.59 thf(fact_9734_DERIV__arctan__series,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.59 => ( has_fi5821293074295781190e_real
% 0.23/0.59 @ ^ [X9: real] :
% 0.23/0.59 ( suminf_real
% 0.23/0.59 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 0.23/0.59 @ ( suminf_real
% 0.23/0.59 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X4 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_arctan_series
% 0.23/0.59 thf(fact_9735_isCont__Lb__Ub,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ( ord_less_eq_real @ A @ X3 )
% 0.23/0.59 & ( ord_less_eq_real @ X3 @ B2 ) )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 0.23/0.59 => ? [L6: real,M9: real] :
% 0.23/0.59 ( ! [X5: real] :
% 0.23/0.59 ( ( ( ord_less_eq_real @ A @ X5 )
% 0.23/0.59 & ( ord_less_eq_real @ X5 @ B2 ) )
% 0.23/0.59 => ( ( ord_less_eq_real @ L6 @ ( F @ X5 ) )
% 0.23/0.59 & ( ord_less_eq_real @ ( F @ X5 ) @ M9 ) ) )
% 0.23/0.59 & ! [Y5: real] :
% 0.23/0.59 ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 0.23/0.59 & ( ord_less_eq_real @ Y5 @ M9 ) )
% 0.23/0.59 => ? [X3: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ X3 )
% 0.23/0.59 & ( ord_less_eq_real @ X3 @ B2 )
% 0.23/0.59 & ( ( F @ X3 )
% 0.23/0.59 = Y5 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % isCont_Lb_Ub
% 0.23/0.59 thf(fact_9736_LIM__fun__less__zero,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,C: real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.23/0.59 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.23/0.59 => ? [R3: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.23/0.59 & ! [X5: real] :
% 0.23/0.59 ( ( ( X5 != C )
% 0.23/0.59 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 0.23/0.59 => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIM_fun_less_zero
% 0.23/0.59 thf(fact_9737_LIM__fun__not__zero,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,C: real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.23/0.59 => ( ( L2 != zero_zero_real )
% 0.23/0.59 => ? [R3: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.23/0.59 & ! [X5: real] :
% 0.23/0.59 ( ( ( X5 != C )
% 0.23/0.59 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 0.23/0.59 => ( ( F @ X5 )
% 0.23/0.59 != zero_zero_real ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIM_fun_not_zero
% 0.23/0.59 thf(fact_9738_LIM__fun__gt__zero,axiom,
% 0.23/0.59 ! [F: real > real,L2: real,C: real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.23/0.59 => ? [R3: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.23/0.59 & ! [X5: real] :
% 0.23/0.59 ( ( ( X5 != C )
% 0.23/0.59 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 0.23/0.59 => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIM_fun_gt_zero
% 0.23/0.59 thf(fact_9739_isCont__real__sqrt,axiom,
% 0.23/0.59 ! [X4: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ sqrt ) ).
% 0.23/0.59
% 0.23/0.59 % isCont_real_sqrt
% 0.23/0.59 thf(fact_9740_isCont__real__root,axiom,
% 0.23/0.59 ! [X4: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ ( root @ N ) ) ).
% 0.23/0.59
% 0.23/0.59 % isCont_real_root
% 0.23/0.59 thf(fact_9741_continuous__frac,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ~ ( member_real @ X4 @ ring_1_Ints_real )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ archim2898591450579166408c_real ) ) ).
% 0.23/0.59
% 0.23/0.59 % continuous_frac
% 0.23/0.59 thf(fact_9742_isCont__inverse__function2,axiom,
% 0.23/0.59 ! [A: real,X4: real,B2: real,G: real > real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ X4 )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ B2 )
% 0.23/0.59 => ( ! [Z2: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ Z2 )
% 0.23/0.59 => ( ( ord_less_eq_real @ Z2 @ B2 )
% 0.23/0.59 => ( ( G @ ( F @ Z2 ) )
% 0.23/0.59 = Z2 ) ) )
% 0.23/0.59 => ( ! [Z2: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ Z2 )
% 0.23/0.59 => ( ( ord_less_eq_real @ Z2 @ B2 )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X4 ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % isCont_inverse_function2
% 0.23/0.59 thf(fact_9743_isCont__arcosh,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arcosh_real ) ) ).
% 0.23/0.59
% 0.23/0.59 % isCont_arcosh
% 0.23/0.59 thf(fact_9744_LIM__cos__div__sin,axiom,
% 0.23/0.59 ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIM_cos_div_sin
% 0.23/0.59 thf(fact_9745_continuous__floor,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ~ ( member_real @ X4 @ ring_1_Ints_real )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ ( comp_int_real_real @ ring_1_of_int_real @ archim6058952711729229775r_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % continuous_floor
% 0.23/0.59 thf(fact_9746_DERIV__inverse__function,axiom,
% 0.23/0.59 ! [F: real > real,D4: real,G: real > real,X4: real,A: real,B2: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X4 ) @ top_top_set_real ) )
% 0.23/0.59 => ( ( D4 != zero_zero_real )
% 0.23/0.59 => ( ( ord_less_real @ A @ X4 )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ B2 )
% 0.23/0.59 => ( ! [Y4: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ Y4 )
% 0.23/0.59 => ( ( ord_less_real @ Y4 @ B2 )
% 0.23/0.59 => ( ( F @ ( G @ Y4 ) )
% 0.23/0.59 = Y4 ) ) )
% 0.23/0.59 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_inverse_function
% 0.23/0.59 thf(fact_9747_isCont__arccos,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arccos ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % isCont_arccos
% 0.23/0.59 thf(fact_9748_isCont__arcsin,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arcsin ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % isCont_arcsin
% 0.23/0.59 thf(fact_9749_LIM__less__bound,axiom,
% 0.23/0.59 ! [B2: real,X4: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ B2 @ X4 )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B2 @ X4 ) )
% 0.23/0.59 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 0.23/0.59 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F )
% 0.23/0.59 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIM_less_bound
% 0.23/0.59 thf(fact_9750_isCont__artanh,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ artanh_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % isCont_artanh
% 0.23/0.59 thf(fact_9751_isCont__inverse__function,axiom,
% 0.23/0.59 ! [D: real,X4: real,G: real > real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ D )
% 0.23/0.59 => ( ! [Z2: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X4 ) ) @ D )
% 0.23/0.59 => ( ( G @ ( F @ Z2 ) )
% 0.23/0.59 = Z2 ) )
% 0.23/0.59 => ( ! [Z2: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X4 ) ) @ D )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X4 ) @ top_top_set_real ) @ G ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % isCont_inverse_function
% 0.23/0.59 thf(fact_9752_GMVT_H,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ! [Z2: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ Z2 )
% 0.23/0.59 => ( ( ord_less_eq_real @ Z2 @ B2 )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 0.23/0.59 => ( ! [Z2: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ Z2 )
% 0.23/0.59 => ( ( ord_less_eq_real @ Z2 @ B2 )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
% 0.23/0.59 => ( ! [Z2: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ Z2 )
% 0.23/0.59 => ( ( ord_less_real @ Z2 @ B2 )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 0.23/0.59 => ( ! [Z2: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ Z2 )
% 0.23/0.59 => ( ( ord_less_real @ Z2 @ B2 )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 0.23/0.59 => ? [C3: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ C3 )
% 0.23/0.59 & ( ord_less_real @ C3 @ B2 )
% 0.23/0.59 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 0.23/0.59 = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ ( F4 @ C3 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % GMVT'
% 0.23/0.59 thf(fact_9753_summable__Leibniz_I2_J,axiom,
% 0.23/0.59 ! [A: nat > real] :
% 0.23/0.59 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.23/0.59 => ( ( topolo6980174941875973593q_real @ A )
% 0.23/0.59 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 0.23/0.59 => ! [N6: nat] :
% 0.23/0.59 ( member_real
% 0.23/0.59 @ ( suminf_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 0.23/0.59 @ ( set_or1222579329274155063t_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % summable_Leibniz(2)
% 0.23/0.59 thf(fact_9754_summable__Leibniz_I3_J,axiom,
% 0.23/0.59 ! [A: nat > real] :
% 0.23/0.59 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.23/0.59 => ( ( topolo6980174941875973593q_real @ A )
% 0.23/0.59 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 0.23/0.59 => ! [N6: nat] :
% 0.23/0.59 ( member_real
% 0.23/0.59 @ ( suminf_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 0.23/0.59 @ ( set_or1222579329274155063t_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % summable_Leibniz(3)
% 0.23/0.59 thf(fact_9755_incseq__convergent,axiom,
% 0.23/0.59 ! [X8: nat > real,B4: real] :
% 0.23/0.59 ( ( order_mono_nat_real @ X8 )
% 0.23/0.59 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X8 @ I3 ) @ B4 )
% 0.23/0.59 => ~ ! [L6: real] :
% 0.23/0.59 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.23/0.59 => ~ ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ L6 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % incseq_convergent
% 0.23/0.59 thf(fact_9756_mult__nat__left__at__top,axiom,
% 0.23/0.59 ! [C: nat] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.23/0.59 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % mult_nat_left_at_top
% 0.23/0.59 thf(fact_9757_mult__nat__right__at__top,axiom,
% 0.23/0.59 ! [C: nat] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.23/0.59 => ( filterlim_nat_nat
% 0.23/0.59 @ ^ [X: nat] : ( times_times_nat @ X @ C )
% 0.23/0.59 @ at_top_nat
% 0.23/0.59 @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % mult_nat_right_at_top
% 0.23/0.59 thf(fact_9758_monoseq__convergent,axiom,
% 0.23/0.59 ! [X8: nat > real,B4: real] :
% 0.23/0.59 ( ( topolo6980174941875973593q_real @ X8 )
% 0.23/0.59 => ( ! [I3: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I3 ) ) @ B4 )
% 0.23/0.59 => ~ ! [L6: real] :
% 0.23/0.59 ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % monoseq_convergent
% 0.23/0.59 thf(fact_9759_LIMSEQ__root,axiom,
% 0.23/0.59 ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.23/0.59 @ at_top_nat ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_root
% 0.23/0.59 thf(fact_9760_nested__sequence__unique,axiom,
% 0.23/0.59 ! [F: nat > real,G: nat > real] :
% 0.23/0.59 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N4 ) ) @ ( G @ N4 ) )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.23/0.59 => ( ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.23/0.59 @ at_top_nat )
% 0.23/0.59 => ? [L3: real] :
% 0.23/0.59 ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L3 )
% 0.23/0.59 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat )
% 0.23/0.59 & ! [N6: nat] : ( ord_less_eq_real @ L3 @ ( G @ N6 ) )
% 0.23/0.59 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % nested_sequence_unique
% 0.23/0.59 thf(fact_9761_LIMSEQ__inverse__zero,axiom,
% 0.23/0.59 ! [X8: nat > real] :
% 0.23/0.59 ( ! [R3: real] :
% 0.23/0.59 ? [N7: nat] :
% 0.23/0.59 ! [N4: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ N7 @ N4 )
% 0.23/0.59 => ( ord_less_real @ R3 @ ( X8 @ N4 ) ) )
% 0.23/0.59 => ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( inverse_inverse_real @ ( X8 @ N2 ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.23/0.59 @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_inverse_zero
% 0.23/0.59 thf(fact_9762_lim__inverse__n_H,axiom,
% 0.23/0.59 ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.23/0.59 @ at_top_nat ) ).
% 0.23/0.59
% 0.23/0.59 % lim_inverse_n'
% 0.23/0.59 thf(fact_9763_LIMSEQ__root__const,axiom,
% 0.23/0.59 ! [C: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ C )
% 0.23/0.59 => ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( root @ N2 @ C )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.23/0.59 @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_root_const
% 0.23/0.59 thf(fact_9764_LIMSEQ__inverse__real__of__nat,axiom,
% 0.23/0.59 ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.23/0.59 @ at_top_nat ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_inverse_real_of_nat
% 0.23/0.59 thf(fact_9765_LIMSEQ__inverse__real__of__nat__add,axiom,
% 0.23/0.59 ! [R2: real] :
% 0.23/0.59 ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ R2 )
% 0.23/0.59 @ at_top_nat ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_inverse_real_of_nat_add
% 0.23/0.59 thf(fact_9766_increasing__LIMSEQ,axiom,
% 0.23/0.59 ! [F: nat > real,L2: real] :
% 0.23/0.59 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ L2 )
% 0.23/0.59 => ( ! [E2: real] :
% 0.23/0.59 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.23/0.59 => ? [N6: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N6 ) @ E2 ) ) )
% 0.23/0.59 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % increasing_LIMSEQ
% 0.23/0.59 thf(fact_9767_LIMSEQ__realpow__zero,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ one_one_real )
% 0.23/0.59 => ( filterlim_nat_real @ ( power_power_real @ X4 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_realpow_zero
% 0.23/0.59 thf(fact_9768_LIMSEQ__divide__realpow__zero,axiom,
% 0.23/0.59 ! [X4: real,A: real] :
% 0.23/0.59 ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.59 => ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X4 @ N2 ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.23/0.59 @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_divide_realpow_zero
% 0.23/0.59 thf(fact_9769_LIMSEQ__abs__realpow__zero,axiom,
% 0.23/0.59 ! [C: real] :
% 0.23/0.59 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.23/0.59 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_abs_realpow_zero
% 0.23/0.59 thf(fact_9770_LIMSEQ__abs__realpow__zero2,axiom,
% 0.23/0.59 ! [C: real] :
% 0.23/0.59 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.23/0.59 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_abs_realpow_zero2
% 0.23/0.59 thf(fact_9771_LIMSEQ__inverse__realpow__zero,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_real @ one_one_real @ X4 )
% 0.23/0.59 => ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X4 @ N2 ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.23/0.59 @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_inverse_realpow_zero
% 0.23/0.59 thf(fact_9772_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 0.23/0.59 ! [R2: real] :
% 0.23/0.59 ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ R2 )
% 0.23/0.59 @ at_top_nat ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_inverse_real_of_nat_add_minus
% 0.23/0.59 thf(fact_9773_tendsto__exp__limit__sequentially,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
% 0.23/0.59 @ at_top_nat ) ).
% 0.23/0.59
% 0.23/0.59 % tendsto_exp_limit_sequentially
% 0.23/0.59 thf(fact_9774_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 0.23/0.59 ! [R2: real] :
% 0.23/0.59 ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ R2 )
% 0.23/0.59 @ at_top_nat ) ).
% 0.23/0.59
% 0.23/0.59 % LIMSEQ_inverse_real_of_nat_add_minus_mult
% 0.23/0.59 thf(fact_9775_summable__Leibniz_I1_J,axiom,
% 0.23/0.59 ! [A: nat > real] :
% 0.23/0.59 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.23/0.59 => ( ( topolo6980174941875973593q_real @ A )
% 0.23/0.59 => ( summable_real
% 0.23/0.59 @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % summable_Leibniz(1)
% 0.23/0.59 thf(fact_9776_summable,axiom,
% 0.23/0.59 ! [A: nat > real] :
% 0.23/0.59 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.23/0.59 => ( summable_real
% 0.23/0.59 @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % summable
% 0.23/0.59 thf(fact_9777_cos__diff__limit__1,axiom,
% 0.23/0.59 ! [Theta: nat > real,Theta2: real] :
% 0.23/0.59 ( ( filterlim_nat_real
% 0.23/0.59 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.23/0.59 @ at_top_nat )
% 0.23/0.59 => ~ ! [K2: nat > int] :
% 0.23/0.59 ~ ( filterlim_nat_real
% 0.23/0.59 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ Theta2 )
% 0.23/0.59 @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % cos_diff_limit_1
% 0.23/0.59 thf(fact_9778_cos__limit__1,axiom,
% 0.23/0.59 ! [Theta: nat > real] :
% 0.23/0.59 ( ( filterlim_nat_real
% 0.23/0.59 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.23/0.59 @ at_top_nat )
% 0.23/0.59 => ? [K2: nat > int] :
% 0.23/0.59 ( filterlim_nat_real
% 0.23/0.59 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.23/0.59 @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % cos_limit_1
% 0.23/0.59 thf(fact_9779_summable__Leibniz_I4_J,axiom,
% 0.23/0.59 ! [A: nat > real] :
% 0.23/0.59 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.23/0.59 => ( ( topolo6980174941875973593q_real @ A )
% 0.23/0.59 => ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] :
% 0.23/0.59 ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real
% 0.23/0.59 @ ( suminf_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 0.23/0.59 @ at_top_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % summable_Leibniz(4)
% 0.23/0.59 thf(fact_9780_zeroseq__arctan__series,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 0.23/0.59 => ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.23/0.59 @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % zeroseq_arctan_series
% 0.23/0.59 thf(fact_9781_summable__Leibniz_H_I2_J,axiom,
% 0.23/0.59 ! [A: nat > real,N: nat] :
% 0.23/0.59 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.23/0.59 => ( ord_less_eq_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.59 @ ( suminf_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % summable_Leibniz'(2)
% 0.23/0.59 thf(fact_9782_summable__Leibniz_H_I3_J,axiom,
% 0.23/0.59 ! [A: nat > real] :
% 0.23/0.59 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.23/0.59 => ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] :
% 0.23/0.59 ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real
% 0.23/0.59 @ ( suminf_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 0.23/0.59 @ at_top_nat ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % summable_Leibniz'(3)
% 0.23/0.59 thf(fact_9783_sums__alternating__upper__lower,axiom,
% 0.23/0.59 ! [A: nat > real] :
% 0.23/0.59 ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.23/0.59 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.23/0.59 => ? [L3: real] :
% 0.23/0.59 ( ! [N6: nat] :
% 0.23/0.59 ( ord_less_eq_real
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 0.23/0.59 @ L3 )
% 0.23/0.59 & ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] :
% 0.23/0.59 ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ L3 )
% 0.23/0.59 @ at_top_nat )
% 0.23/0.59 & ! [N6: nat] :
% 0.23/0.59 ( ord_less_eq_real @ L3
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
% 0.23/0.59 & ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] :
% 0.23/0.59 ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ L3 )
% 0.23/0.59 @ at_top_nat ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % sums_alternating_upper_lower
% 0.23/0.59 thf(fact_9784_summable__Leibniz_I5_J,axiom,
% 0.23/0.59 ! [A: nat > real] :
% 0.23/0.59 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.23/0.59 => ( ( topolo6980174941875973593q_real @ A )
% 0.23/0.59 => ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] :
% 0.23/0.59 ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real
% 0.23/0.59 @ ( suminf_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 0.23/0.59 @ at_top_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % summable_Leibniz(5)
% 0.23/0.59 thf(fact_9785_summable__Leibniz_H_I5_J,axiom,
% 0.23/0.59 ! [A: nat > real] :
% 0.23/0.59 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.23/0.59 => ( filterlim_nat_real
% 0.23/0.59 @ ^ [N2: nat] :
% 0.23/0.59 ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real
% 0.23/0.59 @ ( suminf_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 0.23/0.59 @ at_top_nat ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % summable_Leibniz'(5)
% 0.23/0.59 thf(fact_9786_summable__Leibniz_H_I4_J,axiom,
% 0.23/0.59 ! [A: nat > real,N: nat] :
% 0.23/0.59 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.23/0.59 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.23/0.59 => ( ord_less_eq_real
% 0.23/0.59 @ ( suminf_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 0.23/0.59 @ ( groups6591440286371151544t_real
% 0.23/0.59 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.23/0.59 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % summable_Leibniz'(4)
% 0.23/0.59 thf(fact_9787_eventually__sequentially__Suc,axiom,
% 0.23/0.59 ! [P: nat > $o] :
% 0.23/0.59 ( ( eventually_nat
% 0.23/0.59 @ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
% 0.23/0.59 @ at_top_nat )
% 0.23/0.59 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % eventually_sequentially_Suc
% 0.23/0.59 thf(fact_9788_eventually__sequentially__seg,axiom,
% 0.23/0.59 ! [P: nat > $o,K: nat] :
% 0.23/0.59 ( ( eventually_nat
% 0.23/0.59 @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
% 0.23/0.59 @ at_top_nat )
% 0.23/0.59 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % eventually_sequentially_seg
% 0.23/0.59 thf(fact_9789_sequentially__offset,axiom,
% 0.23/0.59 ! [P: nat > $o,K: nat] :
% 0.23/0.59 ( ( eventually_nat @ P @ at_top_nat )
% 0.23/0.59 => ( eventually_nat
% 0.23/0.59 @ ^ [I2: nat] : ( P @ ( plus_plus_nat @ I2 @ K ) )
% 0.23/0.59 @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % sequentially_offset
% 0.23/0.59 thf(fact_9790_eventually__sequentially,axiom,
% 0.23/0.59 ! [P: nat > $o] :
% 0.23/0.59 ( ( eventually_nat @ P @ at_top_nat )
% 0.23/0.59 = ( ? [N8: nat] :
% 0.23/0.59 ! [N2: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ N8 @ N2 )
% 0.23/0.59 => ( P @ N2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % eventually_sequentially
% 0.23/0.59 thf(fact_9791_eventually__sequentiallyI,axiom,
% 0.23/0.59 ! [C: nat,P: nat > $o] :
% 0.23/0.59 ( ! [X3: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ C @ X3 )
% 0.23/0.59 => ( P @ X3 ) )
% 0.23/0.59 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % eventually_sequentiallyI
% 0.23/0.59 thf(fact_9792_le__sequentially,axiom,
% 0.23/0.59 ! [F5: filter_nat] :
% 0.23/0.59 ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
% 0.23/0.59 = ( ! [N8: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N8 ) @ F5 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % le_sequentially
% 0.23/0.59 thf(fact_9793_filterlim__Suc,axiom,
% 0.23/0.59 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 0.23/0.59
% 0.23/0.59 % filterlim_Suc
% 0.23/0.59 thf(fact_9794_real__bounded__linear,axiom,
% 0.23/0.59 ( real_V5970128139526366754l_real
% 0.23/0.59 = ( ^ [F2: real > real] :
% 0.23/0.59 ? [C2: real] :
% 0.23/0.59 ( F2
% 0.23/0.59 = ( ^ [X: real] : ( times_times_real @ X @ C2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % real_bounded_linear
% 0.23/0.59 thf(fact_9795_sqrt__at__top,axiom,
% 0.23/0.59 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 0.23/0.59
% 0.23/0.59 % sqrt_at_top
% 0.23/0.59 thf(fact_9796_lhopital__left__at__top__at__top,axiom,
% 0.23/0.59 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.23/0.59 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ at_top_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ at_top_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_left_at_top_at_top
% 0.23/0.59 thf(fact_9797_lhopital__at__top__at__top,axiom,
% 0.23/0.59 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.23/0.59 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ at_top_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ at_top_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_at_top_at_top
% 0.23/0.59 thf(fact_9798_eventually__at__left__real,axiom,
% 0.23/0.59 ! [B2: real,A: real] :
% 0.23/0.59 ( ( ord_less_real @ B2 @ A )
% 0.23/0.59 => ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B2 @ A ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % eventually_at_left_real
% 0.23/0.59 thf(fact_9799_lhopital__left__at__top,axiom,
% 0.23/0.59 ! [G: real > real,X4: real,G2: real > real,F: real > real,F4: real > real,Y3: real] :
% 0.23/0.59 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G2 @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_left_at_top
% 0.23/0.59 thf(fact_9800_lhospital__at__top__at__top,axiom,
% 0.23/0.59 ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X4: real] :
% 0.23/0.59 ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G2 @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ at_top_real )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ at_top_real )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ at_top_real )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ X4 )
% 0.23/0.59 @ at_top_real )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ X4 )
% 0.23/0.59 @ at_top_real ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhospital_at_top_at_top
% 0.23/0.59 thf(fact_9801_lhopital__at__top,axiom,
% 0.23/0.59 ! [G: real > real,X4: real,G2: real > real,F: real > real,F4: real > real,Y3: real] :
% 0.23/0.59 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G2 @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_at_top
% 0.23/0.59 thf(fact_9802_tanh__real__at__top,axiom,
% 0.23/0.59 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 0.23/0.59
% 0.23/0.59 % tanh_real_at_top
% 0.23/0.59 thf(fact_9803_artanh__real__at__left__1,axiom,
% 0.23/0.59 filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 0.23/0.59
% 0.23/0.59 % artanh_real_at_left_1
% 0.23/0.59 thf(fact_9804_ln__x__over__x__tendsto__0,axiom,
% 0.23/0.59 ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ X )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.23/0.59 @ at_top_real ) ).
% 0.23/0.59
% 0.23/0.59 % ln_x_over_x_tendsto_0
% 0.23/0.59 thf(fact_9805_tendsto__power__div__exp__0,axiom,
% 0.23/0.59 ! [K: nat] :
% 0.23/0.59 ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( power_power_real @ X @ K ) @ ( exp_real @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.23/0.59 @ at_top_real ) ).
% 0.23/0.59
% 0.23/0.59 % tendsto_power_div_exp_0
% 0.23/0.59 thf(fact_9806_lhopital__left,axiom,
% 0.23/0.59 ! [F: real > real,X4: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G2 @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ F5
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ F5
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_left
% 0.23/0.59 thf(fact_9807_lhopital,axiom,
% 0.23/0.59 ! [F: real > real,X4: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G2 @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ F5
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ F5
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital
% 0.23/0.59 thf(fact_9808_tendsto__exp__limit__at__top,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( filterlim_real_real
% 0.23/0.59 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ Y ) ) @ Y )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
% 0.23/0.59 @ at_top_real ) ).
% 0.23/0.59
% 0.23/0.59 % tendsto_exp_limit_at_top
% 0.23/0.59 thf(fact_9809_filterlim__tan__at__left,axiom,
% 0.23/0.59 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 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % filterlim_tan_at_left
% 0.23/0.59 thf(fact_9810_DERIV__neg__imp__decreasing__at__top,axiom,
% 0.23/0.59 ! [B2: real,F: real > real,Flim: real] :
% 0.23/0.59 ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ B2 @ X3 )
% 0.23/0.59 => ? [Y5: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 0.23/0.59 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 0.23/0.59 => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_neg_imp_decreasing_at_top
% 0.23/0.59 thf(fact_9811_tendsto__arctan__at__top,axiom,
% 0.23/0.59 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 0.23/0.59
% 0.23/0.59 % tendsto_arctan_at_top
% 0.23/0.59 thf(fact_9812_filterlim__pow__at__bot__even,axiom,
% 0.23/0.59 ! [N: nat,F: real > real,F5: filter_real] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 0.23/0.59 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 0.23/0.59 @ at_top_real
% 0.23/0.59 @ F5 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % filterlim_pow_at_bot_even
% 0.23/0.59 thf(fact_9813_dist__real__def,axiom,
% 0.23/0.59 ( real_V975177566351809787t_real
% 0.23/0.59 = ( ^ [X: real,Y: real] : ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % dist_real_def
% 0.23/0.59 thf(fact_9814_dist__complex__def,axiom,
% 0.23/0.59 ( real_V3694042436643373181omplex
% 0.23/0.59 = ( ^ [X: complex,Y: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % dist_complex_def
% 0.23/0.59 thf(fact_9815_tanh__real__at__bot,axiom,
% 0.23/0.59 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 0.23/0.59
% 0.23/0.59 % tanh_real_at_bot
% 0.23/0.59 thf(fact_9816_lhopital__at__top__at__bot,axiom,
% 0.23/0.59 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.23/0.59 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ at_bot_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ at_bot_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_at_top_at_bot
% 0.23/0.59 thf(fact_9817_DERIV__pos__imp__increasing__at__bot,axiom,
% 0.23/0.59 ! [B2: real,F: real > real,Flim: real] :
% 0.23/0.59 ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ X3 @ B2 )
% 0.23/0.59 => ? [Y5: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 0.23/0.59 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 0.23/0.59 => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_pos_imp_increasing_at_bot
% 0.23/0.59 thf(fact_9818_filterlim__pow__at__bot__odd,axiom,
% 0.23/0.59 ! [N: nat,F: real > real,F5: filter_real] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 0.23/0.59 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 0.23/0.59 @ at_bot_real
% 0.23/0.59 @ F5 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % filterlim_pow_at_bot_odd
% 0.23/0.59 thf(fact_9819_lhopital__left__at__top__at__bot,axiom,
% 0.23/0.59 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.23/0.59 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ at_bot_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ at_bot_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_left_at_top_at_bot
% 0.23/0.59 thf(fact_9820_tendsto__arctan__at__bot,axiom,
% 0.23/0.59 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 0.23/0.59
% 0.23/0.59 % tendsto_arctan_at_bot
% 0.23/0.59 thf(fact_9821_incseq__bounded,axiom,
% 0.23/0.59 ! [X8: nat > real,B4: real] :
% 0.23/0.59 ( ( order_mono_nat_real @ X8 )
% 0.23/0.59 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X8 @ I3 ) @ B4 )
% 0.23/0.59 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % incseq_bounded
% 0.23/0.59 thf(fact_9822_Bseq__realpow,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.59 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.59 => ( bfun_nat_real @ ( power_power_real @ X4 ) @ at_top_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Bseq_realpow
% 0.23/0.59 thf(fact_9823_tendsto__exp__limit__at__right,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( filterlim_real_real
% 0.23/0.59 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X4 @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % tendsto_exp_limit_at_right
% 0.23/0.59 thf(fact_9824_filterlim__tan__at__right,axiom,
% 0.23/0.59 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 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % filterlim_tan_at_right
% 0.23/0.59 thf(fact_9825_eventually__at__right__to__0,axiom,
% 0.23/0.59 ! [P: real > $o,A: real] :
% 0.23/0.59 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.59 = ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( P @ ( plus_plus_real @ X @ A ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % eventually_at_right_to_0
% 0.23/0.59 thf(fact_9826_eventually__at__right__real,axiom,
% 0.23/0.59 ! [A: real,B2: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % eventually_at_right_real
% 0.23/0.59 thf(fact_9827_tendsto__arcosh__at__left__1,axiom,
% 0.23/0.59 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 0.23/0.59
% 0.23/0.59 % tendsto_arcosh_at_left_1
% 0.23/0.59 thf(fact_9828_artanh__real__at__right__1,axiom,
% 0.23/0.59 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 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % artanh_real_at_right_1
% 0.23/0.59 thf(fact_9829_lhopital__right__at__top__at__top,axiom,
% 0.23/0.59 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.59 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ at_top_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ at_top_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_right_at_top_at_top
% 0.23/0.59 thf(fact_9830_lhopital__right__0,axiom,
% 0.23/0.59 ! [F0: real > real,G0: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 0.23/0.59 ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G0 @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G2 @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ F5
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F0 @ X ) @ ( G0 @ X ) )
% 0.23/0.59 @ F5
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_right_0
% 0.23/0.59 thf(fact_9831_lhopital__right,axiom,
% 0.23/0.59 ! [F: real > real,X4: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G2 @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ F5
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ F5
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_right
% 0.23/0.59 thf(fact_9832_lhopital__right__at__top__at__bot,axiom,
% 0.23/0.59 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.23/0.59 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.59 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ at_bot_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ at_bot_real
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_right_at_top_at_bot
% 0.23/0.59 thf(fact_9833_lhopital__right__at__top,axiom,
% 0.23/0.59 ! [G: real > real,X4: real,G2: real > real,F: real > real,F4: real > real,Y3: real] :
% 0.23/0.59 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G2 @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_right_at_top
% 0.23/0.59 thf(fact_9834_lhopital__right__0__at__top,axiom,
% 0.23/0.59 ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X4: real] :
% 0.23/0.59 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] :
% 0.23/0.59 ( ( G2 @ X )
% 0.23/0.59 != zero_zero_real )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( ( eventually_real
% 0.23/0.59 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ X4 )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.23/0.59 => ( filterlim_real_real
% 0.23/0.59 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 0.23/0.59 @ ( topolo2815343760600316023s_real @ X4 )
% 0.23/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % lhopital_right_0_at_top
% 0.23/0.59 thf(fact_9835_atLeast__0,axiom,
% 0.23/0.59 ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 0.23/0.59 = top_top_set_nat ) ).
% 0.23/0.59
% 0.23/0.59 % atLeast_0
% 0.23/0.59 thf(fact_9836_atLeast__Suc__greaterThan,axiom,
% 0.23/0.59 ! [K: nat] :
% 0.23/0.59 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.23/0.59 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 0.23/0.59
% 0.23/0.59 % atLeast_Suc_greaterThan
% 0.23/0.59 thf(fact_9837_decseq__bounded,axiom,
% 0.23/0.59 ! [X8: nat > real,B4: real] :
% 0.23/0.59 ( ( order_9091379641038594480t_real @ X8 )
% 0.23/0.59 => ( ! [I3: nat] : ( ord_less_eq_real @ B4 @ ( X8 @ I3 ) )
% 0.23/0.59 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % decseq_bounded
% 0.23/0.59 thf(fact_9838_greaterThan__0,axiom,
% 0.23/0.59 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 0.23/0.59 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % greaterThan_0
% 0.23/0.59 thf(fact_9839_greaterThan__Suc,axiom,
% 0.23/0.59 ! [K: nat] :
% 0.23/0.59 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 0.23/0.59 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % greaterThan_Suc
% 0.23/0.59 thf(fact_9840_decseq__convergent,axiom,
% 0.23/0.59 ! [X8: nat > real,B4: real] :
% 0.23/0.59 ( ( order_9091379641038594480t_real @ X8 )
% 0.23/0.59 => ( ! [I3: nat] : ( ord_less_eq_real @ B4 @ ( X8 @ I3 ) )
% 0.23/0.59 => ~ ! [L6: real] :
% 0.23/0.59 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.23/0.59 => ~ ! [I4: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I4 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % decseq_convergent
% 0.23/0.59 thf(fact_9841_atLeast__Suc,axiom,
% 0.23/0.59 ! [K: nat] :
% 0.23/0.59 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.23/0.59 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % atLeast_Suc
% 0.23/0.59 thf(fact_9842_GMVT,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real,G: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ( ord_less_eq_real @ A @ X3 )
% 0.23/0.59 & ( ord_less_eq_real @ X3 @ B2 ) )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ( ord_less_real @ A @ X3 )
% 0.23/0.59 & ( ord_less_real @ X3 @ B2 ) )
% 0.23/0.59 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ( ord_less_eq_real @ A @ X3 )
% 0.23/0.59 & ( ord_less_eq_real @ X3 @ B2 ) )
% 0.23/0.59 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ( ord_less_real @ A @ X3 )
% 0.23/0.59 & ( ord_less_real @ X3 @ B2 ) )
% 0.23/0.59 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.23/0.59 => ? [G_c: real,F_c: real,C3: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.23/0.59 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.23/0.59 & ( ord_less_real @ A @ C3 )
% 0.23/0.59 & ( ord_less_real @ C3 @ B2 )
% 0.23/0.59 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ G_c )
% 0.23/0.59 = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % GMVT
% 0.23/0.59 thf(fact_9843_real__differentiableE,axiom,
% 0.23/0.59 ! [F: real > real,X4: real,S: set_real] :
% 0.23/0.59 ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ S ) )
% 0.23/0.59 => ~ ! [Df: real] :
% 0.23/0.59 ~ ( has_fi5821293074295781190e_real @ F @ Df @ ( topolo2177554685111907308n_real @ X4 @ S ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % real_differentiableE
% 0.23/0.59 thf(fact_9844_real__differentiable__def,axiom,
% 0.23/0.59 ! [F: real > real,X4: real,S: set_real] :
% 0.23/0.59 ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ S ) )
% 0.23/0.59 = ( ? [D6: real] : ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ X4 @ S ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % real_differentiable_def
% 0.23/0.59 thf(fact_9845_MVT,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_real @ X3 @ B2 )
% 0.23/0.59 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.23/0.59 => ? [L3: real,Z2: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ Z2 )
% 0.23/0.59 & ( ord_less_real @ Z2 @ B2 )
% 0.23/0.59 & ( has_fi5821293074295781190e_real @ F @ L3 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
% 0.23/0.59 & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 0.23/0.59 = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ L3 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % MVT
% 0.23/0.59 thf(fact_9846_suminf__eq__SUP__real,axiom,
% 0.23/0.59 ! [X8: nat > real] :
% 0.23/0.59 ( ( summable_real @ X8 )
% 0.23/0.59 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I3 ) )
% 0.23/0.59 => ( ( suminf_real @ X8 )
% 0.23/0.59 = ( comple1385675409528146559p_real
% 0.23/0.59 @ ( image_nat_real
% 0.23/0.59 @ ^ [I2: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I2 ) )
% 0.23/0.59 @ top_top_set_nat ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % suminf_eq_SUP_real
% 0.23/0.59 thf(fact_9847_continuous__on__arcosh,axiom,
% 0.23/0.59 ! [A3: set_real] :
% 0.23/0.59 ( ( ord_less_eq_set_real @ A3 @ ( set_ord_atLeast_real @ one_one_real ) )
% 0.23/0.59 => ( topolo5044208981011980120l_real @ A3 @ arcosh_real ) ) ).
% 0.23/0.59
% 0.23/0.59 % continuous_on_arcosh
% 0.23/0.59 thf(fact_9848_continuous__on__arcosh_H,axiom,
% 0.23/0.59 ! [A3: set_real,F: real > real] :
% 0.23/0.59 ( ( topolo5044208981011980120l_real @ A3 @ F )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( member_real @ X3 @ A3 )
% 0.23/0.59 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.23/0.59 => ( topolo5044208981011980120l_real @ A3
% 0.23/0.59 @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % continuous_on_arcosh'
% 0.23/0.59 thf(fact_9849_continuous__image__closed__interval,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_eq_real @ A @ B2 )
% 0.23/0.59 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.23/0.59 => ? [C3: real,D3: real] :
% 0.23/0.59 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 0.23/0.59 = ( set_or1222579329274155063t_real @ C3 @ D3 ) )
% 0.23/0.59 & ( ord_less_eq_real @ C3 @ D3 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % continuous_image_closed_interval
% 0.23/0.59 thf(fact_9850_continuous__on__arccos_H,axiom,
% 0.23/0.59 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 0.23/0.59
% 0.23/0.59 % continuous_on_arccos'
% 0.23/0.59 thf(fact_9851_continuous__on__arcsin_H,axiom,
% 0.23/0.59 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 0.23/0.59
% 0.23/0.59 % continuous_on_arcsin'
% 0.23/0.59 thf(fact_9852_continuous__on__artanh,axiom,
% 0.23/0.59 ! [A3: set_real] :
% 0.23/0.59 ( ( ord_less_eq_set_real @ A3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 0.23/0.59 => ( topolo5044208981011980120l_real @ A3 @ artanh_real ) ) ).
% 0.23/0.59
% 0.23/0.59 % continuous_on_artanh
% 0.23/0.59 thf(fact_9853_continuous__on__artanh_H,axiom,
% 0.23/0.59 ! [A3: set_real,F: real > real] :
% 0.23/0.59 ( ( topolo5044208981011980120l_real @ A3 @ F )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( member_real @ X3 @ A3 )
% 0.23/0.59 => ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 0.23/0.59 => ( topolo5044208981011980120l_real @ A3
% 0.23/0.59 @ ^ [X: real] : ( artanh_real @ ( F @ X ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % continuous_on_artanh'
% 0.23/0.59 thf(fact_9854_Rolle__deriv,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real,F4: real > real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ( ( F @ A )
% 0.23/0.59 = ( F @ B2 ) )
% 0.23/0.59 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_real @ X3 @ B2 )
% 0.23/0.59 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.23/0.59 => ? [Z2: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ Z2 )
% 0.23/0.59 & ( ord_less_real @ Z2 @ B2 )
% 0.23/0.59 & ( ( F4 @ Z2 )
% 0.23/0.59 = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Rolle_deriv
% 0.23/0.59 thf(fact_9855_mvt,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real,F4: real > real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_real @ X3 @ B2 )
% 0.23/0.59 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.23/0.59 => ~ ! [Xi: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ Xi )
% 0.23/0.59 => ( ( ord_less_real @ Xi @ B2 )
% 0.23/0.59 => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 0.23/0.59 != ( F4 @ Xi @ ( minus_minus_real @ B2 @ A ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % mvt
% 0.23/0.59 thf(fact_9856_DERIV__pos__imp__increasing__open,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_real @ X3 @ B2 )
% 0.23/0.59 => ? [Y5: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.23/0.59 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.23/0.59 => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_pos_imp_increasing_open
% 0.23/0.59 thf(fact_9857_DERIV__neg__imp__decreasing__open,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_real @ X3 @ B2 )
% 0.23/0.59 => ? [Y5: real] :
% 0.23/0.59 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.23/0.59 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.23/0.59 => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_neg_imp_decreasing_open
% 0.23/0.59 thf(fact_9858_DERIV__isconst__end,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_real @ X3 @ B2 )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.23/0.59 => ( ( F @ B2 )
% 0.23/0.59 = ( F @ A ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_isconst_end
% 0.23/0.59 thf(fact_9859_DERIV__isconst2,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real,X4: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_real @ X3 @ B2 )
% 0.23/0.59 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.23/0.59 => ( ( ord_less_eq_real @ A @ X4 )
% 0.23/0.59 => ( ( ord_less_eq_real @ X4 @ B2 )
% 0.23/0.59 => ( ( F @ X4 )
% 0.23/0.59 = ( F @ A ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DERIV_isconst2
% 0.23/0.59 thf(fact_9860_Rolle,axiom,
% 0.23/0.59 ! [A: real,B2: real,F: real > real] :
% 0.23/0.59 ( ( ord_less_real @ A @ B2 )
% 0.23/0.59 => ( ( ( F @ A )
% 0.23/0.59 = ( F @ B2 ) )
% 0.23/0.59 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.23/0.59 => ( ! [X3: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ X3 )
% 0.23/0.59 => ( ( ord_less_real @ X3 @ B2 )
% 0.23/0.59 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.23/0.59 => ? [Z2: real] :
% 0.23/0.59 ( ( ord_less_real @ A @ Z2 )
% 0.23/0.59 & ( ord_less_real @ Z2 @ B2 )
% 0.23/0.59 & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Rolle
% 0.23/0.59 thf(fact_9861_UN__lessThan__UNIV,axiom,
% 0.23/0.59 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
% 0.23/0.59 = top_top_set_nat ) ).
% 0.23/0.59
% 0.23/0.59 % UN_lessThan_UNIV
% 0.23/0.59 thf(fact_9862_UN__atMost__UNIV,axiom,
% 0.23/0.59 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
% 0.23/0.59 = top_top_set_nat ) ).
% 0.23/0.59
% 0.23/0.59 % UN_atMost_UNIV
% 0.23/0.59 thf(fact_9863_UN__atLeast__UNIV,axiom,
% 0.23/0.59 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
% 0.23/0.59 = top_top_set_nat ) ).
% 0.23/0.59
% 0.23/0.59 % UN_atLeast_UNIV
% 0.23/0.59 thf(fact_9864_INT__greaterThan__UNIV,axiom,
% 0.23/0.59 ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 0.23/0.59 = bot_bot_set_nat ) ).
% 0.23/0.59
% 0.23/0.59 % INT_greaterThan_UNIV
% 0.23/0.59 thf(fact_9865_Inf__real__def,axiom,
% 0.23/0.59 ( comple4887499456419720421f_real
% 0.23/0.59 = ( ^ [X6: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X6 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Inf_real_def
% 0.23/0.59 thf(fact_9866_inf__enat__def,axiom,
% 0.23/0.59 inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% 0.23/0.59
% 0.23/0.59 % inf_enat_def
% 0.23/0.59 thf(fact_9867_inf__nat__def,axiom,
% 0.23/0.59 inf_inf_nat = ord_min_nat ).
% 0.23/0.59
% 0.23/0.59 % inf_nat_def
% 0.23/0.59 thf(fact_9868_rat__less__code,axiom,
% 0.23/0.59 ( ord_less_rat
% 0.23/0.59 = ( ^ [P3: rat,Q4: rat] :
% 0.23/0.59 ( produc4947309494688390418_int_o
% 0.23/0.59 @ ^ [A4: int,C2: int] :
% 0.23/0.59 ( produc4947309494688390418_int_o
% 0.23/0.59 @ ^ [B3: int,D2: int] : ( ord_less_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) )
% 0.23/0.59 @ ( quotient_of @ Q4 ) )
% 0.23/0.59 @ ( quotient_of @ P3 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % rat_less_code
% 0.23/0.59 thf(fact_9869_rat__less__eq__code,axiom,
% 0.23/0.59 ( ord_less_eq_rat
% 0.23/0.59 = ( ^ [P3: rat,Q4: rat] :
% 0.23/0.59 ( produc4947309494688390418_int_o
% 0.23/0.59 @ ^ [A4: int,C2: int] :
% 0.23/0.59 ( produc4947309494688390418_int_o
% 0.23/0.59 @ ^ [B3: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) )
% 0.23/0.59 @ ( quotient_of @ Q4 ) )
% 0.23/0.59 @ ( quotient_of @ P3 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % rat_less_eq_code
% 0.23/0.59 thf(fact_9870_int__ge__less__than2__def,axiom,
% 0.23/0.59 ( int_ge_less_than2
% 0.23/0.59 = ( ^ [D2: int] :
% 0.23/0.59 ( collec213857154873943460nt_int
% 0.23/0.59 @ ( produc4947309494688390418_int_o
% 0.23/0.59 @ ^ [Z7: int,Z5: int] :
% 0.23/0.59 ( ( ord_less_eq_int @ D2 @ Z5 )
% 0.23/0.59 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % int_ge_less_than2_def
% 0.23/0.59 thf(fact_9871_int__ge__less__than__def,axiom,
% 0.23/0.59 ( int_ge_less_than
% 0.23/0.59 = ( ^ [D2: int] :
% 0.23/0.59 ( collec213857154873943460nt_int
% 0.23/0.59 @ ( produc4947309494688390418_int_o
% 0.23/0.59 @ ^ [Z7: int,Z5: int] :
% 0.23/0.59 ( ( ord_less_eq_int @ D2 @ Z7 )
% 0.23/0.59 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % int_ge_less_than_def
% 0.23/0.59 thf(fact_9872_uniformity__real__def,axiom,
% 0.23/0.59 ( topolo1511823702728130853y_real
% 0.23/0.59 = ( comple2936214249959783750l_real
% 0.23/0.59 @ ( image_2178119161166701260l_real
% 0.23/0.59 @ ^ [E3: real] :
% 0.23/0.59 ( princi6114159922880469582l_real
% 0.23/0.59 @ ( collec3799799289383736868l_real
% 0.23/0.59 @ ( produc5414030515140494994real_o
% 0.23/0.59 @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
% 0.23/0.59 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % uniformity_real_def
% 0.23/0.59 thf(fact_9873_uniformity__complex__def,axiom,
% 0.23/0.59 ( topolo896644834953643431omplex
% 0.23/0.59 = ( comple8358262395181532106omplex
% 0.23/0.59 @ ( image_5971271580939081552omplex
% 0.23/0.59 @ ^ [E3: real] :
% 0.23/0.59 ( princi3496590319149328850omplex
% 0.23/0.59 @ ( collec8663557070575231912omplex
% 0.23/0.59 @ ( produc6771430404735790350plex_o
% 0.23/0.59 @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
% 0.23/0.59 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % uniformity_complex_def
% 0.23/0.59 thf(fact_9874_nonneg__incseq__Bseq__subseq__iff,axiom,
% 0.23/0.59 ! [F: nat > real,G: nat > nat] :
% 0.23/0.59 ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.23/0.59 => ( ( order_mono_nat_real @ F )
% 0.23/0.59 => ( ( order_5726023648592871131at_nat @ G )
% 0.23/0.59 => ( ( bfun_nat_real
% 0.23/0.59 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 0.23/0.59 @ at_top_nat )
% 0.23/0.59 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % nonneg_incseq_Bseq_subseq_iff
% 0.23/0.59 thf(fact_9875_inj__sgn__power,axiom,
% 0.23/0.59 ! [N: nat] :
% 0.23/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.59 => ( inj_on_real_real
% 0.23/0.59 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 0.23/0.59 @ top_top_set_real ) ) ).
% 0.23/0.59
% 0.23/0.59 % inj_sgn_power
% 0.23/0.59 thf(fact_9876_strict__mono__imp__increasing,axiom,
% 0.23/0.59 ! [F: nat > nat,N: nat] :
% 0.23/0.59 ( ( order_5726023648592871131at_nat @ F )
% 0.23/0.59 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % strict_mono_imp_increasing
% 0.23/0.59 thf(fact_9877_log__inj,axiom,
% 0.23/0.59 ! [B2: real] :
% 0.23/0.59 ( ( ord_less_real @ one_one_real @ B2 )
% 0.23/0.59 => ( inj_on_real_real @ ( log @ B2 ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % log_inj
% 0.23/0.59 thf(fact_9878_inj__Suc,axiom,
% 0.23/0.59 ! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).
% 0.23/0.59
% 0.23/0.59 % inj_Suc
% 0.23/0.59 thf(fact_9879_inj__on__diff__nat,axiom,
% 0.23/0.59 ! [N3: set_nat,K: nat] :
% 0.23/0.59 ( ! [N4: nat] :
% 0.23/0.59 ( ( member_nat @ N4 @ N3 )
% 0.23/0.59 => ( ord_less_eq_nat @ K @ N4 ) )
% 0.23/0.59 => ( inj_on_nat_nat
% 0.23/0.59 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
% 0.23/0.59 @ N3 ) ) ).
% 0.23/0.59
% 0.23/0.59 % inj_on_diff_nat
% 0.23/0.59 thf(fact_9880_summable__reindex,axiom,
% 0.23/0.59 ! [F: nat > real,G: nat > nat] :
% 0.23/0.59 ( ( summable_real @ F )
% 0.23/0.59 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.23/0.59 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.23/0.59 => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % summable_reindex
% 0.23/0.59 thf(fact_9881_suminf__reindex__mono,axiom,
% 0.23/0.59 ! [F: nat > real,G: nat > nat] :
% 0.23/0.59 ( ( summable_real @ F )
% 0.23/0.59 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.23/0.59 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.23/0.59 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % suminf_reindex_mono
% 0.23/0.59 thf(fact_9882_powr__real__of__int_H,axiom,
% 0.23/0.59 ! [X4: real,N: int] :
% 0.23/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.59 => ( ( ( X4 != zero_zero_real )
% 0.23/0.59 | ( ord_less_int @ zero_zero_int @ N ) )
% 0.23/0.59 => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N ) )
% 0.23/0.59 = ( power_int_real @ X4 @ N ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % powr_real_of_int'
% 0.23/0.59 thf(fact_9883_inj__on__char__of__nat,axiom,
% 0.23/0.59 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % inj_on_char_of_nat
% 0.23/0.59 thf(fact_9884_suminf__reindex,axiom,
% 0.23/0.59 ! [F: nat > real,G: nat > nat] :
% 0.23/0.59 ( ( summable_real @ F )
% 0.23/0.59 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.23/0.59 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.23/0.59 => ( ! [X3: nat] :
% 0.23/0.59 ( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 0.23/0.59 => ( ( F @ X3 )
% 0.23/0.59 = zero_zero_real ) )
% 0.23/0.59 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 0.23/0.59 = ( suminf_real @ F ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % suminf_reindex
% 0.23/0.59 thf(fact_9885_pos__deriv__imp__strict__mono,axiom,
% 0.23/0.59 ! [F: real > real,F4: real > real] :
% 0.23/0.59 ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.23/0.59 => ( ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( F4 @ X3 ) )
% 0.23/0.59 => ( order_7092887310737990675l_real @ F ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % pos_deriv_imp_strict_mono
% 0.23/0.59 thf(fact_9886_pred__nat__def,axiom,
% 0.23/0.59 ( pred_nat
% 0.23/0.59 = ( collec3392354462482085612at_nat
% 0.23/0.59 @ ( produc6081775807080527818_nat_o
% 0.23/0.59 @ ^ [M6: nat,N2: nat] :
% 0.23/0.59 ( N2
% 0.23/0.59 = ( suc @ M6 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % pred_nat_def
% 0.23/0.59 thf(fact_9887_atLeastLessThan__add__Un,axiom,
% 0.23/0.59 ! [I: nat,J: nat,K: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ I @ J )
% 0.23/0.59 => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 0.23/0.59 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % atLeastLessThan_add_Un
% 0.23/0.59 thf(fact_9888_sup__enat__def,axiom,
% 0.23/0.59 sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 0.23/0.59
% 0.23/0.59 % sup_enat_def
% 0.23/0.59 thf(fact_9889_sup__nat__def,axiom,
% 0.23/0.59 sup_sup_nat = ord_max_nat ).
% 0.23/0.59
% 0.23/0.59 % sup_nat_def
% 0.23/0.59 thf(fact_9890_VEBT_Osize_I3_J,axiom,
% 0.23/0.59 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.23/0.59 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.23/0.59 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % VEBT.size(3)
% 0.23/0.59 thf(fact_9891_VEBT_Osize__gen_I1_J,axiom,
% 0.23/0.59 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.23/0.59 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.23/0.59 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % VEBT.size_gen(1)
% 0.23/0.59 thf(fact_9892_tl__upt,axiom,
% 0.23/0.59 ! [M: nat,N: nat] :
% 0.23/0.59 ( ( tl_nat @ ( upt @ M @ N ) )
% 0.23/0.59 = ( upt @ ( suc @ M ) @ N ) ) ).
% 0.23/0.59
% 0.23/0.59 % tl_upt
% 0.23/0.59 thf(fact_9893_vanishes__mult__bounded,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ? [A7: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ A7 )
% 0.23/0.59 & ! [N4: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N4 ) ) @ A7 ) )
% 0.23/0.59 => ( ( vanishes @ Y7 )
% 0.23/0.59 => ( vanishes
% 0.23/0.59 @ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % vanishes_mult_bounded
% 0.23/0.59 thf(fact_9894_vanishes__const,axiom,
% 0.23/0.59 ! [C: rat] :
% 0.23/0.59 ( ( vanishes
% 0.23/0.59 @ ^ [N2: nat] : C )
% 0.23/0.59 = ( C = zero_zero_rat ) ) ).
% 0.23/0.59
% 0.23/0.59 % vanishes_const
% 0.23/0.59 thf(fact_9895_vanishes__minus,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ( vanishes @ X8 )
% 0.23/0.59 => ( vanishes
% 0.23/0.59 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % vanishes_minus
% 0.23/0.59 thf(fact_9896_vanishes__add,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( vanishes @ X8 )
% 0.23/0.59 => ( ( vanishes @ Y7 )
% 0.23/0.59 => ( vanishes
% 0.23/0.59 @ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % vanishes_add
% 0.23/0.59 thf(fact_9897_vanishes__diff,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( vanishes @ X8 )
% 0.23/0.59 => ( ( vanishes @ Y7 )
% 0.23/0.59 => ( vanishes
% 0.23/0.59 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % vanishes_diff
% 0.23/0.59 thf(fact_9898_vanishesD,axiom,
% 0.23/0.59 ! [X8: nat > rat,R2: rat] :
% 0.23/0.59 ( ( vanishes @ X8 )
% 0.23/0.59 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.23/0.59 => ? [K2: nat] :
% 0.23/0.59 ! [N6: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K2 @ N6 )
% 0.23/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N6 ) ) @ R2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % vanishesD
% 0.23/0.59 thf(fact_9899_vanishesI,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ! [R3: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 0.23/0.59 => ? [K4: nat] :
% 0.23/0.59 ! [N4: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K4 @ N4 )
% 0.23/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N4 ) ) @ R3 ) ) )
% 0.23/0.59 => ( vanishes @ X8 ) ) ).
% 0.23/0.59
% 0.23/0.59 % vanishesI
% 0.23/0.59 thf(fact_9900_vanishes__def,axiom,
% 0.23/0.59 ( vanishes
% 0.23/0.59 = ( ^ [X6: nat > rat] :
% 0.23/0.59 ! [R5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.23/0.59 => ? [K3: nat] :
% 0.23/0.59 ! [N2: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.23/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N2 ) ) @ R5 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % vanishes_def
% 0.23/0.59 thf(fact_9901_and__not__num_Opelims,axiom,
% 0.23/0.59 ! [X4: num,Xa: num,Y3: option_num] :
% 0.23/0.59 ( ( ( bit_and_not_num @ X4 @ Xa )
% 0.23/0.59 = Y3 )
% 0.23/0.59 => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3 = none_num )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ one ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ( ( Y3 = none_num )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.23/0.59 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.23/0.59 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ~ ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % and_not_num.pelims
% 0.23/0.59 thf(fact_9902_and__num_Opelims,axiom,
% 0.23/0.59 ! [X4: num,Xa: num,Y3: option_num] :
% 0.23/0.59 ( ( ( bit_un7362597486090784418nd_num @ X4 @ Xa )
% 0.23/0.59 = Y3 )
% 0.23/0.59 => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ one ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ( ( Y3 = none_num )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ one ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3 = none_num )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ one ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ~ ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.23/0.59 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.23/0.59 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % and_num.pelims
% 0.23/0.59 thf(fact_9903_xor__num_Opelims,axiom,
% 0.23/0.59 ! [X4: num,Xa: num,Y3: option_num] :
% 0.23/0.59 ( ( ( bit_un2480387367778600638or_num @ X4 @ Xa )
% 0.23/0.59 = Y3 )
% 0.23/0.59 => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3 = none_num )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ ( bit1 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ ( bit0 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ ( bit1 @ M4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.23/0.59 => ( ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.23/0.59 => ~ ! [M4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ! [N4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % xor_num.pelims
% 0.23/0.59 thf(fact_9904_or__not__num__neg_Opelims,axiom,
% 0.23/0.59 ! [X4: num,Xa: num,Y3: num] :
% 0.23/0.59 ( ( ( bit_or_not_num_neg @ X4 @ Xa )
% 0.23/0.59 = Y3 )
% 0.23/0.59 => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3 = one )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ! [M4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
% 0.23/0.59 => ( ( ( X4 = one )
% 0.23/0.59 => ! [M4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [N4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( bit0 @ one ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ one ) ) ) ) )
% 0.23/0.59 => ( ! [N4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ! [M4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [N4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit0 @ N4 ) )
% 0.23/0.59 => ! [M4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( bit0 @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit1 @ M4 ) ) ) ) ) )
% 0.23/0.59 => ( ! [N4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ( ( Xa = one )
% 0.23/0.59 => ( ( Y3 = one )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ one ) ) ) ) )
% 0.23/0.59 => ( ! [N4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ! [M4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit0 @ M4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 0.23/0.59 => ~ ! [N4: num] :
% 0.23/0.59 ( ( X4
% 0.23/0.59 = ( bit1 @ N4 ) )
% 0.23/0.59 => ! [M4: num] :
% 0.23/0.59 ( ( Xa
% 0.23/0.59 = ( bit1 @ M4 ) )
% 0.23/0.59 => ( ( Y3
% 0.23/0.59 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 0.23/0.59 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % or_not_num_neg.pelims
% 0.23/0.59 thf(fact_9905_xor__num__rel__dict,axiom,
% 0.23/0.59 bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 0.23/0.59
% 0.23/0.59 % xor_num_rel_dict
% 0.23/0.59 thf(fact_9906_and__num__rel__dict,axiom,
% 0.23/0.59 bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 0.23/0.59
% 0.23/0.59 % and_num_rel_dict
% 0.23/0.59 thf(fact_9907_rcis__inverse,axiom,
% 0.23/0.59 ! [R2: real,A: real] :
% 0.23/0.59 ( ( invers8013647133539491842omplex @ ( rcis @ R2 @ A ) )
% 0.23/0.59 = ( rcis @ ( divide_divide_real @ one_one_real @ R2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % rcis_inverse
% 0.23/0.59 thf(fact_9908_Re__rcis,axiom,
% 0.23/0.59 ! [R2: real,A: real] :
% 0.23/0.59 ( ( re @ ( rcis @ R2 @ A ) )
% 0.23/0.59 = ( times_times_real @ R2 @ ( cos_real @ A ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Re_rcis
% 0.23/0.59 thf(fact_9909_Im__rcis,axiom,
% 0.23/0.59 ! [R2: real,A: real] :
% 0.23/0.59 ( ( im @ ( rcis @ R2 @ A ) )
% 0.23/0.59 = ( times_times_real @ R2 @ ( sin_real @ A ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Im_rcis
% 0.23/0.59 thf(fact_9910_cis__rcis__eq,axiom,
% 0.23/0.59 ( cis
% 0.23/0.59 = ( rcis @ one_one_real ) ) ).
% 0.23/0.59
% 0.23/0.59 % cis_rcis_eq
% 0.23/0.59 thf(fact_9911_rcis__mult,axiom,
% 0.23/0.59 ! [R1: real,A: real,R22: real,B2: real] :
% 0.23/0.59 ( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B2 ) )
% 0.23/0.59 = ( rcis @ ( times_times_real @ R1 @ R22 ) @ ( plus_plus_real @ A @ B2 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % rcis_mult
% 0.23/0.59 thf(fact_9912_rcis__divide,axiom,
% 0.23/0.59 ! [R1: real,A: real,R22: real,B2: real] :
% 0.23/0.59 ( ( divide1717551699836669952omplex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B2 ) )
% 0.23/0.59 = ( rcis @ ( divide_divide_real @ R1 @ R22 ) @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % rcis_divide
% 0.23/0.59 thf(fact_9913_rcis__def,axiom,
% 0.23/0.59 ( rcis
% 0.23/0.59 = ( ^ [R5: real,A4: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) @ ( cis @ A4 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % rcis_def
% 0.23/0.59 thf(fact_9914_DeMoivre2,axiom,
% 0.23/0.59 ! [R2: real,A: real,N: nat] :
% 0.23/0.59 ( ( power_power_complex @ ( rcis @ R2 @ A ) @ N )
% 0.23/0.59 = ( rcis @ ( power_power_real @ R2 @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % DeMoivre2
% 0.23/0.59 thf(fact_9915_nat__of__integer__code,axiom,
% 0.23/0.59 ( code_nat_of_integer
% 0.23/0.59 = ( ^ [K3: code_integer] :
% 0.23/0.59 ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 0.23/0.59 @ ( produc1555791787009142072er_nat
% 0.23/0.59 @ ^ [L: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
% 0.23/0.59 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % nat_of_integer_code
% 0.23/0.59 thf(fact_9916_nat__of__integer__code__post_I3_J,axiom,
% 0.23/0.59 ! [K: num] :
% 0.23/0.59 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.23/0.59 = ( numeral_numeral_nat @ K ) ) ).
% 0.23/0.59
% 0.23/0.59 % nat_of_integer_code_post(3)
% 0.23/0.59 thf(fact_9917_nat__of__integer__code__post_I2_J,axiom,
% 0.23/0.59 ( ( code_nat_of_integer @ one_one_Code_integer )
% 0.23/0.59 = one_one_nat ) ).
% 0.23/0.59
% 0.23/0.59 % nat_of_integer_code_post(2)
% 0.23/0.59 thf(fact_9918_cauchy__def,axiom,
% 0.23/0.59 ( cauchy
% 0.23/0.59 = ( ^ [X6: nat > rat] :
% 0.23/0.59 ! [R5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.23/0.59 => ? [K3: nat] :
% 0.23/0.59 ! [M6: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K3 @ M6 )
% 0.23/0.59 => ! [N2: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.23/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cauchy_def
% 0.23/0.59 thf(fact_9919_cauchyI,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ! [R3: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 0.23/0.59 => ? [K4: nat] :
% 0.23/0.59 ! [M4: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K4 @ M4 )
% 0.23/0.59 => ! [N4: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K4 @ N4 )
% 0.23/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ R3 ) ) ) )
% 0.23/0.59 => ( cauchy @ X8 ) ) ).
% 0.23/0.59
% 0.23/0.59 % cauchyI
% 0.23/0.59 thf(fact_9920_cauchy__inverse,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ~ ( vanishes @ X8 )
% 0.23/0.59 => ( cauchy
% 0.23/0.59 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X8 @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cauchy_inverse
% 0.23/0.59 thf(fact_9921_cauchy__diff,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( cauchy @ Y7 )
% 0.23/0.59 => ( cauchy
% 0.23/0.59 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cauchy_diff
% 0.23/0.59 thf(fact_9922_cauchy__mult,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( cauchy @ Y7 )
% 0.23/0.59 => ( cauchy
% 0.23/0.59 @ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cauchy_mult
% 0.23/0.59 thf(fact_9923_cauchy__const,axiom,
% 0.23/0.59 ! [X4: rat] :
% 0.23/0.59 ( cauchy
% 0.23/0.59 @ ^ [N2: nat] : X4 ) ).
% 0.23/0.59
% 0.23/0.59 % cauchy_const
% 0.23/0.59 thf(fact_9924_cauchy__minus,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( cauchy
% 0.23/0.59 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cauchy_minus
% 0.23/0.59 thf(fact_9925_cauchy__add,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( cauchy @ Y7 )
% 0.23/0.59 => ( cauchy
% 0.23/0.59 @ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cauchy_add
% 0.23/0.59 thf(fact_9926_cauchy__imp__bounded,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ? [B5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ B5 )
% 0.23/0.59 & ! [N6: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N6 ) ) @ B5 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cauchy_imp_bounded
% 0.23/0.59 thf(fact_9927_vanishes__diff__inverse,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ~ ( vanishes @ X8 )
% 0.23/0.59 => ( ( cauchy @ Y7 )
% 0.23/0.59 => ( ~ ( vanishes @ Y7 )
% 0.23/0.59 => ( ( vanishes
% 0.23/0.59 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) )
% 0.23/0.59 => ( vanishes
% 0.23/0.59 @ ^ [N2: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X8 @ N2 ) ) @ ( inverse_inverse_rat @ ( Y7 @ N2 ) ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % vanishes_diff_inverse
% 0.23/0.59 thf(fact_9928_cauchy__not__vanishes__cases,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ~ ( vanishes @ X8 )
% 0.23/0.59 => ? [B5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ B5 )
% 0.23/0.59 & ? [K2: nat] :
% 0.23/0.59 ( ! [N6: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K2 @ N6 )
% 0.23/0.59 => ( ord_less_rat @ B5 @ ( uminus_uminus_rat @ ( X8 @ N6 ) ) ) )
% 0.23/0.59 | ! [N6: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K2 @ N6 )
% 0.23/0.59 => ( ord_less_rat @ B5 @ ( X8 @ N6 ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cauchy_not_vanishes_cases
% 0.23/0.59 thf(fact_9929_cauchy__not__vanishes,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ~ ( vanishes @ X8 )
% 0.23/0.59 => ? [B5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ B5 )
% 0.23/0.59 & ? [K2: nat] :
% 0.23/0.59 ! [N6: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K2 @ N6 )
% 0.23/0.59 => ( ord_less_rat @ B5 @ ( abs_abs_rat @ ( X8 @ N6 ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cauchy_not_vanishes
% 0.23/0.59 thf(fact_9930_cauchyD,axiom,
% 0.23/0.59 ! [X8: nat > rat,R2: rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.23/0.59 => ? [K2: nat] :
% 0.23/0.59 ! [M5: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K2 @ M5 )
% 0.23/0.59 => ! [N6: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K2 @ N6 )
% 0.23/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ R2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cauchyD
% 0.23/0.59 thf(fact_9931_le__Real,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( cauchy @ Y7 )
% 0.23/0.59 => ( ( ord_less_eq_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
% 0.23/0.59 = ( ! [R5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.23/0.59 => ? [K3: nat] :
% 0.23/0.59 ! [N2: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.23/0.59 => ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % le_Real
% 0.23/0.59 thf(fact_9932_inverse__Real,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( ( vanishes @ X8 )
% 0.23/0.59 => ( ( inverse_inverse_real @ ( real2 @ X8 ) )
% 0.23/0.59 = zero_zero_real ) )
% 0.23/0.59 & ( ~ ( vanishes @ X8 )
% 0.23/0.59 => ( ( inverse_inverse_real @ ( real2 @ X8 ) )
% 0.23/0.59 = ( real2
% 0.23/0.59 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X8 @ N2 ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % inverse_Real
% 0.23/0.59 thf(fact_9933_Real__induct,axiom,
% 0.23/0.59 ! [P: real > $o,X4: real] :
% 0.23/0.59 ( ! [X10: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X10 )
% 0.23/0.59 => ( P @ ( real2 @ X10 ) ) )
% 0.23/0.59 => ( P @ X4 ) ) ).
% 0.23/0.59
% 0.23/0.59 % Real_induct
% 0.23/0.59 thf(fact_9934_one__real__def,axiom,
% 0.23/0.59 ( one_one_real
% 0.23/0.59 = ( real2
% 0.23/0.59 @ ^ [N2: nat] : one_one_rat ) ) ).
% 0.23/0.59
% 0.23/0.59 % one_real_def
% 0.23/0.59 thf(fact_9935_of__nat__Real,axiom,
% 0.23/0.59 ( semiri5074537144036343181t_real
% 0.23/0.59 = ( ^ [X: nat] :
% 0.23/0.59 ( real2
% 0.23/0.59 @ ^ [N2: nat] : ( semiri681578069525770553at_rat @ X ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % of_nat_Real
% 0.23/0.59 thf(fact_9936_zero__real__def,axiom,
% 0.23/0.59 ( zero_zero_real
% 0.23/0.59 = ( real2
% 0.23/0.59 @ ^ [N2: nat] : zero_zero_rat ) ) ).
% 0.23/0.59
% 0.23/0.59 % zero_real_def
% 0.23/0.59 thf(fact_9937_of__int__Real,axiom,
% 0.23/0.59 ( ring_1_of_int_real
% 0.23/0.59 = ( ^ [X: int] :
% 0.23/0.59 ( real2
% 0.23/0.59 @ ^ [N2: nat] : ( ring_1_of_int_rat @ X ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % of_int_Real
% 0.23/0.59 thf(fact_9938_minus__Real,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( uminus_uminus_real @ ( real2 @ X8 ) )
% 0.23/0.59 = ( real2
% 0.23/0.59 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % minus_Real
% 0.23/0.59 thf(fact_9939_add__Real,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( cauchy @ Y7 )
% 0.23/0.59 => ( ( plus_plus_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
% 0.23/0.59 = ( real2
% 0.23/0.59 @ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % add_Real
% 0.23/0.59 thf(fact_9940_mult__Real,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( cauchy @ Y7 )
% 0.23/0.59 => ( ( times_times_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
% 0.23/0.59 = ( real2
% 0.23/0.59 @ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % mult_Real
% 0.23/0.59 thf(fact_9941_diff__Real,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( cauchy @ Y7 )
% 0.23/0.59 => ( ( minus_minus_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
% 0.23/0.59 = ( real2
% 0.23/0.59 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % diff_Real
% 0.23/0.59 thf(fact_9942_eq__Real,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( cauchy @ Y7 )
% 0.23/0.59 => ( ( ( real2 @ X8 )
% 0.23/0.59 = ( real2 @ Y7 ) )
% 0.23/0.59 = ( vanishes
% 0.23/0.59 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % eq_Real
% 0.23/0.59 thf(fact_9943_not__positive__Real,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( ~ ( positive2 @ ( real2 @ X8 ) ) )
% 0.23/0.59 = ( ! [R5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.23/0.59 => ? [K3: nat] :
% 0.23/0.59 ! [N2: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.23/0.59 => ( ord_less_eq_rat @ ( X8 @ N2 ) @ R5 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % not_positive_Real
% 0.23/0.59 thf(fact_9944_positive__Real,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( positive2 @ ( real2 @ X8 ) )
% 0.23/0.59 = ( ? [R5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.23/0.59 & ? [K3: nat] :
% 0.23/0.59 ! [N2: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.23/0.59 => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % positive_Real
% 0.23/0.59 thf(fact_9945_Real_Opositive__mult,axiom,
% 0.23/0.59 ! [X4: real,Y3: real] :
% 0.23/0.59 ( ( positive2 @ X4 )
% 0.23/0.59 => ( ( positive2 @ Y3 )
% 0.23/0.59 => ( positive2 @ ( times_times_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Real.positive_mult
% 0.23/0.59 thf(fact_9946_Real_Opositive__add,axiom,
% 0.23/0.59 ! [X4: real,Y3: real] :
% 0.23/0.59 ( ( positive2 @ X4 )
% 0.23/0.59 => ( ( positive2 @ Y3 )
% 0.23/0.59 => ( positive2 @ ( plus_plus_real @ X4 @ Y3 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Real.positive_add
% 0.23/0.59 thf(fact_9947_Real_Opositive__zero,axiom,
% 0.23/0.59 ~ ( positive2 @ zero_zero_real ) ).
% 0.23/0.59
% 0.23/0.59 % Real.positive_zero
% 0.23/0.59 thf(fact_9948_Real_Opositive__minus,axiom,
% 0.23/0.59 ! [X4: real] :
% 0.23/0.59 ( ~ ( positive2 @ X4 )
% 0.23/0.59 => ( ( X4 != zero_zero_real )
% 0.23/0.59 => ( positive2 @ ( uminus_uminus_real @ X4 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Real.positive_minus
% 0.23/0.59 thf(fact_9949_less__real__def,axiom,
% 0.23/0.59 ( ord_less_real
% 0.23/0.59 = ( ^ [X: real,Y: real] : ( positive2 @ ( minus_minus_real @ Y @ X ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % less_real_def
% 0.23/0.59 thf(fact_9950_Real_Opositive_Orep__eq,axiom,
% 0.23/0.59 ( positive2
% 0.23/0.59 = ( ^ [X: real] :
% 0.23/0.59 ? [R5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.23/0.59 & ? [K3: nat] :
% 0.23/0.59 ! [N2: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.23/0.59 => ( ord_less_rat @ R5 @ ( rep_real @ X @ N2 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Real.positive.rep_eq
% 0.23/0.59 thf(fact_9951_inverse__real_Oabs__eq,axiom,
% 0.23/0.59 ! [X4: nat > rat] :
% 0.23/0.59 ( ( realrel @ X4 @ X4 )
% 0.23/0.59 => ( ( inverse_inverse_real @ ( real2 @ X4 ) )
% 0.23/0.59 = ( real2
% 0.23/0.59 @ ( if_nat_rat @ ( vanishes @ X4 )
% 0.23/0.59 @ ^ [N2: nat] : zero_zero_rat
% 0.23/0.59 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X4 @ N2 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % inverse_real.abs_eq
% 0.23/0.59 thf(fact_9952_realrel__refl,axiom,
% 0.23/0.59 ! [X8: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( realrel @ X8 @ X8 ) ) ).
% 0.23/0.59
% 0.23/0.59 % realrel_refl
% 0.23/0.59 thf(fact_9953_one__real_Orsp,axiom,
% 0.23/0.59 ( realrel
% 0.23/0.59 @ ^ [N2: nat] : one_one_rat
% 0.23/0.59 @ ^ [N2: nat] : one_one_rat ) ).
% 0.23/0.59
% 0.23/0.59 % one_real.rsp
% 0.23/0.59 thf(fact_9954_zero__real_Orsp,axiom,
% 0.23/0.59 ( realrel
% 0.23/0.59 @ ^ [N2: nat] : zero_zero_rat
% 0.23/0.59 @ ^ [N2: nat] : zero_zero_rat ) ).
% 0.23/0.59
% 0.23/0.59 % zero_real.rsp
% 0.23/0.59 thf(fact_9955_real_Oabs__induct,axiom,
% 0.23/0.59 ! [P: real > $o,X4: real] :
% 0.23/0.59 ( ! [Y4: nat > rat] :
% 0.23/0.59 ( ( realrel @ Y4 @ Y4 )
% 0.23/0.59 => ( P @ ( real2 @ Y4 ) ) )
% 0.23/0.59 => ( P @ X4 ) ) ).
% 0.23/0.59
% 0.23/0.59 % real.abs_induct
% 0.23/0.59 thf(fact_9956_uminus__real_Oabs__eq,axiom,
% 0.23/0.59 ! [X4: nat > rat] :
% 0.23/0.59 ( ( realrel @ X4 @ X4 )
% 0.23/0.59 => ( ( uminus_uminus_real @ ( real2 @ X4 ) )
% 0.23/0.59 = ( real2
% 0.23/0.59 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X4 @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % uminus_real.abs_eq
% 0.23/0.59 thf(fact_9957_plus__real_Oabs__eq,axiom,
% 0.23/0.59 ! [Xa: nat > rat,X4: nat > rat] :
% 0.23/0.59 ( ( realrel @ Xa @ Xa )
% 0.23/0.59 => ( ( realrel @ X4 @ X4 )
% 0.23/0.59 => ( ( plus_plus_real @ ( real2 @ Xa ) @ ( real2 @ X4 ) )
% 0.23/0.59 = ( real2
% 0.23/0.59 @ ^ [N2: nat] : ( plus_plus_rat @ ( Xa @ N2 ) @ ( X4 @ N2 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % plus_real.abs_eq
% 0.23/0.59 thf(fact_9958_times__real_Oabs__eq,axiom,
% 0.23/0.59 ! [Xa: nat > rat,X4: nat > rat] :
% 0.23/0.59 ( ( realrel @ Xa @ Xa )
% 0.23/0.59 => ( ( realrel @ X4 @ X4 )
% 0.23/0.59 => ( ( times_times_real @ ( real2 @ Xa ) @ ( real2 @ X4 ) )
% 0.23/0.59 = ( real2
% 0.23/0.59 @ ^ [N2: nat] : ( times_times_rat @ ( Xa @ N2 ) @ ( X4 @ N2 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % times_real.abs_eq
% 0.23/0.59 thf(fact_9959_realrelI,axiom,
% 0.23/0.59 ! [X8: nat > rat,Y7: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X8 )
% 0.23/0.59 => ( ( cauchy @ Y7 )
% 0.23/0.59 => ( ( vanishes
% 0.23/0.59 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) )
% 0.23/0.59 => ( realrel @ X8 @ Y7 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % realrelI
% 0.23/0.59 thf(fact_9960_realrel__def,axiom,
% 0.23/0.59 ( realrel
% 0.23/0.59 = ( ^ [X6: nat > rat,Y8: nat > rat] :
% 0.23/0.59 ( ( cauchy @ X6 )
% 0.23/0.59 & ( cauchy @ Y8 )
% 0.23/0.59 & ( vanishes
% 0.23/0.59 @ ^ [N2: nat] : ( minus_minus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % realrel_def
% 0.23/0.59 thf(fact_9961_Real_Opositive_Oabs__eq,axiom,
% 0.23/0.59 ! [X4: nat > rat] :
% 0.23/0.59 ( ( realrel @ X4 @ X4 )
% 0.23/0.59 => ( ( positive2 @ ( real2 @ X4 ) )
% 0.23/0.59 = ( ? [R5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.23/0.59 & ? [K3: nat] :
% 0.23/0.59 ! [N2: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.23/0.59 => ( ord_less_rat @ R5 @ ( X4 @ N2 ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Real.positive.abs_eq
% 0.23/0.59 thf(fact_9962_inverse__real__def,axiom,
% 0.23/0.59 ( inverse_inverse_real
% 0.23/0.59 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 0.23/0.59 @ ^ [X6: nat > rat] :
% 0.23/0.59 ( if_nat_rat @ ( vanishes @ X6 )
% 0.23/0.59 @ ^ [N2: nat] : zero_zero_rat
% 0.23/0.59 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % inverse_real_def
% 0.23/0.59 thf(fact_9963_cr__real__def,axiom,
% 0.23/0.59 ( cr_real
% 0.23/0.59 = ( ^ [X: nat > rat,Y: real] :
% 0.23/0.59 ( ( realrel @ X @ X )
% 0.23/0.59 & ( ( real2 @ X )
% 0.23/0.59 = Y ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cr_real_def
% 0.23/0.59 thf(fact_9964_uminus__real__def,axiom,
% 0.23/0.59 ( uminus_uminus_real
% 0.23/0.59 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 0.23/0.59 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % uminus_real_def
% 0.23/0.59 thf(fact_9965_times__real__def,axiom,
% 0.23/0.59 ( times_times_real
% 0.23/0.59 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 0.23/0.59 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % times_real_def
% 0.23/0.59 thf(fact_9966_plus__real__def,axiom,
% 0.23/0.59 ( plus_plus_real
% 0.23/0.59 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 0.23/0.59 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % plus_real_def
% 0.23/0.59 thf(fact_9967_Real_Opositive_Orsp,axiom,
% 0.23/0.59 ( bNF_re728719798268516973at_o_o @ realrel
% 0.23/0.59 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [X6: nat > rat] :
% 0.23/0.59 ? [R5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.23/0.59 & ? [K3: nat] :
% 0.23/0.59 ! [N2: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.23/0.59 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 0.23/0.59 @ ^ [X6: nat > rat] :
% 0.23/0.59 ? [R5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.23/0.59 & ? [K3: nat] :
% 0.23/0.59 ! [N2: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.23/0.59 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Real.positive.rsp
% 0.23/0.59 thf(fact_9968_plus__real_Orsp,axiom,
% 0.23/0.59 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 0.23/0.59 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 0.23/0.59 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % plus_real.rsp
% 0.23/0.59 thf(fact_9969_uminus__real_Orsp,axiom,
% 0.23/0.59 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 0.23/0.59 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
% 0.23/0.59 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % uminus_real.rsp
% 0.23/0.59 thf(fact_9970_times__real_Orsp,axiom,
% 0.23/0.59 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 0.23/0.59 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 0.23/0.59 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % times_real.rsp
% 0.23/0.59 thf(fact_9971_sub_Orsp,axiom,
% 0.23/0.59 ( bNF_re8402795839162346335um_int
% 0.23/0.59 @ ^ [Y6: num,Z4: num] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re1822329894187522285nt_int
% 0.23/0.59 @ ^ [Y6: num,Z4: num] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.23/0.59 @ ^ [M6: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) )
% 0.23/0.59 @ ^ [M6: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % sub.rsp
% 0.23/0.59 thf(fact_9972_times__natural_Orsp,axiom,
% 0.23/0.59 ( bNF_re1345281282404953727at_nat
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re5653821019739307937at_nat
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.23/0.59 @ times_times_nat
% 0.23/0.59 @ times_times_nat ) ).
% 0.23/0.59
% 0.23/0.59 % times_natural.rsp
% 0.23/0.59 thf(fact_9973_times__integer_Orsp,axiom,
% 0.23/0.59 ( bNF_re711492959462206631nt_int
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re4712519889275205905nt_int
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.23/0.59 @ times_times_int
% 0.23/0.59 @ times_times_int ) ).
% 0.23/0.59
% 0.23/0.59 % times_integer.rsp
% 0.23/0.59 thf(fact_9974_minus__integer_Orsp,axiom,
% 0.23/0.59 ( bNF_re711492959462206631nt_int
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re4712519889275205905nt_int
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.23/0.59 @ minus_minus_int
% 0.23/0.59 @ minus_minus_int ) ).
% 0.23/0.59
% 0.23/0.59 % minus_integer.rsp
% 0.23/0.59 thf(fact_9975_minus__natural_Orsp,axiom,
% 0.23/0.59 ( bNF_re1345281282404953727at_nat
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re5653821019739307937at_nat
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.23/0.59 @ minus_minus_nat
% 0.23/0.59 @ minus_minus_nat ) ).
% 0.23/0.59
% 0.23/0.59 % minus_natural.rsp
% 0.23/0.59 thf(fact_9976_divide__natural_Orsp,axiom,
% 0.23/0.59 ( bNF_re1345281282404953727at_nat
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re5653821019739307937at_nat
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.23/0.59 @ divide_divide_nat
% 0.23/0.59 @ divide_divide_nat ) ).
% 0.23/0.59
% 0.23/0.59 % divide_natural.rsp
% 0.23/0.59 thf(fact_9977_divide__integer_Orsp,axiom,
% 0.23/0.59 ( bNF_re711492959462206631nt_int
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re4712519889275205905nt_int
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.23/0.59 @ divide_divide_int
% 0.23/0.59 @ divide_divide_int ) ).
% 0.23/0.59
% 0.23/0.59 % divide_integer.rsp
% 0.23/0.59 thf(fact_9978_modulo__natural_Orsp,axiom,
% 0.23/0.59 ( bNF_re1345281282404953727at_nat
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re5653821019739307937at_nat
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.23/0.59 @ modulo_modulo_nat
% 0.23/0.59 @ modulo_modulo_nat ) ).
% 0.23/0.59
% 0.23/0.59 % modulo_natural.rsp
% 0.23/0.59 thf(fact_9979_modulo__integer_Orsp,axiom,
% 0.23/0.59 ( bNF_re711492959462206631nt_int
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re4712519889275205905nt_int
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.23/0.59 @ modulo_modulo_int
% 0.23/0.59 @ modulo_modulo_int ) ).
% 0.23/0.59
% 0.23/0.59 % modulo_integer.rsp
% 0.23/0.59 thf(fact_9980_Suc_Orsp,axiom,
% 0.23/0.59 ( bNF_re5653821019739307937at_nat
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ suc
% 0.23/0.59 @ suc ) ).
% 0.23/0.59
% 0.23/0.59 % Suc.rsp
% 0.23/0.59 thf(fact_9981_dup_Orsp,axiom,
% 0.23/0.59 ( bNF_re4712519889275205905nt_int
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
% 0.23/0.59 @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ).
% 0.23/0.59
% 0.23/0.59 % dup.rsp
% 0.23/0.59 thf(fact_9982_plus__natural_Orsp,axiom,
% 0.23/0.59 ( bNF_re1345281282404953727at_nat
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re5653821019739307937at_nat
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.23/0.59 @ plus_plus_nat
% 0.23/0.59 @ plus_plus_nat ) ).
% 0.23/0.59
% 0.23/0.59 % plus_natural.rsp
% 0.23/0.59 thf(fact_9983_plus__integer_Orsp,axiom,
% 0.23/0.59 ( bNF_re711492959462206631nt_int
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re4712519889275205905nt_int
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.23/0.59 @ plus_plus_int
% 0.23/0.59 @ plus_plus_int ) ).
% 0.23/0.59
% 0.23/0.59 % plus_integer.rsp
% 0.23/0.59 thf(fact_9984_less__integer_Orsp,axiom,
% 0.23/0.59 ( bNF_re3403563459893282935_int_o
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re5089333283451836215nt_o_o
% 0.23/0.59 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.23/0.59 @ ord_less_int
% 0.23/0.59 @ ord_less_int ) ).
% 0.23/0.59
% 0.23/0.59 % less_integer.rsp
% 0.23/0.59 thf(fact_9985_less__natural_Orsp,axiom,
% 0.23/0.59 ( bNF_re578469030762574527_nat_o
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re4705727531993890431at_o_o
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.23/0.59 @ ord_less_nat
% 0.23/0.59 @ ord_less_nat ) ).
% 0.23/0.59
% 0.23/0.59 % less_natural.rsp
% 0.23/0.59 thf(fact_9986_less__eq__natural_Orsp,axiom,
% 0.23/0.59 ( bNF_re578469030762574527_nat_o
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ( bNF_re4705727531993890431at_o_o
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.23/0.59 @ ord_less_eq_nat
% 0.23/0.59 @ ord_less_eq_nat ) ).
% 0.23/0.59
% 0.23/0.59 % less_eq_natural.rsp
% 0.23/0.59 thf(fact_9987_le__enumerate,axiom,
% 0.23/0.59 ! [S2: set_nat,N: nat] :
% 0.23/0.59 ( ~ ( finite_finite_nat @ S2 )
% 0.23/0.59 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % le_enumerate
% 0.23/0.59 thf(fact_9988_inverse__real_Orsp,axiom,
% 0.23/0.59 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 0.23/0.59 @ ^ [X6: nat > rat] :
% 0.23/0.59 ( if_nat_rat @ ( vanishes @ X6 )
% 0.23/0.59 @ ^ [N2: nat] : zero_zero_rat
% 0.23/0.59 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
% 0.23/0.59 @ ^ [X6: nat > rat] :
% 0.23/0.59 ( if_nat_rat @ ( vanishes @ X6 )
% 0.23/0.59 @ ^ [N2: nat] : zero_zero_rat
% 0.23/0.59 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % inverse_real.rsp
% 0.23/0.59 thf(fact_9989_finite__le__enumerate,axiom,
% 0.23/0.59 ! [S2: set_nat,N: nat] :
% 0.23/0.59 ( ( finite_finite_nat @ S2 )
% 0.23/0.59 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
% 0.23/0.59 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % finite_le_enumerate
% 0.23/0.59 thf(fact_9990_Real_Opositive_Otransfer,axiom,
% 0.23/0.59 ( bNF_re4297313714947099218al_o_o @ pcr_real
% 0.23/0.59 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [X6: nat > rat] :
% 0.23/0.59 ? [R5: rat] :
% 0.23/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.23/0.59 & ? [K3: nat] :
% 0.23/0.59 ! [N2: nat] :
% 0.23/0.59 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.23/0.59 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 0.23/0.59 @ positive2 ) ).
% 0.23/0.59
% 0.23/0.59 % Real.positive.transfer
% 0.23/0.59 thf(fact_9991_plus__rat_Otransfer,axiom,
% 0.23/0.59 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.23/0.59 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 0.23/0.59 @ plus_plus_rat ) ).
% 0.23/0.59
% 0.23/0.59 % plus_rat.transfer
% 0.23/0.59 thf(fact_9992_real_Orel__eq__transfer,axiom,
% 0.23/0.59 ( bNF_re4521903465945308077real_o @ pcr_real
% 0.23/0.59 @ ( bNF_re4297313714947099218al_o_o @ pcr_real
% 0.23/0.59 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.23/0.59 @ realrel
% 0.23/0.59 @ ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) ) ).
% 0.23/0.59
% 0.23/0.59 % real.rel_eq_transfer
% 0.23/0.59 thf(fact_9993_real_Opcr__cr__eq,axiom,
% 0.23/0.59 pcr_real = cr_real ).
% 0.23/0.59
% 0.23/0.59 % real.pcr_cr_eq
% 0.23/0.59 thf(fact_9994_zero__real_Otransfer,axiom,
% 0.23/0.59 ( pcr_real
% 0.23/0.59 @ ^ [N2: nat] : zero_zero_rat
% 0.23/0.59 @ zero_zero_real ) ).
% 0.23/0.59
% 0.23/0.59 % zero_real.transfer
% 0.23/0.59 thf(fact_9995_one__real_Otransfer,axiom,
% 0.23/0.59 ( pcr_real
% 0.23/0.59 @ ^ [N2: nat] : one_one_rat
% 0.23/0.59 @ one_one_real ) ).
% 0.23/0.59
% 0.23/0.59 % one_real.transfer
% 0.23/0.59 thf(fact_9996_cr__real__eq,axiom,
% 0.23/0.59 ( pcr_real
% 0.23/0.59 = ( ^ [X: nat > rat,Y: real] :
% 0.23/0.59 ( ( cauchy @ X )
% 0.23/0.59 & ( ( real2 @ X )
% 0.23/0.59 = Y ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % cr_real_eq
% 0.23/0.59 thf(fact_9997_uminus__real_Otransfer,axiom,
% 0.23/0.59 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 0.23/0.59 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
% 0.23/0.59 @ uminus_uminus_real ) ).
% 0.23/0.59
% 0.23/0.59 % uminus_real.transfer
% 0.23/0.59 thf(fact_9998_plus__real_Otransfer,axiom,
% 0.23/0.59 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 0.23/0.59 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 0.23/0.59 @ plus_plus_real ) ).
% 0.23/0.59
% 0.23/0.59 % plus_real.transfer
% 0.23/0.59 thf(fact_9999_times__real_Otransfer,axiom,
% 0.23/0.59 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 0.23/0.59 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 0.23/0.59 @ times_times_real ) ).
% 0.23/0.59
% 0.23/0.59 % times_real.transfer
% 0.23/0.59 thf(fact_10000_inverse__real_Otransfer,axiom,
% 0.23/0.59 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 0.23/0.59 @ ^ [X6: nat > rat] :
% 0.23/0.59 ( if_nat_rat @ ( vanishes @ X6 )
% 0.23/0.59 @ ^ [N2: nat] : zero_zero_rat
% 0.23/0.59 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
% 0.23/0.59 @ inverse_inverse_real ) ).
% 0.23/0.59
% 0.23/0.59 % inverse_real.transfer
% 0.23/0.59 thf(fact_10001_times__rat_Otransfer,axiom,
% 0.23/0.59 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.23/0.59 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 0.23/0.59 @ times_times_rat ) ).
% 0.23/0.59
% 0.23/0.59 % times_rat.transfer
% 0.23/0.59 thf(fact_10002_Rat_Opositive_Otransfer,axiom,
% 0.23/0.59 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 0.23/0.59 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 0.23/0.59 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 0.23/0.59 @ positive ) ).
% 0.23/0.59
% 0.23/0.59 % Rat.positive.transfer
% 0.23/0.59 thf(fact_10003_times__int_Otransfer,axiom,
% 0.23/0.59 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.23/0.59 @ ( produc27273713700761075at_nat
% 0.23/0.59 @ ^ [X: nat,Y: nat] :
% 0.23/0.59 ( produc2626176000494625587at_nat
% 0.23/0.59 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) )
% 0.23/0.59 @ times_times_int ) ).
% 0.23/0.59
% 0.23/0.59 % times_int.transfer
% 0.23/0.59 thf(fact_10004_Least__eq__0,axiom,
% 0.23/0.59 ! [P: nat > $o] :
% 0.23/0.59 ( ( P @ zero_zero_nat )
% 0.23/0.59 => ( ( ord_Least_nat @ P )
% 0.23/0.59 = zero_zero_nat ) ) ).
% 0.23/0.59
% 0.23/0.59 % Least_eq_0
% 0.23/0.59 thf(fact_10005_Least__Suc,axiom,
% 0.23/0.59 ! [P: nat > $o,N: nat] :
% 0.23/0.59 ( ( P @ N )
% 0.23/0.59 => ( ~ ( P @ zero_zero_nat )
% 0.23/0.59 => ( ( ord_Least_nat @ P )
% 0.23/0.59 = ( suc
% 0.23/0.59 @ ( ord_Least_nat
% 0.23/0.59 @ ^ [M6: nat] : ( P @ ( suc @ M6 ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Least_Suc
% 0.23/0.59 thf(fact_10006_Least__Suc2,axiom,
% 0.23/0.59 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 0.23/0.59 ( ( P @ N )
% 0.23/0.59 => ( ( Q @ M )
% 0.23/0.59 => ( ~ ( P @ zero_zero_nat )
% 0.23/0.59 => ( ! [K2: nat] :
% 0.23/0.59 ( ( P @ ( suc @ K2 ) )
% 0.23/0.59 = ( Q @ K2 ) )
% 0.23/0.59 => ( ( ord_Least_nat @ P )
% 0.23/0.59 = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Least_Suc2
% 0.23/0.59 thf(fact_10007_Sup__real__def,axiom,
% 0.23/0.59 ( comple1385675409528146559p_real
% 0.23/0.59 = ( ^ [X6: set_real] :
% 0.23/0.59 ( ord_Least_real
% 0.23/0.59 @ ^ [Z5: real] :
% 0.23/0.59 ! [X: real] :
% 0.23/0.59 ( ( member_real @ X @ X6 )
% 0.23/0.59 => ( ord_less_eq_real @ X @ Z5 ) ) ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % Sup_real_def
% 0.23/0.59 thf(fact_10008_nat_Otransfer,axiom,
% 0.23/0.59 ( bNF_re4555766996558763186at_nat @ pcr_int
% 0.23/0.59 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.59 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 0.23/0.59 @ nat2 ) ).
% 0.23/0.59
% 0.23/0.59 % nat.transfer
% 0.23/0.59 thf(fact_10009_one__int_Otransfer,axiom,
% 0.23/0.59 pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 0.23/0.59
% 0.23/0.59 % one_int.transfer
% 0.23/0.59 thf(fact_10010_less__int_Otransfer,axiom,
% 0.23/0.59 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.23/0.59 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.23/0.59 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.23/0.59 @ ( produc8739625826339149834_nat_o
% 0.23/0.59 @ ^ [X: nat,Y: nat] :
% 0.23/0.59 ( produc6081775807080527818_nat_o
% 0.23/0.59 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
% 0.23/0.59 @ ord_less_int ) ).
% 0.23/0.59
% 0.23/0.59 % less_int.transfer
% 0.23/0.59 thf(fact_10011_less__eq__int_Otransfer,axiom,
% 0.23/0.59 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.23/0.59 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.23/0.59 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.23/0.59 @ ( produc8739625826339149834_nat_o
% 0.23/0.59 @ ^ [X: nat,Y: nat] :
% 0.23/0.59 ( produc6081775807080527818_nat_o
% 0.23/0.59 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
% 0.23/0.59 @ ord_less_eq_int ) ).
% 0.23/0.59
% 0.23/0.59 % less_eq_int.transfer
% 0.23/0.59 thf(fact_10012_plus__int_Otransfer,axiom,
% 0.23/0.59 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.23/0.59 @ ( produc27273713700761075at_nat
% 0.23/0.59 @ ^ [X: nat,Y: nat] :
% 0.23/0.59 ( produc2626176000494625587at_nat
% 0.23/0.59 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 0.23/0.59 @ plus_plus_int ) ).
% 0.23/0.59
% 0.23/0.59 % plus_int.transfer
% 0.23/0.59 thf(fact_10013_minus__int_Otransfer,axiom,
% 0.23/0.59 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.23/0.59 @ ( produc27273713700761075at_nat
% 0.23/0.59 @ ^ [X: nat,Y: nat] :
% 0.23/0.59 ( produc2626176000494625587at_nat
% 0.23/0.59 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) )
% 0.23/0.59 @ minus_minus_int ) ).
% 0.23/0.59
% 0.23/0.59 % minus_int.transfer
% 0.23/0.59 thf(fact_10014_of__rat__Real,axiom,
% 0.23/0.59 ( field_7254667332652039916t_real
% 0.23/0.59 = ( ^ [X: rat] :
% 0.23/0.59 ( real2
% 0.23/0.59 @ ^ [N2: nat] : X ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % of_rat_Real
% 0.23/0.59 thf(fact_10015_of__rat__dense,axiom,
% 0.23/0.59 ! [X4: real,Y3: real] :
% 0.23/0.59 ( ( ord_less_real @ X4 @ Y3 )
% 0.23/0.59 => ? [Q3: rat] :
% 0.23/0.59 ( ( ord_less_real @ X4 @ ( field_7254667332652039916t_real @ Q3 ) )
% 0.23/0.59 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q3 ) @ Y3 ) ) ) ).
% 0.23/0.59
% 0.23/0.59 % of_rat_dense
% 0.23/0.59 thf(fact_10016_less__RealD,axiom,
% 0.23/0.59 ! [Y7: nat > rat,X4: real] :
% 0.23/0.59 ( ( cauchy @ Y7 )
% 0.23/0.59 => ( ( ord_less_real @ X4 @ ( real2 @ Y7 ) )
% 0.23/0.59 => ? [N4: nat] : ( ord_less_real @ X4 @ ( field_7254667332652039916t_real @ ( Y7 @ N4 ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % less_RealD
% 0.23/0.60 thf(fact_10017_le__RealI,axiom,
% 0.23/0.60 ! [Y7: nat > rat,X4: real] :
% 0.23/0.60 ( ( cauchy @ Y7 )
% 0.23/0.60 => ( ! [N4: nat] : ( ord_less_eq_real @ X4 @ ( field_7254667332652039916t_real @ ( Y7 @ N4 ) ) )
% 0.23/0.60 => ( ord_less_eq_real @ X4 @ ( real2 @ Y7 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % le_RealI
% 0.23/0.60 thf(fact_10018_Real__leI,axiom,
% 0.23/0.60 ! [X8: nat > rat,Y3: real] :
% 0.23/0.60 ( ( cauchy @ X8 )
% 0.23/0.60 => ( ! [N4: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X8 @ N4 ) ) @ Y3 )
% 0.23/0.60 => ( ord_less_eq_real @ ( real2 @ X8 ) @ Y3 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % Real_leI
% 0.23/0.60 thf(fact_10019_times__int_Orsp,axiom,
% 0.23/0.60 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.23/0.60 @ ( produc27273713700761075at_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc2626176000494625587at_nat
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) )
% 0.23/0.60 @ ( produc27273713700761075at_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc2626176000494625587at_nat
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % times_int.rsp
% 0.23/0.60 thf(fact_10020_intrel__iff,axiom,
% 0.23/0.60 ! [X4: nat,Y3: nat,U: nat,V: nat] :
% 0.23/0.60 ( ( intrel @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( product_Pair_nat_nat @ U @ V ) )
% 0.23/0.60 = ( ( plus_plus_nat @ X4 @ V )
% 0.23/0.60 = ( plus_plus_nat @ U @ Y3 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % intrel_iff
% 0.23/0.60 thf(fact_10021_nat_Orsp,axiom,
% 0.23/0.60 ( bNF_re8246922863344978751at_nat @ intrel
% 0.23/0.60 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.23/0.60 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 0.23/0.60 @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% 0.23/0.60
% 0.23/0.60 % nat.rsp
% 0.23/0.60 thf(fact_10022_one__int_Orsp,axiom,
% 0.23/0.60 intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.23/0.60
% 0.23/0.60 % one_int.rsp
% 0.23/0.60 thf(fact_10023_intrel__def,axiom,
% 0.23/0.60 ( intrel
% 0.23/0.60 = ( produc8739625826339149834_nat_o
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc6081775807080527818_nat_o
% 0.23/0.60 @ ^ [U4: nat,V4: nat] :
% 0.23/0.60 ( ( plus_plus_nat @ X @ V4 )
% 0.23/0.60 = ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % intrel_def
% 0.23/0.60 thf(fact_10024_less__int_Orsp,axiom,
% 0.23/0.60 ( bNF_re4202695980764964119_nat_o @ intrel
% 0.23/0.60 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.23/0.60 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.23/0.60 @ ( produc8739625826339149834_nat_o
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc6081775807080527818_nat_o
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
% 0.23/0.60 @ ( produc8739625826339149834_nat_o
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc6081775807080527818_nat_o
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % less_int.rsp
% 0.23/0.60 thf(fact_10025_less__eq__int_Orsp,axiom,
% 0.23/0.60 ( bNF_re4202695980764964119_nat_o @ intrel
% 0.23/0.60 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.23/0.60 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.23/0.60 @ ( produc8739625826339149834_nat_o
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc6081775807080527818_nat_o
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
% 0.23/0.60 @ ( produc8739625826339149834_nat_o
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc6081775807080527818_nat_o
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % less_eq_int.rsp
% 0.23/0.60 thf(fact_10026_plus__int_Orsp,axiom,
% 0.23/0.60 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.23/0.60 @ ( produc27273713700761075at_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc2626176000494625587at_nat
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 0.23/0.60 @ ( produc27273713700761075at_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc2626176000494625587at_nat
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % plus_int.rsp
% 0.23/0.60 thf(fact_10027_minus__int_Orsp,axiom,
% 0.23/0.60 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.23/0.60 @ ( produc27273713700761075at_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc2626176000494625587at_nat
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) )
% 0.23/0.60 @ ( produc27273713700761075at_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc2626176000494625587at_nat
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % minus_int.rsp
% 0.23/0.60 thf(fact_10028_has__vector__derivative__id,axiom,
% 0.23/0.60 ! [Net: filter_real] :
% 0.23/0.60 ( has_ve631408500373753343e_real
% 0.23/0.60 @ ^ [X: real] : X
% 0.23/0.60 @ one_one_real
% 0.23/0.60 @ Net ) ).
% 0.23/0.60
% 0.23/0.60 % has_vector_derivative_id
% 0.23/0.60 thf(fact_10029_has__field__derivative__iff__has__vector__derivative,axiom,
% 0.23/0.60 has_fi5821293074295781190e_real = has_ve631408500373753343e_real ).
% 0.23/0.60
% 0.23/0.60 % has_field_derivative_iff_has_vector_derivative
% 0.23/0.60 thf(fact_10030_plus__rat_Orsp,axiom,
% 0.23/0.60 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.23/0.60 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 0.23/0.60 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % plus_rat.rsp
% 0.23/0.60 thf(fact_10031_ratrel__iff,axiom,
% 0.23/0.60 ( ratrel
% 0.23/0.60 = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.23/0.60 ( ( ( product_snd_int_int @ X )
% 0.23/0.60 != zero_zero_int )
% 0.23/0.60 & ( ( product_snd_int_int @ Y )
% 0.23/0.60 != zero_zero_int )
% 0.23/0.60 & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 0.23/0.60 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % ratrel_iff
% 0.23/0.60 thf(fact_10032_ratrel__def,axiom,
% 0.23/0.60 ( ratrel
% 0.23/0.60 = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.23/0.60 ( ( ( product_snd_int_int @ X )
% 0.23/0.60 != zero_zero_int )
% 0.23/0.60 & ( ( product_snd_int_int @ Y )
% 0.23/0.60 != zero_zero_int )
% 0.23/0.60 & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 0.23/0.60 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % ratrel_def
% 0.23/0.60 thf(fact_10033_times__rat_Orsp,axiom,
% 0.23/0.60 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.23/0.60 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 0.23/0.60 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % times_rat.rsp
% 0.23/0.60 thf(fact_10034_Rat_Opositive_Orsp,axiom,
% 0.23/0.60 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 0.23/0.60 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 0.23/0.60 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 0.23/0.60 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % Rat.positive.rsp
% 0.23/0.60 thf(fact_10035_plus__rat_Oabs__eq,axiom,
% 0.23/0.60 ! [Xa: product_prod_int_int,X4: product_prod_int_int] :
% 0.23/0.60 ( ( ratrel @ Xa @ Xa )
% 0.23/0.60 => ( ( ratrel @ X4 @ X4 )
% 0.23/0.60 => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X4 ) )
% 0.23/0.60 = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X4 ) ) @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % plus_rat.abs_eq
% 0.23/0.60 thf(fact_10036_Rat_Opositive_Oabs__eq,axiom,
% 0.23/0.60 ! [X4: product_prod_int_int] :
% 0.23/0.60 ( ( ratrel @ X4 @ X4 )
% 0.23/0.60 => ( ( positive @ ( abs_Rat @ X4 ) )
% 0.23/0.60 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % Rat.positive.abs_eq
% 0.23/0.60 thf(fact_10037_times__rat_Oabs__eq,axiom,
% 0.23/0.60 ! [Xa: product_prod_int_int,X4: product_prod_int_int] :
% 0.23/0.60 ( ( ratrel @ Xa @ Xa )
% 0.23/0.60 => ( ( ratrel @ X4 @ X4 )
% 0.23/0.60 => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X4 ) )
% 0.23/0.60 = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X4 ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % times_rat.abs_eq
% 0.23/0.60 thf(fact_10038_last__upt,axiom,
% 0.23/0.60 ! [I: nat,J: nat] :
% 0.23/0.60 ( ( ord_less_nat @ I @ J )
% 0.23/0.60 => ( ( last_nat @ ( upt @ I @ J ) )
% 0.23/0.60 = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % last_upt
% 0.23/0.60 thf(fact_10039_Bseq__monoseq__convergent_H__dec,axiom,
% 0.23/0.60 ! [F: nat > real,M7: nat] :
% 0.23/0.60 ( ( bfun_nat_real
% 0.23/0.60 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M7 ) )
% 0.23/0.60 @ at_top_nat )
% 0.23/0.60 => ( ! [M4: nat,N4: nat] :
% 0.23/0.60 ( ( ord_less_eq_nat @ M7 @ M4 )
% 0.23/0.60 => ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.23/0.60 => ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ M4 ) ) ) )
% 0.23/0.60 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % Bseq_monoseq_convergent'_dec
% 0.23/0.60 thf(fact_10040_Bseq__monoseq__convergent_H__inc,axiom,
% 0.23/0.60 ! [F: nat > real,M7: nat] :
% 0.23/0.60 ( ( bfun_nat_real
% 0.23/0.60 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M7 ) )
% 0.23/0.60 @ at_top_nat )
% 0.23/0.60 => ( ! [M4: nat,N4: nat] :
% 0.23/0.60 ( ( ord_less_eq_nat @ M7 @ M4 )
% 0.23/0.60 => ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.23/0.60 => ( ord_less_eq_real @ ( F @ M4 ) @ ( F @ N4 ) ) ) )
% 0.23/0.60 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % Bseq_monoseq_convergent'_inc
% 0.23/0.60 thf(fact_10041_Bseq__mono__convergent,axiom,
% 0.23/0.60 ! [X8: nat > real] :
% 0.23/0.60 ( ( bfun_nat_real @ X8 @ at_top_nat )
% 0.23/0.60 => ( ! [M4: nat,N4: nat] :
% 0.23/0.60 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.23/0.60 => ( ord_less_eq_real @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) )
% 0.23/0.60 => ( topolo7531315842566124627t_real @ X8 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % Bseq_mono_convergent
% 0.23/0.60 thf(fact_10042_convergent__realpow,axiom,
% 0.23/0.60 ! [X4: real] :
% 0.23/0.60 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.23/0.60 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 0.23/0.60 => ( topolo7531315842566124627t_real @ ( power_power_real @ X4 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % convergent_realpow
% 0.23/0.60 thf(fact_10043_filtermap__at__right__shift,axiom,
% 0.23/0.60 ! [D: real,A: real] :
% 0.23/0.60 ( ( filtermap_real_real
% 0.23/0.60 @ ^ [X: real] : ( minus_minus_real @ X @ D )
% 0.23/0.60 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.23/0.60 = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % filtermap_at_right_shift
% 0.23/0.60 thf(fact_10044_at__right__to__0,axiom,
% 0.23/0.60 ! [A: real] :
% 0.23/0.60 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 0.23/0.60 = ( filtermap_real_real
% 0.23/0.60 @ ^ [X: real] : ( plus_plus_real @ X @ A )
% 0.23/0.60 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % at_right_to_0
% 0.23/0.60 thf(fact_10045_pair__lessI2,axiom,
% 0.23/0.60 ! [A: nat,B2: nat,S: nat,T: nat] :
% 0.23/0.60 ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.60 => ( ( ord_less_nat @ S @ T )
% 0.23/0.60 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % pair_lessI2
% 0.23/0.60 thf(fact_10046_pair__less__iff1,axiom,
% 0.23/0.60 ! [X4: nat,Y3: nat,Z: nat] :
% 0.23/0.60 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( product_Pair_nat_nat @ X4 @ Z ) ) @ fun_pair_less )
% 0.23/0.60 = ( ord_less_nat @ Y3 @ Z ) ) ).
% 0.23/0.60
% 0.23/0.60 % pair_less_iff1
% 0.23/0.60 thf(fact_10047_pair__lessI1,axiom,
% 0.23/0.60 ! [A: nat,B2: nat,S: nat,T: nat] :
% 0.23/0.60 ( ( ord_less_nat @ A @ B2 )
% 0.23/0.60 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ).
% 0.23/0.60
% 0.23/0.60 % pair_lessI1
% 0.23/0.60 thf(fact_10048_pair__leqI2,axiom,
% 0.23/0.60 ! [A: nat,B2: nat,S: nat,T: nat] :
% 0.23/0.60 ( ( ord_less_eq_nat @ A @ B2 )
% 0.23/0.60 => ( ( ord_less_eq_nat @ S @ T )
% 0.23/0.60 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % pair_leqI2
% 0.23/0.60 thf(fact_10049_pair__leqI1,axiom,
% 0.23/0.60 ! [A: nat,B2: nat,S: nat,T: nat] :
% 0.23/0.60 ( ( ord_less_nat @ A @ B2 )
% 0.23/0.60 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ).
% 0.23/0.60
% 0.23/0.60 % pair_leqI1
% 0.23/0.60 thf(fact_10050_bot__nat__0_Oordering__top__axioms,axiom,
% 0.23/0.60 ( ordering_top_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 0.23/0.60 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 0.23/0.60 @ zero_zero_nat ) ).
% 0.23/0.60
% 0.23/0.60 % bot_nat_0.ordering_top_axioms
% 0.23/0.60 thf(fact_10051_set__encode__vimage__Suc,axiom,
% 0.23/0.60 ! [A3: set_nat] :
% 0.23/0.60 ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A3 ) )
% 0.23/0.60 = ( divide_divide_nat @ ( nat_set_encode @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % set_encode_vimage_Suc
% 0.23/0.60 thf(fact_10052_euclidean__size__int__def,axiom,
% 0.23/0.60 ( euclid4774559944035922753ze_int
% 0.23/0.60 = ( comp_int_nat_int @ nat2 @ abs_abs_int ) ) ).
% 0.23/0.60
% 0.23/0.60 % euclidean_size_int_def
% 0.23/0.60 thf(fact_10053_finite__vimage__Suc__iff,axiom,
% 0.23/0.60 ! [F5: set_nat] :
% 0.23/0.60 ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F5 ) )
% 0.23/0.60 = ( finite_finite_nat @ F5 ) ) ).
% 0.23/0.60
% 0.23/0.60 % finite_vimage_Suc_iff
% 0.23/0.60 thf(fact_10054_vimage__Suc__insert__Suc,axiom,
% 0.23/0.60 ! [N: nat,A3: set_nat] :
% 0.23/0.60 ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A3 ) )
% 0.23/0.60 = ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A3 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % vimage_Suc_insert_Suc
% 0.23/0.60 thf(fact_10055_vimage__Suc__insert__0,axiom,
% 0.23/0.60 ! [A3: set_nat] :
% 0.23/0.60 ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A3 ) )
% 0.23/0.60 = ( vimage_nat_nat @ suc @ A3 ) ) ).
% 0.23/0.60
% 0.23/0.60 % vimage_Suc_insert_0
% 0.23/0.60 thf(fact_10056_set__decode__div__2,axiom,
% 0.23/0.60 ! [X4: nat] :
% 0.23/0.60 ( ( nat_set_decode @ ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.60 = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X4 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % set_decode_div_2
% 0.23/0.60 thf(fact_10057_abs__division__segment,axiom,
% 0.23/0.60 ! [K: int] :
% 0.23/0.60 ( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
% 0.23/0.60 = one_one_int ) ).
% 0.23/0.60
% 0.23/0.60 % abs_division_segment
% 0.23/0.60 thf(fact_10058_division__segment__nat__def,axiom,
% 0.23/0.60 ( euclid3398187327856392827nt_nat
% 0.23/0.60 = ( ^ [N2: nat] : one_one_nat ) ) ).
% 0.23/0.60
% 0.23/0.60 % division_segment_nat_def
% 0.23/0.60 thf(fact_10059_division__segment__eq__sgn,axiom,
% 0.23/0.60 ! [K: int] :
% 0.23/0.60 ( ( K != zero_zero_int )
% 0.23/0.60 => ( ( euclid3395696857347342551nt_int @ K )
% 0.23/0.60 = ( sgn_sgn_int @ K ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % division_segment_eq_sgn
% 0.23/0.60 thf(fact_10060_division__segment__int__def,axiom,
% 0.23/0.60 ( euclid3395696857347342551nt_int
% 0.23/0.60 = ( ^ [K3: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % division_segment_int_def
% 0.23/0.60 thf(fact_10061_less__eq__enat__def,axiom,
% 0.23/0.60 ( ord_le2932123472753598470d_enat
% 0.23/0.60 = ( ^ [M6: extended_enat] :
% 0.23/0.60 ( extended_case_enat_o
% 0.23/0.60 @ ^ [N1: nat] :
% 0.23/0.60 ( extended_case_enat_o
% 0.23/0.60 @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
% 0.23/0.60 @ $false
% 0.23/0.60 @ M6 )
% 0.23/0.60 @ $true ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % less_eq_enat_def
% 0.23/0.60 thf(fact_10062_transp__realrel,axiom,
% 0.23/0.60 transp_nat_rat @ realrel ).
% 0.23/0.60
% 0.23/0.60 % transp_realrel
% 0.23/0.60 thf(fact_10063_less__enat__def,axiom,
% 0.23/0.60 ( ord_le72135733267957522d_enat
% 0.23/0.60 = ( ^ [M6: extended_enat,N2: extended_enat] :
% 0.23/0.60 ( extended_case_enat_o
% 0.23/0.60 @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N2 )
% 0.23/0.60 @ $false
% 0.23/0.60 @ M6 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % less_enat_def
% 0.23/0.60 thf(fact_10064_Divides_Oadjust__div__def,axiom,
% 0.23/0.60 ( adjust_div
% 0.23/0.60 = ( produc8211389475949308722nt_int
% 0.23/0.60 @ ^ [Q4: int,R5: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % Divides.adjust_div_def
% 0.23/0.60 thf(fact_10065_rat__floor__code,axiom,
% 0.23/0.60 ( archim3151403230148437115or_rat
% 0.23/0.60 = ( ^ [P3: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P3 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % rat_floor_code
% 0.23/0.60 thf(fact_10066_prod__decode__triangle__add,axiom,
% 0.23/0.60 ! [K: nat,M: nat] :
% 0.23/0.60 ( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) )
% 0.23/0.60 = ( nat_prod_decode_aux @ K @ M ) ) ).
% 0.23/0.60
% 0.23/0.60 % prod_decode_triangle_add
% 0.23/0.60 thf(fact_10067_list__decode_Opinduct,axiom,
% 0.23/0.60 ! [A0: nat,P: nat > $o] :
% 0.23/0.60 ( ( accp_nat @ nat_list_decode_rel @ A0 )
% 0.23/0.60 => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
% 0.23/0.60 => ( P @ zero_zero_nat ) )
% 0.23/0.60 => ( ! [N4: nat] :
% 0.23/0.60 ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N4 ) )
% 0.23/0.60 => ( ! [X5: nat,Y5: nat] :
% 0.23/0.60 ( ( ( product_Pair_nat_nat @ X5 @ Y5 )
% 0.23/0.60 = ( nat_prod_decode @ N4 ) )
% 0.23/0.60 => ( P @ Y5 ) )
% 0.23/0.60 => ( P @ ( suc @ N4 ) ) ) )
% 0.23/0.60 => ( P @ A0 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % list_decode.pinduct
% 0.23/0.60 thf(fact_10068_list__decode_Oelims,axiom,
% 0.23/0.60 ! [X4: nat,Y3: list_nat] :
% 0.23/0.60 ( ( ( nat_list_decode @ X4 )
% 0.23/0.60 = Y3 )
% 0.23/0.60 => ( ( ( X4 = zero_zero_nat )
% 0.23/0.60 => ( Y3 != nil_nat ) )
% 0.23/0.60 => ~ ! [N4: nat] :
% 0.23/0.60 ( ( X4
% 0.23/0.60 = ( suc @ N4 ) )
% 0.23/0.60 => ( Y3
% 0.23/0.60 != ( produc2761476792215241774st_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 0.23/0.60 @ ( nat_prod_decode @ N4 ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % list_decode.elims
% 0.23/0.60 thf(fact_10069_list__decode_Opsimps_I2_J,axiom,
% 0.23/0.60 ! [N: nat] :
% 0.23/0.60 ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) )
% 0.23/0.60 => ( ( nat_list_decode @ ( suc @ N ) )
% 0.23/0.60 = ( produc2761476792215241774st_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 0.23/0.60 @ ( nat_prod_decode @ N ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % list_decode.psimps(2)
% 0.23/0.60 thf(fact_10070_list__decode_Osimps_I2_J,axiom,
% 0.23/0.60 ! [N: nat] :
% 0.23/0.60 ( ( nat_list_decode @ ( suc @ N ) )
% 0.23/0.60 = ( produc2761476792215241774st_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 0.23/0.60 @ ( nat_prod_decode @ N ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % list_decode.simps(2)
% 0.23/0.60 thf(fact_10071_list__decode_Opelims,axiom,
% 0.23/0.60 ! [X4: nat,Y3: list_nat] :
% 0.23/0.60 ( ( ( nat_list_decode @ X4 )
% 0.23/0.60 = Y3 )
% 0.23/0.60 => ( ( accp_nat @ nat_list_decode_rel @ X4 )
% 0.23/0.60 => ( ( ( X4 = zero_zero_nat )
% 0.23/0.60 => ( ( Y3 = nil_nat )
% 0.23/0.60 => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
% 0.23/0.60 => ~ ! [N4: nat] :
% 0.23/0.60 ( ( X4
% 0.23/0.60 = ( suc @ N4 ) )
% 0.23/0.60 => ( ( Y3
% 0.23/0.60 = ( produc2761476792215241774st_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 0.23/0.60 @ ( nat_prod_decode @ N4 ) ) )
% 0.23/0.60 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N4 ) ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % list_decode.pelims
% 0.23/0.60 thf(fact_10072_compute__powr__real,axiom,
% 0.23/0.60 ( powr_real2
% 0.23/0.60 = ( ^ [B3: real,I2: real] :
% 0.23/0.60 ( if_real @ ( ord_less_eq_real @ B3 @ zero_zero_real )
% 0.23/0.60 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.23/0.60 @ ^ [Uu: product_unit] : ( powr_real2 @ B3 @ I2 ) )
% 0.23/0.60 @ ( if_real
% 0.23/0.60 @ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I2 ) )
% 0.23/0.60 = I2 )
% 0.23/0.60 @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I2 ) @ ( power_power_real @ B3 @ ( nat2 @ ( archim6058952711729229775r_real @ I2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B3 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I2 ) ) ) ) ) )
% 0.23/0.60 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.23/0.60 @ ^ [Uu: product_unit] : ( powr_real2 @ B3 @ I2 ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % compute_powr_real
% 0.23/0.60 thf(fact_10073_pairs__le__eq__Sigma,axiom,
% 0.23/0.60 ! [M: nat] :
% 0.23/0.60 ( ( collec3392354462482085612at_nat
% 0.23/0.60 @ ( produc6081775807080527818_nat_o
% 0.23/0.60 @ ^ [I2: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ J3 ) @ M ) ) )
% 0.23/0.60 = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 0.23/0.60 @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % pairs_le_eq_Sigma
% 0.23/0.60 thf(fact_10074_product__atMost__eq__Un,axiom,
% 0.23/0.60 ! [A3: set_nat,M: nat] :
% 0.23/0.60 ( ( produc457027306803732586at_nat @ A3
% 0.23/0.60 @ ^ [Uu: nat] : ( set_ord_atMost_nat @ M ) )
% 0.23/0.60 = ( sup_su6327502436637775413at_nat
% 0.23/0.60 @ ( produc457027306803732586at_nat @ A3
% 0.23/0.60 @ ^ [I2: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ I2 ) ) )
% 0.23/0.60 @ ( produc457027306803732586at_nat @ A3
% 0.23/0.60 @ ^ [I2: nat] : ( set_or6659071591806873216st_nat @ ( minus_minus_nat @ M @ I2 ) @ M ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % product_atMost_eq_Un
% 0.23/0.60 thf(fact_10075_of__nat__eq__id,axiom,
% 0.23/0.60 semiri1316708129612266289at_nat = id_nat ).
% 0.23/0.60
% 0.23/0.60 % of_nat_eq_id
% 0.23/0.60 thf(fact_10076_Real_Opositive__def,axiom,
% 0.23/0.60 ( positive2
% 0.23/0.60 = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
% 0.23/0.60 @ ^ [X6: nat > rat] :
% 0.23/0.60 ? [R5: rat] :
% 0.23/0.60 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.23/0.60 & ? [K3: nat] :
% 0.23/0.60 ! [N2: nat] :
% 0.23/0.60 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.23/0.60 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % Real.positive_def
% 0.23/0.60 thf(fact_10077_euclidean__size__nat__def,axiom,
% 0.23/0.60 euclid4777050414544973029ze_nat = id_nat ).
% 0.23/0.60
% 0.23/0.60 % euclidean_size_nat_def
% 0.23/0.60 thf(fact_10078_times__rat__def,axiom,
% 0.23/0.60 ( times_times_rat
% 0.23/0.60 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.23/0.60 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % times_rat_def
% 0.23/0.60 thf(fact_10079_plus__rat__def,axiom,
% 0.23/0.60 ( plus_plus_rat
% 0.23/0.60 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.23/0.60 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % plus_rat_def
% 0.23/0.60 thf(fact_10080_Rat_Opositive__def,axiom,
% 0.23/0.60 ( positive
% 0.23/0.60 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 0.23/0.60 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % Rat.positive_def
% 0.23/0.60 thf(fact_10081_nat__def,axiom,
% 0.23/0.60 ( nat2
% 0.23/0.60 = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % nat_def
% 0.23/0.60 thf(fact_10082_less__int__def,axiom,
% 0.23/0.60 ( ord_less_int
% 0.23/0.60 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.23/0.60 @ ( produc8739625826339149834_nat_o
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc6081775807080527818_nat_o
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % less_int_def
% 0.23/0.60 thf(fact_10083_less__eq__int__def,axiom,
% 0.23/0.60 ( ord_less_eq_int
% 0.23/0.60 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.23/0.60 @ ( produc8739625826339149834_nat_o
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc6081775807080527818_nat_o
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % less_eq_int_def
% 0.23/0.60 thf(fact_10084_plus__int__def,axiom,
% 0.23/0.60 ( plus_plus_int
% 0.23/0.60 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.23/0.60 @ ( produc27273713700761075at_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc2626176000494625587at_nat
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % plus_int_def
% 0.23/0.60 thf(fact_10085_minus__int__def,axiom,
% 0.23/0.60 ( minus_minus_int
% 0.23/0.60 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.23/0.60 @ ( produc27273713700761075at_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc2626176000494625587at_nat
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % minus_int_def
% 0.23/0.60 thf(fact_10086_times__int__def,axiom,
% 0.23/0.60 ( times_times_int
% 0.23/0.60 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.23/0.60 @ ( produc27273713700761075at_nat
% 0.23/0.60 @ ^ [X: nat,Y: nat] :
% 0.23/0.60 ( produc2626176000494625587at_nat
% 0.23/0.60 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % times_int_def
% 0.23/0.60 thf(fact_10087_eventually__prod__sequentially,axiom,
% 0.23/0.60 ! [P: product_prod_nat_nat > $o] :
% 0.23/0.60 ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 0.23/0.60 = ( ? [N8: nat] :
% 0.23/0.60 ! [M6: nat] :
% 0.23/0.60 ( ( ord_less_eq_nat @ N8 @ M6 )
% 0.23/0.60 => ! [N2: nat] :
% 0.23/0.60 ( ( ord_less_eq_nat @ N8 @ N2 )
% 0.23/0.60 => ( P @ ( product_Pair_nat_nat @ N2 @ M6 ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % eventually_prod_sequentially
% 0.23/0.60 thf(fact_10088_less__eq,axiom,
% 0.23/0.60 ! [M: nat,N: nat] :
% 0.23/0.60 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 0.23/0.60 = ( ord_less_nat @ M @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % less_eq
% 0.23/0.60 thf(fact_10089_numeral__le__enat__iff,axiom,
% 0.23/0.60 ! [M: num,N: nat] :
% 0.23/0.60 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 0.23/0.60 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % numeral_le_enat_iff
% 0.23/0.60 thf(fact_10090_enat_Oinject,axiom,
% 0.23/0.60 ! [Nat: nat,Nat2: nat] :
% 0.23/0.60 ( ( ( extended_enat2 @ Nat )
% 0.23/0.60 = ( extended_enat2 @ Nat2 ) )
% 0.23/0.60 = ( Nat = Nat2 ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat.inject
% 0.23/0.60 thf(fact_10091_enat__ord__simps_I2_J,axiom,
% 0.23/0.60 ! [M: nat,N: nat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.23/0.60 = ( ord_less_nat @ M @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_ord_simps(2)
% 0.23/0.60 thf(fact_10092_enat__ord__simps_I1_J,axiom,
% 0.23/0.60 ! [M: nat,N: nat] :
% 0.23/0.60 ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.23/0.60 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_ord_simps(1)
% 0.23/0.60 thf(fact_10093_plus__enat__simps_I1_J,axiom,
% 0.23/0.60 ! [M: nat,N: nat] :
% 0.23/0.60 ( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.23/0.60 = ( extended_enat2 @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % plus_enat_simps(1)
% 0.23/0.60 thf(fact_10094_idiff__enat__0__right,axiom,
% 0.23/0.60 ! [N: extended_enat] :
% 0.23/0.60 ( ( minus_3235023915231533773d_enat @ N @ ( extended_enat2 @ zero_zero_nat ) )
% 0.23/0.60 = N ) ).
% 0.23/0.60
% 0.23/0.60 % idiff_enat_0_right
% 0.23/0.60 thf(fact_10095_idiff__enat__0,axiom,
% 0.23/0.60 ! [N: extended_enat] :
% 0.23/0.60 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N )
% 0.23/0.60 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 0.23/0.60
% 0.23/0.60 % idiff_enat_0
% 0.23/0.60 thf(fact_10096_idiff__enat__enat,axiom,
% 0.23/0.60 ! [A: nat,B2: nat] :
% 0.23/0.60 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ ( extended_enat2 @ B2 ) )
% 0.23/0.60 = ( extended_enat2 @ ( minus_minus_nat @ A @ B2 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % idiff_enat_enat
% 0.23/0.60 thf(fact_10097_times__enat__simps_I1_J,axiom,
% 0.23/0.60 ! [M: nat,N: nat] :
% 0.23/0.60 ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.23/0.60 = ( extended_enat2 @ ( times_times_nat @ M @ N ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % times_enat_simps(1)
% 0.23/0.60 thf(fact_10098_max__enat__simps_I1_J,axiom,
% 0.23/0.60 ! [M: nat,N: nat] :
% 0.23/0.60 ( ( ord_ma741700101516333627d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.23/0.60 = ( extended_enat2 @ ( ord_max_nat @ M @ N ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % max_enat_simps(1)
% 0.23/0.60 thf(fact_10099_min__enat__simps_I1_J,axiom,
% 0.23/0.60 ! [M: nat,N: nat] :
% 0.23/0.60 ( ( ord_mi8085742599997312461d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.23/0.60 = ( extended_enat2 @ ( ord_min_nat @ M @ N ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % min_enat_simps(1)
% 0.23/0.60 thf(fact_10100_numeral__less__enat__iff,axiom,
% 0.23/0.60 ! [M: num,N: nat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 0.23/0.60 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % numeral_less_enat_iff
% 0.23/0.60 thf(fact_10101_zero__enat__def,axiom,
% 0.23/0.60 ( zero_z5237406670263579293d_enat
% 0.23/0.60 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 0.23/0.60
% 0.23/0.60 % zero_enat_def
% 0.23/0.60 thf(fact_10102_enat__0__iff_I1_J,axiom,
% 0.23/0.60 ! [X4: nat] :
% 0.23/0.60 ( ( ( extended_enat2 @ X4 )
% 0.23/0.60 = zero_z5237406670263579293d_enat )
% 0.23/0.60 = ( X4 = zero_zero_nat ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_0_iff(1)
% 0.23/0.60 thf(fact_10103_enat__0__iff_I2_J,axiom,
% 0.23/0.60 ! [X4: nat] :
% 0.23/0.60 ( ( zero_z5237406670263579293d_enat
% 0.23/0.60 = ( extended_enat2 @ X4 ) )
% 0.23/0.60 = ( X4 = zero_zero_nat ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_0_iff(2)
% 0.23/0.60 thf(fact_10104_of__nat__eq__enat,axiom,
% 0.23/0.60 semiri4216267220026989637d_enat = extended_enat2 ).
% 0.23/0.60
% 0.23/0.60 % of_nat_eq_enat
% 0.23/0.60 thf(fact_10105_enat__iless,axiom,
% 0.23/0.60 ! [N: extended_enat,M: nat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
% 0.23/0.60 => ? [K2: nat] :
% 0.23/0.60 ( N
% 0.23/0.60 = ( extended_enat2 @ K2 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_iless
% 0.23/0.60 thf(fact_10106_less__enatE,axiom,
% 0.23/0.60 ! [N: extended_enat,M: nat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
% 0.23/0.60 => ~ ! [K2: nat] :
% 0.23/0.60 ( ( N
% 0.23/0.60 = ( extended_enat2 @ K2 ) )
% 0.23/0.60 => ~ ( ord_less_nat @ K2 @ M ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % less_enatE
% 0.23/0.60 thf(fact_10107_one__enat__def,axiom,
% 0.23/0.60 ( one_on7984719198319812577d_enat
% 0.23/0.60 = ( extended_enat2 @ one_one_nat ) ) ).
% 0.23/0.60
% 0.23/0.60 % one_enat_def
% 0.23/0.60 thf(fact_10108_enat__1__iff_I1_J,axiom,
% 0.23/0.60 ! [X4: nat] :
% 0.23/0.60 ( ( ( extended_enat2 @ X4 )
% 0.23/0.60 = one_on7984719198319812577d_enat )
% 0.23/0.60 = ( X4 = one_one_nat ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_1_iff(1)
% 0.23/0.60 thf(fact_10109_enat__1__iff_I2_J,axiom,
% 0.23/0.60 ! [X4: nat] :
% 0.23/0.60 ( ( one_on7984719198319812577d_enat
% 0.23/0.60 = ( extended_enat2 @ X4 ) )
% 0.23/0.60 = ( X4 = one_one_nat ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_1_iff(2)
% 0.23/0.60 thf(fact_10110_numeral__eq__enat,axiom,
% 0.23/0.60 ( numera1916890842035813515d_enat
% 0.23/0.60 = ( ^ [K3: num] : ( extended_enat2 @ ( numeral_numeral_nat @ K3 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % numeral_eq_enat
% 0.23/0.60 thf(fact_10111_Suc__ile__eq,axiom,
% 0.23/0.60 ! [M: nat,N: extended_enat] :
% 0.23/0.60 ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
% 0.23/0.60 = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % Suc_ile_eq
% 0.23/0.60 thf(fact_10112_finite__enat__bounded,axiom,
% 0.23/0.60 ! [A3: set_Extended_enat,N: nat] :
% 0.23/0.60 ( ! [Y4: extended_enat] :
% 0.23/0.60 ( ( member_Extended_enat @ Y4 @ A3 )
% 0.23/0.60 => ( ord_le2932123472753598470d_enat @ Y4 @ ( extended_enat2 @ N ) ) )
% 0.23/0.60 => ( finite4001608067531595151d_enat @ A3 ) ) ).
% 0.23/0.60
% 0.23/0.60 % finite_enat_bounded
% 0.23/0.60 thf(fact_10113_enat__ile,axiom,
% 0.23/0.60 ! [N: extended_enat,M: nat] :
% 0.23/0.60 ( ( ord_le2932123472753598470d_enat @ N @ ( extended_enat2 @ M ) )
% 0.23/0.60 => ? [K2: nat] :
% 0.23/0.60 ( N
% 0.23/0.60 = ( extended_enat2 @ K2 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_ile
% 0.23/0.60 thf(fact_10114_pred__nat__trancl__eq__le,axiom,
% 0.23/0.60 ! [M: nat,N: nat] :
% 0.23/0.60 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 0.23/0.60 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % pred_nat_trancl_eq_le
% 0.23/0.60 thf(fact_10115_iadd__le__enat__iff,axiom,
% 0.23/0.60 ! [X4: extended_enat,Y3: extended_enat,N: nat] :
% 0.23/0.60 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y3 ) @ ( extended_enat2 @ N ) )
% 0.23/0.60 = ( ? [Y9: nat,X9: nat] :
% 0.23/0.60 ( ( X4
% 0.23/0.60 = ( extended_enat2 @ X9 ) )
% 0.23/0.60 & ( Y3
% 0.23/0.60 = ( extended_enat2 @ Y9 ) )
% 0.23/0.60 & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y9 ) @ N ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % iadd_le_enat_iff
% 0.23/0.60 thf(fact_10116_elimnum,axiom,
% 0.23/0.60 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.23/0.60 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 0.23/0.60 => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.23/0.60 = ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % elimnum
% 0.23/0.60 thf(fact_10117_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
% 0.23/0.60 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,L2: nat] :
% 0.23/0.60 ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ L2 ) )
% 0.23/0.60 = ( vEBT_Node @ Info @ Deg
% 0.23/0.60 @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.60 @ ( map_VE8901447254227204932T_VEBT
% 0.23/0.60 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.60 @ TreeList2 ) )
% 0.23/0.60 @ ( vEBT_VEBT_elim_dead @ Summary @ ( extended_enat2 @ ( divide_divide_nat @ L2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % VEBT_internal.elim_dead.simps(3)
% 0.23/0.60 thf(fact_10118_VEBT__internal_Oelim__dead_Oelims,axiom,
% 0.23/0.60 ! [X4: vEBT_VEBT,Xa: extended_enat,Y3: vEBT_VEBT] :
% 0.23/0.60 ( ( ( vEBT_VEBT_elim_dead @ X4 @ Xa )
% 0.23/0.60 = Y3 )
% 0.23/0.60 => ( ! [A5: $o,B5: $o] :
% 0.23/0.60 ( ( X4
% 0.23/0.60 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.60 => ( Y3
% 0.23/0.60 != ( vEBT_Leaf @ A5 @ B5 ) ) )
% 0.23/0.60 => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.60 ( ( X4
% 0.23/0.60 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.60 => ( ( Xa = extend5688581933313929465d_enat )
% 0.23/0.60 => ( Y3
% 0.23/0.60 != ( vEBT_Node @ Info2 @ Deg2
% 0.23/0.60 @ ( map_VE8901447254227204932T_VEBT
% 0.23/0.60 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.60 @ TreeList3 )
% 0.23/0.60 @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) ) ) )
% 0.23/0.60 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.60 ( ( X4
% 0.23/0.60 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.60 => ! [L3: nat] :
% 0.23/0.60 ( ( Xa
% 0.23/0.60 = ( extended_enat2 @ L3 ) )
% 0.23/0.60 => ( Y3
% 0.23/0.60 != ( vEBT_Node @ Info2 @ Deg2
% 0.23/0.60 @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.60 @ ( map_VE8901447254227204932T_VEBT
% 0.23/0.60 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.60 @ TreeList3 ) )
% 0.23/0.60 @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % VEBT_internal.elim_dead.elims
% 0.23/0.60 thf(fact_10119_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
% 0.23/0.60 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.23/0.60 ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
% 0.23/0.60 = ( vEBT_Node @ Info @ Deg
% 0.23/0.60 @ ( map_VE8901447254227204932T_VEBT
% 0.23/0.60 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.60 @ TreeList2 )
% 0.23/0.60 @ ( vEBT_VEBT_elim_dead @ Summary @ extend5688581933313929465d_enat ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % VEBT_internal.elim_dead.simps(2)
% 0.23/0.60 thf(fact_10120_elimcomplete,axiom,
% 0.23/0.60 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.23/0.60 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 0.23/0.60 => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
% 0.23/0.60 = ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % elimcomplete
% 0.23/0.60 thf(fact_10121_not__infinity__eq,axiom,
% 0.23/0.60 ! [X4: extended_enat] :
% 0.23/0.60 ( ( X4 != extend5688581933313929465d_enat )
% 0.23/0.60 = ( ? [I2: nat] :
% 0.23/0.60 ( X4
% 0.23/0.60 = ( extended_enat2 @ I2 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % not_infinity_eq
% 0.23/0.60 thf(fact_10122_not__enat__eq,axiom,
% 0.23/0.60 ! [X4: extended_enat] :
% 0.23/0.60 ( ( ! [Y: nat] :
% 0.23/0.60 ( X4
% 0.23/0.60 != ( extended_enat2 @ Y ) ) )
% 0.23/0.60 = ( X4 = extend5688581933313929465d_enat ) ) ).
% 0.23/0.60
% 0.23/0.60 % not_enat_eq
% 0.23/0.60 thf(fact_10123_enat__ord__simps_I4_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ Q2 @ extend5688581933313929465d_enat )
% 0.23/0.60 = ( Q2 != extend5688581933313929465d_enat ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_ord_simps(4)
% 0.23/0.60 thf(fact_10124_enat__ord__simps_I6_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] :
% 0.23/0.60 ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ Q2 ) ).
% 0.23/0.60
% 0.23/0.60 % enat_ord_simps(6)
% 0.23/0.60 thf(fact_10125_plus__enat__simps_I2_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] :
% 0.23/0.60 ( ( plus_p3455044024723400733d_enat @ extend5688581933313929465d_enat @ Q2 )
% 0.23/0.60 = extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % plus_enat_simps(2)
% 0.23/0.60 thf(fact_10126_plus__enat__simps_I3_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] :
% 0.23/0.60 ( ( plus_p3455044024723400733d_enat @ Q2 @ extend5688581933313929465d_enat )
% 0.23/0.60 = extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % plus_enat_simps(3)
% 0.23/0.60 thf(fact_10127_enat__ord__code_I3_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] : ( ord_le2932123472753598470d_enat @ Q2 @ extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % enat_ord_code(3)
% 0.23/0.60 thf(fact_10128_enat__ord__simps_I5_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] :
% 0.23/0.60 ( ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ Q2 )
% 0.23/0.60 = ( Q2 = extend5688581933313929465d_enat ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_ord_simps(5)
% 0.23/0.60 thf(fact_10129_idiff__infinity,axiom,
% 0.23/0.60 ! [N: extended_enat] :
% 0.23/0.60 ( ( minus_3235023915231533773d_enat @ extend5688581933313929465d_enat @ N )
% 0.23/0.60 = extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % idiff_infinity
% 0.23/0.60 thf(fact_10130_times__enat__simps_I2_J,axiom,
% 0.23/0.60 ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ extend5688581933313929465d_enat )
% 0.23/0.60 = extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % times_enat_simps(2)
% 0.23/0.60 thf(fact_10131_max__enat__simps_I4_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] :
% 0.23/0.60 ( ( ord_ma741700101516333627d_enat @ Q2 @ extend5688581933313929465d_enat )
% 0.23/0.60 = extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % max_enat_simps(4)
% 0.23/0.60 thf(fact_10132_max__enat__simps_I5_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] :
% 0.23/0.60 ( ( ord_ma741700101516333627d_enat @ extend5688581933313929465d_enat @ Q2 )
% 0.23/0.60 = extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % max_enat_simps(5)
% 0.23/0.60 thf(fact_10133_min__enat__simps_I4_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] :
% 0.23/0.60 ( ( ord_mi8085742599997312461d_enat @ Q2 @ extend5688581933313929465d_enat )
% 0.23/0.60 = Q2 ) ).
% 0.23/0.60
% 0.23/0.60 % min_enat_simps(4)
% 0.23/0.60 thf(fact_10134_min__enat__simps_I5_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] :
% 0.23/0.60 ( ( ord_mi8085742599997312461d_enat @ extend5688581933313929465d_enat @ Q2 )
% 0.23/0.60 = Q2 ) ).
% 0.23/0.60
% 0.23/0.60 % min_enat_simps(5)
% 0.23/0.60 thf(fact_10135_idiff__self,axiom,
% 0.23/0.60 ! [N: extended_enat] :
% 0.23/0.60 ( ( N != extend5688581933313929465d_enat )
% 0.23/0.60 => ( ( minus_3235023915231533773d_enat @ N @ N )
% 0.23/0.60 = zero_z5237406670263579293d_enat ) ) ).
% 0.23/0.60
% 0.23/0.60 % idiff_self
% 0.23/0.60 thf(fact_10136_add__diff__cancel__enat,axiom,
% 0.23/0.60 ! [X4: extended_enat,Y3: extended_enat] :
% 0.23/0.60 ( ( X4 != extend5688581933313929465d_enat )
% 0.23/0.60 => ( ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y3 ) @ X4 )
% 0.23/0.60 = Y3 ) ) ).
% 0.23/0.60
% 0.23/0.60 % add_diff_cancel_enat
% 0.23/0.60 thf(fact_10137_idiff__infinity__right,axiom,
% 0.23/0.60 ! [A: nat] :
% 0.23/0.60 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ extend5688581933313929465d_enat )
% 0.23/0.60 = zero_z5237406670263579293d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % idiff_infinity_right
% 0.23/0.60 thf(fact_10138_times__enat__simps_I3_J,axiom,
% 0.23/0.60 ! [N: nat] :
% 0.23/0.60 ( ( ( N = zero_zero_nat )
% 0.23/0.60 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 0.23/0.60 = zero_z5237406670263579293d_enat ) )
% 0.23/0.60 & ( ( N != zero_zero_nat )
% 0.23/0.60 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 0.23/0.60 = extend5688581933313929465d_enat ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % times_enat_simps(3)
% 0.23/0.60 thf(fact_10139_times__enat__simps_I4_J,axiom,
% 0.23/0.60 ! [M: nat] :
% 0.23/0.60 ( ( ( M = zero_zero_nat )
% 0.23/0.60 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 0.23/0.60 = zero_z5237406670263579293d_enat ) )
% 0.23/0.60 & ( ( M != zero_zero_nat )
% 0.23/0.60 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 0.23/0.60 = extend5688581933313929465d_enat ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % times_enat_simps(4)
% 0.23/0.60 thf(fact_10140_Sup__enat__def,axiom,
% 0.23/0.60 ( comple4398354569131411667d_enat
% 0.23/0.60 = ( ^ [A6: set_Extended_enat] : ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ zero_z5237406670263579293d_enat @ ( if_Extended_enat @ ( finite4001608067531595151d_enat @ A6 ) @ ( lattic921264341876707157d_enat @ A6 ) @ extend5688581933313929465d_enat ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % Sup_enat_def
% 0.23/0.60 thf(fact_10141_Inf__enat__def,axiom,
% 0.23/0.60 ( comple2295165028678016749d_enat
% 0.23/0.60 = ( ^ [A6: set_Extended_enat] :
% 0.23/0.60 ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ extend5688581933313929465d_enat
% 0.23/0.60 @ ( ord_Le1955565732374568822d_enat
% 0.23/0.60 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % Inf_enat_def
% 0.23/0.60 thf(fact_10142_imult__is__infinity,axiom,
% 0.23/0.60 ! [A: extended_enat,B2: extended_enat] :
% 0.23/0.60 ( ( ( times_7803423173614009249d_enat @ A @ B2 )
% 0.23/0.60 = extend5688581933313929465d_enat )
% 0.23/0.60 = ( ( ( A = extend5688581933313929465d_enat )
% 0.23/0.60 & ( B2 != zero_z5237406670263579293d_enat ) )
% 0.23/0.60 | ( ( B2 = extend5688581933313929465d_enat )
% 0.23/0.60 & ( A != zero_z5237406670263579293d_enat ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % imult_is_infinity
% 0.23/0.60 thf(fact_10143_infinity__ne__i0,axiom,
% 0.23/0.60 extend5688581933313929465d_enat != zero_z5237406670263579293d_enat ).
% 0.23/0.60
% 0.23/0.60 % infinity_ne_i0
% 0.23/0.60 thf(fact_10144_enat__add__left__cancel__less,axiom,
% 0.23/0.60 ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
% 0.23/0.60 = ( ( A != extend5688581933313929465d_enat )
% 0.23/0.60 & ( ord_le72135733267957522d_enat @ B2 @ C ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_add_left_cancel_less
% 0.23/0.60 thf(fact_10145_infinity__ne__i1,axiom,
% 0.23/0.60 extend5688581933313929465d_enat != one_on7984719198319812577d_enat ).
% 0.23/0.60
% 0.23/0.60 % infinity_ne_i1
% 0.23/0.60 thf(fact_10146_plus__eq__infty__iff__enat,axiom,
% 0.23/0.60 ! [M: extended_enat,N: extended_enat] :
% 0.23/0.60 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 0.23/0.60 = extend5688581933313929465d_enat )
% 0.23/0.60 = ( ( M = extend5688581933313929465d_enat )
% 0.23/0.60 | ( N = extend5688581933313929465d_enat ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % plus_eq_infty_iff_enat
% 0.23/0.60 thf(fact_10147_enat__add__left__cancel,axiom,
% 0.23/0.60 ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
% 0.23/0.60 ( ( ( plus_p3455044024723400733d_enat @ A @ B2 )
% 0.23/0.60 = ( plus_p3455044024723400733d_enat @ A @ C ) )
% 0.23/0.60 = ( ( A = extend5688581933313929465d_enat )
% 0.23/0.60 | ( B2 = C ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_add_left_cancel
% 0.23/0.60 thf(fact_10148_numeral__ne__infinity,axiom,
% 0.23/0.60 ! [K: num] :
% 0.23/0.60 ( ( numera1916890842035813515d_enat @ K )
% 0.23/0.60 != extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % numeral_ne_infinity
% 0.23/0.60 thf(fact_10149_top__enat__def,axiom,
% 0.23/0.60 top_to3028658606643905974d_enat = extend5688581933313929465d_enat ).
% 0.23/0.60
% 0.23/0.60 % top_enat_def
% 0.23/0.60 thf(fact_10150_enat__ord__simps_I3_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] : ( ord_le2932123472753598470d_enat @ Q2 @ extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % enat_ord_simps(3)
% 0.23/0.60 thf(fact_10151_enat__add__left__cancel__le,axiom,
% 0.23/0.60 ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
% 0.23/0.60 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
% 0.23/0.60 = ( ( A = extend5688581933313929465d_enat )
% 0.23/0.60 | ( ord_le2932123472753598470d_enat @ B2 @ C ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_add_left_cancel_le
% 0.23/0.60 thf(fact_10152_enat__ord__code_I5_J,axiom,
% 0.23/0.60 ! [N: nat] :
% 0.23/0.60 ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_ord_code(5)
% 0.23/0.60 thf(fact_10153_infinity__ileE,axiom,
% 0.23/0.60 ! [M: nat] :
% 0.23/0.60 ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).
% 0.23/0.60
% 0.23/0.60 % infinity_ileE
% 0.23/0.60 thf(fact_10154_enat__ex__split,axiom,
% 0.23/0.60 ( ( ^ [P5: extended_enat > $o] :
% 0.23/0.60 ? [X7: extended_enat] : ( P5 @ X7 ) )
% 0.23/0.60 = ( ^ [P6: extended_enat > $o] :
% 0.23/0.60 ( ( P6 @ extend5688581933313929465d_enat )
% 0.23/0.60 | ? [X: nat] : ( P6 @ ( extended_enat2 @ X ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_ex_split
% 0.23/0.60 thf(fact_10155_enat3__cases,axiom,
% 0.23/0.60 ! [Y3: extended_enat,Ya: extended_enat,Yb: extended_enat] :
% 0.23/0.60 ( ( ? [Nat3: nat] :
% 0.23/0.60 ( Y3
% 0.23/0.60 = ( extended_enat2 @ Nat3 ) )
% 0.23/0.60 => ( ? [Nata: nat] :
% 0.23/0.60 ( Ya
% 0.23/0.60 = ( extended_enat2 @ Nata ) )
% 0.23/0.60 => ! [Natb: nat] :
% 0.23/0.60 ( Yb
% 0.23/0.60 != ( extended_enat2 @ Natb ) ) ) )
% 0.23/0.60 => ( ( ? [Nat3: nat] :
% 0.23/0.60 ( Y3
% 0.23/0.60 = ( extended_enat2 @ Nat3 ) )
% 0.23/0.60 => ( ? [Nata: nat] :
% 0.23/0.60 ( Ya
% 0.23/0.60 = ( extended_enat2 @ Nata ) )
% 0.23/0.60 => ( Yb != extend5688581933313929465d_enat ) ) )
% 0.23/0.60 => ( ( ? [Nat3: nat] :
% 0.23/0.60 ( Y3
% 0.23/0.60 = ( extended_enat2 @ Nat3 ) )
% 0.23/0.60 => ( ( Ya = extend5688581933313929465d_enat )
% 0.23/0.60 => ! [Nata: nat] :
% 0.23/0.60 ( Yb
% 0.23/0.60 != ( extended_enat2 @ Nata ) ) ) )
% 0.23/0.60 => ( ( ? [Nat3: nat] :
% 0.23/0.60 ( Y3
% 0.23/0.60 = ( extended_enat2 @ Nat3 ) )
% 0.23/0.60 => ( ( Ya = extend5688581933313929465d_enat )
% 0.23/0.60 => ( Yb != extend5688581933313929465d_enat ) ) )
% 0.23/0.60 => ( ( ( Y3 = extend5688581933313929465d_enat )
% 0.23/0.60 => ( ? [Nat3: nat] :
% 0.23/0.60 ( Ya
% 0.23/0.60 = ( extended_enat2 @ Nat3 ) )
% 0.23/0.60 => ! [Nata: nat] :
% 0.23/0.60 ( Yb
% 0.23/0.60 != ( extended_enat2 @ Nata ) ) ) )
% 0.23/0.60 => ( ( ( Y3 = extend5688581933313929465d_enat )
% 0.23/0.60 => ( ? [Nat3: nat] :
% 0.23/0.60 ( Ya
% 0.23/0.60 = ( extended_enat2 @ Nat3 ) )
% 0.23/0.60 => ( Yb != extend5688581933313929465d_enat ) ) )
% 0.23/0.60 => ( ( ( Y3 = extend5688581933313929465d_enat )
% 0.23/0.60 => ( ( Ya = extend5688581933313929465d_enat )
% 0.23/0.60 => ! [Nat3: nat] :
% 0.23/0.60 ( Yb
% 0.23/0.60 != ( extended_enat2 @ Nat3 ) ) ) )
% 0.23/0.60 => ~ ( ( Y3 = extend5688581933313929465d_enat )
% 0.23/0.60 => ( ( Ya = extend5688581933313929465d_enat )
% 0.23/0.60 => ( Yb != extend5688581933313929465d_enat ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat3_cases
% 0.23/0.60 thf(fact_10156_enat2__cases,axiom,
% 0.23/0.60 ! [Y3: extended_enat,Ya: extended_enat] :
% 0.23/0.60 ( ( ? [Nat3: nat] :
% 0.23/0.60 ( Y3
% 0.23/0.60 = ( extended_enat2 @ Nat3 ) )
% 0.23/0.60 => ! [Nata: nat] :
% 0.23/0.60 ( Ya
% 0.23/0.60 != ( extended_enat2 @ Nata ) ) )
% 0.23/0.60 => ( ( ? [Nat3: nat] :
% 0.23/0.60 ( Y3
% 0.23/0.60 = ( extended_enat2 @ Nat3 ) )
% 0.23/0.60 => ( Ya != extend5688581933313929465d_enat ) )
% 0.23/0.60 => ( ( ( Y3 = extend5688581933313929465d_enat )
% 0.23/0.60 => ! [Nat3: nat] :
% 0.23/0.60 ( Ya
% 0.23/0.60 != ( extended_enat2 @ Nat3 ) ) )
% 0.23/0.60 => ~ ( ( Y3 = extend5688581933313929465d_enat )
% 0.23/0.60 => ( Ya != extend5688581933313929465d_enat ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat2_cases
% 0.23/0.60 thf(fact_10157_enat_Oexhaust,axiom,
% 0.23/0.60 ! [Y3: extended_enat] :
% 0.23/0.60 ( ! [Nat3: nat] :
% 0.23/0.60 ( Y3
% 0.23/0.60 != ( extended_enat2 @ Nat3 ) )
% 0.23/0.60 => ( Y3 = extend5688581933313929465d_enat ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat.exhaust
% 0.23/0.60 thf(fact_10158_enat_Odistinct_I1_J,axiom,
% 0.23/0.60 ! [Nat: nat] :
% 0.23/0.60 ( ( extended_enat2 @ Nat )
% 0.23/0.60 != extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % enat.distinct(1)
% 0.23/0.60 thf(fact_10159_enat_Odistinct_I2_J,axiom,
% 0.23/0.60 ! [Nat: nat] :
% 0.23/0.60 ( extend5688581933313929465d_enat
% 0.23/0.60 != ( extended_enat2 @ Nat ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat.distinct(2)
% 0.23/0.60 thf(fact_10160_infinity__ilessE,axiom,
% 0.23/0.60 ! [M: nat] :
% 0.23/0.60 ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).
% 0.23/0.60
% 0.23/0.60 % infinity_ilessE
% 0.23/0.60 thf(fact_10161_less__infinityE,axiom,
% 0.23/0.60 ! [N: extended_enat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ N @ extend5688581933313929465d_enat )
% 0.23/0.60 => ~ ! [K2: nat] :
% 0.23/0.60 ( N
% 0.23/0.60 != ( extended_enat2 @ K2 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % less_infinityE
% 0.23/0.60 thf(fact_10162_enat__ord__code_I4_J,axiom,
% 0.23/0.60 ! [M: nat] : ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % enat_ord_code(4)
% 0.23/0.60 thf(fact_10163_imult__infinity,axiom,
% 0.23/0.60 ! [N: extended_enat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.23/0.60 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ N )
% 0.23/0.60 = extend5688581933313929465d_enat ) ) ).
% 0.23/0.60
% 0.23/0.60 % imult_infinity
% 0.23/0.60 thf(fact_10164_imult__infinity__right,axiom,
% 0.23/0.60 ! [N: extended_enat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.23/0.60 => ( ( times_7803423173614009249d_enat @ N @ extend5688581933313929465d_enat )
% 0.23/0.60 = extend5688581933313929465d_enat ) ) ).
% 0.23/0.60
% 0.23/0.60 % imult_infinity_right
% 0.23/0.60 thf(fact_10165_plus__enat__def,axiom,
% 0.23/0.60 ( plus_p3455044024723400733d_enat
% 0.23/0.60 = ( ^ [M6: extended_enat,N2: extended_enat] :
% 0.23/0.60 ( extend3600170679010898289d_enat
% 0.23/0.60 @ ^ [O: nat] :
% 0.23/0.60 ( extend3600170679010898289d_enat
% 0.23/0.60 @ ^ [P3: nat] : ( extended_enat2 @ ( plus_plus_nat @ O @ P3 ) )
% 0.23/0.60 @ extend5688581933313929465d_enat
% 0.23/0.60 @ N2 )
% 0.23/0.60 @ extend5688581933313929465d_enat
% 0.23/0.60 @ M6 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % plus_enat_def
% 0.23/0.60 thf(fact_10166_diff__enat__def,axiom,
% 0.23/0.60 ( minus_3235023915231533773d_enat
% 0.23/0.60 = ( ^ [A4: extended_enat,B3: extended_enat] :
% 0.23/0.60 ( extend3600170679010898289d_enat
% 0.23/0.60 @ ^ [X: nat] :
% 0.23/0.60 ( extend3600170679010898289d_enat
% 0.23/0.60 @ ^ [Y: nat] : ( extended_enat2 @ ( minus_minus_nat @ X @ Y ) )
% 0.23/0.60 @ zero_z5237406670263579293d_enat
% 0.23/0.60 @ B3 )
% 0.23/0.60 @ extend5688581933313929465d_enat
% 0.23/0.60 @ A4 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % diff_enat_def
% 0.23/0.60 thf(fact_10167_times__enat__def,axiom,
% 0.23/0.60 ( times_7803423173614009249d_enat
% 0.23/0.60 = ( ^ [M6: extended_enat,N2: extended_enat] :
% 0.23/0.60 ( extend3600170679010898289d_enat
% 0.23/0.60 @ ^ [O: nat] :
% 0.23/0.60 ( extend3600170679010898289d_enat
% 0.23/0.60 @ ^ [P3: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P3 ) )
% 0.23/0.60 @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 0.23/0.60 @ N2 )
% 0.23/0.60 @ ( if_Extended_enat @ ( N2 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 0.23/0.60 @ M6 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % times_enat_def
% 0.23/0.60 thf(fact_10168_VEBT__internal_Oelim__dead_Opelims,axiom,
% 0.23/0.60 ! [X4: vEBT_VEBT,Xa: extended_enat,Y3: vEBT_VEBT] :
% 0.23/0.60 ( ( ( vEBT_VEBT_elim_dead @ X4 @ Xa )
% 0.23/0.60 = Y3 )
% 0.23/0.60 => ( ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ X4 @ Xa ) )
% 0.23/0.60 => ( ! [A5: $o,B5: $o] :
% 0.23/0.60 ( ( X4
% 0.23/0.60 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.60 => ( ( Y3
% 0.23/0.60 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.23/0.60 => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
% 0.23/0.60 => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.60 ( ( X4
% 0.23/0.60 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.60 => ( ( Xa = extend5688581933313929465d_enat )
% 0.23/0.60 => ( ( Y3
% 0.23/0.60 = ( vEBT_Node @ Info2 @ Deg2
% 0.23/0.60 @ ( map_VE8901447254227204932T_VEBT
% 0.23/0.60 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.60 @ TreeList3 )
% 0.23/0.60 @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) )
% 0.23/0.60 => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ extend5688581933313929465d_enat ) ) ) ) )
% 0.23/0.60 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.23/0.60 ( ( X4
% 0.23/0.60 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.23/0.60 => ! [L3: nat] :
% 0.23/0.60 ( ( Xa
% 0.23/0.60 = ( extended_enat2 @ L3 ) )
% 0.23/0.60 => ( ( Y3
% 0.23/0.60 = ( vEBT_Node @ Info2 @ Deg2
% 0.23/0.60 @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.23/0.60 @ ( map_VE8901447254227204932T_VEBT
% 0.23/0.60 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.23/0.60 @ TreeList3 ) )
% 0.23/0.60 @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.23/0.60 => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ ( extended_enat2 @ L3 ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % VEBT_internal.elim_dead.pelims
% 0.23/0.60 thf(fact_10169_the__enat_Osimps,axiom,
% 0.23/0.60 ! [N: nat] :
% 0.23/0.60 ( ( extended_the_enat @ ( extended_enat2 @ N ) )
% 0.23/0.60 = N ) ).
% 0.23/0.60
% 0.23/0.60 % the_enat.simps
% 0.23/0.60 thf(fact_10170_eSuc__Max,axiom,
% 0.23/0.60 ! [A3: set_Extended_enat] :
% 0.23/0.60 ( ( finite4001608067531595151d_enat @ A3 )
% 0.23/0.60 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 0.23/0.60 => ( ( extended_eSuc @ ( lattic921264341876707157d_enat @ A3 ) )
% 0.23/0.60 = ( lattic921264341876707157d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_Max
% 0.23/0.60 thf(fact_10171_eSuc__inject,axiom,
% 0.23/0.60 ! [M: extended_enat,N: extended_enat] :
% 0.23/0.60 ( ( ( extended_eSuc @ M )
% 0.23/0.60 = ( extended_eSuc @ N ) )
% 0.23/0.60 = ( M = N ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_inject
% 0.23/0.60 thf(fact_10172_eSuc__infinity,axiom,
% 0.23/0.60 ( ( extended_eSuc @ extend5688581933313929465d_enat )
% 0.23/0.60 = extend5688581933313929465d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_infinity
% 0.23/0.60 thf(fact_10173_eSuc__mono,axiom,
% 0.23/0.60 ! [N: extended_enat,M: extended_enat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 0.23/0.60 = ( ord_le72135733267957522d_enat @ N @ M ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_mono
% 0.23/0.60 thf(fact_10174_eSuc__ile__mono,axiom,
% 0.23/0.60 ! [N: extended_enat,M: extended_enat] :
% 0.23/0.60 ( ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 0.23/0.60 = ( ord_le2932123472753598470d_enat @ N @ M ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_ile_mono
% 0.23/0.60 thf(fact_10175_eSuc__minus__eSuc,axiom,
% 0.23/0.60 ! [N: extended_enat,M: extended_enat] :
% 0.23/0.60 ( ( minus_3235023915231533773d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 0.23/0.60 = ( minus_3235023915231533773d_enat @ N @ M ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_minus_eSuc
% 0.23/0.60 thf(fact_10176_iless__eSuc0,axiom,
% 0.23/0.60 ! [N: extended_enat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ N @ ( extended_eSuc @ zero_z5237406670263579293d_enat ) )
% 0.23/0.60 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 0.23/0.60
% 0.23/0.60 % iless_eSuc0
% 0.23/0.60 thf(fact_10177_eSuc__minus__1,axiom,
% 0.23/0.60 ! [N: extended_enat] :
% 0.23/0.60 ( ( minus_3235023915231533773d_enat @ ( extended_eSuc @ N ) @ one_on7984719198319812577d_enat )
% 0.23/0.60 = N ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_minus_1
% 0.23/0.60 thf(fact_10178_eSuc__numeral,axiom,
% 0.23/0.60 ! [K: num] :
% 0.23/0.60 ( ( extended_eSuc @ ( numera1916890842035813515d_enat @ K ) )
% 0.23/0.60 = ( numera1916890842035813515d_enat @ ( plus_plus_num @ K @ one ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_numeral
% 0.23/0.60 thf(fact_10179_iless__Suc__eq,axiom,
% 0.23/0.60 ! [M: nat,N: extended_enat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N ) )
% 0.23/0.60 = ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % iless_Suc_eq
% 0.23/0.60 thf(fact_10180_one__eSuc,axiom,
% 0.23/0.60 ( one_on7984719198319812577d_enat
% 0.23/0.60 = ( extended_eSuc @ zero_z5237406670263579293d_enat ) ) ).
% 0.23/0.60
% 0.23/0.60 % one_eSuc
% 0.23/0.60 thf(fact_10181_i0__iless__eSuc,axiom,
% 0.23/0.60 ! [N: extended_enat] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( extended_eSuc @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % i0_iless_eSuc
% 0.23/0.60 thf(fact_10182_zero__ne__eSuc,axiom,
% 0.23/0.60 ! [N: extended_enat] :
% 0.23/0.60 ( zero_z5237406670263579293d_enat
% 0.23/0.60 != ( extended_eSuc @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % zero_ne_eSuc
% 0.23/0.60 thf(fact_10183_eSuc__plus__1,axiom,
% 0.23/0.60 ( extended_eSuc
% 0.23/0.60 = ( ^ [N2: extended_enat] : ( plus_p3455044024723400733d_enat @ N2 @ one_on7984719198319812577d_enat ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_plus_1
% 0.23/0.60 thf(fact_10184_plus__1__eSuc_I1_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] :
% 0.23/0.60 ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ Q2 )
% 0.23/0.60 = ( extended_eSuc @ Q2 ) ) ).
% 0.23/0.60
% 0.23/0.60 % plus_1_eSuc(1)
% 0.23/0.60 thf(fact_10185_plus__1__eSuc_I2_J,axiom,
% 0.23/0.60 ! [Q2: extended_enat] :
% 0.23/0.60 ( ( plus_p3455044024723400733d_enat @ Q2 @ one_on7984719198319812577d_enat )
% 0.23/0.60 = ( extended_eSuc @ Q2 ) ) ).
% 0.23/0.60
% 0.23/0.60 % plus_1_eSuc(2)
% 0.23/0.60 thf(fact_10186_mono__eSuc,axiom,
% 0.23/0.60 order_4130057895858720880d_enat @ extended_eSuc ).
% 0.23/0.60
% 0.23/0.60 % mono_eSuc
% 0.23/0.60 thf(fact_10187_iadd__Suc__right,axiom,
% 0.23/0.60 ! [M: extended_enat,N: extended_enat] :
% 0.23/0.60 ( ( plus_p3455044024723400733d_enat @ M @ ( extended_eSuc @ N ) )
% 0.23/0.60 = ( extended_eSuc @ ( plus_p3455044024723400733d_enat @ M @ N ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % iadd_Suc_right
% 0.23/0.60 thf(fact_10188_iadd__Suc,axiom,
% 0.23/0.60 ! [M: extended_enat,N: extended_enat] :
% 0.23/0.60 ( ( plus_p3455044024723400733d_enat @ ( extended_eSuc @ M ) @ N )
% 0.23/0.60 = ( extended_eSuc @ ( plus_p3455044024723400733d_enat @ M @ N ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % iadd_Suc
% 0.23/0.60 thf(fact_10189_eSuc__max,axiom,
% 0.23/0.60 ! [X4: extended_enat,Y3: extended_enat] :
% 0.23/0.60 ( ( extended_eSuc @ ( ord_ma741700101516333627d_enat @ X4 @ Y3 ) )
% 0.23/0.60 = ( ord_ma741700101516333627d_enat @ ( extended_eSuc @ X4 ) @ ( extended_eSuc @ Y3 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_max
% 0.23/0.60 thf(fact_10190_mult__eSuc__right,axiom,
% 0.23/0.60 ! [M: extended_enat,N: extended_enat] :
% 0.23/0.60 ( ( times_7803423173614009249d_enat @ M @ ( extended_eSuc @ N ) )
% 0.23/0.60 = ( plus_p3455044024723400733d_enat @ M @ ( times_7803423173614009249d_enat @ M @ N ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % mult_eSuc_right
% 0.23/0.60 thf(fact_10191_mult__eSuc,axiom,
% 0.23/0.60 ! [M: extended_enat,N: extended_enat] :
% 0.23/0.60 ( ( times_7803423173614009249d_enat @ ( extended_eSuc @ M ) @ N )
% 0.23/0.60 = ( plus_p3455044024723400733d_enat @ N @ ( times_7803423173614009249d_enat @ M @ N ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % mult_eSuc
% 0.23/0.60 thf(fact_10192_ileI1,axiom,
% 0.23/0.60 ! [M: extended_enat,N: extended_enat] :
% 0.23/0.60 ( ( ord_le72135733267957522d_enat @ M @ N )
% 0.23/0.60 => ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ M ) @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % ileI1
% 0.23/0.60 thf(fact_10193_ile__eSuc,axiom,
% 0.23/0.60 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ N @ ( extended_eSuc @ N ) ) ).
% 0.23/0.60
% 0.23/0.60 % ile_eSuc
% 0.23/0.60 thf(fact_10194_not__eSuc__ilei0,axiom,
% 0.23/0.60 ! [N: extended_enat] :
% 0.23/0.60 ~ ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ zero_z5237406670263579293d_enat ) ).
% 0.23/0.60
% 0.23/0.60 % not_eSuc_ilei0
% 0.23/0.60 thf(fact_10195_eSuc__enat,axiom,
% 0.23/0.60 ! [N: nat] :
% 0.23/0.60 ( ( extended_eSuc @ ( extended_enat2 @ N ) )
% 0.23/0.60 = ( extended_enat2 @ ( suc @ N ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_enat
% 0.23/0.60 thf(fact_10196_eSuc__enat__iff,axiom,
% 0.23/0.60 ! [X4: extended_enat,Y3: nat] :
% 0.23/0.60 ( ( ( extended_eSuc @ X4 )
% 0.23/0.60 = ( extended_enat2 @ Y3 ) )
% 0.23/0.60 = ( ? [N2: nat] :
% 0.23/0.60 ( ( Y3
% 0.23/0.60 = ( suc @ N2 ) )
% 0.23/0.60 & ( X4
% 0.23/0.60 = ( extended_enat2 @ N2 ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_enat_iff
% 0.23/0.60 thf(fact_10197_enat__eSuc__iff,axiom,
% 0.23/0.60 ! [Y3: nat,X4: extended_enat] :
% 0.23/0.60 ( ( ( extended_enat2 @ Y3 )
% 0.23/0.60 = ( extended_eSuc @ X4 ) )
% 0.23/0.60 = ( ? [N2: nat] :
% 0.23/0.60 ( ( Y3
% 0.23/0.60 = ( suc @ N2 ) )
% 0.23/0.60 & ( ( extended_enat2 @ N2 )
% 0.23/0.60 = X4 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % enat_eSuc_iff
% 0.23/0.60 thf(fact_10198_eSuc__Sup,axiom,
% 0.23/0.60 ! [A3: set_Extended_enat] :
% 0.23/0.60 ( ( A3 != bot_bo7653980558646680370d_enat )
% 0.23/0.60 => ( ( extended_eSuc @ ( comple4398354569131411667d_enat @ A3 ) )
% 0.23/0.60 = ( comple4398354569131411667d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_Sup
% 0.23/0.60 thf(fact_10199_eSuc__def,axiom,
% 0.23/0.60 ( extended_eSuc
% 0.23/0.60 = ( extend3600170679010898289d_enat
% 0.23/0.60 @ ^ [N2: nat] : ( extended_enat2 @ ( suc @ N2 ) )
% 0.23/0.60 @ extend5688581933313929465d_enat ) ) ).
% 0.23/0.60
% 0.23/0.60 % eSuc_def
% 0.23/0.60 thf(fact_10200_less__than__iff,axiom,
% 0.23/0.60 ! [X4: nat,Y3: nat] :
% 0.23/0.60 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ less_than )
% 0.23/0.60 = ( ord_less_nat @ X4 @ Y3 ) ) ).
% 0.23/0.60
% 0.23/0.60 % less_than_iff
% 0.23/0.60 thf(fact_10201_Quotient__real,axiom,
% 0.23/0.60 quotie3684837364556693515t_real @ realrel @ real2 @ rep_real @ cr_real ).
% 0.23/0.60
% 0.23/0.60 % Quotient_real
% 0.23/0.60 thf(fact_10202_lcm__code__integer,axiom,
% 0.23/0.60 ( gcd_lcm_Code_integer
% 0.23/0.60 = ( ^ [A4: code_integer,B3: code_integer] : ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A4 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( gcd_gcd_Code_integer @ A4 @ B3 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % lcm_code_integer
% 0.23/0.60 thf(fact_10203_lcm__1__iff__nat,axiom,
% 0.23/0.60 ! [M: nat,N: nat] :
% 0.23/0.60 ( ( ( gcd_lcm_nat @ M @ N )
% 0.23/0.60 = ( suc @ zero_zero_nat ) )
% 0.23/0.60 = ( ( M
% 0.23/0.60 = ( suc @ zero_zero_nat ) )
% 0.23/0.60 & ( N
% 0.23/0.60 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % lcm_1_iff_nat
% 0.23/0.60 thf(fact_10204_prod__gcd__lcm__int,axiom,
% 0.23/0.60 ! [M: int,N: int] :
% 0.23/0.60 ( ( times_times_int @ ( abs_abs_int @ M ) @ ( abs_abs_int @ N ) )
% 0.23/0.60 = ( times_times_int @ ( gcd_gcd_int @ M @ N ) @ ( gcd_lcm_int @ M @ N ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % prod_gcd_lcm_int
% 0.23/0.60 thf(fact_10205_prod__gcd__lcm__nat,axiom,
% 0.23/0.60 ( times_times_nat
% 0.23/0.60 = ( ^ [M6: nat,N2: nat] : ( times_times_nat @ ( gcd_gcd_nat @ M6 @ N2 ) @ ( gcd_lcm_nat @ M6 @ N2 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % prod_gcd_lcm_nat
% 0.23/0.60 thf(fact_10206_lcm__nat__def,axiom,
% 0.23/0.60 ( gcd_lcm_nat
% 0.23/0.60 = ( ^ [X: nat,Y: nat] : ( divide_divide_nat @ ( times_times_nat @ X @ Y ) @ ( gcd_gcd_nat @ X @ Y ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % lcm_nat_def
% 0.23/0.60 thf(fact_10207_lcm__pos__nat,axiom,
% 0.23/0.60 ! [M: nat,N: nat] :
% 0.23/0.60 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.23/0.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.23/0.60 => ( ord_less_nat @ zero_zero_nat @ ( gcd_lcm_nat @ M @ N ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % lcm_pos_nat
% 0.23/0.60 thf(fact_10208_lcm__pos__int,axiom,
% 0.23/0.60 ! [M: int,N: int] :
% 0.23/0.60 ( ( M != zero_zero_int )
% 0.23/0.60 => ( ( N != zero_zero_int )
% 0.23/0.60 => ( ord_less_int @ zero_zero_int @ ( gcd_lcm_int @ M @ N ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % lcm_pos_int
% 0.23/0.60 thf(fact_10209_lcm__altdef__int,axiom,
% 0.23/0.60 ( gcd_lcm_int
% 0.23/0.60 = ( ^ [A4: int,B3: int] : ( divide_divide_int @ ( times_times_int @ ( abs_abs_int @ A4 ) @ ( abs_abs_int @ B3 ) ) @ ( gcd_gcd_int @ A4 @ B3 ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 % lcm_altdef_int
% 0.23/0.60
% 0.23/0.60 % Helper facts (46)
% 0.23/0.60 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.23/0.60 ! [X4: int,Y3: int] :
% 0.23/0.60 ( ( if_int @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.23/0.60 ! [X4: int,Y3: int] :
% 0.23/0.60 ( ( if_int @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.23/0.60 ! [X4: nat,Y3: nat] :
% 0.23/0.60 ( ( if_nat @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.23/0.60 ! [X4: nat,Y3: nat] :
% 0.23/0.60 ( ( if_nat @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 0.23/0.60 ! [X4: num,Y3: num] :
% 0.23/0.60 ( ( if_num @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 0.23/0.60 ! [X4: num,Y3: num] :
% 0.23/0.60 ( ( if_num @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.23/0.60 ! [X4: rat,Y3: rat] :
% 0.23/0.60 ( ( if_rat @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.23/0.60 ! [X4: rat,Y3: rat] :
% 0.23/0.60 ( ( if_rat @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.23/0.60 ! [X4: real,Y3: real] :
% 0.23/0.60 ( ( if_real @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.23/0.60 ! [X4: real,Y3: real] :
% 0.23/0.60 ( ( if_real @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 0.23/0.60 ! [P: real > $o] :
% 0.23/0.60 ( ( P @ ( fChoice_real @ P ) )
% 0.23/0.60 = ( ? [X6: real] : ( P @ X6 ) ) ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.23/0.60 ! [X4: complex,Y3: complex] :
% 0.23/0.60 ( ( if_complex @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.23/0.60 ! [X4: complex,Y3: complex] :
% 0.23/0.60 ( ( if_complex @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.23/0.60 ! [X4: extended_enat,Y3: extended_enat] :
% 0.23/0.60 ( ( if_Extended_enat @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.23/0.60 ! [X4: extended_enat,Y3: extended_enat] :
% 0.23/0.60 ( ( if_Extended_enat @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.23/0.60 ! [X4: code_integer,Y3: code_integer] :
% 0.23/0.60 ( ( if_Code_integer @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.23/0.60 ! [X4: code_integer,Y3: code_integer] :
% 0.23/0.60 ( ( if_Code_integer @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.23/0.60 ! [X4: set_int,Y3: set_int] :
% 0.23/0.60 ( ( if_set_int @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.23/0.60 ! [X4: set_int,Y3: set_int] :
% 0.23/0.60 ( ( if_set_int @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.23/0.60 ! [X4: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.23/0.60 ( ( if_VEBT_VEBT @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.23/0.60 ! [X4: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.23/0.60 ( ( if_VEBT_VEBT @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.23/0.60 ! [X4: list_int,Y3: list_int] :
% 0.23/0.60 ( ( if_list_int @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.23/0.60 ! [X4: list_int,Y3: list_int] :
% 0.23/0.60 ( ( if_list_int @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.23/0.60 ! [X4: list_nat,Y3: list_nat] :
% 0.23/0.60 ( ( if_list_nat @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.23/0.60 ! [X4: list_nat,Y3: list_nat] :
% 0.23/0.60 ( ( if_list_nat @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.23/0.60 ! [X4: int > int,Y3: int > int] :
% 0.23/0.60 ( ( if_int_int @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.23/0.60 ! [X4: int > int,Y3: int > int] :
% 0.23/0.60 ( ( if_int_int @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 0.23/0.60 ! [X4: nat > rat,Y3: nat > rat] :
% 0.23/0.60 ( ( if_nat_rat @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 0.23/0.60 ! [X4: nat > rat,Y3: nat > rat] :
% 0.23/0.60 ( ( if_nat_rat @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.23/0.60 ! [X4: option_nat,Y3: option_nat] :
% 0.23/0.60 ( ( if_option_nat @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.23/0.60 ! [X4: option_nat,Y3: option_nat] :
% 0.23/0.60 ( ( if_option_nat @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.23/0.60 ! [X4: option_num,Y3: option_num] :
% 0.23/0.60 ( ( if_option_num @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.23/0.60 ! [X4: option_num,Y3: option_num] :
% 0.23/0.60 ( ( if_option_num @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.23/0.60 ! [X4: product_prod_int_int,Y3: product_prod_int_int] :
% 0.23/0.60 ( ( if_Pro3027730157355071871nt_int @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.23/0.60 ! [X4: product_prod_int_int,Y3: product_prod_int_int] :
% 0.23/0.60 ( ( if_Pro3027730157355071871nt_int @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.23/0.60 ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.23/0.60 ( ( if_Pro6206227464963214023at_nat @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.23/0.60 ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.23/0.60 ( ( if_Pro6206227464963214023at_nat @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 0.23/0.60 ! [X4: nat > int > int,Y3: nat > int > int] :
% 0.23/0.60 ( ( if_nat_int_int @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 0.23/0.60 ! [X4: nat > int > int,Y3: nat > int > int] :
% 0.23/0.60 ( ( if_nat_int_int @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 0.23/0.60 ! [X4: nat > nat > nat,Y3: nat > nat > nat] :
% 0.23/0.60 ( ( if_nat_nat_nat @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 0.23/0.60 ! [X4: nat > nat > nat,Y3: nat > nat > nat] :
% 0.23/0.60 ( ( if_nat_nat_nat @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.23/0.60 ! [X4: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.23/0.60 ( ( if_Pro5737122678794959658eger_o @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.23/0.60 ! [X4: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.23/0.60 ( ( if_Pro5737122678794959658eger_o @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.23/0.60 ! [P: $o] :
% 0.23/0.60 ( ( P = $true )
% 0.23/0.60 | ( P = $false ) ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.23/0.60 ! [X4: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.23/0.60 ( ( if_Pro6119634080678213985nteger @ $false @ X4 @ Y3 )
% 0.23/0.60 = Y3 ) ).
% 0.23/0.60
% 0.23/0.60 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.23/0.60 ! [X4: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.23/0.60 ( ( if_Pro6119634080678213985nteger @ $true @ X4 @ Y3 )
% 0.23/0.60 = X4 ) ).
% 0.23/0.60
% 0.23/0.60 % Conjectures (1)
% 0.23/0.60 thf(conj_0,conjecture,
% 0.23/0.60 ( ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.23/0.60 = ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.23/0.60
% 0.23/0.60 %------------------------------------------------------------------------------
% 0.23/0.60 ------------------------
% 8.61/3.29 ---- Embedded in THF ---
% 8.61/3.29 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 8.61/3.29 %%% Generated on Thu Feb 26 21:45:21 EST 2026
% 8.61/3.29 %%% using '$$dhol' embedding, version 1.3.0.
% 8.61/3.29 %%% Logic specification used:
% 8.61/3.29 %%% thf(spec, logic, $$dhol).
% 8.61/3.29
% 8.61/3.29 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.5KBf3uDrBi/DTF2DTF21372.p
% See solution above
% 8.91/3.33 ------------------------
% 12.39/6.84 ---- Cleaned THF ---
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J_J,type,
% 12.39/6.85 produc3368934014287244435at_num: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 12.39/6.85 produc4471711990508489141at_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 12.39/6.85 set_Pr8693737435421807431at_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 produc859450856879609959at_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_J,type,
% 12.39/6.85 set_fi4554929511873752355omplex: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 12.39/6.85 list_P7413028617227757229T_VEBT: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
% 12.39/6.85 produc2963631642982155120at_num: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 produc7248412053542808358at_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
% 12.39/6.85 set_fi7789364187291644575l_real: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 12.39/6.85 filter6041513312241820739omplex: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 list_P7037539587688870467BT_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 12.39/6.85 list_P4547456442757143711BT_int: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 12.39/6.85 list_P5647936690300460905T_VEBT: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 12.39/6.85 produc8243902056947475879T_VEBT: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 12.39/6.85 set_Pr5085853215250843933omplex: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 12.39/6.85 produc8923325533196201883nteger: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
% 12.39/6.85 produc7272778201969148633d_enat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 12.39/6.85 list_P3126845725202233233VEBT_o: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 12.39/6.85 list_P7495141550334521929T_VEBT: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 12.39/6.85 filter2146258269922977983l_real: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 option4927543243414619207at_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 filter1242075044329608583at_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 12.39/6.85 set_Pr6218003697084177305l_real: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
% 12.39/6.85 list_P3744719386663036955um_num: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
% 12.39/6.85 list_P1726324292696863441at_num: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 12.39/6.85 produc9072475918466114483BT_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 12.39/6.85 produc4894624898956917775BT_int: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 set_Pr1261947904930325089at_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 12.39/6.85 set_Pr958786334691620121nt_int: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 12.39/6.85 produc4411394909380815293omplex: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 12.39/6.85 list_P7333126701944960589_nat_o: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 list_P6285523579766656935_o_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 12.39/6.85 list_P3795440434834930179_o_int: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 12.39/6.85 produc334124729049499915VEBT_o: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 12.39/6.85 produc2504756804600209347T_VEBT: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 12.39/6.85 produc6271795597528267376eger_o: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 12.39/6.85 produc2422161461964618553l_real: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 12.39/6.85 product_prod_num_num: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 12.39/6.85 product_prod_nat_num: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 product_prod_nat_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.39/6.85 product_prod_int_int: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 12.39/6.85 list_P4002435161011370285od_o_o: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 12.39/6.85 list_list_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 12.39/6.85 list_VEBT_VEBT: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 12.39/6.85 set_list_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 12.39/6.85 product_prod_o_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 12.39/6.85 product_prod_o_int: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 12.39/6.85 set_VEBT_VEBT: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 12.39/6.85 set_set_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 12.39/6.85 set_Code_integer: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Extended____Nat__Oenat_J,type,
% 12.39/6.85 list_Extended_enat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
% 12.39/6.85 set_Product_unit: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
% 12.39/6.85 set_Extended_enat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
% 12.39/6.85 list_complex: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 12.39/6.85 product_prod_o_o: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
% 12.39/6.85 set_complex: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 12.39/6.85 filter_real: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
% 12.39/6.85 option_num: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
% 12.39/6.85 option_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
% 12.39/6.85 filter_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
% 12.39/6.85 set_char: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
% 12.39/6.85 list_real: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 12.39/6.85 set_real: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
% 12.39/6.85 list_num: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 12.39/6.85 list_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
% 12.39/6.85 list_int: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 vEBT_VEBT: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
% 12.39/6.85 set_rat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
% 12.39/6.85 set_num: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 set_nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 12.39/6.85 set_int: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Code____Numeral__Ointeger,type,
% 12.39/6.85 code_integer: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Product____Type__Ounit,type,
% 12.39/6.85 product_unit: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Extended____Nat__Oenat,type,
% 12.39/6.85 extended_enat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__List__Olist_I_Eo_J,type,
% 12.39/6.85 list_o: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Complex__Ocomplex,type,
% 12.39/6.85 complex: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__String__Oliteral,type,
% 12.39/6.85 literal: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Set__Oset_I_Eo_J,type,
% 12.39/6.85 set_o: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__String__Ochar,type,
% 12.39/6.85 char: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Real__Oreal,type,
% 12.39/6.85 real: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Rat__Orat,type,
% 12.39/6.85 rat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Num__Onum,type,
% 12.39/6.85 num: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Nat__Onat,type,
% 12.39/6.85 nat: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(ty_n_t__Int__Oint,type,
% 12.39/6.85 int: $tType ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
% 12.39/6.85 archim2889992004027027881ng_rat: rat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
% 12.39/6.85 archim7802044766580827645g_real: real > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
% 12.39/6.85 archim3151403230148437115or_rat: rat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
% 12.39/6.85 archim6058952711729229775r_real: real > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
% 12.39/6.85 archimedean_frac_rat: rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
% 12.39/6.85 archim2898591450579166408c_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
% 12.39/6.85 archim7778729529865785530nd_rat: rat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
% 12.39/6.85 archim8280529875227126926d_real: real > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
% 12.39/6.85 bNF_re1962705104956426057at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
% 12.39/6.85 bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
% 12.39/6.85 bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 12.39/6.85 bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_Eo_J_001_062_It__Real__Oreal_M_Eo_J,type,
% 12.39/6.85 bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
% 12.39/6.85 bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
% 12.39/6.85 bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 12.39/6.85 bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.39/6.85 bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
% 12.39/6.85 bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 12.39/6.85 bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
% 12.39/6.85 bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
% 12.39/6.85 bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
% 12.39/6.85 bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
% 12.39/6.85 bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 12.39/6.85 bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 12.39/6.85 bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.39/6.85 bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 12.39/6.85 bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
% 12.39/6.85 bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 12.39/6.85 bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 12.39/6.85 bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.39/6.85 bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
% 12.39/6.85 bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 12.39/6.85 bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 12.39/6.85 bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 12.39/6.85 bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Binomial_Obinomial,type,
% 12.39/6.85 binomial: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
% 12.39/6.85 gbinomial_complex: complex > nat > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
% 12.39/6.85 gbinomial_int: int > nat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
% 12.39/6.85 gbinomial_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
% 12.39/6.85 gbinomial_rat: rat > nat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
% 12.39/6.85 gbinomial_real: real > nat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 12.39/6.85 bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 12.39/6.85 bit_and_not_num: num > num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 12.39/6.85 bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 12.39/6.85 bit_concat_bit: nat > int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 12.39/6.85 bit_or_not_num_neg: num > num > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
% 12.39/6.85 bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 bit_ri7632146776885996613nteger: code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
% 12.39/6.85 bit_ri7919022796975470100ot_int: int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
% 12.39/6.85 bit_ri631733984087533419it_int: nat > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
% 12.39/6.85 bit_se725231765392027082nd_int: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
% 12.39/6.85 bit_se727722235901077358nd_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
% 12.39/6.85 bit_se8568078237143864401it_int: nat > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
% 12.39/6.85 bit_se8570568707652914677it_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
% 12.39/6.85 bit_se2159334234014336723it_int: nat > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
% 12.39/6.85 bit_se2161824704523386999it_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 bit_se2119862282449309892nteger: nat > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
% 12.39/6.85 bit_se2000444600071755411sk_int: nat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
% 12.39/6.85 bit_se2002935070580805687sk_nat: nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
% 12.39/6.85 bit_se1409905431419307370or_int: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
% 12.39/6.85 bit_se1412395901928357646or_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 bit_se7788150548672797655nteger: nat > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
% 12.39/6.85 bit_se545348938243370406it_int: nat > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
% 12.39/6.85 bit_se547839408752420682it_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
% 12.39/6.85 bit_se7879613467334960850it_int: nat > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
% 12.39/6.85 bit_se7882103937844011126it_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 bit_se1745604003318907178nteger: nat > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
% 12.39/6.85 bit_se2923211474154528505it_int: nat > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
% 12.39/6.85 bit_se2925701944663578781it_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
% 12.39/6.85 bit_se4203085406695923979it_int: nat > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
% 12.39/6.85 bit_se4205575877204974255it_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
% 12.39/6.85 bit_se6526347334894502574or_int: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
% 12.39/6.85 bit_se6528837805403552850or_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 bit_se9216721137139052372nteger: code_integer > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
% 12.39/6.85 bit_se1146084159140164899it_int: int > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
% 12.39/6.85 bit_se1148574629649215175it_nat: nat > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Otake__bit__num,type,
% 12.39/6.85 bit_take_bit_num: nat > num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
% 12.39/6.85 bit_un1837492267222099188nd_num: num > num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num__rel,type,
% 12.39/6.85 bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
% 12.39/6.85 bit_un6178654185764691216or_num: num > num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num__rel,type,
% 12.39/6.85 bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
% 12.39/6.85 bit_un7362597486090784418nd_num: num > num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
% 12.39/6.85 bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
% 12.39/6.85 bit_un2480387367778600638or_num: num > num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
% 12.39/6.85 bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Code__Numeral_Obit__cut__integer,type,
% 12.39/6.85 code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Code__Numeral_Odivmod__abs,type,
% 12.39/6.85 code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Code__Numeral_Odivmod__integer,type,
% 12.39/6.85 code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
% 12.39/6.85 code_int_of_integer: code_integer > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
% 12.39/6.85 code_integer_of_int: int > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Code__Numeral_Ointeger__of__num,type,
% 12.39/6.85 code_integer_of_num: num > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Code__Numeral_Onat__of__integer,type,
% 12.39/6.85 code_nat_of_integer: code_integer > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Code__Numeral_Onum__of__integer,type,
% 12.39/6.85 code_num_of_integer: code_integer > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 comple2295165028678016749d_enat: set_Extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 12.39/6.85 comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 12.39/6.85 comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
% 12.39/6.85 comple4887499456419720421f_real: set_real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 comple7806235888213564991et_nat: set_set_nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 comple4398354569131411667d_enat: set_Extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
% 12.39/6.85 comple1385675409528146559p_real: set_real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 comple7399068483239264473et_nat: set_set_nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complex_OArg,type,
% 12.39/6.85 arg: complex > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complex_Ocis,type,
% 12.39/6.85 cis: real > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complex_Ocnj,type,
% 12.39/6.85 cnj: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complex_Ocomplex_OComplex,type,
% 12.39/6.85 complex2: real > real > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complex_Ocomplex_OIm,type,
% 12.39/6.85 im: complex > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complex_Ocomplex_ORe,type,
% 12.39/6.85 re: complex > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complex_Ocsqrt,type,
% 12.39/6.85 csqrt: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complex_Oimaginary__unit,type,
% 12.39/6.85 imaginary_unit: complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Complex_Orcis,type,
% 12.39/6.85 rcis: real > real > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
% 12.39/6.85 has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Deriv_Ohas__vector__derivative_001t__Real__Oreal,type,
% 12.39/6.85 has_ve631408500373753343e_real: ( real > real ) > real > filter_real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Oadjust__div,type,
% 12.39/6.85 adjust_div: product_prod_int_int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Oadjust__mod,type,
% 12.39/6.85 adjust_mod: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Odivmod__nat,type,
% 12.39/6.85 divmod_nat: nat > nat > product_prod_nat_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Oeucl__rel__int,type,
% 12.39/6.85 eucl_rel_int: int > int > product_prod_int_int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
% 12.39/6.85 unique6319869463603278526ux_int: product_prod_int_int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
% 12.39/6.85 unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
% 12.39/6.85 unique5052692396658037445od_int: num > num > product_prod_int_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
% 12.39/6.85 unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
% 12.39/6.85 unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
% 12.39/6.85 unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Int__Oint,type,
% 12.39/6.85 euclid4774559944035922753ze_int: int > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Nat__Onat,type,
% 12.39/6.85 euclid4777050414544973029ze_nat: nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Int__Oint,type,
% 12.39/6.85 euclid3395696857347342551nt_int: int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Nat__Onat,type,
% 12.39/6.85 euclid3398187327856392827nt_nat: nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Extended__Nat_OeSuc,type,
% 12.39/6.85 extended_eSuc: extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Extended__Nat_Oenat,type,
% 12.39/6.85 extended_enat2: nat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001_Eo,type,
% 12.39/6.85 extended_case_enat_o: ( nat > $o ) > $o > extended_enat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 extend3600170679010898289d_enat: ( nat > extended_enat ) > extended_enat > extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 extend5688581933313929465d_enat: extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Extended__Nat_Othe__enat,type,
% 12.39/6.85 extended_the_enat: extended_enat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
% 12.39/6.85 comm_s2602460028002588243omplex: complex > nat > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
% 12.39/6.85 comm_s4660882817536571857er_int: int > nat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
% 12.39/6.85 comm_s4663373288045622133er_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
% 12.39/6.85 comm_s4028243227959126397er_rat: rat > nat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
% 12.39/6.85 comm_s7457072308508201937r_real: real > nat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 semiri3624122377584611663nteger: nat > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
% 12.39/6.85 semiri5044797733671781792omplex: nat > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
% 12.39/6.85 semiri1406184849735516958ct_int: nat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
% 12.39/6.85 semiri1408675320244567234ct_nat: nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
% 12.39/6.85 semiri773545260158071498ct_rat: nat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
% 12.39/6.85 semiri2265585572941072030t_real: nat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
% 12.39/6.85 invers8013647133539491842omplex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
% 12.39/6.85 inverse_inverse_rat: rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 12.39/6.85 inverse_inverse_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 12.39/6.85 at_bot_real: filter_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 12.39/6.85 at_top_nat: filter_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 12.39/6.85 at_top_real: filter_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 12.39/6.85 eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
% 12.39/6.85 eventually_real: ( real > $o ) > filter_real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 12.39/6.85 filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Ofiltermap_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 filtermap_real_real: ( real > real ) > filter_real > filter_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 12.39/6.85 princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 12.39/6.85 princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Finite__Set_Ocard_001_Eo,type,
% 12.39/6.85 finite_card_o: set_o > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
% 12.39/6.85 finite_card_complex: set_complex > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
% 12.39/6.85 finite_card_int: set_int > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 12.39/6.85 finite_card_list_nat: set_list_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 12.39/6.85 finite_card_nat: set_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
% 12.39/6.85 finite410649719033368117t_unit: set_Product_unit > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
% 12.39/6.85 finite_card_char: set_char > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
% 12.39/6.85 finite3207457112153483333omplex: set_complex > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Finite__Set_Ofinite_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 finite4001608067531595151d_enat: set_Extended_enat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
% 12.39/6.85 finite_finite_int: set_int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 12.39/6.85 finite_finite_nat: set_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 12.39/6.85 bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 12.39/6.85 bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 12.39/6.85
% 12.39/6.85 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,
% 12.39/6.85 comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 12.39/6.85
% 12.39/6.85 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,
% 12.39/6.85 comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
% 12.39/6.85 comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 comp_int_real_real: ( int > real ) > ( real > int ) > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
% 12.39/6.85 comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Oid_001_Eo,type,
% 12.39/6.85 id_o: $o > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
% 12.39/6.85 id_nat: nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
% 12.39/6.85 inj_on_nat_char: ( nat > char ) > set_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 inj_on_real_real: ( real > real ) > set_real > $o ).
% 12.39/6.85
% 12.39/6.85 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,
% 12.39/6.85 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 ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.39/6.85 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 ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 12.39/6.85 map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 12.39/6.85 map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 12.39/6.85 map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 12.39/6.85 map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 12.39/6.85 map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 12.39/6.85 map_fu1532550112467129777l_real: ( real > nat > rat ) > ( ( ( nat > rat ) > nat > rat ) > real > real ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > real > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
% 12.39/6.85 map_fu7146612038024189824t_real: ( real > nat > rat ) > ( ( nat > rat ) > real ) > ( ( nat > rat ) > nat > rat ) > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
% 12.39/6.85 map_fu1856342031159181835at_o_o: ( real > nat > rat ) > ( $o > $o ) > ( ( nat > rat ) > $o ) > real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun__Def_Opair__leq,type,
% 12.39/6.85 fun_pair_leq: set_Pr8693737435421807431at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Fun__Def_Opair__less,type,
% 12.39/6.85 fun_pair_less: set_Pr8693737435421807431at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
% 12.39/6.85 gcd_Gcd_nat: set_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_GCD_Obezw,type,
% 12.39/6.85 bezw: nat > nat > product_prod_int_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_GCD_Obezw__rel,type,
% 12.39/6.85 bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 gcd_gcd_Code_integer: code_integer > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
% 12.39/6.85 gcd_gcd_int: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
% 12.39/6.85 gcd_gcd_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_GCD_Ogcd__class_Olcm_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 gcd_lcm_Code_integer: code_integer > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_GCD_Ogcd__class_Olcm_001t__Int__Oint,type,
% 12.39/6.85 gcd_lcm_int: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_GCD_Ogcd__class_Olcm_001t__Nat__Onat,type,
% 12.39/6.85 gcd_lcm_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_GCD_Ogcd__nat__rel,type,
% 12.39/6.85 gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 abs_abs_Code_integer: code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
% 12.39/6.85 abs_abs_complex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
% 12.39/6.85 abs_abs_int: int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
% 12.39/6.85 abs_abs_rat: rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 12.39/6.85 abs_abs_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
% 12.39/6.85 minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Extended____Nat__Oenat_M_Eo_J,type,
% 12.39/6.85 minus_2020553357622893040enat_o: ( extended_enat > $o ) > ( extended_enat > $o ) > extended_enat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
% 12.39/6.85 minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
% 12.39/6.85 minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
% 12.39/6.85 minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
% 12.39/6.85 minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
% 12.39/6.85 minus_minus_complex: complex > complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 12.39/6.85 minus_minus_int: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 12.39/6.85 minus_minus_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
% 12.39/6.85 minus_minus_rat: rat > rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 12.39/6.85 minus_minus_real: real > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 12.39/6.85 minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
% 12.39/6.85 minus_925952699566721837d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
% 12.39/6.85 minus_minus_set_int: set_int > set_int > set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 12.39/6.85 minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 minus_minus_set_nat: set_nat > set_nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
% 12.39/6.85 minus_minus_set_real: set_real > set_real > set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 one_one_Code_integer: code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
% 12.39/6.85 one_one_complex: complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 one_on7984719198319812577d_enat: extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
% 12.39/6.85 one_one_int: int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 12.39/6.85 one_one_nat: nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
% 12.39/6.85 one_one_rat: rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 12.39/6.85 one_one_real: real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
% 12.39/6.85 plus_plus_complex: complex > complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 12.39/6.85 plus_plus_int: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 12.39/6.85 plus_plus_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
% 12.39/6.85 plus_plus_num: num > num > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
% 12.39/6.85 plus_plus_rat: rat > rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 12.39/6.85 plus_plus_real: real > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Oplus__class_Oplus_001t__String__Oliteral,type,
% 12.39/6.85 plus_plus_literal: literal > literal > literal ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 sgn_sgn_Code_integer: code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
% 12.39/6.85 sgn_sgn_complex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
% 12.39/6.85 sgn_sgn_int: int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
% 12.39/6.85 sgn_sgn_rat: rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
% 12.39/6.85 sgn_sgn_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
% 12.39/6.85 times_times_complex: complex > complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
% 12.39/6.85 times_times_int: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 12.39/6.85 times_times_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
% 12.39/6.85 times_times_num: num > num > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
% 12.39/6.85 times_times_rat: rat > rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
% 12.39/6.85 times_times_real: real > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 uminus1351360451143612070nteger: code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
% 12.39/6.85 uminus1482373934393186551omplex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 12.39/6.85 uminus_uminus_int: int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
% 12.39/6.85 uminus_uminus_rat: rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 12.39/6.85 uminus_uminus_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 uminus5710092332889474511et_nat: set_nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 zero_z3403309356797280102nteger: code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
% 12.39/6.85 zero_zero_complex: complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 zero_z5237406670263579293d_enat: extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 12.39/6.85 zero_zero_int: int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 12.39/6.85 zero_zero_nat: nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
% 12.39/6.85 zero_zero_rat: rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 12.39/6.85 zero_zero_real: real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups_Ozero__class_Ozero_001t__String__Oliteral,type,
% 12.39/6.85 zero_zero_literal: literal ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 12.39/6.85 groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
% 12.39/6.85 groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
% 12.39/6.85 groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 12.39/6.85 groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Rat__Orat,type,
% 12.39/6.85 groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Real__Oreal,type,
% 12.39/6.85 groups766887009212190081x_real: ( complex > real ) > set_complex > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Complex__Ocomplex,type,
% 12.39/6.85 groups4622424608036095791omplex: ( extended_enat > complex ) > set_Extended_enat > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Int__Oint,type,
% 12.39/6.85 groups2878480467620962989at_int: ( extended_enat > int ) > set_Extended_enat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
% 12.39/6.85 groups2880970938130013265at_nat: ( extended_enat > nat ) > set_Extended_enat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Rat__Orat,type,
% 12.39/6.85 groups2245840878043517529at_rat: ( extended_enat > rat ) > set_Extended_enat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
% 12.39/6.85 groups97031904164794029t_real: ( extended_enat > real ) > set_Extended_enat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Complex__Ocomplex,type,
% 12.39/6.85 groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
% 12.39/6.85 groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Rat__Orat,type,
% 12.39/6.85 groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Real__Oreal,type,
% 12.39/6.85 groups2316167850115554303t_real: ( int > real ) > set_int > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 12.39/6.85 groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
% 12.39/6.85 groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Rat__Orat,type,
% 12.39/6.85 groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Real__Oreal,type,
% 12.39/6.85 groups129246275422532515t_real: ( nat > real ) > set_nat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Complex__Ocomplex,type,
% 12.39/6.85 groups713298508707869441omplex: ( real > complex ) > set_real > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Int__Oint,type,
% 12.39/6.85 groups4694064378042380927al_int: ( real > int ) > set_real > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Nat__Onat,type,
% 12.39/6.85 groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Rat__Orat,type,
% 12.39/6.85 groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 groups1681761925125756287l_real: ( real > real ) > set_real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
% 12.39/6.85 groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
% 12.39/6.85 groups4561878855575611511st_nat: list_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_HOL_OThe_001t__Int__Oint,type,
% 12.39/6.85 the_int: ( int > $o ) > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.39/6.85 the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
% 12.39/6.85 the_real: ( real > $o ) > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.39/6.85 if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 12.39/6.85 if_nat_int_int: $o > ( nat > int > int ) > ( nat > int > int ) > nat > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 if_nat_nat_nat: $o > ( nat > nat > nat ) > ( nat > nat > nat ) > nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
% 12.39/6.85 if_nat_rat: $o > ( nat > rat ) > ( nat > rat ) > nat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 if_Code_integer: $o > code_integer > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Complex__Ocomplex,type,
% 12.39/6.85 if_complex: $o > complex > complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Int__Oint,type,
% 12.39/6.85 if_int: $o > int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
% 12.39/6.85 if_list_int: $o > list_int > list_int > list_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
% 12.39/6.85 if_list_nat: $o > list_nat > list_nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Nat__Onat,type,
% 12.39/6.85 if_nat: $o > nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Num__Onum,type,
% 12.39/6.85 if_num: $o > num > num > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
% 12.39/6.85 if_option_nat: $o > option_nat > option_nat > option_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
% 12.39/6.85 if_option_num: $o > option_num > option_num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 12.39/6.85 if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 12.39/6.85 if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.39/6.85 if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Rat__Orat,type,
% 12.39/6.85 if_rat: $o > rat > rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Real__Oreal,type,
% 12.39/6.85 if_real: $o > real > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
% 12.39/6.85 if_set_int: $o > set_int > set_int > set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
% 12.39/6.85 infini8530281810654367211te_nat: set_nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_OAbs__Integ,type,
% 12.39/6.85 abs_Integ: product_prod_nat_nat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_ORep__Integ,type,
% 12.39/6.85 rep_Integ: int > product_prod_nat_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oint__ge__less__than,type,
% 12.39/6.85 int_ge_less_than: int > set_Pr958786334691620121nt_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oint__ge__less__than2,type,
% 12.39/6.85 int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Ointrel,type,
% 12.39/6.85 intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Onat,type,
% 12.39/6.85 nat2: int > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Opcr__int,type,
% 12.39/6.85 pcr_int: product_prod_nat_nat > int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
% 12.39/6.85 power_int_real: real > int > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 ring_11222124179247155820nteger: set_Code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
% 12.39/6.85 ring_1_Ints_complex: set_complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
% 12.39/6.85 ring_1_Ints_int: set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
% 12.39/6.85 ring_1_Ints_rat: set_rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 12.39/6.85 ring_1_Ints_real: set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 ring_18347121197199848620nteger: int > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
% 12.39/6.85 ring_17405671764205052669omplex: int > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 12.39/6.85 ring_1_of_int_int: int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
% 12.39/6.85 ring_1_of_int_rat: int > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 12.39/6.85 ring_1_of_int_real: int > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 12.39/6.85 inf_inf_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
% 12.39/6.85 semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 12.39/6.85 sup_sup_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 lattic921264341876707157d_enat: set_Extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 12.39/6.85 lattic8265883725875713057ax_nat: set_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Lifting_OQuotient_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
% 12.39/6.85 quotie3684837364556693515t_real: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > real ) > ( real > nat > rat ) > ( ( nat > rat ) > real > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
% 12.39/6.85 bfun_nat_real: ( nat > real ) > filter_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 12.39/6.85 append_int: list_int > list_int > list_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 12.39/6.85 append_nat: list_nat > list_nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Odrop_001t__Nat__Onat,type,
% 12.39/6.85 drop_nat: nat > list_nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olast_001t__Nat__Onat,type,
% 12.39/6.85 last_nat: list_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 12.39/6.85 linord2614967742042102400et_nat: set_nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 12.39/6.85 cons_int: int > list_int > list_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 12.39/6.85 cons_nat: nat > list_nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 12.39/6.85 nil_int: list_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 12.39/6.85 nil_nat: list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
% 12.39/6.85 hd_nat: list_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 map_VE8901447254227204932T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > list_VEBT_VEBT > list_VEBT_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Oset_001_Eo,type,
% 12.39/6.85 set_o2: list_o > set_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 12.39/6.85 set_complex2: list_complex > set_complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Oset_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 set_Extended_enat2: list_Extended_enat > set_Extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 12.39/6.85 set_int2: list_int > set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
% 12.39/6.85 set_list_nat2: list_list_nat > set_list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 12.39/6.85 set_nat2: list_nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 12.39/6.85 set_real2: list_real > set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
% 12.39/6.85 tl_nat: list_nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001_Eo,type,
% 12.39/6.85 nth_o: list_o > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 12.39/6.85 nth_complex: list_complex > nat > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 nth_Extended_enat: list_Extended_enat > nat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Int__Oint,type,
% 12.39/6.85 nth_int: list_int > nat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
% 12.39/6.85 nth_list_nat: list_list_nat > nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 12.39/6.85 nth_nat: list_nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Num__Onum,type,
% 12.39/6.85 nth_num: list_num > nat > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 12.39/6.85 nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 12.39/6.85 nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 12.39/6.85 nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 12.39/6.85 nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 12.39/6.85 nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 12.39/6.85 nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 12.39/6.85 nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 12.39/6.85 nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 12.39/6.85 nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 12.39/6.85 nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 12.39/6.85 nth_real: list_real > nat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 12.39/6.85 product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 12.39/6.85 product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
% 12.39/6.85 product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 12.39/6.85 product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
% 12.39/6.85 product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
% 12.39/6.85 product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 12.39/6.85 product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 12.39/6.85 produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 12.39/6.85 produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 12.39/6.85 sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 12.39/6.85 sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 12.39/6.85 take_nat: nat > list_nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oupt,type,
% 12.39/6.85 upt: nat > nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oupto,type,
% 12.39/6.85 upto: int > int > list_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oupto__aux,type,
% 12.39/6.85 upto_aux: int > int > list_int > list_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_List_Oupto__rel,type,
% 12.39/6.85 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_OSuc,type,
% 12.39/6.85 suc: nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 12.39/6.85 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 12.39/6.85 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 12.39/6.85 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Onat_Opred,type,
% 12.39/6.85 pred: nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 semiri4939895301339042750nteger: nat > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 12.39/6.85 semiri8010041392384452111omplex: nat > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 semiri4216267220026989637d_enat: nat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 12.39/6.85 semiri1314217659103216013at_int: nat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 12.39/6.85 semiri1316708129612266289at_nat: nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 12.39/6.85 semiri681578069525770553at_rat: nat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 12.39/6.85 semiri5074537144036343181t_real: nat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 12.39/6.85 size_size_list_o: list_o > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 12.39/6.85 size_s3451745648224563538omplex: list_complex > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
% 12.39/6.85 size_s3941691890525107288d_enat: list_Extended_enat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 12.39/6.85 size_size_list_int: list_int > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 12.39/6.85 size_s3023201423986296836st_nat: list_list_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 12.39/6.85 size_size_list_nat: list_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 12.39/6.85 size_size_list_num: list_num > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 12.39/6.85 size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 12.39/6.85 size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 12.39/6.85 size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 12.39/6.85 size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 12.39/6.85 size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 12.39/6.85 size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 12.39/6.85 size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
% 12.39/6.85
% 12.39/6.85 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,
% 12.39/6.85 size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 12.39/6.85 size_size_list_real: list_real > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 12.39/6.85 size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 12.39/6.85 size_size_num: num > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat__Bijection_Olist__decode,type,
% 12.39/6.85 nat_list_decode: nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
% 12.39/6.85 nat_list_decode_rel: nat > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 12.39/6.85 nat_list_encode: list_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 12.39/6.85 nat_list_encode_rel: list_nat > list_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat__Bijection_Oprod__decode,type,
% 12.39/6.85 nat_prod_decode: nat > product_prod_nat_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 12.39/6.85 nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 12.39/6.85 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 12.39/6.85 nat_prod_encode: product_prod_nat_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 12.39/6.85 nat_set_decode: nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 12.39/6.85 nat_set_encode: set_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Nat__Bijection_Otriangle,type,
% 12.39/6.85 nat_triangle: nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_NthRoot_Oroot,type,
% 12.39/6.85 root: nat > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_NthRoot_Osqrt,type,
% 12.39/6.85 sqrt: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_OBitM,type,
% 12.39/6.85 bitM: num > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oinc,type,
% 12.39/6.85 inc: num > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 12.39/6.85 neg_nu7009210354673126013omplex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 12.39/6.85 neg_numeral_dbl_int: int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 12.39/6.85 neg_numeral_dbl_rat: rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 12.39/6.85 neg_numeral_dbl_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 12.39/6.85 neg_nu6511756317524482435omplex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 12.39/6.85 neg_nu3811975205180677377ec_int: int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 12.39/6.85 neg_nu3179335615603231917ec_rat: rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 12.39/6.85 neg_nu6075765906172075777c_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 neg_nu5831290666863070958nteger: code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 12.39/6.85 neg_nu8557863876264182079omplex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
% 12.39/6.85 neg_nu5851722552734809277nc_int: int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 12.39/6.85 neg_nu5219082963157363817nc_rat: rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 12.39/6.85 neg_nu8295874005876285629c_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 12.39/6.85 neg_numeral_sub_int: num > num > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onum_OBit0,type,
% 12.39/6.85 bit0: num > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onum_OBit1,type,
% 12.39/6.85 bit1: num > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onum_OOne,type,
% 12.39/6.85 one: num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 12.39/6.85 case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onum_Osize__num,type,
% 12.39/6.85 size_num: num > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onum__of__nat,type,
% 12.39/6.85 num_of_nat: nat > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 numera6620942414471956472nteger: num > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 12.39/6.85 numera6690914467698888265omplex: num > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 numera1916890842035813515d_enat: num > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 12.39/6.85 numeral_numeral_int: num > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 12.39/6.85 numeral_numeral_nat: num > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 12.39/6.85 numeral_numeral_rat: num > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 12.39/6.85 numeral_numeral_real: num > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Opow,type,
% 12.39/6.85 pow: num > num > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Opred__numeral,type,
% 12.39/6.85 pred_numeral: num > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Num_Osqr,type,
% 12.39/6.85 sqr: num > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
% 12.39/6.85 none_nat: option_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 12.39/6.85 none_num: option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
% 12.39/6.85 some_nat: nat > option_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 12.39/6.85 some_num: num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
% 12.39/6.85 case_option_int_num: int > ( num > int ) > option_num > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
% 12.39/6.85 case_option_num_num: num > ( num > num ) > option_num > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
% 12.39/6.85 case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
% 12.39/6.85 map_option_num_num: ( num > num ) > option_num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
% 12.39/6.85 the_nat: option_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
% 12.39/6.85 bot_bot_nat_o: nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 bot_bo4199563552545308370d_enat: extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 12.39/6.85 bot_bot_nat: nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
% 12.39/6.85 bot_bo7653980558646680370d_enat: set_Extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
% 12.39/6.85 bot_bot_set_int: set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 bot_bot_set_nat: set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
% 12.39/6.85 bot_bot_set_num: set_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
% 12.39/6.85 bot_bot_set_rat: set_rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
% 12.39/6.85 bot_bot_set_real: set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 12.39/6.85 bot_bot_set_set_nat: set_set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_OLeast_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 ord_Le1955565732374568822d_enat: ( extended_enat > $o ) > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
% 12.39/6.85 ord_Least_nat: ( nat > $o ) > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
% 12.39/6.85 ord_Least_real: ( real > $o ) > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Complex__Ocomplex_M_Eo_J,type,
% 12.39/6.85 ord_less_complex_o: ( complex > $o ) > ( complex > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Extended____Nat__Oenat_M_Eo_J,type,
% 12.39/6.85 ord_le8499522857272258027enat_o: ( extended_enat > $o ) > ( extended_enat > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
% 12.39/6.85 ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
% 12.39/6.85 ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
% 12.39/6.85 ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 12.39/6.85 ord_less_int: int > int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 12.39/6.85 ord_less_nat: nat > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
% 12.39/6.85 ord_less_num: num > num > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
% 12.39/6.85 ord_less_rat: rat > rat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 12.39/6.85 ord_less_real: real > real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 12.39/6.85 ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 12.39/6.85 ord_less_set_complex: set_complex > set_complex > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
% 12.39/6.85 ord_le2529575680413868914d_enat: set_Extended_enat > set_Extended_enat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
% 12.39/6.85 ord_less_set_int: set_int > set_int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 ord_less_set_nat: set_nat > set_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
% 12.39/6.85 ord_less_set_num: set_num > set_num > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
% 12.39/6.85 ord_less_set_rat: set_rat > set_rat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
% 12.39/6.85 ord_less_set_real: set_real > set_real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 12.39/6.85 ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
% 12.39/6.85 ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 12.39/6.85 ord_less_eq_int: int > int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 12.39/6.85 ord_less_eq_nat: nat > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
% 12.39/6.85 ord_less_eq_num: num > num > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
% 12.39/6.85 ord_less_eq_rat: rat > rat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 12.39/6.85 ord_less_eq_real: real > real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 12.39/6.85 ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 12.39/6.85 ord_le211207098394363844omplex: set_complex > set_complex > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
% 12.39/6.85 ord_le7203529160286727270d_enat: set_Extended_enat > set_Extended_enat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
% 12.39/6.85 ord_less_eq_set_int: set_int > set_int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
% 12.39/6.85 ord_less_eq_set_num: set_num > set_num > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
% 12.39/6.85 ord_less_eq_set_rat: set_rat > set_rat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
% 12.39/6.85 ord_less_eq_set_real: set_real > set_real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 12.39/6.85 ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
% 12.39/6.85 ord_max_int: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
% 12.39/6.85 ord_max_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
% 12.39/6.85 ord_min_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
% 12.39/6.85 order_Greatest_nat: ( nat > $o ) > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
% 12.39/6.85 order_9091379641038594480t_real: ( nat > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oorder__class_Omono_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 order_4130057895858720880d_enat: ( extended_enat > extended_enat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 order_mono_nat_nat: ( nat > nat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
% 12.39/6.85 order_mono_nat_real: ( nat > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 order_5726023648592871131at_nat: ( nat > nat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 order_7092887310737990675l_real: ( real > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
% 12.39/6.85 ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 top_to3028658606643905974d_enat: extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
% 12.39/6.85 top_top_set_o: set_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 top_top_set_nat: set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 12.39/6.85 top_to1996260823553986621t_unit: set_Product_unit ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 12.39/6.85 top_top_set_real: set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
% 12.39/6.85 top_top_set_char: set_char ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 power_8256067586552552935nteger: code_integer > nat > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
% 12.39/6.85 power_power_complex: complex > nat > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 12.39/6.85 power_power_int: int > nat > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 12.39/6.85 power_power_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
% 12.39/6.85 power_power_rat: rat > nat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 12.39/6.85 power_power_real: real > nat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
% 12.39/6.85 produc851828971589881931at_num: ( nat > num > num ) > produc2963631642982155120at_num > produc3368934014287244435at_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
% 12.39/6.85 product_Pair_o_o: $o > $o > product_prod_o_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
% 12.39/6.85 product_Pair_o_int: $o > int > product_prod_o_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
% 12.39/6.85 product_Pair_o_nat: $o > nat > product_prod_o_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
% 12.39/6.85 produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
% 12.39/6.85 product_Pair_int_int: int > int > product_prod_int_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
% 12.39/6.85 product_Pair_nat_num: nat > num > product_prod_nat_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 12.39/6.85 produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
% 12.39/6.85 product_Pair_num_num: num > num > product_prod_num_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 12.39/6.85 produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 produc581526299967858633d_enat: vEBT_VEBT > extended_enat > produc7272778201969148633d_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 12.39/6.85 produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 12.39/6.85 produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 12.39/6.85 produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 12.39/6.85 produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 12.39/6.85 produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 12.39/6.85 produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 12.39/6.85 produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 12.39/6.85 produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 12.39/6.85 produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 12.39/6.85 produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.39/6.85 produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 12.39/6.85 produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.39/6.85 produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 12.39/6.85 produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
% 12.39/6.85 produc2761476792215241774st_nat: ( nat > nat > list_nat ) > product_prod_nat_nat > list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 12.39/6.85 produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 12.39/6.85 produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ofst_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 12.39/6.85 product_fst_int_int: product_prod_int_int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 product_fst_nat_nat: product_prod_nat_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 produc6174133586879617921nteger: produc8923325533196201883nteger > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 12.39/6.85 product_snd_int_int: product_prod_int_int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 product_snd_nat_nat: product_prod_nat_nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rat_OAbs__Rat,type,
% 12.39/6.85 abs_Rat: product_prod_int_int > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rat_OFract,type,
% 12.39/6.85 fract: int > int > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rat_OFrct,type,
% 12.39/6.85 frct: product_prod_int_int > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rat_ORep__Rat,type,
% 12.39/6.85 rep_Rat: rat > product_prod_int_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 12.39/6.85 field_5140801741446780682s_real: set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
% 12.39/6.85 field_7254667332652039916t_real: rat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rat_Onormalize,type,
% 12.39/6.85 normalize: product_prod_int_int > product_prod_int_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rat_Opcr__rat,type,
% 12.39/6.85 pcr_rat: product_prod_int_int > rat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rat_Opositive,type,
% 12.39/6.85 positive: rat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rat_Oquotient__of,type,
% 12.39/6.85 quotient_of: rat > product_prod_int_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rat_Oratrel,type,
% 12.39/6.85 ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real_OReal,type,
% 12.39/6.85 real2: ( nat > rat ) > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real_Ocauchy,type,
% 12.39/6.85 cauchy: ( nat > rat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real_Ocr__real,type,
% 12.39/6.85 cr_real: ( nat > rat ) > real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real_Opcr__real,type,
% 12.39/6.85 pcr_real: ( nat > rat ) > real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real_Opositive,type,
% 12.39/6.85 positive2: real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real_Orealrel,type,
% 12.39/6.85 realrel: ( nat > rat ) > ( nat > rat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real_Orep__real,type,
% 12.39/6.85 rep_real: real > nat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real_Ovanishes,type,
% 12.39/6.85 vanishes: ( nat > rat ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
% 12.39/6.85 real_V2521375963428798218omplex: set_complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 real_V5970128139526366754l_real: ( real > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 12.39/6.85 real_V3694042436643373181omplex: complex > complex > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 12.39/6.85 real_V975177566351809787t_real: real > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 12.39/6.85 real_V1022390504157884413omplex: complex > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 12.39/6.85 real_V7735802525324610683m_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 12.39/6.85 real_V4546457046886955230omplex: real > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
% 12.39/6.85 real_V1803761363581548252l_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
% 12.39/6.85 real_V2046097035970521341omplex: real > complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 12.39/6.85 real_V1485227260804924795R_real: real > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Relation_Otransp_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
% 12.39/6.85 transp_nat_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
% 12.39/6.85 algebr932160517623751201me_int: int > int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
% 12.39/6.85 algebr934650988132801477me_nat: nat > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
% 12.39/6.85 divide1717551699836669952omplex: complex > complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 12.39/6.85 divide_divide_int: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 12.39/6.85 divide_divide_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
% 12.39/6.85 divide_divide_rat: rat > rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 12.39/6.85 divide_divide_real: real > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 dvd_dvd_Code_integer: code_integer > code_integer > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
% 12.39/6.85 dvd_dvd_complex: complex > complex > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
% 12.39/6.85 dvd_dvd_int: int > int > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 12.39/6.85 dvd_dvd_nat: nat > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
% 12.39/6.85 dvd_dvd_rat: rat > rat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
% 12.39/6.85 dvd_dvd_real: real > real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
% 12.39/6.85 modulo_modulo_int: int > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
% 12.39/6.85 modulo_modulo_nat: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 zero_n356916108424825756nteger: $o > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
% 12.39/6.85 zero_n1201886186963655149omplex: $o > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
% 12.39/6.85 zero_n2684676970156552555ol_int: $o > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
% 12.39/6.85 zero_n2687167440665602831ol_nat: $o > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
% 12.39/6.85 zero_n2052037380579107095ol_rat: $o > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
% 12.39/6.85 zero_n3304061248610475627l_real: $o > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
% 12.39/6.85 suminf_complex: ( nat > complex ) > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
% 12.39/6.85 suminf_real: ( nat > real ) > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
% 12.39/6.85 summable_complex: ( nat > complex ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
% 12.39/6.85 summable_real: ( nat > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Series_Osums_001t__Real__Oreal,type,
% 12.39/6.85 sums_real: ( nat > real ) > real > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001_Eo,type,
% 12.39/6.85 collect_o: ( $o > $o ) > set_o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
% 12.39/6.85 collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
% 12.39/6.85 collect_complex: ( complex > $o ) > set_complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 collec4429806609662206161d_enat: ( extended_enat > $o ) > set_Extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__Int__Oint,type,
% 12.39/6.85 collect_int: ( int > $o ) > set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
% 12.39/6.85 collect_list_nat: ( list_nat > $o ) > set_list_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 12.39/6.85 collect_nat: ( nat > $o ) > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 12.39/6.85 collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.39/6.85 collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.85 collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 12.39/6.85 collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 12.39/6.85 collect_real: ( real > $o ) > set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.85 collect_VEBT_VEBT: ( vEBT_VEBT > $o ) > set_VEBT_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oimage_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 image_80655429650038917d_enat: ( extended_enat > extended_enat ) > set_Extended_enat > set_Extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
% 12.39/6.85 image_int_int: ( int > int ) > set_int > set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
% 12.39/6.85 image_nat_int: ( nat > int ) > set_nat > set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
% 12.39/6.85 image_nat_real: ( nat > real ) > set_nat > set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
% 12.39/6.85 image_nat_char: ( nat > char ) > set_nat > set_char ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 12.39/6.85 image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 12.39/6.85 image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 image_real_real: ( real > real ) > set_real > set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
% 12.39/6.85 image_char_nat: ( char > nat ) > set_char > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
% 12.39/6.85 insert_int: int > set_int > set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 12.39/6.85 insert_nat: nat > set_nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
% 12.39/6.85 insert_real: real > set_real > set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.39/6.85 vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
% 12.39/6.85 set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
% 12.39/6.85 set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
% 12.39/6.85 set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
% 12.39/6.85 set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
% 12.39/6.85 set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Extended____Nat__Oenat,type,
% 12.39/6.85 set_or5403411693681687835d_enat: extended_enat > extended_enat > set_Extended_enat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
% 12.39/6.85 set_or1266510415728281911st_int: int > int > set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
% 12.39/6.85 set_or1269000886237332187st_nat: nat > nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
% 12.39/6.85 set_or7049704709247886629st_num: num > num > set_num ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
% 12.39/6.85 set_or633870826150836451st_rat: rat > rat > set_rat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
% 12.39/6.85 set_or1222579329274155063t_real: real > real > set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.85 set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
% 12.39/6.85 set_or4662586982721622107an_int: int > int > set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
% 12.39/6.85 set_or4665077453230672383an_nat: nat > nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
% 12.39/6.85 set_ord_atLeast_nat: nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
% 12.39/6.85 set_ord_atLeast_real: real > set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 12.39/6.85 set_ord_atMost_nat: nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
% 12.39/6.85 set_or6656581121297822940st_int: int > int > set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
% 12.39/6.85 set_or6659071591806873216st_nat: nat > nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 12.39/6.85 set_or5832277885323065728an_int: int > int > set_int ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 12.39/6.85 set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 12.39/6.85 set_or1633881224788618240n_real: real > real > set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 12.39/6.85 set_or1210151606488870762an_nat: nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 12.39/6.85 set_or5849166863359141190n_real: real > set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 12.39/6.85 set_ord_lessThan_nat: nat > set_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 12.39/6.85 set_or5984915006950818249n_real: real > set_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_String_OCode_Oabort_001t__Real__Oreal,type,
% 12.39/6.85 abort_real: literal > ( product_unit > real ) > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_String_OLiteral,type,
% 12.39/6.85 literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_String_Oascii__of,type,
% 12.39/6.85 ascii_of: char > char ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_String_Ochar_OChar,type,
% 12.39/6.85 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 12.39/6.85 comm_s629917340098488124ar_nat: char > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_String_Ointeger__of__char,type,
% 12.39/6.85 integer_of_char: char > code_integer ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 12.39/6.85 unique3096191561947761185of_nat: nat > char ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.39/6.85 topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 12.39/6.85 topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 12.39/6.85 topolo2177554685111907308n_real: real > set_real > filter_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
% 12.39/6.85 topolo7531315842566124627t_real: ( nat > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 12.39/6.85 topolo2815343760600316023s_real: real > filter_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
% 12.39/6.85 topolo6517432010174082258omplex: ( nat > complex ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 12.39/6.85 topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 12.39/6.85 topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 12.39/6.85 topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Oarccos,type,
% 12.39/6.85 arccos: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 12.39/6.85 arcosh_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Oarcsin,type,
% 12.39/6.85 arcsin: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Oarctan,type,
% 12.39/6.85 arctan: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 12.39/6.85 arsinh_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 12.39/6.85 artanh_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 12.39/6.85 cos_complex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 12.39/6.85 cos_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Ocos__coeff,type,
% 12.39/6.85 cos_coeff: nat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
% 12.39/6.85 cosh_complex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 12.39/6.85 cosh_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
% 12.39/6.85 cot_complex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 12.39/6.85 cot_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 12.39/6.85 exp_complex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 12.39/6.85 exp_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 12.39/6.85 ln_ln_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Olog,type,
% 12.39/6.85 log: real > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Opi,type,
% 12.39/6.85 pi: real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 12.39/6.85 powr_real: real > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Opowr__real,type,
% 12.39/6.85 powr_real2: real > real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 12.39/6.85 sin_complex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 12.39/6.85 sin_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Osin__coeff,type,
% 12.39/6.85 sin_coeff: nat > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
% 12.39/6.85 sinh_complex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 12.39/6.85 sinh_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 12.39/6.85 tan_complex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 12.39/6.85 tan_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 12.39/6.85 tanh_complex: complex > complex ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 12.39/6.85 tanh_real: real > real ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 12.39/6.85 transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
% 12.39/6.85 transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 12.39/6.85 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 12.39/6.85 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 12.39/6.85 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 12.39/6.85 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
% 12.39/6.85 vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
% 12.39/6.85 vEBT_V312737461966249ad_rel: produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 12.39/6.85 vEBT_VEBT_high: nat > nat > nat ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 12.39/6.85 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 12.39/6.85
% 12.39/6.85 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 12.39/6.85 vEBT_VEBT_low: nat > nat > nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 12.39/6.86 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 12.39/6.86 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 12.39/6.86 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 12.39/6.86 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 12.39/6.86 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 12.39/6.86 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 12.39/6.86 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 12.39/6.86 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 12.39/6.86 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 12.39/6.86 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 12.39/6.86 vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 12.39/6.86 vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 12.39/6.86 vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 12.39/6.86 vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 12.39/6.86 vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Member_Ovebt__member,type,
% 12.39/6.86 vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 12.39/6.86 vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 12.39/6.86 vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 12.39/6.86 vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 12.39/6.86 vEBT_VEBT_less: option_nat > option_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 12.39/6.86 vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 12.39/6.86 vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 12.39/6.86 vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 12.39/6.86 vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 12.39/6.86 vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 12.39/6.86 vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 12.39/6.86 vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 12.39/6.86 vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 12.39/6.86 vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 12.39/6.86 vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
% 12.39/6.86 vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Succ_Ovebt__succ,type,
% 12.39/6.86 vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
% 12.39/6.86 vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
% 12.39/6.86 accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 12.39/6.86 accp_nat: ( nat > nat > $o ) > nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.39/6.86 accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.86 accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 12.39/6.86 accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
% 12.39/6.86 accp_P6183159247885693666d_enat: ( produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ) > produc7272778201969148633d_enat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 12.39/6.86 accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.86 accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_Wellfounded_Oless__than,type,
% 12.39/6.86 less_than: set_Pr1261947904930325089at_nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_Wellfounded_Opred__nat,type,
% 12.39/6.86 pred_nat: set_Pr1261947904930325089at_nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_fChoice_001t__Real__Oreal,type,
% 12.39/6.86 fChoice_real: ( real > $o ) > real ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001_Eo,type,
% 12.39/6.86 member_o: $o > set_o > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
% 12.39/6.86 member_Code_integer: code_integer > set_Code_integer > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 12.39/6.86 member_complex: complex > set_complex > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__Extended____Nat__Oenat,type,
% 12.39/6.86 member_Extended_enat: extended_enat > set_Extended_enat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__Int__Oint,type,
% 12.39/6.86 member_int: int > set_int > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 12.39/6.86 member_list_nat: list_nat > set_list_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__Nat__Onat,type,
% 12.39/6.86 member_nat: nat > set_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__Num__Onum,type,
% 12.39/6.86 member_num: num > set_num > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.39/6.86 member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.39/6.86 member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__Rat__Orat,type,
% 12.39/6.86 member_rat: rat > set_rat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__Real__Oreal,type,
% 12.39/6.86 member_real: real > set_real > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.39/6.86 member_set_nat: set_nat > set_set_nat > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 12.39/6.86 member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_deg____,type,
% 12.39/6.86 deg: nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_m____,type,
% 12.39/6.86 m: nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_ma____,type,
% 12.39/6.86 ma: nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_mi____,type,
% 12.39/6.86 mi: nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_miny____,type,
% 12.39/6.86 miny: nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_na____,type,
% 12.39/6.86 na: nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_res____,type,
% 12.39/6.86 res: nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_sc____,type,
% 12.39/6.86 sc: nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_summary____,type,
% 12.39/6.86 summary: vEBT_VEBT ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_treeList____,type,
% 12.39/6.86 treeList: list_VEBT_VEBT ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_xa____,type,
% 12.39/6.86 xa: nat ).
% 12.39/6.86
% 12.39/6.86 thf(sy_v_za____,type,
% 12.39/6.86 za: nat ).
% 12.39/6.86
% 12.39/6.86 thf(fact_0_bbbb,axiom,
% 12.39/6.86 ord_less_eq_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_1__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
% 12.39/6.86 ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 12.39/6.86
% 12.39/6.86 thf(fact_2_aaaa,axiom,
% 12.39/6.86 ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_3_max__in__set__def,axiom,
% 12.39/6.86 ( vEBT_VEBT_max_in_set
% 12.39/6.86 = ( ^ [Xs: set_nat,X: nat] :
% 12.39/6.86 ( ( member_nat @ X @ Xs )
% 12.39/6.86 & ! [Y: nat] :
% 12.39/6.86 ( ( member_nat @ Y @ Xs )
% 12.39/6.86 => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_4_min__in__set__def,axiom,
% 12.39/6.86 ( vEBT_VEBT_min_in_set
% 12.39/6.86 = ( ^ [Xs: set_nat,X: nat] :
% 12.39/6.86 ( ( member_nat @ X @ Xs )
% 12.39/6.86 & ! [Y: nat] :
% 12.39/6.86 ( ( member_nat @ Y @ Xs )
% 12.39/6.86 => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_5__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
% 12.39/6.86 ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = na ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_6_semiring__norm_I85_J,axiom,
% 12.39/6.86 ! [M: num] :
% 12.39/6.86 ( ( bit0 @ M )
% 12.39/6.86 != one ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_7_semiring__norm_I83_J,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( one
% 12.39/6.86 != ( bit0 @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_8_numeral__Bit0__div__2,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = ( numeral_numeral_nat @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_9_numeral__Bit0__div__2,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.86 = ( numeral_numeral_int @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_10_high__def,axiom,
% 12.39/6.86 ( vEBT_VEBT_high
% 12.39/6.86 = ( ^ [X: nat,N2: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_11__092_060open_062high_Az_A_Ideg_Adiv_A2_J_A_060_Asc_092_060close_062,axiom,
% 12.39/6.86 ord_less_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ).
% 12.39/6.86
% 12.39/6.86 thf(fact_12_divide__numeral__1,axiom,
% 12.39/6.86 ! [A: complex] :
% 12.39/6.86 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_13_divide__numeral__1,axiom,
% 12.39/6.86 ! [A: real] :
% 12.39/6.86 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_14_divide__numeral__1,axiom,
% 12.39/6.86 ! [A: rat] :
% 12.39/6.86 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_15_verit__eq__simplify_I8_J,axiom,
% 12.39/6.86 ! [X2: num,Y2: num] :
% 12.39/6.86 ( ( ( bit0 @ X2 )
% 12.39/6.86 = ( bit0 @ Y2 ) )
% 12.39/6.86 = ( X2 = Y2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_16_semiring__norm_I87_J,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ( bit0 @ M )
% 12.39/6.86 = ( bit0 @ N ) )
% 12.39/6.86 = ( M = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_17_numeral__eq__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ( numera6690914467698888265omplex @ M )
% 12.39/6.86 = ( numera6690914467698888265omplex @ N ) )
% 12.39/6.86 = ( M = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_18_numeral__eq__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ( numeral_numeral_real @ M )
% 12.39/6.86 = ( numeral_numeral_real @ N ) )
% 12.39/6.86 = ( M = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_19_numeral__eq__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ( numeral_numeral_rat @ M )
% 12.39/6.86 = ( numeral_numeral_rat @ N ) )
% 12.39/6.86 = ( M = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_20_numeral__eq__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ( numeral_numeral_nat @ M )
% 12.39/6.86 = ( numeral_numeral_nat @ N ) )
% 12.39/6.86 = ( M = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_21_numeral__eq__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ( numeral_numeral_int @ M )
% 12.39/6.86 = ( numeral_numeral_int @ N ) )
% 12.39/6.86 = ( M = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_22_numeral__le__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 12.39/6.86 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_23_numeral__le__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 12.39/6.86 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_24_numeral__le__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.39/6.86 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_25_numeral__le__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.39/6.86 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_26_numeral__less__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 12.39/6.86 = ( ord_less_num @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_27_numeral__less__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 12.39/6.86 = ( ord_less_num @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_28_numeral__less__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.39/6.86 = ( ord_less_num @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_29_numeral__less__iff,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.39/6.86 = ( ord_less_num @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_30__092_060open_062z_A_060_A2_A_094_Adeg_092_060close_062,axiom,
% 12.39/6.86 ord_less_nat @ za @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_31__C5_Ohyps_C_I10_J,axiom,
% 12.39/6.86 ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_32_verit__comp__simplify1_I3_J,axiom,
% 12.39/6.86 ! [B: real,A2: real] :
% 12.39/6.86 ( ( ~ ( ord_less_eq_real @ B @ A2 ) )
% 12.39/6.86 = ( ord_less_real @ A2 @ B ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_33_verit__comp__simplify1_I3_J,axiom,
% 12.39/6.86 ! [B: rat,A2: rat] :
% 12.39/6.86 ( ( ~ ( ord_less_eq_rat @ B @ A2 ) )
% 12.39/6.86 = ( ord_less_rat @ A2 @ B ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_34_verit__comp__simplify1_I3_J,axiom,
% 12.39/6.86 ! [B: num,A2: num] :
% 12.39/6.86 ( ( ~ ( ord_less_eq_num @ B @ A2 ) )
% 12.39/6.86 = ( ord_less_num @ A2 @ B ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_35_verit__comp__simplify1_I3_J,axiom,
% 12.39/6.86 ! [B: nat,A2: nat] :
% 12.39/6.86 ( ( ~ ( ord_less_eq_nat @ B @ A2 ) )
% 12.39/6.86 = ( ord_less_nat @ A2 @ B ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_36_verit__comp__simplify1_I3_J,axiom,
% 12.39/6.86 ! [B: int,A2: int] :
% 12.39/6.86 ( ( ~ ( ord_less_eq_int @ B @ A2 ) )
% 12.39/6.86 = ( ord_less_int @ A2 @ B ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_37_verit__comp__simplify1_I2_J,axiom,
% 12.39/6.86 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_38_verit__comp__simplify1_I2_J,axiom,
% 12.39/6.86 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_39_verit__comp__simplify1_I2_J,axiom,
% 12.39/6.86 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_40_verit__comp__simplify1_I2_J,axiom,
% 12.39/6.86 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_41_verit__comp__simplify1_I2_J,axiom,
% 12.39/6.86 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_42_verit__comp__simplify1_I1_J,axiom,
% 12.39/6.86 ! [A: real] :
% 12.39/6.86 ~ ( ord_less_real @ A @ A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_43_verit__comp__simplify1_I1_J,axiom,
% 12.39/6.86 ! [A: rat] :
% 12.39/6.86 ~ ( ord_less_rat @ A @ A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_44_verit__comp__simplify1_I1_J,axiom,
% 12.39/6.86 ! [A: num] :
% 12.39/6.86 ~ ( ord_less_num @ A @ A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_45_verit__comp__simplify1_I1_J,axiom,
% 12.39/6.86 ! [A: nat] :
% 12.39/6.86 ~ ( ord_less_nat @ A @ A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_46_verit__comp__simplify1_I1_J,axiom,
% 12.39/6.86 ! [A: int] :
% 12.39/6.86 ~ ( ord_less_int @ A @ A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_47_verit__la__disequality,axiom,
% 12.39/6.86 ! [A: rat,B2: rat] :
% 12.39/6.86 ( ( A = B2 )
% 12.39/6.86 | ~ ( ord_less_eq_rat @ A @ B2 )
% 12.39/6.86 | ~ ( ord_less_eq_rat @ B2 @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_48_verit__la__disequality,axiom,
% 12.39/6.86 ! [A: num,B2: num] :
% 12.39/6.86 ( ( A = B2 )
% 12.39/6.86 | ~ ( ord_less_eq_num @ A @ B2 )
% 12.39/6.86 | ~ ( ord_less_eq_num @ B2 @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_49_verit__la__disequality,axiom,
% 12.39/6.86 ! [A: nat,B2: nat] :
% 12.39/6.86 ( ( A = B2 )
% 12.39/6.86 | ~ ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.86 | ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_50_verit__la__disequality,axiom,
% 12.39/6.86 ! [A: int,B2: int] :
% 12.39/6.86 ( ( A = B2 )
% 12.39/6.86 | ~ ( ord_less_eq_int @ A @ B2 )
% 12.39/6.86 | ~ ( ord_less_eq_int @ B2 @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_51_div__le__dividend,axiom,
% 12.39/6.86 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_52_div__le__mono,axiom,
% 12.39/6.86 ! [M: nat,N: nat,K: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.86 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_53_verit__eq__simplify_I10_J,axiom,
% 12.39/6.86 ! [X2: num] :
% 12.39/6.86 ( one
% 12.39/6.86 != ( bit0 @ X2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_54__092_060open_062high_Ares_A_Ideg_Adiv_A2_J_A_061_Asc_092_060close_062,axiom,
% 12.39/6.86 ( ( vEBT_VEBT_high @ res @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.86 = sc ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_55__092_060open_062x_A_060_Ares_092_060close_062,axiom,
% 12.39/6.86 ord_less_nat @ xa @ res ).
% 12.39/6.86
% 12.39/6.86 thf(fact_56__092_060open_062sc_A_060_A2_A_094_Am_092_060close_062,axiom,
% 12.39/6.86 ord_less_nat @ sc @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_57_power2__nat__le__imp__le,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 12.39/6.86 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_58_power2__nat__le__eq__le,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.86 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_59_self__le__ge2__pow,axiom,
% 12.39/6.86 ! [K: nat,M: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 12.39/6.86 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_60_less__exp,axiom,
% 12.39/6.86 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_61_high__bound__aux,axiom,
% 12.39/6.86 ! [Ma: nat,N: nat,M: nat] :
% 12.39/6.86 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 12.39/6.86 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_62_power__minus__is__div,axiom,
% 12.39/6.86 ! [B2: nat,A: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ B2 @ A )
% 12.39/6.86 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B2 ) )
% 12.39/6.86 = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_63_pow__sum,axiom,
% 12.39/6.86 ! [A: nat,B2: nat] :
% 12.39/6.86 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 12.39/6.86 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_64__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
% 12.39/6.86 ord_less_eq_nat @ one_one_nat @ na ).
% 12.39/6.86
% 12.39/6.86 thf(fact_65__092_060open_062mi_A_092_060le_062_Ax_092_060close_062,axiom,
% 12.39/6.86 ord_less_eq_nat @ mi @ xa ).
% 12.39/6.86
% 12.39/6.86 thf(fact_66__C5_Ohyps_C_I9_J,axiom,
% 12.39/6.86 ord_less_eq_nat @ mi @ ma ).
% 12.39/6.86
% 12.39/6.86 thf(fact_67__C5_Ohyps_C_I6_J,axiom,
% 12.39/6.86 ( deg
% 12.39/6.86 = ( plus_plus_nat @ na @ m ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_68__092_060open_062z_A_092_060noteq_062_Ami_092_060close_062,axiom,
% 12.39/6.86 za != mi ).
% 12.39/6.86
% 12.39/6.86 thf(fact_69__092_060open_062mi_A_092_060noteq_062_Ama_092_060close_062,axiom,
% 12.39/6.86 mi != ma ).
% 12.39/6.86
% 12.39/6.86 thf(fact_70__092_060open_062res_A_092_060le_062_Ama_092_060close_062,axiom,
% 12.39/6.86 ord_less_eq_nat @ res @ ma ).
% 12.39/6.86
% 12.39/6.86 thf(fact_71_mem__Collect__eq,axiom,
% 12.39/6.86 ! [A: extended_enat,P: extended_enat > $o] :
% 12.39/6.86 ( ( member_Extended_enat @ A @ ( collec4429806609662206161d_enat @ P ) )
% 12.39/6.86 = ( P @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_72_mem__Collect__eq,axiom,
% 12.39/6.86 ! [A: complex,P: complex > $o] :
% 12.39/6.86 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 12.39/6.86 = ( P @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_73_mem__Collect__eq,axiom,
% 12.39/6.86 ! [A: real,P: real > $o] :
% 12.39/6.86 ( ( member_real @ A @ ( collect_real @ P ) )
% 12.39/6.86 = ( P @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_74_mem__Collect__eq,axiom,
% 12.39/6.86 ! [A: list_nat,P: list_nat > $o] :
% 12.39/6.86 ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 12.39/6.86 = ( P @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_75_mem__Collect__eq,axiom,
% 12.39/6.86 ! [A: nat,P: nat > $o] :
% 12.39/6.86 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 12.39/6.86 = ( P @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_76_mem__Collect__eq,axiom,
% 12.39/6.86 ! [A: int,P: int > $o] :
% 12.39/6.86 ( ( member_int @ A @ ( collect_int @ P ) )
% 12.39/6.86 = ( P @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_77_Collect__mem__eq,axiom,
% 12.39/6.86 ! [A3: set_Extended_enat] :
% 12.39/6.86 ( ( collec4429806609662206161d_enat
% 12.39/6.86 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A3 ) )
% 12.39/6.86 = A3 ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_78_Collect__mem__eq,axiom,
% 12.39/6.86 ! [A3: set_complex] :
% 12.39/6.86 ( ( collect_complex
% 12.39/6.86 @ ^ [X: complex] : ( member_complex @ X @ A3 ) )
% 12.39/6.86 = A3 ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_79_Collect__mem__eq,axiom,
% 12.39/6.86 ! [A3: set_real] :
% 12.39/6.86 ( ( collect_real
% 12.39/6.86 @ ^ [X: real] : ( member_real @ X @ A3 ) )
% 12.39/6.86 = A3 ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_80_Collect__mem__eq,axiom,
% 12.39/6.86 ! [A3: set_list_nat] :
% 12.39/6.86 ( ( collect_list_nat
% 12.39/6.86 @ ^ [X: list_nat] : ( member_list_nat @ X @ A3 ) )
% 12.39/6.86 = A3 ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_81_Collect__mem__eq,axiom,
% 12.39/6.86 ! [A3: set_nat] :
% 12.39/6.86 ( ( collect_nat
% 12.39/6.86 @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
% 12.39/6.86 = A3 ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_82_Collect__mem__eq,axiom,
% 12.39/6.86 ! [A3: set_int] :
% 12.39/6.86 ( ( collect_int
% 12.39/6.86 @ ^ [X: int] : ( member_int @ X @ A3 ) )
% 12.39/6.86 = A3 ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_83_Collect__cong,axiom,
% 12.39/6.86 ! [P: complex > $o,Q: complex > $o] :
% 12.39/6.86 ( ! [X3: complex] :
% 12.39/6.86 ( ( P @ X3 )
% 12.39/6.86 = ( Q @ X3 ) )
% 12.39/6.86 => ( ( collect_complex @ P )
% 12.39/6.86 = ( collect_complex @ Q ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_84_Collect__cong,axiom,
% 12.39/6.86 ! [P: real > $o,Q: real > $o] :
% 12.39/6.86 ( ! [X3: real] :
% 12.39/6.86 ( ( P @ X3 )
% 12.39/6.86 = ( Q @ X3 ) )
% 12.39/6.86 => ( ( collect_real @ P )
% 12.39/6.86 = ( collect_real @ Q ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_85_Collect__cong,axiom,
% 12.39/6.86 ! [P: list_nat > $o,Q: list_nat > $o] :
% 12.39/6.86 ( ! [X3: list_nat] :
% 12.39/6.86 ( ( P @ X3 )
% 12.39/6.86 = ( Q @ X3 ) )
% 12.39/6.86 => ( ( collect_list_nat @ P )
% 12.39/6.86 = ( collect_list_nat @ Q ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_86_Collect__cong,axiom,
% 12.39/6.86 ! [P: nat > $o,Q: nat > $o] :
% 12.39/6.86 ( ! [X3: nat] :
% 12.39/6.86 ( ( P @ X3 )
% 12.39/6.86 = ( Q @ X3 ) )
% 12.39/6.86 => ( ( collect_nat @ P )
% 12.39/6.86 = ( collect_nat @ Q ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_87_Collect__cong,axiom,
% 12.39/6.86 ! [P: int > $o,Q: int > $o] :
% 12.39/6.86 ( ! [X3: int] :
% 12.39/6.86 ( ( P @ X3 )
% 12.39/6.86 = ( Q @ X3 ) )
% 12.39/6.86 => ( ( collect_int @ P )
% 12.39/6.86 = ( collect_int @ Q ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_88__092_060open_062mi_A_060_Ares_092_060close_062,axiom,
% 12.39/6.86 ord_less_nat @ mi @ res ).
% 12.39/6.86
% 12.39/6.86 thf(fact_89_add__numeral__left,axiom,
% 12.39/6.86 ! [V: num,W: num,Z: complex] :
% 12.39/6.86 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 12.39/6.86 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_90_add__numeral__left,axiom,
% 12.39/6.86 ! [V: num,W: num,Z: real] :
% 12.39/6.86 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 12.39/6.86 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_91_add__numeral__left,axiom,
% 12.39/6.86 ! [V: num,W: num,Z: rat] :
% 12.39/6.86 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 12.39/6.86 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_92_add__numeral__left,axiom,
% 12.39/6.86 ! [V: num,W: num,Z: nat] :
% 12.39/6.86 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 12.39/6.86 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_93_add__numeral__left,axiom,
% 12.39/6.86 ! [V: num,W: num,Z: int] :
% 12.39/6.86 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 12.39/6.86 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_94_numeral__plus__numeral,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 12.39/6.86 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_95_numeral__plus__numeral,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 12.39/6.86 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_96_numeral__plus__numeral,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 12.39/6.86 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_97_numeral__plus__numeral,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.39/6.86 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_98_numeral__plus__numeral,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.39/6.86 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_99_power__one,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ( ( power_power_rat @ one_one_rat @ N )
% 12.39/6.86 = one_one_rat ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_100_power__one,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ( ( power_power_nat @ one_one_nat @ N )
% 12.39/6.86 = one_one_nat ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_101_power__one,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ( ( power_power_real @ one_one_real @ N )
% 12.39/6.86 = one_one_real ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_102_power__one,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ( ( power_power_int @ one_one_int @ N )
% 12.39/6.86 = one_one_int ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_103_power__one,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ( ( power_power_complex @ one_one_complex @ N )
% 12.39/6.86 = one_one_complex ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_104_power__one__right,axiom,
% 12.39/6.86 ! [A: nat] :
% 12.39/6.86 ( ( power_power_nat @ A @ one_one_nat )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_105_power__one__right,axiom,
% 12.39/6.86 ! [A: real] :
% 12.39/6.86 ( ( power_power_real @ A @ one_one_nat )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_106_power__one__right,axiom,
% 12.39/6.86 ! [A: int] :
% 12.39/6.86 ( ( power_power_int @ A @ one_one_nat )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_107_power__one__right,axiom,
% 12.39/6.86 ! [A: complex] :
% 12.39/6.86 ( ( power_power_complex @ A @ one_one_nat )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_108_semiring__norm_I78_J,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.39/6.86 = ( ord_less_num @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_109_semiring__norm_I71_J,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.39/6.86 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_110_semiring__norm_I75_J,axiom,
% 12.39/6.86 ! [M: num] :
% 12.39/6.86 ~ ( ord_less_num @ M @ one ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_111_semiring__norm_I68_J,axiom,
% 12.39/6.86 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_112_numeral__eq__one__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ( numera6690914467698888265omplex @ N )
% 12.39/6.86 = one_one_complex )
% 12.39/6.86 = ( N = one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_113_numeral__eq__one__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ( numeral_numeral_real @ N )
% 12.39/6.86 = one_one_real )
% 12.39/6.86 = ( N = one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_114_numeral__eq__one__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ( numeral_numeral_rat @ N )
% 12.39/6.86 = one_one_rat )
% 12.39/6.86 = ( N = one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_115_numeral__eq__one__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ( numeral_numeral_nat @ N )
% 12.39/6.86 = one_one_nat )
% 12.39/6.86 = ( N = one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_116_numeral__eq__one__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ( numeral_numeral_int @ N )
% 12.39/6.86 = one_one_int )
% 12.39/6.86 = ( N = one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_117_one__eq__numeral__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( one_one_complex
% 12.39/6.86 = ( numera6690914467698888265omplex @ N ) )
% 12.39/6.86 = ( one = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_118_one__eq__numeral__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( one_one_real
% 12.39/6.86 = ( numeral_numeral_real @ N ) )
% 12.39/6.86 = ( one = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_119_one__eq__numeral__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( one_one_rat
% 12.39/6.86 = ( numeral_numeral_rat @ N ) )
% 12.39/6.86 = ( one = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_120_one__eq__numeral__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( one_one_nat
% 12.39/6.86 = ( numeral_numeral_nat @ N ) )
% 12.39/6.86 = ( one = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_121_one__eq__numeral__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( one_one_int
% 12.39/6.86 = ( numeral_numeral_int @ N ) )
% 12.39/6.86 = ( one = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_122_power__inject__exp,axiom,
% 12.39/6.86 ! [A: real,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_real @ one_one_real @ A )
% 12.39/6.86 => ( ( ( power_power_real @ A @ M )
% 12.39/6.86 = ( power_power_real @ A @ N ) )
% 12.39/6.86 = ( M = N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_123_power__inject__exp,axiom,
% 12.39/6.86 ! [A: rat,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_rat @ one_one_rat @ A )
% 12.39/6.86 => ( ( ( power_power_rat @ A @ M )
% 12.39/6.86 = ( power_power_rat @ A @ N ) )
% 12.39/6.86 = ( M = N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_124_power__inject__exp,axiom,
% 12.39/6.86 ! [A: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ one_one_nat @ A )
% 12.39/6.86 => ( ( ( power_power_nat @ A @ M )
% 12.39/6.86 = ( power_power_nat @ A @ N ) )
% 12.39/6.86 = ( M = N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_125_power__inject__exp,axiom,
% 12.39/6.86 ! [A: int,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_int @ one_one_int @ A )
% 12.39/6.86 => ( ( ( power_power_int @ A @ M )
% 12.39/6.86 = ( power_power_int @ A @ N ) )
% 12.39/6.86 = ( M = N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_126_semiring__norm_I76_J,axiom,
% 12.39/6.86 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_127_semiring__norm_I69_J,axiom,
% 12.39/6.86 ! [M: num] :
% 12.39/6.86 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_128__C5_Ohyps_C_I5_J,axiom,
% 12.39/6.86 ( m
% 12.39/6.86 = ( suc @ na ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_129_numeral__plus__one,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 12.39/6.86 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_130_numeral__plus__one,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 12.39/6.86 = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_131_numeral__plus__one,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 12.39/6.86 = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_132_numeral__plus__one,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 12.39/6.86 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_133_numeral__plus__one,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 12.39/6.86 = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_134_one__plus__numeral,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 12.39/6.86 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_135_one__plus__numeral,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 12.39/6.86 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_136_one__plus__numeral,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 12.39/6.86 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_137_one__plus__numeral,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 12.39/6.86 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_138_one__plus__numeral,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 12.39/6.86 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_139_power__strict__increasing__iff,axiom,
% 12.39/6.86 ! [B2: real,X4: nat,Y3: nat] :
% 12.39/6.86 ( ( ord_less_real @ one_one_real @ B2 )
% 12.39/6.86 => ( ( ord_less_real @ ( power_power_real @ B2 @ X4 ) @ ( power_power_real @ B2 @ Y3 ) )
% 12.39/6.86 = ( ord_less_nat @ X4 @ Y3 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_140_power__strict__increasing__iff,axiom,
% 12.39/6.86 ! [B2: rat,X4: nat,Y3: nat] :
% 12.39/6.86 ( ( ord_less_rat @ one_one_rat @ B2 )
% 12.39/6.86 => ( ( ord_less_rat @ ( power_power_rat @ B2 @ X4 ) @ ( power_power_rat @ B2 @ Y3 ) )
% 12.39/6.86 = ( ord_less_nat @ X4 @ Y3 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_141_power__strict__increasing__iff,axiom,
% 12.39/6.86 ! [B2: nat,X4: nat,Y3: nat] :
% 12.39/6.86 ( ( ord_less_nat @ one_one_nat @ B2 )
% 12.39/6.86 => ( ( ord_less_nat @ ( power_power_nat @ B2 @ X4 ) @ ( power_power_nat @ B2 @ Y3 ) )
% 12.39/6.86 = ( ord_less_nat @ X4 @ Y3 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_142_power__strict__increasing__iff,axiom,
% 12.39/6.86 ! [B2: int,X4: nat,Y3: nat] :
% 12.39/6.86 ( ( ord_less_int @ one_one_int @ B2 )
% 12.39/6.86 => ( ( ord_less_int @ ( power_power_int @ B2 @ X4 ) @ ( power_power_int @ B2 @ Y3 ) )
% 12.39/6.86 = ( ord_less_nat @ X4 @ Y3 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_143_one__add__one,axiom,
% 12.39/6.86 ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 12.39/6.86 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_144_one__add__one,axiom,
% 12.39/6.86 ( ( plus_plus_real @ one_one_real @ one_one_real )
% 12.39/6.86 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_145_one__add__one,axiom,
% 12.39/6.86 ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 12.39/6.86 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_146_one__add__one,axiom,
% 12.39/6.86 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 12.39/6.86 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_147_one__add__one,axiom,
% 12.39/6.86 ( ( plus_plus_int @ one_one_int @ one_one_int )
% 12.39/6.86 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_148_power__increasing__iff,axiom,
% 12.39/6.86 ! [B2: real,X4: nat,Y3: nat] :
% 12.39/6.86 ( ( ord_less_real @ one_one_real @ B2 )
% 12.39/6.86 => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ X4 ) @ ( power_power_real @ B2 @ Y3 ) )
% 12.39/6.86 = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_149_power__increasing__iff,axiom,
% 12.39/6.86 ! [B2: rat,X4: nat,Y3: nat] :
% 12.39/6.86 ( ( ord_less_rat @ one_one_rat @ B2 )
% 12.39/6.86 => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ X4 ) @ ( power_power_rat @ B2 @ Y3 ) )
% 12.39/6.86 = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_150_power__increasing__iff,axiom,
% 12.39/6.86 ! [B2: nat,X4: nat,Y3: nat] :
% 12.39/6.86 ( ( ord_less_nat @ one_one_nat @ B2 )
% 12.39/6.86 => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X4 ) @ ( power_power_nat @ B2 @ Y3 ) )
% 12.39/6.86 = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_151_power__increasing__iff,axiom,
% 12.39/6.86 ! [B2: int,X4: nat,Y3: nat] :
% 12.39/6.86 ( ( ord_less_int @ one_one_int @ B2 )
% 12.39/6.86 => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X4 ) @ ( power_power_int @ B2 @ Y3 ) )
% 12.39/6.86 = ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_152_add__self__div__2,axiom,
% 12.39/6.86 ! [M: nat] :
% 12.39/6.86 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = M ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_153_numeral__le__one__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 12.39/6.86 = ( ord_less_eq_num @ N @ one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_154_numeral__le__one__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 12.39/6.86 = ( ord_less_eq_num @ N @ one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_155_numeral__le__one__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 12.39/6.86 = ( ord_less_eq_num @ N @ one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_156_numeral__le__one__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 12.39/6.86 = ( ord_less_eq_num @ N @ one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_157_one__less__numeral__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 12.39/6.86 = ( ord_less_num @ one @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_158_one__less__numeral__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 12.39/6.86 = ( ord_less_num @ one @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_159_one__less__numeral__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 12.39/6.86 = ( ord_less_num @ one @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_160_one__less__numeral__iff,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 12.39/6.86 = ( ord_less_num @ one @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_161_is__num__normalize_I1_J,axiom,
% 12.39/6.86 ! [A: real,B2: real,C: real] :
% 12.39/6.86 ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_162_is__num__normalize_I1_J,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.86 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_163_is__num__normalize_I1_J,axiom,
% 12.39/6.86 ! [A: int,B2: int,C: int] :
% 12.39/6.86 ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_164_one__plus__numeral__commute,axiom,
% 12.39/6.86 ! [X4: num] :
% 12.39/6.86 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X4 ) )
% 12.39/6.86 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X4 ) @ one_one_complex ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_165_one__plus__numeral__commute,axiom,
% 12.39/6.86 ! [X4: num] :
% 12.39/6.86 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X4 ) )
% 12.39/6.86 = ( plus_plus_real @ ( numeral_numeral_real @ X4 ) @ one_one_real ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_166_one__plus__numeral__commute,axiom,
% 12.39/6.86 ! [X4: num] :
% 12.39/6.86 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X4 ) )
% 12.39/6.86 = ( plus_plus_rat @ ( numeral_numeral_rat @ X4 ) @ one_one_rat ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_167_one__plus__numeral__commute,axiom,
% 12.39/6.86 ! [X4: num] :
% 12.39/6.86 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X4 ) )
% 12.39/6.86 = ( plus_plus_nat @ ( numeral_numeral_nat @ X4 ) @ one_one_nat ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_168_one__plus__numeral__commute,axiom,
% 12.39/6.86 ! [X4: num] :
% 12.39/6.86 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X4 ) )
% 12.39/6.86 = ( plus_plus_int @ ( numeral_numeral_int @ X4 ) @ one_one_int ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_169_le__num__One__iff,axiom,
% 12.39/6.86 ! [X4: num] :
% 12.39/6.86 ( ( ord_less_eq_num @ X4 @ one )
% 12.39/6.86 = ( X4 = one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_170_le__numeral__extra_I4_J,axiom,
% 12.39/6.86 ord_less_eq_real @ one_one_real @ one_one_real ).
% 12.39/6.86
% 12.39/6.86 thf(fact_171_le__numeral__extra_I4_J,axiom,
% 12.39/6.86 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 12.39/6.86
% 12.39/6.86 thf(fact_172_le__numeral__extra_I4_J,axiom,
% 12.39/6.86 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 12.39/6.86
% 12.39/6.86 thf(fact_173_le__numeral__extra_I4_J,axiom,
% 12.39/6.86 ord_less_eq_int @ one_one_int @ one_one_int ).
% 12.39/6.86
% 12.39/6.86 thf(fact_174_less__numeral__extra_I4_J,axiom,
% 12.39/6.86 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_175_less__numeral__extra_I4_J,axiom,
% 12.39/6.86 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_176_less__numeral__extra_I4_J,axiom,
% 12.39/6.86 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_177_less__numeral__extra_I4_J,axiom,
% 12.39/6.86 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_178_one__le__power,axiom,
% 12.39/6.86 ! [A: real,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_real @ one_one_real @ A )
% 12.39/6.86 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_179_one__le__power,axiom,
% 12.39/6.86 ! [A: rat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 12.39/6.86 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_180_one__le__power,axiom,
% 12.39/6.86 ! [A: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 12.39/6.86 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_181_one__le__power,axiom,
% 12.39/6.86 ! [A: int,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_int @ one_one_int @ A )
% 12.39/6.86 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_182_power__one__over,axiom,
% 12.39/6.86 ! [A: complex,N: nat] :
% 12.39/6.86 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 12.39/6.86 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_183_power__one__over,axiom,
% 12.39/6.86 ! [A: real,N: nat] :
% 12.39/6.86 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 12.39/6.86 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_184_power__one__over,axiom,
% 12.39/6.86 ! [A: rat,N: nat] :
% 12.39/6.86 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 12.39/6.86 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_185_nat__1__add__1,axiom,
% 12.39/6.86 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 12.39/6.86 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_186_numeral__Bit0,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 12.39/6.86 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_187_numeral__Bit0,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 12.39/6.86 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_188_numeral__Bit0,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 12.39/6.86 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_189_numeral__Bit0,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 12.39/6.86 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_190_numeral__Bit0,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 12.39/6.86 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_191_power__strict__increasing,axiom,
% 12.39/6.86 ! [N: nat,N3: nat,A: real] :
% 12.39/6.86 ( ( ord_less_nat @ N @ N3 )
% 12.39/6.86 => ( ( ord_less_real @ one_one_real @ A )
% 12.39/6.86 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_192_power__strict__increasing,axiom,
% 12.39/6.86 ! [N: nat,N3: nat,A: rat] :
% 12.39/6.86 ( ( ord_less_nat @ N @ N3 )
% 12.39/6.86 => ( ( ord_less_rat @ one_one_rat @ A )
% 12.39/6.86 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_193_power__strict__increasing,axiom,
% 12.39/6.86 ! [N: nat,N3: nat,A: nat] :
% 12.39/6.86 ( ( ord_less_nat @ N @ N3 )
% 12.39/6.86 => ( ( ord_less_nat @ one_one_nat @ A )
% 12.39/6.86 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_194_power__strict__increasing,axiom,
% 12.39/6.86 ! [N: nat,N3: nat,A: int] :
% 12.39/6.86 ( ( ord_less_nat @ N @ N3 )
% 12.39/6.86 => ( ( ord_less_int @ one_one_int @ A )
% 12.39/6.86 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_195_power__less__imp__less__exp,axiom,
% 12.39/6.86 ! [A: real,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_real @ one_one_real @ A )
% 12.39/6.86 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 12.39/6.86 => ( ord_less_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_196_power__less__imp__less__exp,axiom,
% 12.39/6.86 ! [A: rat,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_rat @ one_one_rat @ A )
% 12.39/6.86 => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 12.39/6.86 => ( ord_less_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_197_power__less__imp__less__exp,axiom,
% 12.39/6.86 ! [A: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ one_one_nat @ A )
% 12.39/6.86 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 12.39/6.86 => ( ord_less_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_198_power__less__imp__less__exp,axiom,
% 12.39/6.86 ! [A: int,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_int @ one_one_int @ A )
% 12.39/6.86 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 12.39/6.86 => ( ord_less_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_199_power__increasing,axiom,
% 12.39/6.86 ! [N: nat,N3: nat,A: real] :
% 12.39/6.86 ( ( ord_less_eq_nat @ N @ N3 )
% 12.39/6.86 => ( ( ord_less_eq_real @ one_one_real @ A )
% 12.39/6.86 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_200_power__increasing,axiom,
% 12.39/6.86 ! [N: nat,N3: nat,A: rat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ N @ N3 )
% 12.39/6.86 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 12.39/6.86 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_201_power__increasing,axiom,
% 12.39/6.86 ! [N: nat,N3: nat,A: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ N @ N3 )
% 12.39/6.86 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 12.39/6.86 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_202_power__increasing,axiom,
% 12.39/6.86 ! [N: nat,N3: nat,A: int] :
% 12.39/6.86 ( ( ord_less_eq_nat @ N @ N3 )
% 12.39/6.86 => ( ( ord_less_eq_int @ one_one_int @ A )
% 12.39/6.86 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_203_one__le__numeral,axiom,
% 12.39/6.86 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_204_one__le__numeral,axiom,
% 12.39/6.86 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_205_one__le__numeral,axiom,
% 12.39/6.86 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_206_one__le__numeral,axiom,
% 12.39/6.86 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_207_not__numeral__less__one,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_208_not__numeral__less__one,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_209_not__numeral__less__one,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_210_not__numeral__less__one,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_211_numeral__One,axiom,
% 12.39/6.86 ( ( numera6690914467698888265omplex @ one )
% 12.39/6.86 = one_one_complex ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_212_numeral__One,axiom,
% 12.39/6.86 ( ( numeral_numeral_real @ one )
% 12.39/6.86 = one_one_real ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_213_numeral__One,axiom,
% 12.39/6.86 ( ( numeral_numeral_rat @ one )
% 12.39/6.86 = one_one_rat ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_214_numeral__One,axiom,
% 12.39/6.86 ( ( numeral_numeral_nat @ one )
% 12.39/6.86 = one_one_nat ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_215_numeral__One,axiom,
% 12.39/6.86 ( ( numeral_numeral_int @ one )
% 12.39/6.86 = one_one_int ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_216_numerals_I1_J,axiom,
% 12.39/6.86 ( ( numeral_numeral_nat @ one )
% 12.39/6.86 = one_one_nat ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_217_ex__power__ivl1,axiom,
% 12.39/6.86 ! [B2: nat,K: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 12.39/6.86 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 12.39/6.86 => ? [N4: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N4 ) @ K )
% 12.39/6.86 & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_218_ex__power__ivl2,axiom,
% 12.39/6.86 ! [B2: nat,K: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 12.39/6.86 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 12.39/6.86 => ? [N4: nat] :
% 12.39/6.86 ( ( ord_less_nat @ ( power_power_nat @ B2 @ N4 ) @ K )
% 12.39/6.86 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_219_power2__commute,axiom,
% 12.39/6.86 ! [X4: complex,Y3: complex] :
% 12.39/6.86 ( ( power_power_complex @ ( minus_minus_complex @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = ( power_power_complex @ ( minus_minus_complex @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_220_power2__commute,axiom,
% 12.39/6.86 ! [X4: real,Y3: real] :
% 12.39/6.86 ( ( power_power_real @ ( minus_minus_real @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = ( power_power_real @ ( minus_minus_real @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_221_power2__commute,axiom,
% 12.39/6.86 ! [X4: rat,Y3: rat] :
% 12.39/6.86 ( ( power_power_rat @ ( minus_minus_rat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = ( power_power_rat @ ( minus_minus_rat @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_222_power2__commute,axiom,
% 12.39/6.86 ! [X4: int,Y3: int] :
% 12.39/6.86 ( ( power_power_int @ ( minus_minus_int @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = ( power_power_int @ ( minus_minus_int @ Y3 @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_223_power__le__imp__le__exp,axiom,
% 12.39/6.86 ! [A: real,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_real @ one_one_real @ A )
% 12.39/6.86 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 12.39/6.86 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_224_power__le__imp__le__exp,axiom,
% 12.39/6.86 ! [A: rat,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_rat @ one_one_rat @ A )
% 12.39/6.86 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 12.39/6.86 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_225_power__le__imp__le__exp,axiom,
% 12.39/6.86 ! [A: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ one_one_nat @ A )
% 12.39/6.86 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 12.39/6.86 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_226_power__le__imp__le__exp,axiom,
% 12.39/6.86 ! [A: int,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_int @ one_one_int @ A )
% 12.39/6.86 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 12.39/6.86 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_227_one__power2,axiom,
% 12.39/6.86 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = one_one_rat ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_228_one__power2,axiom,
% 12.39/6.86 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = one_one_nat ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_229_one__power2,axiom,
% 12.39/6.86 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = one_one_real ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_230_one__power2,axiom,
% 12.39/6.86 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = one_one_int ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_231_one__power2,axiom,
% 12.39/6.86 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = one_one_complex ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_232_diff__le__diff__pow,axiom,
% 12.39/6.86 ! [K: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 12.39/6.86 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_233_power__divide,axiom,
% 12.39/6.86 ! [A: complex,B2: complex,N: nat] :
% 12.39/6.86 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B2 ) @ N )
% 12.39/6.86 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_234_power__divide,axiom,
% 12.39/6.86 ! [A: real,B2: real,N: nat] :
% 12.39/6.86 ( ( power_power_real @ ( divide_divide_real @ A @ B2 ) @ N )
% 12.39/6.86 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_235_power__divide,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,N: nat] :
% 12.39/6.86 ( ( power_power_rat @ ( divide_divide_rat @ A @ B2 ) @ N )
% 12.39/6.86 = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_236_Nat_Oadd__diff__assoc,axiom,
% 12.39/6.86 ! [K: nat,J: nat,I: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ K @ J )
% 12.39/6.86 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 12.39/6.86 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_237_Nat_Oadd__diff__assoc2,axiom,
% 12.39/6.86 ! [K: nat,J: nat,I: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ K @ J )
% 12.39/6.86 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 12.39/6.86 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_238_Nat_Odiff__diff__right,axiom,
% 12.39/6.86 ! [K: nat,J: nat,I: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ K @ J )
% 12.39/6.86 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 12.39/6.86 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_239__092_060open_062high_Ax_An_A_060_A2_A_094_Am_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_092_060close_062,axiom,
% 12.39/6.86 ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 12.39/6.86 & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_240_le__add__diff__inverse,axiom,
% 12.39/6.86 ! [B2: real,A: real] :
% 12.39/6.86 ( ( ord_less_eq_real @ B2 @ A )
% 12.39/6.86 => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
% 12.39/6.86 = A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_241_le__add__diff__inverse,axiom,
% 12.39/6.86 ! [B2: rat,A: rat] :
% 12.39/6.86 ( ( ord_less_eq_rat @ B2 @ A )
% 12.39/6.86 => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
% 12.39/6.86 = A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_242_le__add__diff__inverse,axiom,
% 12.39/6.86 ! [B2: nat,A: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ B2 @ A )
% 12.39/6.86 => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
% 12.39/6.86 = A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_243_le__add__diff__inverse,axiom,
% 12.39/6.86 ! [B2: int,A: int] :
% 12.39/6.86 ( ( ord_less_eq_int @ B2 @ A )
% 12.39/6.86 => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
% 12.39/6.86 = A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_244_le__add__diff__inverse2,axiom,
% 12.39/6.86 ! [B2: real,A: real] :
% 12.39/6.86 ( ( ord_less_eq_real @ B2 @ A )
% 12.39/6.86 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
% 12.39/6.86 = A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_245_le__add__diff__inverse2,axiom,
% 12.39/6.86 ! [B2: rat,A: rat] :
% 12.39/6.86 ( ( ord_less_eq_rat @ B2 @ A )
% 12.39/6.86 => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
% 12.39/6.86 = A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_246_le__add__diff__inverse2,axiom,
% 12.39/6.86 ! [B2: nat,A: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ B2 @ A )
% 12.39/6.86 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
% 12.39/6.86 = A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_247_le__add__diff__inverse2,axiom,
% 12.39/6.86 ! [B2: int,A: int] :
% 12.39/6.86 ( ( ord_less_eq_int @ B2 @ A )
% 12.39/6.86 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 12.39/6.86 = A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_248_div__exp__eq,axiom,
% 12.39/6.86 ! [A: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( 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 ) )
% 12.39/6.86 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_249_div__exp__eq,axiom,
% 12.39/6.86 ! [A: int,M: nat,N: nat] :
% 12.39/6.86 ( ( 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 ) )
% 12.39/6.86 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_250_field__less__half__sum,axiom,
% 12.39/6.86 ! [X4: real,Y3: real] :
% 12.39/6.86 ( ( ord_less_real @ X4 @ Y3 )
% 12.39/6.86 => ( ord_less_real @ X4 @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_251_field__less__half__sum,axiom,
% 12.39/6.86 ! [X4: rat,Y3: rat] :
% 12.39/6.86 ( ( ord_less_rat @ X4 @ Y3 )
% 12.39/6.86 => ( ord_less_rat @ X4 @ ( divide_divide_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_252_high__inv,axiom,
% 12.39/6.86 ! [X4: nat,N: nat,Y3: nat] :
% 12.39/6.86 ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.86 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X4 ) @ N )
% 12.39/6.86 = Y3 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_253_diff__diff__left,axiom,
% 12.39/6.86 ! [I: nat,J: nat,K: nat] :
% 12.39/6.86 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 12.39/6.86 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_254_diff__diff__cancel,axiom,
% 12.39/6.86 ! [I: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ I @ N )
% 12.39/6.86 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 12.39/6.86 = I ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_255_nat__add__left__cancel__le,axiom,
% 12.39/6.86 ! [K: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 12.39/6.86 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_256_even__odd__cases,axiom,
% 12.39/6.86 ! [X4: nat] :
% 12.39/6.86 ( ! [N4: nat] :
% 12.39/6.86 ( X4
% 12.39/6.86 != ( plus_plus_nat @ N4 @ N4 ) )
% 12.39/6.86 => ~ ! [N4: nat] :
% 12.39/6.86 ( X4
% 12.39/6.86 != ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_257_bit__split__inv,axiom,
% 12.39/6.86 ! [X4: nat,D: nat] :
% 12.39/6.86 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X4 @ D ) @ ( vEBT_VEBT_low @ X4 @ D ) @ D )
% 12.39/6.86 = X4 ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_258_nat_Oinject,axiom,
% 12.39/6.86 ! [X2: nat,Y2: nat] :
% 12.39/6.86 ( ( ( suc @ X2 )
% 12.39/6.86 = ( suc @ Y2 ) )
% 12.39/6.86 = ( X2 = Y2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_259_old_Onat_Oinject,axiom,
% 12.39/6.86 ! [Nat: nat,Nat2: nat] :
% 12.39/6.86 ( ( ( suc @ Nat )
% 12.39/6.86 = ( suc @ Nat2 ) )
% 12.39/6.86 = ( Nat = Nat2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_260_numeral__times__numeral,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 12.39/6.86 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_261_numeral__times__numeral,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 12.39/6.86 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_262_numeral__times__numeral,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 12.39/6.86 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_263_numeral__times__numeral,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.39/6.86 = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_264_numeral__times__numeral,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.39/6.86 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_265_mult__numeral__left__semiring__numeral,axiom,
% 12.39/6.86 ! [V: num,W: num,Z: complex] :
% 12.39/6.86 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 12.39/6.86 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_266_mult__numeral__left__semiring__numeral,axiom,
% 12.39/6.86 ! [V: num,W: num,Z: real] :
% 12.39/6.86 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 12.39/6.86 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_267_mult__numeral__left__semiring__numeral,axiom,
% 12.39/6.86 ! [V: num,W: num,Z: rat] :
% 12.39/6.86 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 12.39/6.86 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_268_mult__numeral__left__semiring__numeral,axiom,
% 12.39/6.86 ! [V: num,W: num,Z: nat] :
% 12.39/6.86 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 12.39/6.86 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_269_mult__numeral__left__semiring__numeral,axiom,
% 12.39/6.86 ! [V: num,W: num,Z: int] :
% 12.39/6.86 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 12.39/6.86 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_270_low__inv,axiom,
% 12.39/6.86 ! [X4: nat,N: nat,Y3: nat] :
% 12.39/6.86 ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.86 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X4 ) @ N )
% 12.39/6.86 = X4 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_271_bits__div__by__1,axiom,
% 12.39/6.86 ! [A: nat] :
% 12.39/6.86 ( ( divide_divide_nat @ A @ one_one_nat )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_272_bits__div__by__1,axiom,
% 12.39/6.86 ! [A: int] :
% 12.39/6.86 ( ( divide_divide_int @ A @ one_one_int )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_273_div__by__1,axiom,
% 12.39/6.86 ! [A: complex] :
% 12.39/6.86 ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_274_div__by__1,axiom,
% 12.39/6.86 ! [A: real] :
% 12.39/6.86 ( ( divide_divide_real @ A @ one_one_real )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_275_div__by__1,axiom,
% 12.39/6.86 ! [A: rat] :
% 12.39/6.86 ( ( divide_divide_rat @ A @ one_one_rat )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_276_div__by__1,axiom,
% 12.39/6.86 ! [A: nat] :
% 12.39/6.86 ( ( divide_divide_nat @ A @ one_one_nat )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_277_div__by__1,axiom,
% 12.39/6.86 ! [A: int] :
% 12.39/6.86 ( ( divide_divide_int @ A @ one_one_int )
% 12.39/6.86 = A ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_278_lessI,axiom,
% 12.39/6.86 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_279_Suc__mono,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ M @ N )
% 12.39/6.86 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_280_Suc__less__eq,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.39/6.86 = ( ord_less_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_281_Suc__le__mono,axiom,
% 12.39/6.86 ! [N: nat,M: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 12.39/6.86 = ( ord_less_eq_nat @ N @ M ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_282_add__Suc__right,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 12.39/6.86 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_283_nat__add__left__cancel__less,axiom,
% 12.39/6.86 ! [K: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 12.39/6.86 = ( ord_less_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_284_diff__Suc__Suc,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.39/6.86 = ( minus_minus_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_285_Suc__diff__diff,axiom,
% 12.39/6.86 ! [M: nat,N: nat,K: nat] :
% 12.39/6.86 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 12.39/6.86 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_286_nat__1__eq__mult__iff,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( one_one_nat
% 12.39/6.86 = ( times_times_nat @ M @ N ) )
% 12.39/6.86 = ( ( M = one_one_nat )
% 12.39/6.86 & ( N = one_one_nat ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_287_nat__mult__eq__1__iff,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ( times_times_nat @ M @ N )
% 12.39/6.86 = one_one_nat )
% 12.39/6.86 = ( ( M = one_one_nat )
% 12.39/6.86 & ( N = one_one_nat ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_288_semiring__norm_I6_J,axiom,
% 12.39/6.86 ! [M: num,N: num] :
% 12.39/6.86 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.39/6.86 = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_289_bit__concat__def,axiom,
% 12.39/6.86 ( vEBT_VEBT_bit_concat
% 12.39/6.86 = ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_290_distrib__left__numeral,axiom,
% 12.39/6.86 ! [V: num,B2: complex,C: complex] :
% 12.39/6.86 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_291_distrib__left__numeral,axiom,
% 12.39/6.86 ! [V: num,B2: real,C: real] :
% 12.39/6.86 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_292_distrib__left__numeral,axiom,
% 12.39/6.86 ! [V: num,B2: rat,C: rat] :
% 12.39/6.86 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_293_distrib__left__numeral,axiom,
% 12.39/6.86 ! [V: num,B2: nat,C: nat] :
% 12.39/6.86 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_294_distrib__left__numeral,axiom,
% 12.39/6.86 ! [V: num,B2: int,C: int] :
% 12.39/6.86 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_295_distrib__right__numeral,axiom,
% 12.39/6.86 ! [A: complex,B2: complex,V: num] :
% 12.39/6.86 ( ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
% 12.39/6.86 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_296_distrib__right__numeral,axiom,
% 12.39/6.86 ! [A: real,B2: real,V: num] :
% 12.39/6.86 ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_297_distrib__right__numeral,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,V: num] :
% 12.39/6.86 ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_298_distrib__right__numeral,axiom,
% 12.39/6.86 ! [A: nat,B2: nat,V: num] :
% 12.39/6.86 ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ ( numeral_numeral_nat @ V ) )
% 12.39/6.86 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_299_distrib__right__numeral,axiom,
% 12.39/6.86 ! [A: int,B2: int,V: num] :
% 12.39/6.86 ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_300_left__diff__distrib__numeral,axiom,
% 12.39/6.86 ! [A: complex,B2: complex,V: num] :
% 12.39/6.86 ( ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
% 12.39/6.86 = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_301_left__diff__distrib__numeral,axiom,
% 12.39/6.86 ! [A: real,B2: real,V: num] :
% 12.39/6.86 ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
% 12.39/6.86 = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_302_left__diff__distrib__numeral,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,V: num] :
% 12.39/6.86 ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
% 12.39/6.86 = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_303_left__diff__distrib__numeral,axiom,
% 12.39/6.86 ! [A: int,B2: int,V: num] :
% 12.39/6.86 ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
% 12.39/6.86 = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_304_right__diff__distrib__numeral,axiom,
% 12.39/6.86 ! [V: num,B2: complex,C: complex] :
% 12.39/6.86 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B2 @ C ) )
% 12.39/6.86 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_305_right__diff__distrib__numeral,axiom,
% 12.39/6.86 ! [V: num,B2: real,C: real] :
% 12.39/6.86 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B2 @ C ) )
% 12.39/6.86 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_306_right__diff__distrib__numeral,axiom,
% 12.39/6.86 ! [V: num,B2: rat,C: rat] :
% 12.39/6.86 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B2 @ C ) )
% 12.39/6.86 = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_307_right__diff__distrib__numeral,axiom,
% 12.39/6.86 ! [V: num,B2: int,C: int] :
% 12.39/6.86 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B2 @ C ) )
% 12.39/6.86 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_308_mult__Suc__right,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( times_times_nat @ M @ ( suc @ N ) )
% 12.39/6.86 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_309_diff__Suc__1,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 12.39/6.86 = N ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_310_semiring__norm_I2_J,axiom,
% 12.39/6.86 ( ( plus_plus_num @ one @ one )
% 12.39/6.86 = ( bit0 @ one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_311_le__divide__eq__numeral1_I1_J,axiom,
% 12.39/6.86 ! [A: real,B2: real,W: num] :
% 12.39/6.86 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 12.39/6.86 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_312_le__divide__eq__numeral1_I1_J,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,W: num] :
% 12.39/6.86 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 12.39/6.86 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_313_divide__le__eq__numeral1_I1_J,axiom,
% 12.39/6.86 ! [B2: real,W: num,A: real] :
% 12.39/6.86 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
% 12.39/6.86 = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_314_divide__le__eq__numeral1_I1_J,axiom,
% 12.39/6.86 ! [B2: rat,W: num,A: rat] :
% 12.39/6.86 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
% 12.39/6.86 = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_315_less__divide__eq__numeral1_I1_J,axiom,
% 12.39/6.86 ! [A: real,B2: real,W: num] :
% 12.39/6.86 ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 12.39/6.86 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_316_less__divide__eq__numeral1_I1_J,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,W: num] :
% 12.39/6.86 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 12.39/6.86 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_317_divide__less__eq__numeral1_I1_J,axiom,
% 12.39/6.86 ! [B2: real,W: num,A: real] :
% 12.39/6.86 ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
% 12.39/6.86 = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_318_divide__less__eq__numeral1_I1_J,axiom,
% 12.39/6.86 ! [B2: rat,W: num,A: rat] :
% 12.39/6.86 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
% 12.39/6.86 = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_319_diff__Suc__diff__eq2,axiom,
% 12.39/6.86 ! [K: nat,J: nat,I: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ K @ J )
% 12.39/6.86 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 12.39/6.86 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_320_diff__Suc__diff__eq1,axiom,
% 12.39/6.86 ! [K: nat,J: nat,I: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ K @ J )
% 12.39/6.86 => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 12.39/6.86 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_321_power__add__numeral,axiom,
% 12.39/6.86 ! [A: complex,M: num,N: num] :
% 12.39/6.86 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 12.39/6.86 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_322_power__add__numeral,axiom,
% 12.39/6.86 ! [A: real,M: num,N: num] :
% 12.39/6.86 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 12.39/6.86 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_323_power__add__numeral,axiom,
% 12.39/6.86 ! [A: rat,M: num,N: num] :
% 12.39/6.86 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 12.39/6.86 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_324_power__add__numeral,axiom,
% 12.39/6.86 ! [A: nat,M: num,N: num] :
% 12.39/6.86 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 12.39/6.86 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_325_power__add__numeral,axiom,
% 12.39/6.86 ! [A: int,M: num,N: num] :
% 12.39/6.86 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 12.39/6.86 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_326_power__add__numeral2,axiom,
% 12.39/6.86 ! [A: complex,M: num,N: num,B2: complex] :
% 12.39/6.86 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 12.39/6.86 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_327_power__add__numeral2,axiom,
% 12.39/6.86 ! [A: real,M: num,N: num,B2: real] :
% 12.39/6.86 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 12.39/6.86 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_328_power__add__numeral2,axiom,
% 12.39/6.86 ! [A: rat,M: num,N: num,B2: rat] :
% 12.39/6.86 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 12.39/6.86 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_329_power__add__numeral2,axiom,
% 12.39/6.86 ! [A: nat,M: num,N: num,B2: nat] :
% 12.39/6.86 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 12.39/6.86 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_330_power__add__numeral2,axiom,
% 12.39/6.86 ! [A: int,M: num,N: num,B2: int] :
% 12.39/6.86 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 12.39/6.86 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_331_Suc__numeral,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( suc @ ( numeral_numeral_nat @ N ) )
% 12.39/6.86 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_332_add__2__eq__Suc_H,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = ( suc @ ( suc @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_333_add__2__eq__Suc,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.39/6.86 = ( suc @ ( suc @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_334_div2__Suc__Suc,axiom,
% 12.39/6.86 ! [M: nat] :
% 12.39/6.86 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.86 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_335_Suc__1,axiom,
% 12.39/6.86 ( ( suc @ one_one_nat )
% 12.39/6.86 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_336_Suc__mult__le__cancel1,axiom,
% 12.39/6.86 ! [K: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 12.39/6.86 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_337_Suc__mult__less__cancel1,axiom,
% 12.39/6.86 ! [K: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 12.39/6.86 = ( ord_less_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_338_mult__Suc,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( times_times_nat @ ( suc @ M ) @ N )
% 12.39/6.86 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_339_Suc__inject,axiom,
% 12.39/6.86 ! [X4: nat,Y3: nat] :
% 12.39/6.86 ( ( ( suc @ X4 )
% 12.39/6.86 = ( suc @ Y3 ) )
% 12.39/6.86 => ( X4 = Y3 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_340_n__not__Suc__n,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ( N
% 12.39/6.86 != ( suc @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_341_Suc__mult__cancel1,axiom,
% 12.39/6.86 ! [K: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 12.39/6.86 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 12.39/6.86 = ( M = N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_342_power__Suc2,axiom,
% 12.39/6.86 ! [A: complex,N: nat] :
% 12.39/6.86 ( ( power_power_complex @ A @ ( suc @ N ) )
% 12.39/6.86 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_343_power__Suc2,axiom,
% 12.39/6.86 ! [A: real,N: nat] :
% 12.39/6.86 ( ( power_power_real @ A @ ( suc @ N ) )
% 12.39/6.86 = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_344_power__Suc2,axiom,
% 12.39/6.86 ! [A: rat,N: nat] :
% 12.39/6.86 ( ( power_power_rat @ A @ ( suc @ N ) )
% 12.39/6.86 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_345_power__Suc2,axiom,
% 12.39/6.86 ! [A: nat,N: nat] :
% 12.39/6.86 ( ( power_power_nat @ A @ ( suc @ N ) )
% 12.39/6.86 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_346_power__Suc2,axiom,
% 12.39/6.86 ! [A: int,N: nat] :
% 12.39/6.86 ( ( power_power_int @ A @ ( suc @ N ) )
% 12.39/6.86 = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_347_power__Suc,axiom,
% 12.39/6.86 ! [A: complex,N: nat] :
% 12.39/6.86 ( ( power_power_complex @ A @ ( suc @ N ) )
% 12.39/6.86 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_348_power__Suc,axiom,
% 12.39/6.86 ! [A: real,N: nat] :
% 12.39/6.86 ( ( power_power_real @ A @ ( suc @ N ) )
% 12.39/6.86 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_349_power__Suc,axiom,
% 12.39/6.86 ! [A: rat,N: nat] :
% 12.39/6.86 ( ( power_power_rat @ A @ ( suc @ N ) )
% 12.39/6.86 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_350_power__Suc,axiom,
% 12.39/6.86 ! [A: nat,N: nat] :
% 12.39/6.86 ( ( power_power_nat @ A @ ( suc @ N ) )
% 12.39/6.86 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_351_power__Suc,axiom,
% 12.39/6.86 ! [A: int,N: nat] :
% 12.39/6.86 ( ( power_power_int @ A @ ( suc @ N ) )
% 12.39/6.86 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_352_ring__class_Oring__distribs_I2_J,axiom,
% 12.39/6.86 ! [A: real,B2: real,C: real] :
% 12.39/6.86 ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_353_ring__class_Oring__distribs_I2_J,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.86 ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_354_ring__class_Oring__distribs_I2_J,axiom,
% 12.39/6.86 ! [A: int,B2: int,C: int] :
% 12.39/6.86 ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_355_ring__class_Oring__distribs_I1_J,axiom,
% 12.39/6.86 ! [A: real,B2: real,C: real] :
% 12.39/6.86 ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_356_ring__class_Oring__distribs_I1_J,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.86 ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_357_ring__class_Oring__distribs_I1_J,axiom,
% 12.39/6.86 ! [A: int,B2: int,C: int] :
% 12.39/6.86 ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_358_comm__semiring__class_Odistrib,axiom,
% 12.39/6.86 ! [A: real,B2: real,C: real] :
% 12.39/6.86 ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_359_comm__semiring__class_Odistrib,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.86 ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_360_comm__semiring__class_Odistrib,axiom,
% 12.39/6.86 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.86 ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_361_comm__semiring__class_Odistrib,axiom,
% 12.39/6.86 ! [A: int,B2: int,C: int] :
% 12.39/6.86 ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_362_distrib__left,axiom,
% 12.39/6.86 ! [A: real,B2: real,C: real] :
% 12.39/6.86 ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_363_distrib__left,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.86 ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_364_distrib__left,axiom,
% 12.39/6.86 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.86 ( ( times_times_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_365_distrib__left,axiom,
% 12.39/6.86 ! [A: int,B2: int,C: int] :
% 12.39/6.86 ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_366_distrib__right,axiom,
% 12.39/6.86 ! [A: real,B2: real,C: real] :
% 12.39/6.86 ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_367_distrib__right,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.86 ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_368_distrib__right,axiom,
% 12.39/6.86 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.86 ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_369_distrib__right,axiom,
% 12.39/6.86 ! [A: int,B2: int,C: int] :
% 12.39/6.86 ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_370_combine__common__factor,axiom,
% 12.39/6.86 ! [A: real,E: real,B2: real,C: real] :
% 12.39/6.86 ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ C ) )
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ E ) @ C ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_371_combine__common__factor,axiom,
% 12.39/6.86 ! [A: rat,E: rat,B2: rat,C: rat] :
% 12.39/6.86 ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ C ) )
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ E ) @ C ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_372_combine__common__factor,axiom,
% 12.39/6.86 ! [A: nat,E: nat,B2: nat,C: nat] :
% 12.39/6.86 ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
% 12.39/6.86 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ E ) @ C ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_373_combine__common__factor,axiom,
% 12.39/6.86 ! [A: int,E: int,B2: int,C: int] :
% 12.39/6.86 ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ E ) @ C ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_374_left__diff__distrib,axiom,
% 12.39/6.86 ! [A: real,B2: real,C: real] :
% 12.39/6.86 ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 12.39/6.86 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_375_left__diff__distrib,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.86 ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 12.39/6.86 = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_376_left__diff__distrib,axiom,
% 12.39/6.86 ! [A: int,B2: int,C: int] :
% 12.39/6.86 ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 12.39/6.86 = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_377_right__diff__distrib,axiom,
% 12.39/6.86 ! [A: real,B2: real,C: real] :
% 12.39/6.86 ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 12.39/6.86 = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_378_right__diff__distrib,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.86 ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 12.39/6.86 = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_379_right__diff__distrib,axiom,
% 12.39/6.86 ! [A: int,B2: int,C: int] :
% 12.39/6.86 ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 12.39/6.86 = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_380_left__diff__distrib_H,axiom,
% 12.39/6.86 ! [B2: real,C: real,A: real] :
% 12.39/6.86 ( ( times_times_real @ ( minus_minus_real @ B2 @ C ) @ A )
% 12.39/6.86 = ( minus_minus_real @ ( times_times_real @ B2 @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_381_left__diff__distrib_H,axiom,
% 12.39/6.86 ! [B2: rat,C: rat,A: rat] :
% 12.39/6.86 ( ( times_times_rat @ ( minus_minus_rat @ B2 @ C ) @ A )
% 12.39/6.86 = ( minus_minus_rat @ ( times_times_rat @ B2 @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_382_left__diff__distrib_H,axiom,
% 12.39/6.86 ! [B2: nat,C: nat,A: nat] :
% 12.39/6.86 ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A )
% 12.39/6.86 = ( minus_minus_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_383_left__diff__distrib_H,axiom,
% 12.39/6.86 ! [B2: int,C: int,A: int] :
% 12.39/6.86 ( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A )
% 12.39/6.86 = ( minus_minus_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_384_right__diff__distrib_H,axiom,
% 12.39/6.86 ! [A: real,B2: real,C: real] :
% 12.39/6.86 ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 12.39/6.86 = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_385_right__diff__distrib_H,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.86 ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 12.39/6.86 = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_386_right__diff__distrib_H,axiom,
% 12.39/6.86 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.86 ( ( times_times_nat @ A @ ( minus_minus_nat @ B2 @ C ) )
% 12.39/6.86 = ( minus_minus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_387_right__diff__distrib_H,axiom,
% 12.39/6.86 ! [A: int,B2: int,C: int] :
% 12.39/6.86 ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 12.39/6.86 = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_388_Nat_OlessE,axiom,
% 12.39/6.86 ! [I: nat,K: nat] :
% 12.39/6.86 ( ( ord_less_nat @ I @ K )
% 12.39/6.86 => ( ( K
% 12.39/6.86 != ( suc @ I ) )
% 12.39/6.86 => ~ ! [J2: nat] :
% 12.39/6.86 ( ( ord_less_nat @ I @ J2 )
% 12.39/6.86 => ( K
% 12.39/6.86 != ( suc @ J2 ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_389_Suc__lessD,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 12.39/6.86 => ( ord_less_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_390_Suc__lessE,axiom,
% 12.39/6.86 ! [I: nat,K: nat] :
% 12.39/6.86 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 12.39/6.86 => ~ ! [J2: nat] :
% 12.39/6.86 ( ( ord_less_nat @ I @ J2 )
% 12.39/6.86 => ( K
% 12.39/6.86 != ( suc @ J2 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_391_Suc__lessI,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ M @ N )
% 12.39/6.86 => ( ( ( suc @ M )
% 12.39/6.86 != N )
% 12.39/6.86 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_392_less__SucE,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 12.39/6.86 => ( ~ ( ord_less_nat @ M @ N )
% 12.39/6.86 => ( M = N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_393_less__SucI,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ M @ N )
% 12.39/6.86 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_394_Ex__less__Suc,axiom,
% 12.39/6.86 ! [N: nat,P: nat > $o] :
% 12.39/6.86 ( ( ? [I2: nat] :
% 12.39/6.86 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 12.39/6.86 & ( P @ I2 ) ) )
% 12.39/6.86 = ( ( P @ N )
% 12.39/6.86 | ? [I2: nat] :
% 12.39/6.86 ( ( ord_less_nat @ I2 @ N )
% 12.39/6.86 & ( P @ I2 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_395_less__Suc__eq,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 12.39/6.86 = ( ( ord_less_nat @ M @ N )
% 12.39/6.86 | ( M = N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_396_not__less__eq,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ~ ( ord_less_nat @ M @ N ) )
% 12.39/6.86 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_397_All__less__Suc,axiom,
% 12.39/6.86 ! [N: nat,P: nat > $o] :
% 12.39/6.86 ( ( ! [I2: nat] :
% 12.39/6.86 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 12.39/6.86 => ( P @ I2 ) ) )
% 12.39/6.86 = ( ( P @ N )
% 12.39/6.86 & ! [I2: nat] :
% 12.39/6.86 ( ( ord_less_nat @ I2 @ N )
% 12.39/6.86 => ( P @ I2 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_398_Suc__less__eq2,axiom,
% 12.39/6.86 ! [N: nat,M: nat] :
% 12.39/6.86 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 12.39/6.86 = ( ? [M2: nat] :
% 12.39/6.86 ( ( M
% 12.39/6.86 = ( suc @ M2 ) )
% 12.39/6.86 & ( ord_less_nat @ N @ M2 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_399_less__antisym,axiom,
% 12.39/6.86 ! [N: nat,M: nat] :
% 12.39/6.86 ( ~ ( ord_less_nat @ N @ M )
% 12.39/6.86 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 12.39/6.86 => ( M = N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_400_Suc__less__SucD,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.39/6.86 => ( ord_less_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_401_less__trans__Suc,axiom,
% 12.39/6.86 ! [I: nat,J: nat,K: nat] :
% 12.39/6.86 ( ( ord_less_nat @ I @ J )
% 12.39/6.86 => ( ( ord_less_nat @ J @ K )
% 12.39/6.86 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_402_less__Suc__induct,axiom,
% 12.39/6.86 ! [I: nat,J: nat,P: nat > nat > $o] :
% 12.39/6.86 ( ( ord_less_nat @ I @ J )
% 12.39/6.86 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 12.39/6.86 => ( ! [I3: nat,J2: nat,K2: nat] :
% 12.39/6.86 ( ( ord_less_nat @ I3 @ J2 )
% 12.39/6.86 => ( ( ord_less_nat @ J2 @ K2 )
% 12.39/6.86 => ( ( P @ I3 @ J2 )
% 12.39/6.86 => ( ( P @ J2 @ K2 )
% 12.39/6.86 => ( P @ I3 @ K2 ) ) ) ) )
% 12.39/6.86 => ( P @ I @ J ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_403_strict__inc__induct,axiom,
% 12.39/6.86 ! [I: nat,J: nat,P: nat > $o] :
% 12.39/6.86 ( ( ord_less_nat @ I @ J )
% 12.39/6.86 => ( ! [I3: nat] :
% 12.39/6.86 ( ( J
% 12.39/6.86 = ( suc @ I3 ) )
% 12.39/6.86 => ( P @ I3 ) )
% 12.39/6.86 => ( ! [I3: nat] :
% 12.39/6.86 ( ( ord_less_nat @ I3 @ J )
% 12.39/6.86 => ( ( P @ ( suc @ I3 ) )
% 12.39/6.86 => ( P @ I3 ) ) )
% 12.39/6.86 => ( P @ I ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_404_not__less__less__Suc__eq,axiom,
% 12.39/6.86 ! [N: nat,M: nat] :
% 12.39/6.86 ( ~ ( ord_less_nat @ N @ M )
% 12.39/6.86 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 12.39/6.86 = ( N = M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_405_Suc__leD,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 12.39/6.86 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_406_le__SucE,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.39/6.86 => ( ~ ( ord_less_eq_nat @ M @ N )
% 12.39/6.86 => ( M
% 12.39/6.86 = ( suc @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_407_le__SucI,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.86 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_408_Suc__le__D,axiom,
% 12.39/6.86 ! [N: nat,M3: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
% 12.39/6.86 => ? [M4: nat] :
% 12.39/6.86 ( M3
% 12.39/6.86 = ( suc @ M4 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_409_le__Suc__eq,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.39/6.86 = ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.86 | ( M
% 12.39/6.86 = ( suc @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_410_Suc__n__not__le__n,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_411_not__less__eq__eq,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 12.39/6.86 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_412_full__nat__induct,axiom,
% 12.39/6.86 ! [P: nat > $o,N: nat] :
% 12.39/6.86 ( ! [N4: nat] :
% 12.39/6.86 ( ! [M5: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
% 12.39/6.86 => ( P @ M5 ) )
% 12.39/6.86 => ( P @ N4 ) )
% 12.39/6.86 => ( P @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_413_nat__induct__at__least,axiom,
% 12.39/6.86 ! [M: nat,N: nat,P: nat > $o] :
% 12.39/6.86 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.86 => ( ( P @ M )
% 12.39/6.86 => ( ! [N4: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ M @ N4 )
% 12.39/6.86 => ( ( P @ N4 )
% 12.39/6.86 => ( P @ ( suc @ N4 ) ) ) )
% 12.39/6.86 => ( P @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_414_transitive__stepwise__le,axiom,
% 12.39/6.86 ! [M: nat,N: nat,R: nat > nat > $o] :
% 12.39/6.86 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.86 => ( ! [X3: nat] : ( R @ X3 @ X3 )
% 12.39/6.86 => ( ! [X3: nat,Y4: nat,Z2: nat] :
% 12.39/6.86 ( ( R @ X3 @ Y4 )
% 12.39/6.86 => ( ( R @ Y4 @ Z2 )
% 12.39/6.86 => ( R @ X3 @ Z2 ) ) )
% 12.39/6.86 => ( ! [N4: nat] : ( R @ N4 @ ( suc @ N4 ) )
% 12.39/6.86 => ( R @ M @ N ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_415_le__cube,axiom,
% 12.39/6.86 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_416_le__square,axiom,
% 12.39/6.86 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_417_mult__le__mono,axiom,
% 12.39/6.86 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.86 => ( ( ord_less_eq_nat @ K @ L2 )
% 12.39/6.86 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_418_mult__le__mono1,axiom,
% 12.39/6.86 ! [I: nat,J: nat,K: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.86 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_419_mult__le__mono2,axiom,
% 12.39/6.86 ! [I: nat,J: nat,K: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.86 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_420_nat__arith_Osuc1,axiom,
% 12.39/6.86 ! [A3: nat,K: nat,A: nat] :
% 12.39/6.86 ( ( A3
% 12.39/6.86 = ( plus_plus_nat @ K @ A ) )
% 12.39/6.86 => ( ( suc @ A3 )
% 12.39/6.86 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_421_add__Suc,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 12.39/6.86 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_422_add__Suc__shift,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 12.39/6.86 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_423_add__mult__distrib,axiom,
% 12.39/6.86 ! [M: nat,N: nat,K: nat] :
% 12.39/6.86 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 12.39/6.86 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_424_add__mult__distrib2,axiom,
% 12.39/6.86 ! [K: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 12.39/6.86 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_425_zero__induct__lemma,axiom,
% 12.39/6.86 ! [P: nat > $o,K: nat,I: nat] :
% 12.39/6.86 ( ( P @ K )
% 12.39/6.86 => ( ! [N4: nat] :
% 12.39/6.86 ( ( P @ ( suc @ N4 ) )
% 12.39/6.86 => ( P @ N4 ) )
% 12.39/6.86 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_426_diff__mult__distrib,axiom,
% 12.39/6.86 ! [M: nat,N: nat,K: nat] :
% 12.39/6.86 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 12.39/6.86 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_427_diff__mult__distrib2,axiom,
% 12.39/6.86 ! [K: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 12.39/6.86 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_428_nat__mult__1,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ( ( times_times_nat @ one_one_nat @ N )
% 12.39/6.86 = N ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_429_nat__mult__1__right,axiom,
% 12.39/6.86 ! [N: nat] :
% 12.39/6.86 ( ( times_times_nat @ N @ one_one_nat )
% 12.39/6.86 = N ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_430_two__realpow__ge__one,axiom,
% 12.39/6.86 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_431_add__One__commute,axiom,
% 12.39/6.86 ! [N: num] :
% 12.39/6.86 ( ( plus_plus_num @ one @ N )
% 12.39/6.86 = ( plus_plus_num @ N @ one ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_432_power__commutes,axiom,
% 12.39/6.86 ! [A: complex,N: nat] :
% 12.39/6.86 ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 12.39/6.86 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_433_power__commutes,axiom,
% 12.39/6.86 ! [A: real,N: nat] :
% 12.39/6.86 ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 12.39/6.86 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_434_power__commutes,axiom,
% 12.39/6.86 ! [A: rat,N: nat] :
% 12.39/6.86 ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 12.39/6.86 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_435_power__commutes,axiom,
% 12.39/6.86 ! [A: nat,N: nat] :
% 12.39/6.86 ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 12.39/6.86 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_436_power__commutes,axiom,
% 12.39/6.86 ! [A: int,N: nat] :
% 12.39/6.86 ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 12.39/6.86 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_437_power__mult__distrib,axiom,
% 12.39/6.86 ! [A: complex,B2: complex,N: nat] :
% 12.39/6.86 ( ( power_power_complex @ ( times_times_complex @ A @ B2 ) @ N )
% 12.39/6.86 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_438_power__mult__distrib,axiom,
% 12.39/6.86 ! [A: real,B2: real,N: nat] :
% 12.39/6.86 ( ( power_power_real @ ( times_times_real @ A @ B2 ) @ N )
% 12.39/6.86 = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_439_power__mult__distrib,axiom,
% 12.39/6.86 ! [A: rat,B2: rat,N: nat] :
% 12.39/6.86 ( ( power_power_rat @ ( times_times_rat @ A @ B2 ) @ N )
% 12.39/6.86 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_440_power__mult__distrib,axiom,
% 12.39/6.86 ! [A: nat,B2: nat,N: nat] :
% 12.39/6.86 ( ( power_power_nat @ ( times_times_nat @ A @ B2 ) @ N )
% 12.39/6.86 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_441_power__mult__distrib,axiom,
% 12.39/6.86 ! [A: int,B2: int,N: nat] :
% 12.39/6.86 ( ( power_power_int @ ( times_times_int @ A @ B2 ) @ N )
% 12.39/6.86 = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_442_power__commuting__commutes,axiom,
% 12.39/6.86 ! [X4: complex,Y3: complex,N: nat] :
% 12.39/6.86 ( ( ( times_times_complex @ X4 @ Y3 )
% 12.39/6.86 = ( times_times_complex @ Y3 @ X4 ) )
% 12.39/6.86 => ( ( times_times_complex @ ( power_power_complex @ X4 @ N ) @ Y3 )
% 12.39/6.86 = ( times_times_complex @ Y3 @ ( power_power_complex @ X4 @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_443_power__commuting__commutes,axiom,
% 12.39/6.86 ! [X4: real,Y3: real,N: nat] :
% 12.39/6.86 ( ( ( times_times_real @ X4 @ Y3 )
% 12.39/6.86 = ( times_times_real @ Y3 @ X4 ) )
% 12.39/6.86 => ( ( times_times_real @ ( power_power_real @ X4 @ N ) @ Y3 )
% 12.39/6.86 = ( times_times_real @ Y3 @ ( power_power_real @ X4 @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_444_power__commuting__commutes,axiom,
% 12.39/6.86 ! [X4: rat,Y3: rat,N: nat] :
% 12.39/6.86 ( ( ( times_times_rat @ X4 @ Y3 )
% 12.39/6.86 = ( times_times_rat @ Y3 @ X4 ) )
% 12.39/6.86 => ( ( times_times_rat @ ( power_power_rat @ X4 @ N ) @ Y3 )
% 12.39/6.86 = ( times_times_rat @ Y3 @ ( power_power_rat @ X4 @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_445_power__commuting__commutes,axiom,
% 12.39/6.86 ! [X4: nat,Y3: nat,N: nat] :
% 12.39/6.86 ( ( ( times_times_nat @ X4 @ Y3 )
% 12.39/6.86 = ( times_times_nat @ Y3 @ X4 ) )
% 12.39/6.86 => ( ( times_times_nat @ ( power_power_nat @ X4 @ N ) @ Y3 )
% 12.39/6.86 = ( times_times_nat @ Y3 @ ( power_power_nat @ X4 @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_446_power__commuting__commutes,axiom,
% 12.39/6.86 ! [X4: int,Y3: int,N: nat] :
% 12.39/6.86 ( ( ( times_times_int @ X4 @ Y3 )
% 12.39/6.86 = ( times_times_int @ Y3 @ X4 ) )
% 12.39/6.86 => ( ( times_times_int @ ( power_power_int @ X4 @ N ) @ Y3 )
% 12.39/6.86 = ( times_times_int @ Y3 @ ( power_power_int @ X4 @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_447_power__mult,axiom,
% 12.39/6.86 ! [A: nat,M: nat,N: nat] :
% 12.39/6.86 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 12.39/6.86 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_448_power__mult,axiom,
% 12.39/6.86 ! [A: real,M: nat,N: nat] :
% 12.39/6.86 ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 12.39/6.86 = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_449_power__mult,axiom,
% 12.39/6.86 ! [A: int,M: nat,N: nat] :
% 12.39/6.86 ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 12.39/6.86 = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_450_power__mult,axiom,
% 12.39/6.86 ! [A: complex,M: nat,N: nat] :
% 12.39/6.86 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 12.39/6.86 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_451_left__add__mult__distrib,axiom,
% 12.39/6.86 ! [I: nat,U: nat,J: nat,K: nat] :
% 12.39/6.86 ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 12.39/6.86 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_452_div__mult2__eq,axiom,
% 12.39/6.86 ! [M: nat,N: nat,Q2: nat] :
% 12.39/6.86 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 12.39/6.86 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_453_power__odd__eq,axiom,
% 12.39/6.86 ! [A: complex,N: nat] :
% 12.39/6.86 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.39/6.86 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_454_power__odd__eq,axiom,
% 12.39/6.86 ! [A: real,N: nat] :
% 12.39/6.86 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.39/6.86 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_455_power__odd__eq,axiom,
% 12.39/6.86 ! [A: rat,N: nat] :
% 12.39/6.86 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.39/6.86 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_456_power__odd__eq,axiom,
% 12.39/6.86 ! [A: nat,N: nat] :
% 12.39/6.86 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.39/6.86 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_457_power__odd__eq,axiom,
% 12.39/6.86 ! [A: int,N: nat] :
% 12.39/6.86 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.39/6.86 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_458_less__1__mult,axiom,
% 12.39/6.86 ! [M: real,N: real] :
% 12.39/6.86 ( ( ord_less_real @ one_one_real @ M )
% 12.39/6.86 => ( ( ord_less_real @ one_one_real @ N )
% 12.39/6.86 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_459_less__1__mult,axiom,
% 12.39/6.86 ! [M: rat,N: rat] :
% 12.39/6.86 ( ( ord_less_rat @ one_one_rat @ M )
% 12.39/6.86 => ( ( ord_less_rat @ one_one_rat @ N )
% 12.39/6.86 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_460_less__1__mult,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ one_one_nat @ M )
% 12.39/6.86 => ( ( ord_less_nat @ one_one_nat @ N )
% 12.39/6.86 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_461_less__1__mult,axiom,
% 12.39/6.86 ! [M: int,N: int] :
% 12.39/6.86 ( ( ord_less_int @ one_one_int @ M )
% 12.39/6.86 => ( ( ord_less_int @ one_one_int @ N )
% 12.39/6.86 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_462_eq__add__iff1,axiom,
% 12.39/6.86 ! [A: real,E: real,C: real,B2: real,D: real] :
% 12.39/6.86 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 12.39/6.86 = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C )
% 12.39/6.86 = D ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_463_eq__add__iff1,axiom,
% 12.39/6.86 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 12.39/6.86 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 12.39/6.86 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C )
% 12.39/6.86 = D ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_464_eq__add__iff1,axiom,
% 12.39/6.86 ! [A: int,E: int,C: int,B2: int,D: int] :
% 12.39/6.86 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 12.39/6.86 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C )
% 12.39/6.86 = D ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_465_eq__add__iff2,axiom,
% 12.39/6.86 ! [A: real,E: real,C: real,B2: real,D: real] :
% 12.39/6.86 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 12.39/6.86 = ( C
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_466_eq__add__iff2,axiom,
% 12.39/6.86 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 12.39/6.86 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 12.39/6.86 = ( C
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_467_eq__add__iff2,axiom,
% 12.39/6.86 ! [A: int,E: int,C: int,B2: int,D: int] :
% 12.39/6.86 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 12.39/6.86 = ( C
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_468_square__diff__square__factored,axiom,
% 12.39/6.86 ! [X4: real,Y3: real] :
% 12.39/6.86 ( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 12.39/6.86 = ( times_times_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( minus_minus_real @ X4 @ Y3 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_469_square__diff__square__factored,axiom,
% 12.39/6.86 ! [X4: rat,Y3: rat] :
% 12.39/6.86 ( ( minus_minus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 12.39/6.86 = ( times_times_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( minus_minus_rat @ X4 @ Y3 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_470_square__diff__square__factored,axiom,
% 12.39/6.86 ! [X4: int,Y3: int] :
% 12.39/6.86 ( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 12.39/6.86 = ( times_times_int @ ( plus_plus_int @ X4 @ Y3 ) @ ( minus_minus_int @ X4 @ Y3 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_471_mult__diff__mult,axiom,
% 12.39/6.86 ! [X4: real,Y3: real,A: real,B2: real] :
% 12.39/6.86 ( ( minus_minus_real @ ( times_times_real @ X4 @ Y3 ) @ ( times_times_real @ A @ B2 ) )
% 12.39/6.86 = ( plus_plus_real @ ( times_times_real @ X4 @ ( minus_minus_real @ Y3 @ B2 ) ) @ ( times_times_real @ ( minus_minus_real @ X4 @ A ) @ B2 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_472_mult__diff__mult,axiom,
% 12.39/6.86 ! [X4: rat,Y3: rat,A: rat,B2: rat] :
% 12.39/6.86 ( ( minus_minus_rat @ ( times_times_rat @ X4 @ Y3 ) @ ( times_times_rat @ A @ B2 ) )
% 12.39/6.86 = ( plus_plus_rat @ ( times_times_rat @ X4 @ ( minus_minus_rat @ Y3 @ B2 ) ) @ ( times_times_rat @ ( minus_minus_rat @ X4 @ A ) @ B2 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_473_mult__diff__mult,axiom,
% 12.39/6.86 ! [X4: int,Y3: int,A: int,B2: int] :
% 12.39/6.86 ( ( minus_minus_int @ ( times_times_int @ X4 @ Y3 ) @ ( times_times_int @ A @ B2 ) )
% 12.39/6.86 = ( plus_plus_int @ ( times_times_int @ X4 @ ( minus_minus_int @ Y3 @ B2 ) ) @ ( times_times_int @ ( minus_minus_int @ X4 @ A ) @ B2 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_474_lift__Suc__mono__less,axiom,
% 12.39/6.86 ! [F: nat > real,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_475_lift__Suc__mono__less,axiom,
% 12.39/6.86 ! [F: nat > rat,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_476_lift__Suc__mono__less,axiom,
% 12.39/6.86 ! [F: nat > num,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_477_lift__Suc__mono__less,axiom,
% 12.39/6.86 ! [F: nat > nat,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_478_lift__Suc__mono__less,axiom,
% 12.39/6.86 ! [F: nat > int,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_479_lift__Suc__mono__less__iff,axiom,
% 12.39/6.86 ! [F: nat > real,N: nat,M: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 12.39/6.86 = ( ord_less_nat @ N @ M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_480_lift__Suc__mono__less__iff,axiom,
% 12.39/6.86 ! [F: nat > rat,N: nat,M: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 12.39/6.86 = ( ord_less_nat @ N @ M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_481_lift__Suc__mono__less__iff,axiom,
% 12.39/6.86 ! [F: nat > num,N: nat,M: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 12.39/6.86 = ( ord_less_nat @ N @ M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_482_lift__Suc__mono__less__iff,axiom,
% 12.39/6.86 ! [F: nat > nat,N: nat,M: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 12.39/6.86 = ( ord_less_nat @ N @ M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_483_lift__Suc__mono__less__iff,axiom,
% 12.39/6.86 ! [F: nat > int,N: nat,M: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 12.39/6.86 = ( ord_less_nat @ N @ M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_484_lift__Suc__mono__le,axiom,
% 12.39/6.86 ! [F: nat > set_nat,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_485_lift__Suc__mono__le,axiom,
% 12.39/6.86 ! [F: nat > rat,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_486_lift__Suc__mono__le,axiom,
% 12.39/6.86 ! [F: nat > num,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_487_lift__Suc__mono__le,axiom,
% 12.39/6.86 ! [F: nat > nat,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_488_lift__Suc__mono__le,axiom,
% 12.39/6.86 ! [F: nat > int,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.39/6.86 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_489_lift__Suc__antimono__le,axiom,
% 12.39/6.86 ! [F: nat > set_nat,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 12.39/6.86 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_490_lift__Suc__antimono__le,axiom,
% 12.39/6.86 ! [F: nat > rat,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 12.39/6.86 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_491_lift__Suc__antimono__le,axiom,
% 12.39/6.86 ! [F: nat > num,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 12.39/6.86 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_492_lift__Suc__antimono__le,axiom,
% 12.39/6.86 ! [F: nat > nat,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 12.39/6.86 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_493_lift__Suc__antimono__le,axiom,
% 12.39/6.86 ! [F: nat > int,N: nat,N5: nat] :
% 12.39/6.86 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 12.39/6.86 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.39/6.86 => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_494_Suc__leI,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ M @ N )
% 12.39/6.86 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_495_Suc__le__eq,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 12.39/6.86 = ( ord_less_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_496_dec__induct,axiom,
% 12.39/6.86 ! [I: nat,J: nat,P: nat > $o] :
% 12.39/6.86 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.86 => ( ( P @ I )
% 12.39/6.86 => ( ! [N4: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ I @ N4 )
% 12.39/6.86 => ( ( ord_less_nat @ N4 @ J )
% 12.39/6.86 => ( ( P @ N4 )
% 12.39/6.86 => ( P @ ( suc @ N4 ) ) ) ) )
% 12.39/6.86 => ( P @ J ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_497_inc__induct,axiom,
% 12.39/6.86 ! [I: nat,J: nat,P: nat > $o] :
% 12.39/6.86 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.86 => ( ( P @ J )
% 12.39/6.86 => ( ! [N4: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ I @ N4 )
% 12.39/6.86 => ( ( ord_less_nat @ N4 @ J )
% 12.39/6.86 => ( ( P @ ( suc @ N4 ) )
% 12.39/6.86 => ( P @ N4 ) ) ) )
% 12.39/6.86 => ( P @ I ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_498_Suc__le__lessD,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 12.39/6.86 => ( ord_less_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_499_le__less__Suc__eq,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.86 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 12.39/6.86 = ( N = M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_500_less__Suc__eq__le,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 12.39/6.86 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_501_less__eq__Suc__le,axiom,
% 12.39/6.86 ( ord_less_nat
% 12.39/6.86 = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_502_le__imp__less__Suc,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.86 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_503_less__imp__Suc__add,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ M @ N )
% 12.39/6.86 => ? [K2: nat] :
% 12.39/6.86 ( N
% 12.39/6.86 = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_504_less__iff__Suc__add,axiom,
% 12.39/6.86 ( ord_less_nat
% 12.39/6.86 = ( ^ [M6: nat,N2: nat] :
% 12.39/6.86 ? [K3: nat] :
% 12.39/6.86 ( N2
% 12.39/6.86 = ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_505_less__add__Suc2,axiom,
% 12.39/6.86 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_506_less__add__Suc1,axiom,
% 12.39/6.86 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_507_less__natE,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( ord_less_nat @ M @ N )
% 12.39/6.86 => ~ ! [Q3: nat] :
% 12.39/6.86 ( N
% 12.39/6.86 != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_508_diff__less__Suc,axiom,
% 12.39/6.86 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_509_Suc__diff__Suc,axiom,
% 12.39/6.86 ! [N: nat,M: nat] :
% 12.39/6.86 ( ( ord_less_nat @ N @ M )
% 12.39/6.86 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 12.39/6.86 = ( minus_minus_nat @ M @ N ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_510_Suc__diff__le,axiom,
% 12.39/6.86 ! [N: nat,M: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ N @ M )
% 12.39/6.86 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 12.39/6.86 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_511_Suc__eq__plus1,axiom,
% 12.39/6.86 ( suc
% 12.39/6.86 = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_512_plus__1__eq__Suc,axiom,
% 12.39/6.86 ( ( plus_plus_nat @ one_one_nat )
% 12.39/6.86 = suc ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_513_Suc__eq__plus1__left,axiom,
% 12.39/6.86 ( suc
% 12.39/6.86 = ( plus_plus_nat @ one_one_nat ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_514_diff__Suc__eq__diff__pred,axiom,
% 12.39/6.86 ! [M: nat,N: nat] :
% 12.39/6.86 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 12.39/6.86 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_515_Suc__nat__number__of__add,axiom,
% 12.39/6.86 ! [V: num,N: nat] :
% 12.39/6.86 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 12.39/6.86 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_516_div__nat__eqI,axiom,
% 12.39/6.86 ! [N: nat,Q2: nat,M: nat] :
% 12.39/6.86 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 12.39/6.86 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 12.39/6.86 => ( ( divide_divide_nat @ M @ N )
% 12.39/6.86 = Q2 ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_517_ordered__ring__class_Ole__add__iff2,axiom,
% 12.39/6.86 ! [A: real,E: real,C: real,B2: real,D: real] :
% 12.39/6.86 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 12.39/6.86 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 12.39/6.86
% 12.39/6.86 thf(fact_518_ordered__ring__class_Ole__add__iff2,axiom,
% 12.39/6.87 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 12.39/6.87 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_519_ordered__ring__class_Ole__add__iff2,axiom,
% 12.39/6.87 ! [A: int,E: int,C: int,B2: int,D: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 12.39/6.87 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_520_ordered__ring__class_Ole__add__iff1,axiom,
% 12.39/6.87 ! [A: real,E: real,C: real,B2: real,D: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 12.39/6.87 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_521_ordered__ring__class_Ole__add__iff1,axiom,
% 12.39/6.87 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 12.39/6.87 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_522_ordered__ring__class_Ole__add__iff1,axiom,
% 12.39/6.87 ! [A: int,E: int,C: int,B2: int,D: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 12.39/6.87 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_523_less__add__iff2,axiom,
% 12.39/6.87 ! [A: real,E: real,C: real,B2: real,D: real] :
% 12.39/6.87 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 12.39/6.87 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_524_less__add__iff2,axiom,
% 12.39/6.87 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 12.39/6.87 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 12.39/6.87 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_525_less__add__iff2,axiom,
% 12.39/6.87 ! [A: int,E: int,C: int,B2: int,D: int] :
% 12.39/6.87 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 12.39/6.87 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_526_less__add__iff1,axiom,
% 12.39/6.87 ! [A: real,E: real,C: real,B2: real,D: real] :
% 12.39/6.87 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 12.39/6.87 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_527_less__add__iff1,axiom,
% 12.39/6.87 ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 12.39/6.87 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 12.39/6.87 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_528_less__add__iff1,axiom,
% 12.39/6.87 ! [A: int,E: int,C: int,B2: int,D: int] :
% 12.39/6.87 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 12.39/6.87 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_529_square__diff__one__factored,axiom,
% 12.39/6.87 ! [X4: complex] :
% 12.39/6.87 ( ( minus_minus_complex @ ( times_times_complex @ X4 @ X4 ) @ one_one_complex )
% 12.39/6.87 = ( times_times_complex @ ( plus_plus_complex @ X4 @ one_one_complex ) @ ( minus_minus_complex @ X4 @ one_one_complex ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_530_square__diff__one__factored,axiom,
% 12.39/6.87 ! [X4: real] :
% 12.39/6.87 ( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ one_one_real )
% 12.39/6.87 = ( times_times_real @ ( plus_plus_real @ X4 @ one_one_real ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_531_square__diff__one__factored,axiom,
% 12.39/6.87 ! [X4: rat] :
% 12.39/6.87 ( ( minus_minus_rat @ ( times_times_rat @ X4 @ X4 ) @ one_one_rat )
% 12.39/6.87 = ( times_times_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) @ ( minus_minus_rat @ X4 @ one_one_rat ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_532_square__diff__one__factored,axiom,
% 12.39/6.87 ! [X4: int] :
% 12.39/6.87 ( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ one_one_int )
% 12.39/6.87 = ( times_times_int @ ( plus_plus_int @ X4 @ one_one_int ) @ ( minus_minus_int @ X4 @ one_one_int ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_533_mult__numeral__1__right,axiom,
% 12.39/6.87 ! [A: complex] :
% 12.39/6.87 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_534_mult__numeral__1__right,axiom,
% 12.39/6.87 ! [A: real] :
% 12.39/6.87 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_535_mult__numeral__1__right,axiom,
% 12.39/6.87 ! [A: rat] :
% 12.39/6.87 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_536_mult__numeral__1__right,axiom,
% 12.39/6.87 ! [A: nat] :
% 12.39/6.87 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_537_mult__numeral__1__right,axiom,
% 12.39/6.87 ! [A: int] :
% 12.39/6.87 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_538_mult__numeral__1,axiom,
% 12.39/6.87 ! [A: complex] :
% 12.39/6.87 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_539_mult__numeral__1,axiom,
% 12.39/6.87 ! [A: real] :
% 12.39/6.87 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_540_mult__numeral__1,axiom,
% 12.39/6.87 ! [A: rat] :
% 12.39/6.87 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_541_mult__numeral__1,axiom,
% 12.39/6.87 ! [A: nat] :
% 12.39/6.87 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_542_mult__numeral__1,axiom,
% 12.39/6.87 ! [A: int] :
% 12.39/6.87 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_543_left__right__inverse__power,axiom,
% 12.39/6.87 ! [X4: complex,Y3: complex,N: nat] :
% 12.39/6.87 ( ( ( times_times_complex @ X4 @ Y3 )
% 12.39/6.87 = one_one_complex )
% 12.39/6.87 => ( ( times_times_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 12.39/6.87 = one_one_complex ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_544_left__right__inverse__power,axiom,
% 12.39/6.87 ! [X4: real,Y3: real,N: nat] :
% 12.39/6.87 ( ( ( times_times_real @ X4 @ Y3 )
% 12.39/6.87 = one_one_real )
% 12.39/6.87 => ( ( times_times_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 12.39/6.87 = one_one_real ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_545_left__right__inverse__power,axiom,
% 12.39/6.87 ! [X4: rat,Y3: rat,N: nat] :
% 12.39/6.87 ( ( ( times_times_rat @ X4 @ Y3 )
% 12.39/6.87 = one_one_rat )
% 12.39/6.87 => ( ( times_times_rat @ ( power_power_rat @ X4 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 12.39/6.87 = one_one_rat ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_546_left__right__inverse__power,axiom,
% 12.39/6.87 ! [X4: nat,Y3: nat,N: nat] :
% 12.39/6.87 ( ( ( times_times_nat @ X4 @ Y3 )
% 12.39/6.87 = one_one_nat )
% 12.39/6.87 => ( ( times_times_nat @ ( power_power_nat @ X4 @ N ) @ ( power_power_nat @ Y3 @ N ) )
% 12.39/6.87 = one_one_nat ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_547_left__right__inverse__power,axiom,
% 12.39/6.87 ! [X4: int,Y3: int,N: nat] :
% 12.39/6.87 ( ( ( times_times_int @ X4 @ Y3 )
% 12.39/6.87 = one_one_int )
% 12.39/6.87 => ( ( times_times_int @ ( power_power_int @ X4 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 12.39/6.87 = one_one_int ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_548_power__add,axiom,
% 12.39/6.87 ! [A: complex,M: nat,N: nat] :
% 12.39/6.87 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 12.39/6.87 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_549_power__add,axiom,
% 12.39/6.87 ! [A: real,M: nat,N: nat] :
% 12.39/6.87 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 12.39/6.87 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_550_power__add,axiom,
% 12.39/6.87 ! [A: rat,M: nat,N: nat] :
% 12.39/6.87 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 12.39/6.87 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_551_power__add,axiom,
% 12.39/6.87 ! [A: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 12.39/6.87 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_552_power__add,axiom,
% 12.39/6.87 ! [A: int,M: nat,N: nat] :
% 12.39/6.87 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 12.39/6.87 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_553_Suc__div__le__mono,axiom,
% 12.39/6.87 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_554_less__mult__imp__div__less,axiom,
% 12.39/6.87 ! [M: nat,I: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 12.39/6.87 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_555_times__div__less__eq__dividend,axiom,
% 12.39/6.87 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_556_div__times__less__eq__dividend,axiom,
% 12.39/6.87 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_557_power__gt1__lemma,axiom,
% 12.39/6.87 ! [A: real,N: nat] :
% 12.39/6.87 ( ( ord_less_real @ one_one_real @ A )
% 12.39/6.87 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_558_power__gt1__lemma,axiom,
% 12.39/6.87 ! [A: rat,N: nat] :
% 12.39/6.87 ( ( ord_less_rat @ one_one_rat @ A )
% 12.39/6.87 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_559_power__gt1__lemma,axiom,
% 12.39/6.87 ! [A: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_nat @ one_one_nat @ A )
% 12.39/6.87 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_560_power__gt1__lemma,axiom,
% 12.39/6.87 ! [A: int,N: nat] :
% 12.39/6.87 ( ( ord_less_int @ one_one_int @ A )
% 12.39/6.87 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_561_power__less__power__Suc,axiom,
% 12.39/6.87 ! [A: real,N: nat] :
% 12.39/6.87 ( ( ord_less_real @ one_one_real @ A )
% 12.39/6.87 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_562_power__less__power__Suc,axiom,
% 12.39/6.87 ! [A: rat,N: nat] :
% 12.39/6.87 ( ( ord_less_rat @ one_one_rat @ A )
% 12.39/6.87 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_563_power__less__power__Suc,axiom,
% 12.39/6.87 ! [A: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_nat @ one_one_nat @ A )
% 12.39/6.87 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_564_power__less__power__Suc,axiom,
% 12.39/6.87 ! [A: int,N: nat] :
% 12.39/6.87 ( ( ord_less_int @ one_one_int @ A )
% 12.39/6.87 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_565_power__gt1,axiom,
% 12.39/6.87 ! [A: real,N: nat] :
% 12.39/6.87 ( ( ord_less_real @ one_one_real @ A )
% 12.39/6.87 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_566_power__gt1,axiom,
% 12.39/6.87 ! [A: rat,N: nat] :
% 12.39/6.87 ( ( ord_less_rat @ one_one_rat @ A )
% 12.39/6.87 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_567_power__gt1,axiom,
% 12.39/6.87 ! [A: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_nat @ one_one_nat @ A )
% 12.39/6.87 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_568_power__gt1,axiom,
% 12.39/6.87 ! [A: int,N: nat] :
% 12.39/6.87 ( ( ord_less_int @ one_one_int @ A )
% 12.39/6.87 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_569_linorder__neqE__linordered__idom,axiom,
% 12.39/6.87 ! [X4: real,Y3: real] :
% 12.39/6.87 ( ( X4 != Y3 )
% 12.39/6.87 => ( ~ ( ord_less_real @ X4 @ Y3 )
% 12.39/6.87 => ( ord_less_real @ Y3 @ X4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_570_linorder__neqE__linordered__idom,axiom,
% 12.39/6.87 ! [X4: rat,Y3: rat] :
% 12.39/6.87 ( ( X4 != Y3 )
% 12.39/6.87 => ( ~ ( ord_less_rat @ X4 @ Y3 )
% 12.39/6.87 => ( ord_less_rat @ Y3 @ X4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_571_linorder__neqE__linordered__idom,axiom,
% 12.39/6.87 ! [X4: int,Y3: int] :
% 12.39/6.87 ( ( X4 != Y3 )
% 12.39/6.87 => ( ~ ( ord_less_int @ X4 @ Y3 )
% 12.39/6.87 => ( ord_less_int @ Y3 @ X4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_572_nat__diff__add__eq2,axiom,
% 12.39/6.87 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.39/6.87 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_573_nat__diff__add__eq1,axiom,
% 12.39/6.87 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ J @ I )
% 12.39/6.87 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.39/6.87 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_574_nat__le__add__iff2,axiom,
% 12.39/6.87 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.39/6.87 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_575_nat__le__add__iff1,axiom,
% 12.39/6.87 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ J @ I )
% 12.39/6.87 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.39/6.87 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_576_nat__eq__add__iff2,axiom,
% 12.39/6.87 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 12.39/6.87 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.39/6.87 = ( M
% 12.39/6.87 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_577_nat__eq__add__iff1,axiom,
% 12.39/6.87 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ J @ I )
% 12.39/6.87 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 12.39/6.87 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.39/6.87 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 12.39/6.87 = N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_578_nat__neq__iff,axiom,
% 12.39/6.87 ! [M: nat,N: nat] :
% 12.39/6.87 ( ( M != N )
% 12.39/6.87 = ( ( ord_less_nat @ M @ N )
% 12.39/6.87 | ( ord_less_nat @ N @ M ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_579_less__not__refl,axiom,
% 12.39/6.87 ! [N: nat] :
% 12.39/6.87 ~ ( ord_less_nat @ N @ N ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_580_less__not__refl2,axiom,
% 12.39/6.87 ! [N: nat,M: nat] :
% 12.39/6.87 ( ( ord_less_nat @ N @ M )
% 12.39/6.87 => ( M != N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_581_less__not__refl3,axiom,
% 12.39/6.87 ! [S: nat,T: nat] :
% 12.39/6.87 ( ( ord_less_nat @ S @ T )
% 12.39/6.87 => ( S != T ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_582_less__irrefl__nat,axiom,
% 12.39/6.87 ! [N: nat] :
% 12.39/6.87 ~ ( ord_less_nat @ N @ N ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_583_nat__less__induct,axiom,
% 12.39/6.87 ! [P: nat > $o,N: nat] :
% 12.39/6.87 ( ! [N4: nat] :
% 12.39/6.87 ( ! [M5: nat] :
% 12.39/6.87 ( ( ord_less_nat @ M5 @ N4 )
% 12.39/6.87 => ( P @ M5 ) )
% 12.39/6.87 => ( P @ N4 ) )
% 12.39/6.87 => ( P @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_584_infinite__descent,axiom,
% 12.39/6.87 ! [P: nat > $o,N: nat] :
% 12.39/6.87 ( ! [N4: nat] :
% 12.39/6.87 ( ~ ( P @ N4 )
% 12.39/6.87 => ? [M5: nat] :
% 12.39/6.87 ( ( ord_less_nat @ M5 @ N4 )
% 12.39/6.87 & ~ ( P @ M5 ) ) )
% 12.39/6.87 => ( P @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_585_linorder__neqE__nat,axiom,
% 12.39/6.87 ! [X4: nat,Y3: nat] :
% 12.39/6.87 ( ( X4 != Y3 )
% 12.39/6.87 => ( ~ ( ord_less_nat @ X4 @ Y3 )
% 12.39/6.87 => ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_586_le__refl,axiom,
% 12.39/6.87 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_587_le__trans,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 => ( ( ord_less_eq_nat @ J @ K )
% 12.39/6.87 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_588_eq__imp__le,axiom,
% 12.39/6.87 ! [M: nat,N: nat] :
% 12.39/6.87 ( ( M = N )
% 12.39/6.87 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_589_le__antisym,axiom,
% 12.39/6.87 ! [M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.87 => ( ( ord_less_eq_nat @ N @ M )
% 12.39/6.87 => ( M = N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_590_nat__le__linear,axiom,
% 12.39/6.87 ! [M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.87 | ( ord_less_eq_nat @ N @ M ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_591_Nat_Oex__has__greatest__nat,axiom,
% 12.39/6.87 ! [P: nat > $o,K: nat,B2: nat] :
% 12.39/6.87 ( ( P @ K )
% 12.39/6.87 => ( ! [Y4: nat] :
% 12.39/6.87 ( ( P @ Y4 )
% 12.39/6.87 => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 12.39/6.87 => ? [X3: nat] :
% 12.39/6.87 ( ( P @ X3 )
% 12.39/6.87 & ! [Y5: nat] :
% 12.39/6.87 ( ( P @ Y5 )
% 12.39/6.87 => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_592_diff__commute,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat] :
% 12.39/6.87 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 12.39/6.87 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_593_left__add__twice,axiom,
% 12.39/6.87 ! [A: complex,B2: complex] :
% 12.39/6.87 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B2 ) )
% 12.39/6.87 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_594_left__add__twice,axiom,
% 12.39/6.87 ! [A: real,B2: real] :
% 12.39/6.87 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 12.39/6.87 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_595_left__add__twice,axiom,
% 12.39/6.87 ! [A: rat,B2: rat] :
% 12.39/6.87 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 12.39/6.87 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_596_left__add__twice,axiom,
% 12.39/6.87 ! [A: nat,B2: nat] :
% 12.39/6.87 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 12.39/6.87 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_597_left__add__twice,axiom,
% 12.39/6.87 ! [A: int,B2: int] :
% 12.39/6.87 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 12.39/6.87 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_598_mult__2__right,axiom,
% 12.39/6.87 ! [Z: complex] :
% 12.39/6.87 ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( plus_plus_complex @ Z @ Z ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_599_mult__2__right,axiom,
% 12.39/6.87 ! [Z: real] :
% 12.39/6.87 ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( plus_plus_real @ Z @ Z ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_600_mult__2__right,axiom,
% 12.39/6.87 ! [Z: rat] :
% 12.39/6.87 ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( plus_plus_rat @ Z @ Z ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_601_mult__2__right,axiom,
% 12.39/6.87 ! [Z: nat] :
% 12.39/6.87 ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( plus_plus_nat @ Z @ Z ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_602_mult__2__right,axiom,
% 12.39/6.87 ! [Z: int] :
% 12.39/6.87 ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( plus_plus_int @ Z @ Z ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_603_mult__2,axiom,
% 12.39/6.87 ! [Z: complex] :
% 12.39/6.87 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 12.39/6.87 = ( plus_plus_complex @ Z @ Z ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_604_mult__2,axiom,
% 12.39/6.87 ! [Z: real] :
% 12.39/6.87 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 12.39/6.87 = ( plus_plus_real @ Z @ Z ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_605_mult__2,axiom,
% 12.39/6.87 ! [Z: rat] :
% 12.39/6.87 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 12.39/6.87 = ( plus_plus_rat @ Z @ Z ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_606_mult__2,axiom,
% 12.39/6.87 ! [Z: nat] :
% 12.39/6.87 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 12.39/6.87 = ( plus_plus_nat @ Z @ Z ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_607_mult__2,axiom,
% 12.39/6.87 ! [Z: int] :
% 12.39/6.87 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 12.39/6.87 = ( plus_plus_int @ Z @ Z ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_608_power2__eq__square,axiom,
% 12.39/6.87 ! [A: complex] :
% 12.39/6.87 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( times_times_complex @ A @ A ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_609_power2__eq__square,axiom,
% 12.39/6.87 ! [A: real] :
% 12.39/6.87 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( times_times_real @ A @ A ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_610_power2__eq__square,axiom,
% 12.39/6.87 ! [A: rat] :
% 12.39/6.87 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( times_times_rat @ A @ A ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_611_power2__eq__square,axiom,
% 12.39/6.87 ! [A: nat] :
% 12.39/6.87 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( times_times_nat @ A @ A ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_612_power2__eq__square,axiom,
% 12.39/6.87 ! [A: int] :
% 12.39/6.87 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( times_times_int @ A @ A ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_613_power4__eq__xxxx,axiom,
% 12.39/6.87 ! [X4: complex] :
% 12.39/6.87 ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.39/6.87 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_614_power4__eq__xxxx,axiom,
% 12.39/6.87 ! [X4: real] :
% 12.39/6.87 ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.39/6.87 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_615_power4__eq__xxxx,axiom,
% 12.39/6.87 ! [X4: rat] :
% 12.39/6.87 ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.39/6.87 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_616_power4__eq__xxxx,axiom,
% 12.39/6.87 ! [X4: nat] :
% 12.39/6.87 ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.39/6.87 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_617_power4__eq__xxxx,axiom,
% 12.39/6.87 ! [X4: int] :
% 12.39/6.87 ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.39/6.87 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_618_power__even__eq,axiom,
% 12.39/6.87 ! [A: nat,N: nat] :
% 12.39/6.87 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.87 = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_619_power__even__eq,axiom,
% 12.39/6.87 ! [A: real,N: nat] :
% 12.39/6.87 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.87 = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_620_power__even__eq,axiom,
% 12.39/6.87 ! [A: int,N: nat] :
% 12.39/6.87 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.87 = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_621_power__even__eq,axiom,
% 12.39/6.87 ! [A: complex,N: nat] :
% 12.39/6.87 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.87 = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_622_nat__less__add__iff1,axiom,
% 12.39/6.87 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ J @ I )
% 12.39/6.87 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.39/6.87 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_623_nat__less__add__iff2,axiom,
% 12.39/6.87 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.39/6.87 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_624_power2__sum,axiom,
% 12.39/6.87 ! [X4: complex,Y3: complex] :
% 12.39/6.87 ( ( power_power_complex @ ( plus_plus_complex @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_625_power2__sum,axiom,
% 12.39/6.87 ! [X4: real,Y3: real] :
% 12.39/6.87 ( ( power_power_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_626_power2__sum,axiom,
% 12.39/6.87 ! [X4: rat,Y3: rat] :
% 12.39/6.87 ( ( power_power_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_627_power2__sum,axiom,
% 12.39/6.87 ! [X4: nat,Y3: nat] :
% 12.39/6.87 ( ( power_power_nat @ ( plus_plus_nat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_628_power2__sum,axiom,
% 12.39/6.87 ! [X4: int,Y3: int] :
% 12.39/6.87 ( ( power_power_int @ ( plus_plus_int @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_629_add__diff__add,axiom,
% 12.39/6.87 ! [A: real,C: real,B2: real,D: real] :
% 12.39/6.87 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) )
% 12.39/6.87 = ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_630_add__diff__add,axiom,
% 12.39/6.87 ! [A: rat,C: rat,B2: rat,D: rat] :
% 12.39/6.87 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) )
% 12.39/6.87 = ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_631_add__diff__add,axiom,
% 12.39/6.87 ! [A: int,C: int,B2: int,D: int] :
% 12.39/6.87 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) )
% 12.39/6.87 = ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_632_nat__less__le,axiom,
% 12.39/6.87 ( ord_less_nat
% 12.39/6.87 = ( ^ [M6: nat,N2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ M6 @ N2 )
% 12.39/6.87 & ( M6 != N2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_633_less__imp__le__nat,axiom,
% 12.39/6.87 ! [M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_nat @ M @ N )
% 12.39/6.87 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_634_le__eq__less__or__eq,axiom,
% 12.39/6.87 ( ord_less_eq_nat
% 12.39/6.87 = ( ^ [M6: nat,N2: nat] :
% 12.39/6.87 ( ( ord_less_nat @ M6 @ N2 )
% 12.39/6.87 | ( M6 = N2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_635_less__or__eq__imp__le,axiom,
% 12.39/6.87 ! [M: nat,N: nat] :
% 12.39/6.87 ( ( ( ord_less_nat @ M @ N )
% 12.39/6.87 | ( M = N ) )
% 12.39/6.87 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_636_le__neq__implies__less,axiom,
% 12.39/6.87 ! [M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.87 => ( ( M != N )
% 12.39/6.87 => ( ord_less_nat @ M @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_637_less__mono__imp__le__mono,axiom,
% 12.39/6.87 ! [F: nat > nat,I: nat,J: nat] :
% 12.39/6.87 ( ! [I3: nat,J2: nat] :
% 12.39/6.87 ( ( ord_less_nat @ I3 @ J2 )
% 12.39/6.87 => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 12.39/6.87 => ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_638_add__lessD1,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat] :
% 12.39/6.87 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 12.39/6.87 => ( ord_less_nat @ I @ K ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_639_add__less__mono,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.87 ( ( ord_less_nat @ I @ J )
% 12.39/6.87 => ( ( ord_less_nat @ K @ L2 )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_640_not__add__less1,axiom,
% 12.39/6.87 ! [I: nat,J: nat] :
% 12.39/6.87 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_641_not__add__less2,axiom,
% 12.39/6.87 ! [J: nat,I: nat] :
% 12.39/6.87 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_642_add__less__mono1,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat] :
% 12.39/6.87 ( ( ord_less_nat @ I @ J )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_643_trans__less__add1,axiom,
% 12.39/6.87 ! [I: nat,J: nat,M: nat] :
% 12.39/6.87 ( ( ord_less_nat @ I @ J )
% 12.39/6.87 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_644_trans__less__add2,axiom,
% 12.39/6.87 ! [I: nat,J: nat,M: nat] :
% 12.39/6.87 ( ( ord_less_nat @ I @ J )
% 12.39/6.87 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_645_less__add__eq__less,axiom,
% 12.39/6.87 ! [K: nat,L2: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_nat @ K @ L2 )
% 12.39/6.87 => ( ( ( plus_plus_nat @ M @ L2 )
% 12.39/6.87 = ( plus_plus_nat @ K @ N ) )
% 12.39/6.87 => ( ord_less_nat @ M @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_646_add__leE,axiom,
% 12.39/6.87 ! [M: nat,K: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 12.39/6.87 => ~ ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.87 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_647_le__add1,axiom,
% 12.39/6.87 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_648_le__add2,axiom,
% 12.39/6.87 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_649_add__leD1,axiom,
% 12.39/6.87 ! [M: nat,K: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 12.39/6.87 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_650_add__leD2,axiom,
% 12.39/6.87 ! [M: nat,K: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 12.39/6.87 => ( ord_less_eq_nat @ K @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_651_le__Suc__ex,axiom,
% 12.39/6.87 ! [K: nat,L2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ K @ L2 )
% 12.39/6.87 => ? [N4: nat] :
% 12.39/6.87 ( L2
% 12.39/6.87 = ( plus_plus_nat @ K @ N4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_652_add__le__mono,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 => ( ( ord_less_eq_nat @ K @ L2 )
% 12.39/6.87 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_653_add__le__mono1,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_654_trans__le__add1,axiom,
% 12.39/6.87 ! [I: nat,J: nat,M: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_655_trans__le__add2,axiom,
% 12.39/6.87 ! [I: nat,J: nat,M: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_656_nat__le__iff__add,axiom,
% 12.39/6.87 ( ord_less_eq_nat
% 12.39/6.87 = ( ^ [M6: nat,N2: nat] :
% 12.39/6.87 ? [K3: nat] :
% 12.39/6.87 ( N2
% 12.39/6.87 = ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_657_diff__less__mono2,axiom,
% 12.39/6.87 ! [M: nat,N: nat,L2: nat] :
% 12.39/6.87 ( ( ord_less_nat @ M @ N )
% 12.39/6.87 => ( ( ord_less_nat @ M @ L2 )
% 12.39/6.87 => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_658_less__imp__diff__less,axiom,
% 12.39/6.87 ! [J: nat,K: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_nat @ J @ K )
% 12.39/6.87 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_659_eq__diff__iff,axiom,
% 12.39/6.87 ! [K: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ K @ M )
% 12.39/6.87 => ( ( ord_less_eq_nat @ K @ N )
% 12.39/6.87 => ( ( ( minus_minus_nat @ M @ K )
% 12.39/6.87 = ( minus_minus_nat @ N @ K ) )
% 12.39/6.87 = ( M = N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_660_le__diff__iff,axiom,
% 12.39/6.87 ! [K: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ K @ M )
% 12.39/6.87 => ( ( ord_less_eq_nat @ K @ N )
% 12.39/6.87 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 12.39/6.87 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_661_Nat_Odiff__diff__eq,axiom,
% 12.39/6.87 ! [K: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ K @ M )
% 12.39/6.87 => ( ( ord_less_eq_nat @ K @ N )
% 12.39/6.87 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 12.39/6.87 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_662_diff__le__mono,axiom,
% 12.39/6.87 ! [M: nat,N: nat,L2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.87 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_663_diff__le__self,axiom,
% 12.39/6.87 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_664_le__diff__iff_H,axiom,
% 12.39/6.87 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ C )
% 12.39/6.87 => ( ( ord_less_eq_nat @ B2 @ C )
% 12.39/6.87 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_665_diff__le__mono2,axiom,
% 12.39/6.87 ! [M: nat,N: nat,L2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.87 => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_666_power2__diff,axiom,
% 12.39/6.87 ! [X4: complex,Y3: complex] :
% 12.39/6.87 ( ( power_power_complex @ ( minus_minus_complex @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_667_power2__diff,axiom,
% 12.39/6.87 ! [X4: real,Y3: real] :
% 12.39/6.87 ( ( power_power_real @ ( minus_minus_real @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_668_power2__diff,axiom,
% 12.39/6.87 ! [X4: rat,Y3: rat] :
% 12.39/6.87 ( ( power_power_rat @ ( minus_minus_rat @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_669_power2__diff,axiom,
% 12.39/6.87 ! [X4: int,Y3: int] :
% 12.39/6.87 ( ( power_power_int @ ( minus_minus_int @ X4 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.87 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_670_Nat_Odiff__cancel,axiom,
% 12.39/6.87 ! [K: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 12.39/6.87 = ( minus_minus_nat @ M @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_671_diff__cancel2,axiom,
% 12.39/6.87 ! [M: nat,K: nat,N: nat] :
% 12.39/6.87 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 12.39/6.87 = ( minus_minus_nat @ M @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_672_diff__add__inverse,axiom,
% 12.39/6.87 ! [N: nat,M: nat] :
% 12.39/6.87 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 12.39/6.87 = M ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_673_diff__add__inverse2,axiom,
% 12.39/6.87 ! [M: nat,N: nat] :
% 12.39/6.87 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 12.39/6.87 = M ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_674_add__le__add__imp__diff__le,axiom,
% 12.39/6.87 ! [I: real,K: real,N: real,J: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 12.39/6.87 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 12.39/6.87 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 12.39/6.87 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 12.39/6.87 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_675_add__le__add__imp__diff__le,axiom,
% 12.39/6.87 ! [I: rat,K: rat,N: rat,J: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 12.39/6.87 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 12.39/6.87 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 12.39/6.87 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 12.39/6.87 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_676_add__le__add__imp__diff__le,axiom,
% 12.39/6.87 ! [I: nat,K: nat,N: nat,J: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 12.39/6.87 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 12.39/6.87 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 12.39/6.87 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 12.39/6.87 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_677_add__le__add__imp__diff__le,axiom,
% 12.39/6.87 ! [I: int,K: int,N: int,J: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 12.39/6.87 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 12.39/6.87 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 12.39/6.87 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 12.39/6.87 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_678_add__le__imp__le__diff,axiom,
% 12.39/6.87 ! [I: real,K: real,N: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 12.39/6.87 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_679_add__le__imp__le__diff,axiom,
% 12.39/6.87 ! [I: rat,K: rat,N: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 12.39/6.87 => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_680_add__le__imp__le__diff,axiom,
% 12.39/6.87 ! [I: nat,K: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 12.39/6.87 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_681_add__le__imp__le__diff,axiom,
% 12.39/6.87 ! [I: int,K: int,N: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 12.39/6.87 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_682_add__mono1,axiom,
% 12.39/6.87 ! [A: real,B2: real] :
% 12.39/6.87 ( ( ord_less_real @ A @ B2 )
% 12.39/6.87 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B2 @ one_one_real ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_683_add__mono1,axiom,
% 12.39/6.87 ! [A: rat,B2: rat] :
% 12.39/6.87 ( ( ord_less_rat @ A @ B2 )
% 12.39/6.87 => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B2 @ one_one_rat ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_684_add__mono1,axiom,
% 12.39/6.87 ! [A: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_nat @ A @ B2 )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_685_add__mono1,axiom,
% 12.39/6.87 ! [A: int,B2: int] :
% 12.39/6.87 ( ( ord_less_int @ A @ B2 )
% 12.39/6.87 => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_686_less__add__one,axiom,
% 12.39/6.87 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_687_less__add__one,axiom,
% 12.39/6.87 ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_688_less__add__one,axiom,
% 12.39/6.87 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_689_less__add__one,axiom,
% 12.39/6.87 ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_690_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 12.39/6.87 ! [A: real,B2: real] :
% 12.39/6.87 ( ~ ( ord_less_real @ A @ B2 )
% 12.39/6.87 => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
% 12.39/6.87 = A ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_691_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 12.39/6.87 ! [A: rat,B2: rat] :
% 12.39/6.87 ( ~ ( ord_less_rat @ A @ B2 )
% 12.39/6.87 => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
% 12.39/6.87 = A ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_692_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 12.39/6.87 ! [A: nat,B2: nat] :
% 12.39/6.87 ( ~ ( ord_less_nat @ A @ B2 )
% 12.39/6.87 => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
% 12.39/6.87 = A ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_693_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 12.39/6.87 ! [A: int,B2: int] :
% 12.39/6.87 ( ~ ( ord_less_int @ A @ B2 )
% 12.39/6.87 => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
% 12.39/6.87 = A ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_694_mono__nat__linear__lb,axiom,
% 12.39/6.87 ! [F: nat > nat,M: nat,K: nat] :
% 12.39/6.87 ( ! [M4: nat,N4: nat] :
% 12.39/6.87 ( ( ord_less_nat @ M4 @ N4 )
% 12.39/6.87 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
% 12.39/6.87 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_695_less__diff__iff,axiom,
% 12.39/6.87 ! [K: nat,M: nat,N: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ K @ M )
% 12.39/6.87 => ( ( ord_less_eq_nat @ K @ N )
% 12.39/6.87 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 12.39/6.87 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_696_diff__less__mono,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_nat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_nat @ C @ A )
% 12.39/6.87 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_697_add__diff__inverse__nat,axiom,
% 12.39/6.87 ! [M: nat,N: nat] :
% 12.39/6.87 ( ~ ( ord_less_nat @ M @ N )
% 12.39/6.87 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 12.39/6.87 = M ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_698_less__diff__conv,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat] :
% 12.39/6.87 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 12.39/6.87 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_699_Nat_Ole__imp__diff__is__add,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 => ( ( ( minus_minus_nat @ J @ I )
% 12.39/6.87 = K )
% 12.39/6.87 = ( J
% 12.39/6.87 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_700_Nat_Odiff__add__assoc2,axiom,
% 12.39/6.87 ! [K: nat,J: nat,I: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ K @ J )
% 12.39/6.87 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 12.39/6.87 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_701_Nat_Odiff__add__assoc,axiom,
% 12.39/6.87 ! [K: nat,J: nat,I: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ K @ J )
% 12.39/6.87 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 12.39/6.87 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_702_Nat_Ole__diff__conv2,axiom,
% 12.39/6.87 ! [K: nat,J: nat,I: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ K @ J )
% 12.39/6.87 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 12.39/6.87 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_703_le__diff__conv,axiom,
% 12.39/6.87 ! [J: nat,K: nat,I: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 12.39/6.87 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_704_less__diff__conv2,axiom,
% 12.39/6.87 ! [K: nat,J: nat,I: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ K @ J )
% 12.39/6.87 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 12.39/6.87 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_705_field__sum__of__halves,axiom,
% 12.39/6.87 ! [X4: real] :
% 12.39/6.87 ( ( plus_plus_real @ ( divide_divide_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.39/6.87 = X4 ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_706_field__sum__of__halves,axiom,
% 12.39/6.87 ! [X4: rat] :
% 12.39/6.87 ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 12.39/6.87 = X4 ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_707__092_060open_062res_A_061_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Asc_A_L_Aminy_092_060close_062,axiom,
% 12.39/6.87 ( res
% 12.39/6.87 = ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ) @ miny ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_708_sum__squares__bound,axiom,
% 12.39/6.87 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_709_sum__squares__bound,axiom,
% 12.39/6.87 ! [X4: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_710_enat__ord__number_I1_J,axiom,
% 12.39/6.87 ! [M: num,N: num] :
% 12.39/6.87 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 12.39/6.87 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_711_enat__ord__number_I2_J,axiom,
% 12.39/6.87 ! [M: num,N: num] :
% 12.39/6.87 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 12.39/6.87 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_712_double__not__eq__Suc__double,axiom,
% 12.39/6.87 ! [M: nat,N: nat] :
% 12.39/6.87 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 12.39/6.87 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_713_Suc__double__not__eq__double,axiom,
% 12.39/6.87 ! [M: nat,N: nat] :
% 12.39/6.87 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.39/6.87 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_714_add__diff__cancel,axiom,
% 12.39/6.87 ! [A: real,B2: real] :
% 12.39/6.87 ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_715_add__diff__cancel,axiom,
% 12.39/6.87 ! [A: rat,B2: rat] :
% 12.39/6.87 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_716_add__diff__cancel,axiom,
% 12.39/6.87 ! [A: int,B2: int] :
% 12.39/6.87 ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_717_diff__add__cancel,axiom,
% 12.39/6.87 ! [A: real,B2: real] :
% 12.39/6.87 ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_718_diff__add__cancel,axiom,
% 12.39/6.87 ! [A: rat,B2: rat] :
% 12.39/6.87 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_719_diff__add__cancel,axiom,
% 12.39/6.87 ! [A: int,B2: int] :
% 12.39/6.87 ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_720_add__diff__cancel__left,axiom,
% 12.39/6.87 ! [C: real,A: real,B2: real] :
% 12.39/6.87 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 12.39/6.87 = ( minus_minus_real @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_721_add__diff__cancel__left,axiom,
% 12.39/6.87 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.87 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 12.39/6.87 = ( minus_minus_rat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_722_add__diff__cancel__left,axiom,
% 12.39/6.87 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.87 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 12.39/6.87 = ( minus_minus_nat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_723_add__diff__cancel__left,axiom,
% 12.39/6.87 ! [C: int,A: int,B2: int] :
% 12.39/6.87 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 12.39/6.87 = ( minus_minus_int @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_724_add__diff__cancel__left_H,axiom,
% 12.39/6.87 ! [A: real,B2: real] :
% 12.39/6.87 ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ A )
% 12.39/6.87 = B2 ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_725_add__diff__cancel__left_H,axiom,
% 12.39/6.87 ! [A: rat,B2: rat] :
% 12.39/6.87 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ A )
% 12.39/6.87 = B2 ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_726_add__diff__cancel__left_H,axiom,
% 12.39/6.87 ! [A: nat,B2: nat] :
% 12.39/6.87 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
% 12.39/6.87 = B2 ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_727_add__diff__cancel__left_H,axiom,
% 12.39/6.87 ! [A: int,B2: int] :
% 12.39/6.87 ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ A )
% 12.39/6.87 = B2 ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_728_add__diff__cancel__right,axiom,
% 12.39/6.87 ! [A: real,C: real,B2: real] :
% 12.39/6.87 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_real @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_729_add__diff__cancel__right,axiom,
% 12.39/6.87 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.87 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_rat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_730_add__diff__cancel__right,axiom,
% 12.39/6.87 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.87 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_nat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_731_add__diff__cancel__right,axiom,
% 12.39/6.87 ! [A: int,C: int,B2: int] :
% 12.39/6.87 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_int @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_732_add__diff__cancel__right_H,axiom,
% 12.39/6.87 ! [A: real,B2: real] :
% 12.39/6.87 ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_733_add__diff__cancel__right_H,axiom,
% 12.39/6.87 ! [A: rat,B2: rat] :
% 12.39/6.87 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_734_add__diff__cancel__right_H,axiom,
% 12.39/6.87 ! [A: nat,B2: nat] :
% 12.39/6.87 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_735_add__diff__cancel__right_H,axiom,
% 12.39/6.87 ! [A: int,B2: int] :
% 12.39/6.87 ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_736_add__right__cancel,axiom,
% 12.39/6.87 ! [B2: real,A: real,C: real] :
% 12.39/6.87 ( ( ( plus_plus_real @ B2 @ A )
% 12.39/6.87 = ( plus_plus_real @ C @ A ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_737_add__right__cancel,axiom,
% 12.39/6.87 ! [B2: rat,A: rat,C: rat] :
% 12.39/6.87 ( ( ( plus_plus_rat @ B2 @ A )
% 12.39/6.87 = ( plus_plus_rat @ C @ A ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_738_add__right__cancel,axiom,
% 12.39/6.87 ! [B2: nat,A: nat,C: nat] :
% 12.39/6.87 ( ( ( plus_plus_nat @ B2 @ A )
% 12.39/6.87 = ( plus_plus_nat @ C @ A ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_739_add__right__cancel,axiom,
% 12.39/6.87 ! [B2: int,A: int,C: int] :
% 12.39/6.87 ( ( ( plus_plus_int @ B2 @ A )
% 12.39/6.87 = ( plus_plus_int @ C @ A ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_740_add__left__cancel,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ( plus_plus_real @ A @ B2 )
% 12.39/6.87 = ( plus_plus_real @ A @ C ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_741_add__left__cancel,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ( plus_plus_rat @ A @ B2 )
% 12.39/6.87 = ( plus_plus_rat @ A @ C ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_742_add__left__cancel,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ( plus_plus_nat @ A @ B2 )
% 12.39/6.87 = ( plus_plus_nat @ A @ C ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_743_add__left__cancel,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ( plus_plus_int @ A @ B2 )
% 12.39/6.87 = ( plus_plus_int @ A @ C ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_744_add__le__cancel__right,axiom,
% 12.39/6.87 ! [A: real,C: real,B2: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 12.39/6.87 = ( ord_less_eq_real @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_745_add__le__cancel__right,axiom,
% 12.39/6.87 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 12.39/6.87 = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_746_add__le__cancel__right,axiom,
% 12.39/6.87 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 12.39/6.87 = ( ord_less_eq_nat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_747_add__le__cancel__right,axiom,
% 12.39/6.87 ! [A: int,C: int,B2: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 12.39/6.87 = ( ord_less_eq_int @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_748_add__le__cancel__left,axiom,
% 12.39/6.87 ! [C: real,A: real,B2: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_eq_real @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_749_add__le__cancel__left,axiom,
% 12.39/6.87 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_750_add__le__cancel__left,axiom,
% 12.39/6.87 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_eq_nat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_751_add__le__cancel__left,axiom,
% 12.39/6.87 ! [C: int,A: int,B2: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_eq_int @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_752_add__less__cancel__right,axiom,
% 12.39/6.87 ! [A: real,C: real,B2: real] :
% 12.39/6.87 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 12.39/6.87 = ( ord_less_real @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_753_add__less__cancel__right,axiom,
% 12.39/6.87 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.87 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 12.39/6.87 = ( ord_less_rat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_754_add__less__cancel__right,axiom,
% 12.39/6.87 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 12.39/6.87 = ( ord_less_nat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_755_add__less__cancel__right,axiom,
% 12.39/6.87 ! [A: int,C: int,B2: int] :
% 12.39/6.87 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 12.39/6.87 = ( ord_less_int @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_756_add__less__cancel__left,axiom,
% 12.39/6.87 ! [C: real,A: real,B2: real] :
% 12.39/6.87 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_real @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_757_add__less__cancel__left,axiom,
% 12.39/6.87 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.87 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_rat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_758_add__less__cancel__left,axiom,
% 12.39/6.87 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_nat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_759_add__less__cancel__left,axiom,
% 12.39/6.87 ! [C: int,A: int,B2: int] :
% 12.39/6.87 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_int @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_760_mult__1,axiom,
% 12.39/6.87 ! [A: complex] :
% 12.39/6.87 ( ( times_times_complex @ one_one_complex @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_761_mult__1,axiom,
% 12.39/6.87 ! [A: real] :
% 12.39/6.87 ( ( times_times_real @ one_one_real @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_762_mult__1,axiom,
% 12.39/6.87 ! [A: rat] :
% 12.39/6.87 ( ( times_times_rat @ one_one_rat @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_763_mult__1,axiom,
% 12.39/6.87 ! [A: nat] :
% 12.39/6.87 ( ( times_times_nat @ one_one_nat @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_764_mult__1,axiom,
% 12.39/6.87 ! [A: int] :
% 12.39/6.87 ( ( times_times_int @ one_one_int @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_765_mult_Oright__neutral,axiom,
% 12.39/6.87 ! [A: complex] :
% 12.39/6.87 ( ( times_times_complex @ A @ one_one_complex )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_766_mult_Oright__neutral,axiom,
% 12.39/6.87 ! [A: real] :
% 12.39/6.87 ( ( times_times_real @ A @ one_one_real )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_767_mult_Oright__neutral,axiom,
% 12.39/6.87 ! [A: rat] :
% 12.39/6.87 ( ( times_times_rat @ A @ one_one_rat )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_768_mult_Oright__neutral,axiom,
% 12.39/6.87 ! [A: nat] :
% 12.39/6.87 ( ( times_times_nat @ A @ one_one_nat )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_769_mult_Oright__neutral,axiom,
% 12.39/6.87 ! [A: int] :
% 12.39/6.87 ( ( times_times_int @ A @ one_one_int )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_770_semiring__norm_I13_J,axiom,
% 12.39/6.87 ! [M: num,N: num] :
% 12.39/6.87 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.39/6.87 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_771_semiring__norm_I12_J,axiom,
% 12.39/6.87 ! [N: num] :
% 12.39/6.87 ( ( times_times_num @ one @ N )
% 12.39/6.87 = N ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_772_semiring__norm_I11_J,axiom,
% 12.39/6.87 ! [M: num] :
% 12.39/6.87 ( ( times_times_num @ M @ one )
% 12.39/6.87 = M ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_773_num__double,axiom,
% 12.39/6.87 ! [N: num] :
% 12.39/6.87 ( ( times_times_num @ ( bit0 @ one ) @ N )
% 12.39/6.87 = ( bit0 @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_774_power__mult__numeral,axiom,
% 12.39/6.87 ! [A: nat,M: num,N: num] :
% 12.39/6.87 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 12.39/6.87 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_775_power__mult__numeral,axiom,
% 12.39/6.87 ! [A: real,M: num,N: num] :
% 12.39/6.87 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 12.39/6.87 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_776_power__mult__numeral,axiom,
% 12.39/6.87 ! [A: int,M: num,N: num] :
% 12.39/6.87 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 12.39/6.87 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_777_power__mult__numeral,axiom,
% 12.39/6.87 ! [A: complex,M: num,N: num] :
% 12.39/6.87 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 12.39/6.87 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_778_complete__real,axiom,
% 12.39/6.87 ! [S2: set_real] :
% 12.39/6.87 ( ? [X5: real] : ( member_real @ X5 @ S2 )
% 12.39/6.87 => ( ? [Z3: real] :
% 12.39/6.87 ! [X3: real] :
% 12.39/6.87 ( ( member_real @ X3 @ S2 )
% 12.39/6.87 => ( ord_less_eq_real @ X3 @ Z3 ) )
% 12.39/6.87 => ? [Y4: real] :
% 12.39/6.87 ( ! [X5: real] :
% 12.39/6.87 ( ( member_real @ X5 @ S2 )
% 12.39/6.87 => ( ord_less_eq_real @ X5 @ Y4 ) )
% 12.39/6.87 & ! [Z3: real] :
% 12.39/6.87 ( ! [X3: real] :
% 12.39/6.87 ( ( member_real @ X3 @ S2 )
% 12.39/6.87 => ( ord_less_eq_real @ X3 @ Z3 ) )
% 12.39/6.87 => ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_779_real__arch__pow,axiom,
% 12.39/6.87 ! [X4: real,Y3: real] :
% 12.39/6.87 ( ( ord_less_real @ one_one_real @ X4 )
% 12.39/6.87 => ? [N4: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X4 @ N4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_780_less__eq__real__def,axiom,
% 12.39/6.87 ( ord_less_eq_real
% 12.39/6.87 = ( ^ [X: real,Y: real] :
% 12.39/6.87 ( ( ord_less_real @ X @ Y )
% 12.39/6.87 | ( X = Y ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_781_add__diff__assoc__enat,axiom,
% 12.39/6.87 ! [Z: extended_enat,Y3: extended_enat,X4: extended_enat] :
% 12.39/6.87 ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
% 12.39/6.87 => ( ( plus_p3455044024723400733d_enat @ X4 @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
% 12.39/6.87 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y3 ) @ Z ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_782_enat__less__induct,axiom,
% 12.39/6.87 ! [P: extended_enat > $o,N: extended_enat] :
% 12.39/6.87 ( ! [N4: extended_enat] :
% 12.39/6.87 ( ! [M5: extended_enat] :
% 12.39/6.87 ( ( ord_le72135733267957522d_enat @ M5 @ N4 )
% 12.39/6.87 => ( P @ M5 ) )
% 12.39/6.87 => ( P @ N4 ) )
% 12.39/6.87 => ( P @ N ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_783_four__x__squared,axiom,
% 12.39/6.87 ! [X4: real] :
% 12.39/6.87 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.87 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_784_L2__set__mult__ineq__lemma,axiom,
% 12.39/6.87 ! [A: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_785_div__mult2__numeral__eq,axiom,
% 12.39/6.87 ! [A: nat,K: num,L2: num] :
% 12.39/6.87 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
% 12.39/6.87 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_786_div__mult2__numeral__eq,axiom,
% 12.39/6.87 ! [A: int,K: num,L2: num] :
% 12.39/6.87 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
% 12.39/6.87 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_787_mult_Oleft__commute,axiom,
% 12.39/6.87 ! [B2: real,A: real,C: real] :
% 12.39/6.87 ( ( times_times_real @ B2 @ ( times_times_real @ A @ C ) )
% 12.39/6.87 = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_788_mult_Oleft__commute,axiom,
% 12.39/6.87 ! [B2: rat,A: rat,C: rat] :
% 12.39/6.87 ( ( times_times_rat @ B2 @ ( times_times_rat @ A @ C ) )
% 12.39/6.87 = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_789_mult_Oleft__commute,axiom,
% 12.39/6.87 ! [B2: nat,A: nat,C: nat] :
% 12.39/6.87 ( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
% 12.39/6.87 = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_790_mult_Oleft__commute,axiom,
% 12.39/6.87 ! [B2: int,A: int,C: int] :
% 12.39/6.87 ( ( times_times_int @ B2 @ ( times_times_int @ A @ C ) )
% 12.39/6.87 = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_791_mult_Ocommute,axiom,
% 12.39/6.87 ( times_times_real
% 12.39/6.87 = ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_792_mult_Ocommute,axiom,
% 12.39/6.87 ( times_times_rat
% 12.39/6.87 = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ B3 @ A4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_793_mult_Ocommute,axiom,
% 12.39/6.87 ( times_times_nat
% 12.39/6.87 = ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_794_mult_Ocommute,axiom,
% 12.39/6.87 ( times_times_int
% 12.39/6.87 = ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_795_mult_Oassoc,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
% 12.39/6.87 = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_796_mult_Oassoc,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 12.39/6.87 = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_797_mult_Oassoc,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 12.39/6.87 = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_798_mult_Oassoc,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
% 12.39/6.87 = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_799_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
% 12.39/6.87 = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_800_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 12.39/6.87 = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_801_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 12.39/6.87 = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_802_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
% 12.39/6.87 = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_803_one__reorient,axiom,
% 12.39/6.87 ! [X4: complex] :
% 12.39/6.87 ( ( one_one_complex = X4 )
% 12.39/6.87 = ( X4 = one_one_complex ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_804_one__reorient,axiom,
% 12.39/6.87 ! [X4: real] :
% 12.39/6.87 ( ( one_one_real = X4 )
% 12.39/6.87 = ( X4 = one_one_real ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_805_one__reorient,axiom,
% 12.39/6.87 ! [X4: rat] :
% 12.39/6.87 ( ( one_one_rat = X4 )
% 12.39/6.87 = ( X4 = one_one_rat ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_806_one__reorient,axiom,
% 12.39/6.87 ! [X4: nat] :
% 12.39/6.87 ( ( one_one_nat = X4 )
% 12.39/6.87 = ( X4 = one_one_nat ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_807_one__reorient,axiom,
% 12.39/6.87 ! [X4: int] :
% 12.39/6.87 ( ( one_one_int = X4 )
% 12.39/6.87 = ( X4 = one_one_int ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_808_add__right__imp__eq,axiom,
% 12.39/6.87 ! [B2: real,A: real,C: real] :
% 12.39/6.87 ( ( ( plus_plus_real @ B2 @ A )
% 12.39/6.87 = ( plus_plus_real @ C @ A ) )
% 12.39/6.87 => ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_809_add__right__imp__eq,axiom,
% 12.39/6.87 ! [B2: rat,A: rat,C: rat] :
% 12.39/6.87 ( ( ( plus_plus_rat @ B2 @ A )
% 12.39/6.87 = ( plus_plus_rat @ C @ A ) )
% 12.39/6.87 => ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_810_add__right__imp__eq,axiom,
% 12.39/6.87 ! [B2: nat,A: nat,C: nat] :
% 12.39/6.87 ( ( ( plus_plus_nat @ B2 @ A )
% 12.39/6.87 = ( plus_plus_nat @ C @ A ) )
% 12.39/6.87 => ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_811_add__right__imp__eq,axiom,
% 12.39/6.87 ! [B2: int,A: int,C: int] :
% 12.39/6.87 ( ( ( plus_plus_int @ B2 @ A )
% 12.39/6.87 = ( plus_plus_int @ C @ A ) )
% 12.39/6.87 => ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_812_add__left__imp__eq,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ( plus_plus_real @ A @ B2 )
% 12.39/6.87 = ( plus_plus_real @ A @ C ) )
% 12.39/6.87 => ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_813_add__left__imp__eq,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ( plus_plus_rat @ A @ B2 )
% 12.39/6.87 = ( plus_plus_rat @ A @ C ) )
% 12.39/6.87 => ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_814_add__left__imp__eq,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ( plus_plus_nat @ A @ B2 )
% 12.39/6.87 = ( plus_plus_nat @ A @ C ) )
% 12.39/6.87 => ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_815_add__left__imp__eq,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ( plus_plus_int @ A @ B2 )
% 12.39/6.87 = ( plus_plus_int @ A @ C ) )
% 12.39/6.87 => ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_816_add_Oleft__commute,axiom,
% 12.39/6.87 ! [B2: real,A: real,C: real] :
% 12.39/6.87 ( ( plus_plus_real @ B2 @ ( plus_plus_real @ A @ C ) )
% 12.39/6.87 = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_817_add_Oleft__commute,axiom,
% 12.39/6.87 ! [B2: rat,A: rat,C: rat] :
% 12.39/6.87 ( ( plus_plus_rat @ B2 @ ( plus_plus_rat @ A @ C ) )
% 12.39/6.87 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_818_add_Oleft__commute,axiom,
% 12.39/6.87 ! [B2: nat,A: nat,C: nat] :
% 12.39/6.87 ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
% 12.39/6.87 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_819_add_Oleft__commute,axiom,
% 12.39/6.87 ! [B2: int,A: int,C: int] :
% 12.39/6.87 ( ( plus_plus_int @ B2 @ ( plus_plus_int @ A @ C ) )
% 12.39/6.87 = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_820_add_Ocommute,axiom,
% 12.39/6.87 ( plus_plus_real
% 12.39/6.87 = ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_821_add_Ocommute,axiom,
% 12.39/6.87 ( plus_plus_rat
% 12.39/6.87 = ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ B3 @ A4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_822_add_Ocommute,axiom,
% 12.39/6.87 ( plus_plus_nat
% 12.39/6.87 = ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_823_add_Ocommute,axiom,
% 12.39/6.87 ( plus_plus_int
% 12.39/6.87 = ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_824_add_Oright__cancel,axiom,
% 12.39/6.87 ! [B2: real,A: real,C: real] :
% 12.39/6.87 ( ( ( plus_plus_real @ B2 @ A )
% 12.39/6.87 = ( plus_plus_real @ C @ A ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_825_add_Oright__cancel,axiom,
% 12.39/6.87 ! [B2: rat,A: rat,C: rat] :
% 12.39/6.87 ( ( ( plus_plus_rat @ B2 @ A )
% 12.39/6.87 = ( plus_plus_rat @ C @ A ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_826_add_Oright__cancel,axiom,
% 12.39/6.87 ! [B2: int,A: int,C: int] :
% 12.39/6.87 ( ( ( plus_plus_int @ B2 @ A )
% 12.39/6.87 = ( plus_plus_int @ C @ A ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_827_add_Oleft__cancel,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ( plus_plus_real @ A @ B2 )
% 12.39/6.87 = ( plus_plus_real @ A @ C ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_828_add_Oleft__cancel,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ( plus_plus_rat @ A @ B2 )
% 12.39/6.87 = ( plus_plus_rat @ A @ C ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_829_add_Oleft__cancel,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ( plus_plus_int @ A @ B2 )
% 12.39/6.87 = ( plus_plus_int @ A @ C ) )
% 12.39/6.87 = ( B2 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_830_add_Oassoc,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 12.39/6.87 = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_831_add_Oassoc,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 12.39/6.87 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_832_add_Oassoc,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 12.39/6.87 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_833_add_Oassoc,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 12.39/6.87 = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_834_group__cancel_Oadd2,axiom,
% 12.39/6.87 ! [B4: real,K: real,B2: real,A: real] :
% 12.39/6.87 ( ( B4
% 12.39/6.87 = ( plus_plus_real @ K @ B2 ) )
% 12.39/6.87 => ( ( plus_plus_real @ A @ B4 )
% 12.39/6.87 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_835_group__cancel_Oadd2,axiom,
% 12.39/6.87 ! [B4: rat,K: rat,B2: rat,A: rat] :
% 12.39/6.87 ( ( B4
% 12.39/6.87 = ( plus_plus_rat @ K @ B2 ) )
% 12.39/6.87 => ( ( plus_plus_rat @ A @ B4 )
% 12.39/6.87 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_836_group__cancel_Oadd2,axiom,
% 12.39/6.87 ! [B4: nat,K: nat,B2: nat,A: nat] :
% 12.39/6.87 ( ( B4
% 12.39/6.87 = ( plus_plus_nat @ K @ B2 ) )
% 12.39/6.87 => ( ( plus_plus_nat @ A @ B4 )
% 12.39/6.87 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_837_group__cancel_Oadd2,axiom,
% 12.39/6.87 ! [B4: int,K: int,B2: int,A: int] :
% 12.39/6.87 ( ( B4
% 12.39/6.87 = ( plus_plus_int @ K @ B2 ) )
% 12.39/6.87 => ( ( plus_plus_int @ A @ B4 )
% 12.39/6.87 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_838_group__cancel_Oadd1,axiom,
% 12.39/6.87 ! [A3: real,K: real,A: real,B2: real] :
% 12.39/6.87 ( ( A3
% 12.39/6.87 = ( plus_plus_real @ K @ A ) )
% 12.39/6.87 => ( ( plus_plus_real @ A3 @ B2 )
% 12.39/6.87 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_839_group__cancel_Oadd1,axiom,
% 12.39/6.87 ! [A3: rat,K: rat,A: rat,B2: rat] :
% 12.39/6.87 ( ( A3
% 12.39/6.87 = ( plus_plus_rat @ K @ A ) )
% 12.39/6.87 => ( ( plus_plus_rat @ A3 @ B2 )
% 12.39/6.87 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_840_group__cancel_Oadd1,axiom,
% 12.39/6.87 ! [A3: nat,K: nat,A: nat,B2: nat] :
% 12.39/6.87 ( ( A3
% 12.39/6.87 = ( plus_plus_nat @ K @ A ) )
% 12.39/6.87 => ( ( plus_plus_nat @ A3 @ B2 )
% 12.39/6.87 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_841_group__cancel_Oadd1,axiom,
% 12.39/6.87 ! [A3: int,K: int,A: int,B2: int] :
% 12.39/6.87 ( ( A3
% 12.39/6.87 = ( plus_plus_int @ K @ A ) )
% 12.39/6.87 => ( ( plus_plus_int @ A3 @ B2 )
% 12.39/6.87 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_842_add__mono__thms__linordered__semiring_I4_J,axiom,
% 12.39/6.87 ! [I: real,J: real,K: real,L2: real] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ( plus_plus_real @ I @ K )
% 12.39/6.87 = ( plus_plus_real @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_843_add__mono__thms__linordered__semiring_I4_J,axiom,
% 12.39/6.87 ! [I: rat,J: rat,K: rat,L2: rat] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ( plus_plus_rat @ I @ K )
% 12.39/6.87 = ( plus_plus_rat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_844_add__mono__thms__linordered__semiring_I4_J,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ( plus_plus_nat @ I @ K )
% 12.39/6.87 = ( plus_plus_nat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_845_add__mono__thms__linordered__semiring_I4_J,axiom,
% 12.39/6.87 ! [I: int,J: int,K: int,L2: int] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ( plus_plus_int @ I @ K )
% 12.39/6.87 = ( plus_plus_int @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_846_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 12.39/6.87 = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_847_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 12.39/6.87 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_848_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 12.39/6.87 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_849_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 12.39/6.87 = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_850_diff__right__commute,axiom,
% 12.39/6.87 ! [A: real,C: real,B2: real] :
% 12.39/6.87 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 )
% 12.39/6.87 = ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_851_diff__right__commute,axiom,
% 12.39/6.87 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.87 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 )
% 12.39/6.87 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_852_diff__right__commute,axiom,
% 12.39/6.87 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.87 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
% 12.39/6.87 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_853_diff__right__commute,axiom,
% 12.39/6.87 ! [A: int,C: int,B2: int] :
% 12.39/6.87 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 )
% 12.39/6.87 = ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_854_diff__eq__diff__eq,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real,D: real] :
% 12.39/6.87 ( ( ( minus_minus_real @ A @ B2 )
% 12.39/6.87 = ( minus_minus_real @ C @ D ) )
% 12.39/6.87 => ( ( A = B2 )
% 12.39/6.87 = ( C = D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_855_diff__eq__diff__eq,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.39/6.87 ( ( ( minus_minus_rat @ A @ B2 )
% 12.39/6.87 = ( minus_minus_rat @ C @ D ) )
% 12.39/6.87 => ( ( A = B2 )
% 12.39/6.87 = ( C = D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_856_diff__eq__diff__eq,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int,D: int] :
% 12.39/6.87 ( ( ( minus_minus_int @ A @ B2 )
% 12.39/6.87 = ( minus_minus_int @ C @ D ) )
% 12.39/6.87 => ( ( A = B2 )
% 12.39/6.87 = ( C = D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_857_add__le__imp__le__right,axiom,
% 12.39/6.87 ! [A: real,C: real,B2: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 12.39/6.87 => ( ord_less_eq_real @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_858_add__le__imp__le__right,axiom,
% 12.39/6.87 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 12.39/6.87 => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_859_add__le__imp__le__right,axiom,
% 12.39/6.87 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 12.39/6.87 => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_860_add__le__imp__le__right,axiom,
% 12.39/6.87 ! [A: int,C: int,B2: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 12.39/6.87 => ( ord_less_eq_int @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_861_add__le__imp__le__left,axiom,
% 12.39/6.87 ! [C: real,A: real,B2: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 12.39/6.87 => ( ord_less_eq_real @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_862_add__le__imp__le__left,axiom,
% 12.39/6.87 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 12.39/6.87 => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_863_add__le__imp__le__left,axiom,
% 12.39/6.87 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 12.39/6.87 => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_864_add__le__imp__le__left,axiom,
% 12.39/6.87 ! [C: int,A: int,B2: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 12.39/6.87 => ( ord_less_eq_int @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_865_le__iff__add,axiom,
% 12.39/6.87 ( ord_less_eq_nat
% 12.39/6.87 = ( ^ [A4: nat,B3: nat] :
% 12.39/6.87 ? [C2: nat] :
% 12.39/6.87 ( B3
% 12.39/6.87 = ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_866_add__right__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_867_add__right__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_868_add__right__mono,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_869_add__right__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_870_less__eqE,axiom,
% 12.39/6.87 ! [A: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ~ ! [C3: nat] :
% 12.39/6.87 ( B2
% 12.39/6.87 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_871_add__left__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_872_add__left__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_873_add__left__mono,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_874_add__left__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_875_add__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real,D: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_real @ C @ D )
% 12.39/6.87 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_876_add__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_rat @ C @ D )
% 12.39/6.87 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_877_add__mono,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_nat @ C @ D )
% 12.39/6.87 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_878_add__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int,D: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_int @ C @ D )
% 12.39/6.87 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_879_add__mono__thms__linordered__semiring_I1_J,axiom,
% 12.39/6.87 ! [I: real,J: real,K: real,L2: real] :
% 12.39/6.87 ( ( ( ord_less_eq_real @ I @ J )
% 12.39/6.87 & ( ord_less_eq_real @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_880_add__mono__thms__linordered__semiring_I1_J,axiom,
% 12.39/6.87 ! [I: rat,J: rat,K: rat,L2: rat] :
% 12.39/6.87 ( ( ( ord_less_eq_rat @ I @ J )
% 12.39/6.87 & ( ord_less_eq_rat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_881_add__mono__thms__linordered__semiring_I1_J,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.87 ( ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 & ( ord_less_eq_nat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_882_add__mono__thms__linordered__semiring_I1_J,axiom,
% 12.39/6.87 ! [I: int,J: int,K: int,L2: int] :
% 12.39/6.87 ( ( ( ord_less_eq_int @ I @ J )
% 12.39/6.87 & ( ord_less_eq_int @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_883_add__mono__thms__linordered__semiring_I2_J,axiom,
% 12.39/6.87 ! [I: real,J: real,K: real,L2: real] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( ord_less_eq_real @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_884_add__mono__thms__linordered__semiring_I2_J,axiom,
% 12.39/6.87 ! [I: rat,J: rat,K: rat,L2: rat] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( ord_less_eq_rat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_885_add__mono__thms__linordered__semiring_I2_J,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( ord_less_eq_nat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_886_add__mono__thms__linordered__semiring_I2_J,axiom,
% 12.39/6.87 ! [I: int,J: int,K: int,L2: int] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( ord_less_eq_int @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_887_add__mono__thms__linordered__semiring_I3_J,axiom,
% 12.39/6.87 ! [I: real,J: real,K: real,L2: real] :
% 12.39/6.87 ( ( ( ord_less_eq_real @ I @ J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_888_add__mono__thms__linordered__semiring_I3_J,axiom,
% 12.39/6.87 ! [I: rat,J: rat,K: rat,L2: rat] :
% 12.39/6.87 ( ( ( ord_less_eq_rat @ I @ J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_889_add__mono__thms__linordered__semiring_I3_J,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.87 ( ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_890_add__mono__thms__linordered__semiring_I3_J,axiom,
% 12.39/6.87 ! [I: int,J: int,K: int,L2: int] :
% 12.39/6.87 ( ( ( ord_less_eq_int @ I @ J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_891_diff__eq__diff__less__eq,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real,D: real] :
% 12.39/6.87 ( ( ( minus_minus_real @ A @ B2 )
% 12.39/6.87 = ( minus_minus_real @ C @ D ) )
% 12.39/6.87 => ( ( ord_less_eq_real @ A @ B2 )
% 12.39/6.87 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_892_diff__eq__diff__less__eq,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.39/6.87 ( ( ( minus_minus_rat @ A @ B2 )
% 12.39/6.87 = ( minus_minus_rat @ C @ D ) )
% 12.39/6.87 => ( ( ord_less_eq_rat @ A @ B2 )
% 12.39/6.87 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_893_diff__eq__diff__less__eq,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int,D: int] :
% 12.39/6.87 ( ( ( minus_minus_int @ A @ B2 )
% 12.39/6.87 = ( minus_minus_int @ C @ D ) )
% 12.39/6.87 => ( ( ord_less_eq_int @ A @ B2 )
% 12.39/6.87 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_894_diff__right__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_895_diff__right__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_896_diff__right__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_897_diff__left__mono,axiom,
% 12.39/6.87 ! [B2: real,A: real,C: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ B2 @ A )
% 12.39/6.87 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_898_diff__left__mono,axiom,
% 12.39/6.87 ! [B2: rat,A: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ B2 @ A )
% 12.39/6.87 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_899_diff__left__mono,axiom,
% 12.39/6.87 ! [B2: int,A: int,C: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ B2 @ A )
% 12.39/6.87 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_900_diff__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,D: real,C: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_real @ D @ C )
% 12.39/6.87 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_901_diff__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,D: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_rat @ D @ C )
% 12.39/6.87 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_902_diff__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,D: int,C: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_int @ D @ C )
% 12.39/6.87 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_903_add__less__imp__less__right,axiom,
% 12.39/6.87 ! [A: real,C: real,B2: real] :
% 12.39/6.87 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 12.39/6.87 => ( ord_less_real @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_904_add__less__imp__less__right,axiom,
% 12.39/6.87 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.87 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 12.39/6.87 => ( ord_less_rat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_905_add__less__imp__less__right,axiom,
% 12.39/6.87 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 12.39/6.87 => ( ord_less_nat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_906_add__less__imp__less__right,axiom,
% 12.39/6.87 ! [A: int,C: int,B2: int] :
% 12.39/6.87 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 12.39/6.87 => ( ord_less_int @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_907_add__less__imp__less__left,axiom,
% 12.39/6.87 ! [C: real,A: real,B2: real] :
% 12.39/6.87 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 12.39/6.87 => ( ord_less_real @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_908_add__less__imp__less__left,axiom,
% 12.39/6.87 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.87 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 12.39/6.87 => ( ord_less_rat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_909_add__less__imp__less__left,axiom,
% 12.39/6.87 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 12.39/6.87 => ( ord_less_nat @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_910_add__less__imp__less__left,axiom,
% 12.39/6.87 ! [C: int,A: int,B2: int] :
% 12.39/6.87 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 12.39/6.87 => ( ord_less_int @ A @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_911_add__strict__right__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ord_less_real @ A @ B2 )
% 12.39/6.87 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_912_add__strict__right__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_rat @ A @ B2 )
% 12.39/6.87 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_913_add__strict__right__mono,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_nat @ A @ B2 )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_914_add__strict__right__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ord_less_int @ A @ B2 )
% 12.39/6.87 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_915_add__strict__left__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ord_less_real @ A @ B2 )
% 12.39/6.87 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_916_add__strict__left__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_rat @ A @ B2 )
% 12.39/6.87 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_917_add__strict__left__mono,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_nat @ A @ B2 )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_918_add__strict__left__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ord_less_int @ A @ B2 )
% 12.39/6.87 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_919_add__strict__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real,D: real] :
% 12.39/6.87 ( ( ord_less_real @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_real @ C @ D )
% 12.39/6.87 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_920_add__strict__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.39/6.87 ( ( ord_less_rat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_rat @ C @ D )
% 12.39/6.87 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_921_add__strict__mono,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.39/6.87 ( ( ord_less_nat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_nat @ C @ D )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_922_add__strict__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int,D: int] :
% 12.39/6.87 ( ( ord_less_int @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_int @ C @ D )
% 12.39/6.87 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_923_add__mono__thms__linordered__field_I1_J,axiom,
% 12.39/6.87 ! [I: real,J: real,K: real,L2: real] :
% 12.39/6.87 ( ( ( ord_less_real @ I @ J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_924_add__mono__thms__linordered__field_I1_J,axiom,
% 12.39/6.87 ! [I: rat,J: rat,K: rat,L2: rat] :
% 12.39/6.87 ( ( ( ord_less_rat @ I @ J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_925_add__mono__thms__linordered__field_I1_J,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.87 ( ( ( ord_less_nat @ I @ J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_926_add__mono__thms__linordered__field_I1_J,axiom,
% 12.39/6.87 ! [I: int,J: int,K: int,L2: int] :
% 12.39/6.87 ( ( ( ord_less_int @ I @ J )
% 12.39/6.87 & ( K = L2 ) )
% 12.39/6.87 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_927_add__mono__thms__linordered__field_I2_J,axiom,
% 12.39/6.87 ! [I: real,J: real,K: real,L2: real] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( ord_less_real @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_928_add__mono__thms__linordered__field_I2_J,axiom,
% 12.39/6.87 ! [I: rat,J: rat,K: rat,L2: rat] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( ord_less_rat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_929_add__mono__thms__linordered__field_I2_J,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( ord_less_nat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_930_add__mono__thms__linordered__field_I2_J,axiom,
% 12.39/6.87 ! [I: int,J: int,K: int,L2: int] :
% 12.39/6.87 ( ( ( I = J )
% 12.39/6.87 & ( ord_less_int @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_931_add__mono__thms__linordered__field_I5_J,axiom,
% 12.39/6.87 ! [I: real,J: real,K: real,L2: real] :
% 12.39/6.87 ( ( ( ord_less_real @ I @ J )
% 12.39/6.87 & ( ord_less_real @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_932_add__mono__thms__linordered__field_I5_J,axiom,
% 12.39/6.87 ! [I: rat,J: rat,K: rat,L2: rat] :
% 12.39/6.87 ( ( ( ord_less_rat @ I @ J )
% 12.39/6.87 & ( ord_less_rat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_933_add__mono__thms__linordered__field_I5_J,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.87 ( ( ( ord_less_nat @ I @ J )
% 12.39/6.87 & ( ord_less_nat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_934_add__mono__thms__linordered__field_I5_J,axiom,
% 12.39/6.87 ! [I: int,J: int,K: int,L2: int] :
% 12.39/6.87 ( ( ( ord_less_int @ I @ J )
% 12.39/6.87 & ( ord_less_int @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_935_diff__strict__right__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ord_less_real @ A @ B2 )
% 12.39/6.87 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_936_diff__strict__right__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_rat @ A @ B2 )
% 12.39/6.87 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_937_diff__strict__right__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ord_less_int @ A @ B2 )
% 12.39/6.87 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_938_diff__strict__left__mono,axiom,
% 12.39/6.87 ! [B2: real,A: real,C: real] :
% 12.39/6.87 ( ( ord_less_real @ B2 @ A )
% 12.39/6.87 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_939_diff__strict__left__mono,axiom,
% 12.39/6.87 ! [B2: rat,A: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_rat @ B2 @ A )
% 12.39/6.87 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_940_diff__strict__left__mono,axiom,
% 12.39/6.87 ! [B2: int,A: int,C: int] :
% 12.39/6.87 ( ( ord_less_int @ B2 @ A )
% 12.39/6.87 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_941_diff__eq__diff__less,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real,D: real] :
% 12.39/6.87 ( ( ( minus_minus_real @ A @ B2 )
% 12.39/6.87 = ( minus_minus_real @ C @ D ) )
% 12.39/6.87 => ( ( ord_less_real @ A @ B2 )
% 12.39/6.87 = ( ord_less_real @ C @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_942_diff__eq__diff__less,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.39/6.87 ( ( ( minus_minus_rat @ A @ B2 )
% 12.39/6.87 = ( minus_minus_rat @ C @ D ) )
% 12.39/6.87 => ( ( ord_less_rat @ A @ B2 )
% 12.39/6.87 = ( ord_less_rat @ C @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_943_diff__eq__diff__less,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int,D: int] :
% 12.39/6.87 ( ( ( minus_minus_int @ A @ B2 )
% 12.39/6.87 = ( minus_minus_int @ C @ D ) )
% 12.39/6.87 => ( ( ord_less_int @ A @ B2 )
% 12.39/6.87 = ( ord_less_int @ C @ D ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_944_diff__strict__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,D: real,C: real] :
% 12.39/6.87 ( ( ord_less_real @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_real @ D @ C )
% 12.39/6.87 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_945_diff__strict__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,D: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_rat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_rat @ D @ C )
% 12.39/6.87 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_946_diff__strict__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,D: int,C: int] :
% 12.39/6.87 ( ( ord_less_int @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_int @ D @ C )
% 12.39/6.87 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_947_mult_Ocomm__neutral,axiom,
% 12.39/6.87 ! [A: complex] :
% 12.39/6.87 ( ( times_times_complex @ A @ one_one_complex )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_948_mult_Ocomm__neutral,axiom,
% 12.39/6.87 ! [A: real] :
% 12.39/6.87 ( ( times_times_real @ A @ one_one_real )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_949_mult_Ocomm__neutral,axiom,
% 12.39/6.87 ! [A: rat] :
% 12.39/6.87 ( ( times_times_rat @ A @ one_one_rat )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_950_mult_Ocomm__neutral,axiom,
% 12.39/6.87 ! [A: nat] :
% 12.39/6.87 ( ( times_times_nat @ A @ one_one_nat )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_951_mult_Ocomm__neutral,axiom,
% 12.39/6.87 ! [A: int] :
% 12.39/6.87 ( ( times_times_int @ A @ one_one_int )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_952_comm__monoid__mult__class_Omult__1,axiom,
% 12.39/6.87 ! [A: complex] :
% 12.39/6.87 ( ( times_times_complex @ one_one_complex @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_953_comm__monoid__mult__class_Omult__1,axiom,
% 12.39/6.87 ! [A: real] :
% 12.39/6.87 ( ( times_times_real @ one_one_real @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_954_comm__monoid__mult__class_Omult__1,axiom,
% 12.39/6.87 ! [A: rat] :
% 12.39/6.87 ( ( times_times_rat @ one_one_rat @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_955_comm__monoid__mult__class_Omult__1,axiom,
% 12.39/6.87 ! [A: nat] :
% 12.39/6.87 ( ( times_times_nat @ one_one_nat @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_956_comm__monoid__mult__class_Omult__1,axiom,
% 12.39/6.87 ! [A: int] :
% 12.39/6.87 ( ( times_times_int @ one_one_int @ A )
% 12.39/6.87 = A ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_957_diff__diff__eq,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 12.39/6.87 = ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_958_diff__diff__eq,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 12.39/6.87 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_959_diff__diff__eq,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
% 12.39/6.87 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_960_diff__diff__eq,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 12.39/6.87 = ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_961_add__implies__diff,axiom,
% 12.39/6.87 ! [C: real,B2: real,A: real] :
% 12.39/6.87 ( ( ( plus_plus_real @ C @ B2 )
% 12.39/6.87 = A )
% 12.39/6.87 => ( C
% 12.39/6.87 = ( minus_minus_real @ A @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_962_add__implies__diff,axiom,
% 12.39/6.87 ! [C: rat,B2: rat,A: rat] :
% 12.39/6.87 ( ( ( plus_plus_rat @ C @ B2 )
% 12.39/6.87 = A )
% 12.39/6.87 => ( C
% 12.39/6.87 = ( minus_minus_rat @ A @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_963_add__implies__diff,axiom,
% 12.39/6.87 ! [C: nat,B2: nat,A: nat] :
% 12.39/6.87 ( ( ( plus_plus_nat @ C @ B2 )
% 12.39/6.87 = A )
% 12.39/6.87 => ( C
% 12.39/6.87 = ( minus_minus_nat @ A @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_964_add__implies__diff,axiom,
% 12.39/6.87 ! [C: int,B2: int,A: int] :
% 12.39/6.87 ( ( ( plus_plus_int @ C @ B2 )
% 12.39/6.87 = A )
% 12.39/6.87 => ( C
% 12.39/6.87 = ( minus_minus_int @ A @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_965_diff__add__eq__diff__diff__swap,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_966_diff__add__eq__diff__diff__swap,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_967_diff__add__eq__diff__diff__swap,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_968_diff__add__eq,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 12.39/6.87 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_969_diff__add__eq,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 12.39/6.87 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_970_diff__add__eq,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 12.39/6.87 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_971_diff__diff__eq2,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( minus_minus_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_972_diff__diff__eq2,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_973_diff__diff__eq2,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( minus_minus_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_974_add__diff__eq,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( plus_plus_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_975_add__diff__eq,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_976_add__diff__eq,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( plus_plus_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 12.39/6.87 = ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_977_eq__diff__eq,axiom,
% 12.39/6.87 ! [A: real,C: real,B2: real] :
% 12.39/6.87 ( ( A
% 12.39/6.87 = ( minus_minus_real @ C @ B2 ) )
% 12.39/6.87 = ( ( plus_plus_real @ A @ B2 )
% 12.39/6.87 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_978_eq__diff__eq,axiom,
% 12.39/6.87 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.87 ( ( A
% 12.39/6.87 = ( minus_minus_rat @ C @ B2 ) )
% 12.39/6.87 = ( ( plus_plus_rat @ A @ B2 )
% 12.39/6.87 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_979_eq__diff__eq,axiom,
% 12.39/6.87 ! [A: int,C: int,B2: int] :
% 12.39/6.87 ( ( A
% 12.39/6.87 = ( minus_minus_int @ C @ B2 ) )
% 12.39/6.87 = ( ( plus_plus_int @ A @ B2 )
% 12.39/6.87 = C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_980_diff__eq__eq,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ( minus_minus_real @ A @ B2 )
% 12.39/6.87 = C )
% 12.39/6.87 = ( A
% 12.39/6.87 = ( plus_plus_real @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_981_diff__eq__eq,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ( minus_minus_rat @ A @ B2 )
% 12.39/6.87 = C )
% 12.39/6.87 = ( A
% 12.39/6.87 = ( plus_plus_rat @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_982_diff__eq__eq,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ( minus_minus_int @ A @ B2 )
% 12.39/6.87 = C )
% 12.39/6.87 = ( A
% 12.39/6.87 = ( plus_plus_int @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_983_group__cancel_Osub1,axiom,
% 12.39/6.87 ! [A3: real,K: real,A: real,B2: real] :
% 12.39/6.87 ( ( A3
% 12.39/6.87 = ( plus_plus_real @ K @ A ) )
% 12.39/6.87 => ( ( minus_minus_real @ A3 @ B2 )
% 12.39/6.87 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_984_group__cancel_Osub1,axiom,
% 12.39/6.87 ! [A3: rat,K: rat,A: rat,B2: rat] :
% 12.39/6.87 ( ( A3
% 12.39/6.87 = ( plus_plus_rat @ K @ A ) )
% 12.39/6.87 => ( ( minus_minus_rat @ A3 @ B2 )
% 12.39/6.87 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_985_group__cancel_Osub1,axiom,
% 12.39/6.87 ! [A3: int,K: int,A: int,B2: int] :
% 12.39/6.87 ( ( A3
% 12.39/6.87 = ( plus_plus_int @ K @ A ) )
% 12.39/6.87 => ( ( minus_minus_int @ A3 @ B2 )
% 12.39/6.87 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B2 ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_986_add__less__le__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real,D: real] :
% 12.39/6.87 ( ( ord_less_real @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_real @ C @ D )
% 12.39/6.87 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_987_add__less__le__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.39/6.87 ( ( ord_less_rat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_rat @ C @ D )
% 12.39/6.87 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_988_add__less__le__mono,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.39/6.87 ( ( ord_less_nat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_nat @ C @ D )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_989_add__less__le__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int,D: int] :
% 12.39/6.87 ( ( ord_less_int @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_int @ C @ D )
% 12.39/6.87 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_990_add__le__less__mono,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real,D: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_real @ C @ D )
% 12.39/6.87 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_991_add__le__less__mono,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_rat @ C @ D )
% 12.39/6.87 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_992_add__le__less__mono,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_nat @ C @ D )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_993_add__le__less__mono,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int,D: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_int @ C @ D )
% 12.39/6.87 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_994_add__mono__thms__linordered__field_I3_J,axiom,
% 12.39/6.87 ! [I: real,J: real,K: real,L2: real] :
% 12.39/6.87 ( ( ( ord_less_real @ I @ J )
% 12.39/6.87 & ( ord_less_eq_real @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_995_add__mono__thms__linordered__field_I3_J,axiom,
% 12.39/6.87 ! [I: rat,J: rat,K: rat,L2: rat] :
% 12.39/6.87 ( ( ( ord_less_rat @ I @ J )
% 12.39/6.87 & ( ord_less_eq_rat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_996_add__mono__thms__linordered__field_I3_J,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.87 ( ( ( ord_less_nat @ I @ J )
% 12.39/6.87 & ( ord_less_eq_nat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_997_add__mono__thms__linordered__field_I3_J,axiom,
% 12.39/6.87 ! [I: int,J: int,K: int,L2: int] :
% 12.39/6.87 ( ( ( ord_less_int @ I @ J )
% 12.39/6.87 & ( ord_less_eq_int @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_998_add__mono__thms__linordered__field_I4_J,axiom,
% 12.39/6.87 ! [I: real,J: real,K: real,L2: real] :
% 12.39/6.87 ( ( ( ord_less_eq_real @ I @ J )
% 12.39/6.87 & ( ord_less_real @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_999_add__mono__thms__linordered__field_I4_J,axiom,
% 12.39/6.87 ! [I: rat,J: rat,K: rat,L2: rat] :
% 12.39/6.87 ( ( ( ord_less_eq_rat @ I @ J )
% 12.39/6.87 & ( ord_less_rat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1000_add__mono__thms__linordered__field_I4_J,axiom,
% 12.39/6.87 ! [I: nat,J: nat,K: nat,L2: nat] :
% 12.39/6.87 ( ( ( ord_less_eq_nat @ I @ J )
% 12.39/6.87 & ( ord_less_nat @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1001_add__mono__thms__linordered__field_I4_J,axiom,
% 12.39/6.87 ! [I: int,J: int,K: int,L2: int] :
% 12.39/6.87 ( ( ( ord_less_eq_int @ I @ J )
% 12.39/6.87 & ( ord_less_int @ K @ L2 ) )
% 12.39/6.87 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1002_diff__le__eq,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 12.39/6.87 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1003_diff__le__eq,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 12.39/6.87 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1004_diff__le__eq,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 12.39/6.87 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1005_le__diff__eq,axiom,
% 12.39/6.87 ! [A: real,C: real,B2: real] :
% 12.39/6.87 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1006_le__diff__eq,axiom,
% 12.39/6.87 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.87 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1007_le__diff__eq,axiom,
% 12.39/6.87 ! [A: int,C: int,B2: int] :
% 12.39/6.87 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1008_diff__add,axiom,
% 12.39/6.87 ! [A: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
% 12.39/6.87 = B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1009_le__add__diff,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1010_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 12.39/6.87 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1011_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 12.39/6.87 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1012_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
% 12.39/6.87 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1013_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
% 12.39/6.87 = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1014_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
% 12.39/6.87 = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1015_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 12.39/6.87 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1016_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 12.39/6.87 ! [A: nat,B2: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
% 12.39/6.87 = B2 ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1017_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 12.39/6.87 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.87 ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( ord_less_eq_nat @ A @ B2 )
% 12.39/6.87 => ( ( ( minus_minus_nat @ B2 @ A )
% 12.39/6.87 = C )
% 12.39/6.87 = ( B2
% 12.39/6.87 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1018_diff__less__eq,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( ord_less_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 12.39/6.87 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1019_diff__less__eq,axiom,
% 12.39/6.87 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.87 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 12.39/6.87 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1020_diff__less__eq,axiom,
% 12.39/6.87 ! [A: int,B2: int,C: int] :
% 12.39/6.87 ( ( ord_less_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 12.39/6.87 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1021_less__diff__eq,axiom,
% 12.39/6.87 ! [A: real,C: real,B2: real] :
% 12.39/6.87 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1022_less__diff__eq,axiom,
% 12.39/6.87 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.87 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1023_less__diff__eq,axiom,
% 12.39/6.87 ! [A: int,C: int,B2: int] :
% 12.39/6.87 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 12.39/6.87 = ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1024_real__average__minus__second,axiom,
% 12.39/6.87 ! [B2: real,A: real] :
% 12.39/6.87 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 12.39/6.87 = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1025_real__average__minus__first,axiom,
% 12.39/6.87 ! [A: real,B2: real] :
% 12.39/6.87 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 12.39/6.87 = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1026_times__divide__eq__left,axiom,
% 12.39/6.87 ! [B2: complex,C: complex,A: complex] :
% 12.39/6.87 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B2 @ C ) @ A )
% 12.39/6.87 = ( divide1717551699836669952omplex @ ( times_times_complex @ B2 @ A ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1027_times__divide__eq__left,axiom,
% 12.39/6.87 ! [B2: real,C: real,A: real] :
% 12.39/6.87 ( ( times_times_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 12.39/6.87 = ( divide_divide_real @ ( times_times_real @ B2 @ A ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1028_times__divide__eq__left,axiom,
% 12.39/6.87 ! [B2: rat,C: rat,A: rat] :
% 12.39/6.87 ( ( times_times_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 12.39/6.87 = ( divide_divide_rat @ ( times_times_rat @ B2 @ A ) @ C ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1029_divide__divide__eq__left,axiom,
% 12.39/6.87 ! [A: complex,B2: complex,C: complex] :
% 12.39/6.87 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
% 12.39/6.87 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).
% 12.39/6.87
% 12.39/6.87 thf(fact_1030_divide__divide__eq__left,axiom,
% 12.39/6.87 ! [A: real,B2: real,C: real] :
% 12.39/6.87 ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
% 12.39/6.87 = ( divide_divide_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1031_divide__divide__eq__left,axiom,
% 12.39/6.88 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.88 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
% 12.39/6.88 = ( divide_divide_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1032_divide__divide__eq__right,axiom,
% 12.39/6.88 ! [A: complex,B2: complex,C: complex] :
% 12.39/6.88 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 12.39/6.88 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1033_divide__divide__eq__right,axiom,
% 12.39/6.88 ! [A: real,B2: real,C: real] :
% 12.39/6.88 ( ( divide_divide_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 12.39/6.88 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1034_divide__divide__eq__right,axiom,
% 12.39/6.88 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.88 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 12.39/6.88 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1035_times__divide__eq__right,axiom,
% 12.39/6.88 ! [A: complex,B2: complex,C: complex] :
% 12.39/6.88 ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 12.39/6.88 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1036_times__divide__eq__right,axiom,
% 12.39/6.88 ! [A: real,B2: real,C: real] :
% 12.39/6.88 ( ( times_times_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 12.39/6.88 = ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1037_times__divide__eq__right,axiom,
% 12.39/6.88 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.88 ( ( times_times_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 12.39/6.88 = ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1038_less__half__sum,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ord_less_real @ A @ B2 )
% 12.39/6.88 => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1039_less__half__sum,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ord_less_rat @ A @ B2 )
% 12.39/6.88 => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1040_gt__half__sum,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ord_less_real @ A @ B2 )
% 12.39/6.88 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1041_gt__half__sum,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ord_less_rat @ A @ B2 )
% 12.39/6.88 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1042_discrete,axiom,
% 12.39/6.88 ( ord_less_nat
% 12.39/6.88 = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1043_discrete,axiom,
% 12.39/6.88 ( ord_less_int
% 12.39/6.88 = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1044_low__def,axiom,
% 12.39/6.88 ( vEBT_VEBT_low
% 12.39/6.88 = ( ^ [X: nat,N2: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1045_zdiv__numeral__Bit0,axiom,
% 12.39/6.88 ! [V: num,W: num] :
% 12.39/6.88 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 12.39/6.88 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1046_mod__mod__trivial,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1047_mod__mod__trivial,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1048_mod__mod__trivial,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1049_mod__add__self2,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1050_mod__add__self2,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1051_mod__add__self2,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1052_mod__add__self1,axiom,
% 12.39/6.88 ! [B2: nat,A: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1053_mod__add__self1,axiom,
% 12.39/6.88 ! [B2: int,A: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1054_mod__add__self1,axiom,
% 12.39/6.88 ! [B2: code_integer,A: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B2 @ A ) @ B2 )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1055_minus__mod__self2,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1056_minus__mod__self2,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1057_real__divide__square__eq,axiom,
% 12.39/6.88 ! [R2: real,A: real] :
% 12.39/6.88 ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
% 12.39/6.88 = ( divide_divide_real @ A @ R2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1058_mod__less,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( ord_less_nat @ M @ N )
% 12.39/6.88 => ( ( modulo_modulo_nat @ M @ N )
% 12.39/6.88 = M ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1059_mod__mult__self4,axiom,
% 12.39/6.88 ! [B2: nat,C: nat,A: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1060_mod__mult__self4,axiom,
% 12.39/6.88 ! [B2: int,C: int,A: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1061_mod__mult__self4,axiom,
% 12.39/6.88 ! [B2: code_integer,C: code_integer,A: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ C ) @ A ) @ B2 )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1062_mod__mult__self3,axiom,
% 12.39/6.88 ! [C: nat,B2: nat,A: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1063_mod__mult__self3,axiom,
% 12.39/6.88 ! [C: int,B2: int,A: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1064_mod__mult__self3,axiom,
% 12.39/6.88 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B2 ) @ A ) @ B2 )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1065_mod__mult__self2,axiom,
% 12.39/6.88 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1066_mod__mult__self2,axiom,
% 12.39/6.88 ! [A: int,B2: int,C: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1067_mod__mult__self2,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) @ B2 )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1068_mod__mult__self1,axiom,
% 12.39/6.88 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1069_mod__mult__self1,axiom,
% 12.39/6.88 ! [A: int,C: int,B2: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1070_mod__mult__self1,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) @ B2 )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1071_Suc__mod__mult__self1,axiom,
% 12.39/6.88 ! [M: nat,K: nat,N: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 12.39/6.88 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1072_Suc__mod__mult__self2,axiom,
% 12.39/6.88 ! [M: nat,N: nat,K: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 12.39/6.88 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1073_Suc__mod__mult__self3,axiom,
% 12.39/6.88 ! [K: nat,N: nat,M: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 12.39/6.88 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1074_Suc__mod__mult__self4,axiom,
% 12.39/6.88 ! [N: nat,K: nat,M: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 12.39/6.88 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1075_bits__one__mod__two__eq__one,axiom,
% 12.39/6.88 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.88 = one_one_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1076_bits__one__mod__two__eq__one,axiom,
% 12.39/6.88 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.88 = one_one_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1077_bits__one__mod__two__eq__one,axiom,
% 12.39/6.88 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.39/6.88 = one_one_Code_integer ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1078_one__mod__two__eq__one,axiom,
% 12.39/6.88 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.88 = one_one_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1079_one__mod__two__eq__one,axiom,
% 12.39/6.88 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.88 = one_one_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1080_one__mod__two__eq__one,axiom,
% 12.39/6.88 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.39/6.88 = one_one_Code_integer ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1081_mod2__Suc__Suc,axiom,
% 12.39/6.88 ! [M: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.88 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1082_Suc__times__numeral__mod__eq,axiom,
% 12.39/6.88 ! [K: num,N: nat] :
% 12.39/6.88 ( ( ( numeral_numeral_nat @ K )
% 12.39/6.88 != one_one_nat )
% 12.39/6.88 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 12.39/6.88 = one_one_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1083_mod__mult__right__eq,axiom,
% 12.39/6.88 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1084_mod__mult__right__eq,axiom,
% 12.39/6.88 ! [A: int,B2: int,C: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1085_mod__mult__right__eq,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1086_mod__mult__left__eq,axiom,
% 12.39/6.88 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1087_mod__mult__left__eq,axiom,
% 12.39/6.88 ! [A: int,C: int,B2: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1088_mod__mult__left__eq,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1089_mult__mod__right,axiom,
% 12.39/6.88 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.88 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 12.39/6.88 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1090_mult__mod__right,axiom,
% 12.39/6.88 ! [C: int,A: int,B2: int] :
% 12.39/6.88 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 12.39/6.88 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1091_mult__mod__right,axiom,
% 12.39/6.88 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1092_mod__mult__mult2,axiom,
% 12.39/6.88 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 12.39/6.88 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1093_mod__mult__mult2,axiom,
% 12.39/6.88 ! [A: int,C: int,B2: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 12.39/6.88 = ( times_times_int @ ( modulo_modulo_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1094_mod__mult__mult2,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 12.39/6.88 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1095_mod__mult__cong,axiom,
% 12.39/6.88 ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
% 12.39/6.88 ( ( ( modulo_modulo_nat @ A @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ A2 @ C ) )
% 12.39/6.88 => ( ( ( modulo_modulo_nat @ B2 @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ B @ C ) )
% 12.39/6.88 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B ) @ C ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1096_mod__mult__cong,axiom,
% 12.39/6.88 ! [A: int,C: int,A2: int,B2: int,B: int] :
% 12.39/6.88 ( ( ( modulo_modulo_int @ A @ C )
% 12.39/6.88 = ( modulo_modulo_int @ A2 @ C ) )
% 12.39/6.88 => ( ( ( modulo_modulo_int @ B2 @ C )
% 12.39/6.88 = ( modulo_modulo_int @ B @ C ) )
% 12.39/6.88 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( times_times_int @ A2 @ B ) @ C ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1097_mod__mult__cong,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 12.39/6.88 ( ( ( modulo364778990260209775nteger @ A @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A2 @ C ) )
% 12.39/6.88 => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ B @ C ) )
% 12.39/6.88 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ C ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1098_mod__mult__eq,axiom,
% 12.39/6.88 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1099_mod__mult__eq,axiom,
% 12.39/6.88 ! [A: int,C: int,B2: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1100_mod__mult__eq,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1101_mod__add__right__eq,axiom,
% 12.39/6.88 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1102_mod__add__right__eq,axiom,
% 12.39/6.88 ! [A: int,B2: int,C: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1103_mod__add__right__eq,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1104_mod__add__left__eq,axiom,
% 12.39/6.88 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1105_mod__add__left__eq,axiom,
% 12.39/6.88 ! [A: int,C: int,B2: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1106_mod__add__left__eq,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1107_mod__add__cong,axiom,
% 12.39/6.88 ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
% 12.39/6.88 ( ( ( modulo_modulo_nat @ A @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ A2 @ C ) )
% 12.39/6.88 => ( ( ( modulo_modulo_nat @ B2 @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ B @ C ) )
% 12.39/6.88 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B ) @ C ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1108_mod__add__cong,axiom,
% 12.39/6.88 ! [A: int,C: int,A2: int,B2: int,B: int] :
% 12.39/6.88 ( ( ( modulo_modulo_int @ A @ C )
% 12.39/6.88 = ( modulo_modulo_int @ A2 @ C ) )
% 12.39/6.88 => ( ( ( modulo_modulo_int @ B2 @ C )
% 12.39/6.88 = ( modulo_modulo_int @ B @ C ) )
% 12.39/6.88 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1109_mod__add__cong,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 12.39/6.88 ( ( ( modulo364778990260209775nteger @ A @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A2 @ C ) )
% 12.39/6.88 => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ B @ C ) )
% 12.39/6.88 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1110_mod__add__eq,axiom,
% 12.39/6.88 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1111_mod__add__eq,axiom,
% 12.39/6.88 ! [A: int,C: int,B2: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1112_mod__add__eq,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1113_mod__diff__right__eq,axiom,
% 12.39/6.88 ! [A: int,B2: int,C: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1114_mod__diff__right__eq,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1115_mod__diff__left__eq,axiom,
% 12.39/6.88 ! [A: int,C: int,B2: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1116_mod__diff__left__eq,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1117_mod__diff__cong,axiom,
% 12.39/6.88 ! [A: int,C: int,A2: int,B2: int,B: int] :
% 12.39/6.88 ( ( ( modulo_modulo_int @ A @ C )
% 12.39/6.88 = ( modulo_modulo_int @ A2 @ C ) )
% 12.39/6.88 => ( ( ( modulo_modulo_int @ B2 @ C )
% 12.39/6.88 = ( modulo_modulo_int @ B @ C ) )
% 12.39/6.88 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1118_mod__diff__cong,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 12.39/6.88 ( ( ( modulo364778990260209775nteger @ A @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A2 @ C ) )
% 12.39/6.88 => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ B @ C ) )
% 12.39/6.88 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1119_mod__diff__eq,axiom,
% 12.39/6.88 ! [A: int,C: int,B2: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1120_mod__diff__eq,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1121_power__mod,axiom,
% 12.39/6.88 ! [A: nat,B2: nat,N: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B2 ) @ N ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1122_power__mod,axiom,
% 12.39/6.88 ! [A: int,B2: int,N: nat] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B2 ) @ N ) @ B2 )
% 12.39/6.88 = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1123_power__mod,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer,N: nat] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ N ) @ B2 )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1124_mod__Suc__Suc__eq,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 12.39/6.88 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1125_mod__Suc__eq,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 12.39/6.88 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1126_mod__less__eq__dividend,axiom,
% 12.39/6.88 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1127_cong__exp__iff__simps_I9_J,axiom,
% 12.39/6.88 ! [M: num,Q2: num,N: num] :
% 12.39/6.88 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.39/6.88 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 12.39/6.88 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 12.39/6.88 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1128_cong__exp__iff__simps_I9_J,axiom,
% 12.39/6.88 ! [M: num,Q2: num,N: num] :
% 12.39/6.88 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.39/6.88 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 12.39/6.88 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 12.39/6.88 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1129_cong__exp__iff__simps_I9_J,axiom,
% 12.39/6.88 ! [M: num,Q2: num,N: num] :
% 12.39/6.88 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 12.39/6.88 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1130_cong__exp__iff__simps_I4_J,axiom,
% 12.39/6.88 ! [M: num,N: num] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 12.39/6.88 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1131_cong__exp__iff__simps_I4_J,axiom,
% 12.39/6.88 ! [M: num,N: num] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 12.39/6.88 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1132_cong__exp__iff__simps_I4_J,axiom,
% 12.39/6.88 ! [M: num,N: num] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 12.39/6.88 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1133_mod__geq,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ~ ( ord_less_nat @ M @ N )
% 12.39/6.88 => ( ( modulo_modulo_nat @ M @ N )
% 12.39/6.88 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1134_nat__mod__eq__iff,axiom,
% 12.39/6.88 ! [X4: nat,N: nat,Y3: nat] :
% 12.39/6.88 ( ( ( modulo_modulo_nat @ X4 @ N )
% 12.39/6.88 = ( modulo_modulo_nat @ Y3 @ N ) )
% 12.39/6.88 = ( ? [Q1: nat,Q22: nat] :
% 12.39/6.88 ( ( plus_plus_nat @ X4 @ ( times_times_nat @ N @ Q1 ) )
% 12.39/6.88 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1135_cong__exp__iff__simps_I6_J,axiom,
% 12.39/6.88 ! [Q2: num,N: num] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.39/6.88 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1136_cong__exp__iff__simps_I6_J,axiom,
% 12.39/6.88 ! [Q2: num,N: num] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.39/6.88 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1137_cong__exp__iff__simps_I6_J,axiom,
% 12.39/6.88 ! [Q2: num,N: num] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.39/6.88 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1138_cong__exp__iff__simps_I8_J,axiom,
% 12.39/6.88 ! [M: num,Q2: num] :
% 12.39/6.88 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.39/6.88 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1139_cong__exp__iff__simps_I8_J,axiom,
% 12.39/6.88 ! [M: num,Q2: num] :
% 12.39/6.88 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.39/6.88 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1140_cong__exp__iff__simps_I8_J,axiom,
% 12.39/6.88 ! [M: num,Q2: num] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.39/6.88 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1141_mod__eqE,axiom,
% 12.39/6.88 ! [A: int,C: int,B2: int] :
% 12.39/6.88 ( ( ( modulo_modulo_int @ A @ C )
% 12.39/6.88 = ( modulo_modulo_int @ B2 @ C ) )
% 12.39/6.88 => ~ ! [D3: int] :
% 12.39/6.88 ( B2
% 12.39/6.88 != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1142_mod__eqE,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( ( modulo364778990260209775nteger @ A @ C )
% 12.39/6.88 = ( modulo364778990260209775nteger @ B2 @ C ) )
% 12.39/6.88 => ~ ! [D3: code_integer] :
% 12.39/6.88 ( B2
% 12.39/6.88 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1143_div__add1__eq,axiom,
% 12.39/6.88 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.88 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 12.39/6.88 = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1144_div__add1__eq,axiom,
% 12.39/6.88 ! [A: int,B2: int,C: int] :
% 12.39/6.88 ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 12.39/6.88 = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1145_div__add1__eq,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.88 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 12.39/6.88 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1146_mod__induct,axiom,
% 12.39/6.88 ! [P: nat > $o,N: nat,P2: nat,M: nat] :
% 12.39/6.88 ( ( P @ N )
% 12.39/6.88 => ( ( ord_less_nat @ N @ P2 )
% 12.39/6.88 => ( ( ord_less_nat @ M @ P2 )
% 12.39/6.88 => ( ! [N4: nat] :
% 12.39/6.88 ( ( ord_less_nat @ N4 @ P2 )
% 12.39/6.88 => ( ( P @ N4 )
% 12.39/6.88 => ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P2 ) ) ) )
% 12.39/6.88 => ( P @ M ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1147_mod__Suc__le__divisor,axiom,
% 12.39/6.88 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1148_nat__mod__eq__lemma,axiom,
% 12.39/6.88 ! [X4: nat,N: nat,Y3: nat] :
% 12.39/6.88 ( ( ( modulo_modulo_nat @ X4 @ N )
% 12.39/6.88 = ( modulo_modulo_nat @ Y3 @ N ) )
% 12.39/6.88 => ( ( ord_less_eq_nat @ Y3 @ X4 )
% 12.39/6.88 => ? [Q3: nat] :
% 12.39/6.88 ( X4
% 12.39/6.88 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1149_mod__eq__nat2E,axiom,
% 12.39/6.88 ! [M: nat,Q2: nat,N: nat] :
% 12.39/6.88 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 12.39/6.88 = ( modulo_modulo_nat @ N @ Q2 ) )
% 12.39/6.88 => ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.88 => ~ ! [S3: nat] :
% 12.39/6.88 ( N
% 12.39/6.88 != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1150_mod__eq__nat1E,axiom,
% 12.39/6.88 ! [M: nat,Q2: nat,N: nat] :
% 12.39/6.88 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 12.39/6.88 = ( modulo_modulo_nat @ N @ Q2 ) )
% 12.39/6.88 => ( ( ord_less_eq_nat @ N @ M )
% 12.39/6.88 => ~ ! [S3: nat] :
% 12.39/6.88 ( M
% 12.39/6.88 != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1151_mod__if,axiom,
% 12.39/6.88 ( modulo_modulo_nat
% 12.39/6.88 = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N2 ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1152_le__mod__geq,axiom,
% 12.39/6.88 ! [N: nat,M: nat] :
% 12.39/6.88 ( ( ord_less_eq_nat @ N @ M )
% 12.39/6.88 => ( ( modulo_modulo_nat @ M @ N )
% 12.39/6.88 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1153_mult__div__mod__eq,axiom,
% 12.39/6.88 ! [B2: nat,A: nat] :
% 12.39/6.88 ( ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1154_mult__div__mod__eq,axiom,
% 12.39/6.88 ! [B2: int,A: int] :
% 12.39/6.88 ( ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1155_mult__div__mod__eq,axiom,
% 12.39/6.88 ! [B2: code_integer,A: code_integer] :
% 12.39/6.88 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1156_mod__mult__div__eq,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1157_mod__mult__div__eq,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1158_mod__mult__div__eq,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1159_mod__div__mult__eq,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1160_mod__div__mult__eq,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1161_mod__div__mult__eq,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1162_div__mult__mod__eq,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1163_div__mult__mod__eq,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1164_div__mult__mod__eq,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1165_mod__div__decomp,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( A
% 12.39/6.88 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1166_mod__div__decomp,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( A
% 12.39/6.88 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1167_mod__div__decomp,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( A
% 12.39/6.88 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1168_cancel__div__mod__rules_I1_J,axiom,
% 12.39/6.88 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.88 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
% 12.39/6.88 = ( plus_plus_nat @ A @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1169_cancel__div__mod__rules_I1_J,axiom,
% 12.39/6.88 ! [A: int,B2: int,C: int] :
% 12.39/6.88 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
% 12.39/6.88 = ( plus_plus_int @ A @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1170_cancel__div__mod__rules_I1_J,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.88 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
% 12.39/6.88 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1171_cancel__div__mod__rules_I2_J,axiom,
% 12.39/6.88 ! [B2: nat,A: nat,C: nat] :
% 12.39/6.88 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
% 12.39/6.88 = ( plus_plus_nat @ A @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1172_cancel__div__mod__rules_I2_J,axiom,
% 12.39/6.88 ! [B2: int,A: int,C: int] :
% 12.39/6.88 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
% 12.39/6.88 = ( plus_plus_int @ A @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1173_cancel__div__mod__rules_I2_J,axiom,
% 12.39/6.88 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 12.39/6.88 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
% 12.39/6.88 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1174_div__mult1__eq,axiom,
% 12.39/6.88 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.88 ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 12.39/6.88 = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1175_div__mult1__eq,axiom,
% 12.39/6.88 ! [A: int,B2: int,C: int] :
% 12.39/6.88 ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C )
% 12.39/6.88 = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1176_div__mult1__eq,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.88 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 12.39/6.88 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1177_minus__mult__div__eq__mod,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( minus_minus_nat @ A @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
% 12.39/6.88 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1178_minus__mult__div__eq__mod,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( minus_minus_int @ A @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
% 12.39/6.88 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1179_minus__mult__div__eq__mod,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1180_minus__mod__eq__mult__div,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
% 12.39/6.88 = ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1181_minus__mod__eq__mult__div,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
% 12.39/6.88 = ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1182_minus__mod__eq__mult__div,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 12.39/6.88 = ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1183_minus__mod__eq__div__mult,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
% 12.39/6.88 = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1184_minus__mod__eq__div__mult,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
% 12.39/6.88 = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1185_minus__mod__eq__div__mult,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 12.39/6.88 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1186_minus__div__mult__eq__mod,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
% 12.39/6.88 = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1187_minus__div__mult__eq__mod,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
% 12.39/6.88 = ( modulo_modulo_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1188_minus__div__mult__eq__mod,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
% 12.39/6.88 = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1189_mod__mult2__eq,axiom,
% 12.39/6.88 ! [M: nat,N: nat,Q2: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 12.39/6.88 = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1190_div__mod__decomp,axiom,
% 12.39/6.88 ! [A3: nat,N: nat] :
% 12.39/6.88 ( A3
% 12.39/6.88 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ N ) @ N ) @ ( modulo_modulo_nat @ A3 @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1191_modulo__nat__def,axiom,
% 12.39/6.88 ( modulo_modulo_nat
% 12.39/6.88 = ( ^ [M6: nat,N2: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N2 ) @ N2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1192_linordered__field__no__lb,axiom,
% 12.39/6.88 ! [X5: real] :
% 12.39/6.88 ? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1193_linordered__field__no__lb,axiom,
% 12.39/6.88 ! [X5: rat] :
% 12.39/6.88 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X5 ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1194_linordered__field__no__ub,axiom,
% 12.39/6.88 ! [X5: real] :
% 12.39/6.88 ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1195_linordered__field__no__ub,axiom,
% 12.39/6.88 ! [X5: rat] :
% 12.39/6.88 ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1196_bounded__Max__nat,axiom,
% 12.39/6.88 ! [P: nat > $o,X4: nat,M7: nat] :
% 12.39/6.88 ( ( P @ X4 )
% 12.39/6.88 => ( ! [X3: nat] :
% 12.39/6.88 ( ( P @ X3 )
% 12.39/6.88 => ( ord_less_eq_nat @ X3 @ M7 ) )
% 12.39/6.88 => ~ ! [M4: nat] :
% 12.39/6.88 ( ( P @ M4 )
% 12.39/6.88 => ~ ! [X5: nat] :
% 12.39/6.88 ( ( P @ X5 )
% 12.39/6.88 => ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1197_div__exp__mod__exp__eq,axiom,
% 12.39/6.88 ! [A: nat,N: nat,M: nat] :
% 12.39/6.88 ( ( 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 ) )
% 12.39/6.88 = ( 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 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1198_div__exp__mod__exp__eq,axiom,
% 12.39/6.88 ! [A: int,N: nat,M: nat] :
% 12.39/6.88 ( ( 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 ) )
% 12.39/6.88 = ( 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 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1199_div__exp__mod__exp__eq,axiom,
% 12.39/6.88 ! [A: code_integer,N: nat,M: nat] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 12.39/6.88 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1200_mult__exp__mod__exp__eq,axiom,
% 12.39/6.88 ! [M: nat,N: nat,A: nat] :
% 12.39/6.88 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.88 => ( ( 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 ) )
% 12.39/6.88 = ( 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 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1201_mult__exp__mod__exp__eq,axiom,
% 12.39/6.88 ! [M: nat,N: nat,A: int] :
% 12.39/6.88 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.88 => ( ( 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 ) )
% 12.39/6.88 = ( 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 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1202_mult__exp__mod__exp__eq,axiom,
% 12.39/6.88 ! [M: nat,N: nat,A: code_integer] :
% 12.39/6.88 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.88 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.88 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1203_divide__divide__eq__left_H,axiom,
% 12.39/6.88 ! [A: complex,B2: complex,C: complex] :
% 12.39/6.88 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
% 12.39/6.88 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1204_divide__divide__eq__left_H,axiom,
% 12.39/6.88 ! [A: real,B2: real,C: real] :
% 12.39/6.88 ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
% 12.39/6.88 = ( divide_divide_real @ A @ ( times_times_real @ C @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1205_divide__divide__eq__left_H,axiom,
% 12.39/6.88 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.88 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
% 12.39/6.88 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1206_divide__divide__times__eq,axiom,
% 12.39/6.88 ! [X4: complex,Y3: complex,Z: complex,W: complex] :
% 12.39/6.88 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 12.39/6.88 = ( divide1717551699836669952omplex @ ( times_times_complex @ X4 @ W ) @ ( times_times_complex @ Y3 @ Z ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1207_divide__divide__times__eq,axiom,
% 12.39/6.88 ! [X4: real,Y3: real,Z: real,W: real] :
% 12.39/6.88 ( ( divide_divide_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 12.39/6.88 = ( divide_divide_real @ ( times_times_real @ X4 @ W ) @ ( times_times_real @ Y3 @ Z ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1208_divide__divide__times__eq,axiom,
% 12.39/6.88 ! [X4: rat,Y3: rat,Z: rat,W: rat] :
% 12.39/6.88 ( ( divide_divide_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 12.39/6.88 = ( divide_divide_rat @ ( times_times_rat @ X4 @ W ) @ ( times_times_rat @ Y3 @ Z ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1209_times__divide__times__eq,axiom,
% 12.39/6.88 ! [X4: complex,Y3: complex,Z: complex,W: complex] :
% 12.39/6.88 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 12.39/6.88 = ( divide1717551699836669952omplex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ Y3 @ W ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1210_times__divide__times__eq,axiom,
% 12.39/6.88 ! [X4: real,Y3: real,Z: real,W: real] :
% 12.39/6.88 ( ( times_times_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 12.39/6.88 = ( divide_divide_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y3 @ W ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1211_times__divide__times__eq,axiom,
% 12.39/6.88 ! [X4: rat,Y3: rat,Z: rat,W: rat] :
% 12.39/6.88 ( ( times_times_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 12.39/6.88 = ( divide_divide_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y3 @ W ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1212_add__divide__distrib,axiom,
% 12.39/6.88 ! [A: complex,B2: complex,C: complex] :
% 12.39/6.88 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B2 ) @ C )
% 12.39/6.88 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1213_add__divide__distrib,axiom,
% 12.39/6.88 ! [A: real,B2: real,C: real] :
% 12.39/6.88 ( ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 12.39/6.88 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1214_add__divide__distrib,axiom,
% 12.39/6.88 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.88 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 12.39/6.88 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1215_diff__divide__distrib,axiom,
% 12.39/6.88 ! [A: complex,B2: complex,C: complex] :
% 12.39/6.88 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B2 ) @ C )
% 12.39/6.88 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1216_diff__divide__distrib,axiom,
% 12.39/6.88 ! [A: real,B2: real,C: real] :
% 12.39/6.88 ( ( divide_divide_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 12.39/6.88 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1217_diff__divide__distrib,axiom,
% 12.39/6.88 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.88 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 12.39/6.88 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1218_unset__bit__Suc,axiom,
% 12.39/6.88 ! [N: nat,A: code_integer] :
% 12.39/6.88 ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 12.39/6.88 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1219_unset__bit__Suc,axiom,
% 12.39/6.88 ! [N: nat,A: int] :
% 12.39/6.88 ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 12.39/6.88 = ( 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 ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1220_unset__bit__Suc,axiom,
% 12.39/6.88 ! [N: nat,A: nat] :
% 12.39/6.88 ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 12.39/6.88 = ( 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 ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1221_flip__bit__Suc,axiom,
% 12.39/6.88 ! [N: nat,A: code_integer] :
% 12.39/6.88 ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 12.39/6.88 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1222_flip__bit__Suc,axiom,
% 12.39/6.88 ! [N: nat,A: int] :
% 12.39/6.88 ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 12.39/6.88 = ( 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 ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1223_flip__bit__Suc,axiom,
% 12.39/6.88 ! [N: nat,A: nat] :
% 12.39/6.88 ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 12.39/6.88 = ( 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 ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1224_set__bit__Suc,axiom,
% 12.39/6.88 ! [N: nat,A: code_integer] :
% 12.39/6.88 ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 12.39/6.88 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1225_set__bit__Suc,axiom,
% 12.39/6.88 ! [N: nat,A: int] :
% 12.39/6.88 ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 12.39/6.88 = ( 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 ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1226_set__bit__Suc,axiom,
% 12.39/6.88 ! [N: nat,A: nat] :
% 12.39/6.88 ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 12.39/6.88 = ( 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 ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1227_dbl__simps_I3_J,axiom,
% 12.39/6.88 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 12.39/6.88 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1228_dbl__simps_I3_J,axiom,
% 12.39/6.88 ( ( neg_numeral_dbl_real @ one_one_real )
% 12.39/6.88 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1229_dbl__simps_I3_J,axiom,
% 12.39/6.88 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 12.39/6.88 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1230_dbl__simps_I3_J,axiom,
% 12.39/6.88 ( ( neg_numeral_dbl_int @ one_one_int )
% 12.39/6.88 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1231_divmod__digit__1_I1_J,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.39/6.88 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 12.39/6.88 => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 12.39/6.88 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_Code_integer )
% 12.39/6.88 = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1232_divmod__digit__1_I1_J,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.39/6.88 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.39/6.88 => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 12.39/6.88 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_nat )
% 12.39/6.88 = ( divide_divide_nat @ A @ B2 ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1233_divmod__digit__1_I1_J,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.39/6.88 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.39/6.88 => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 12.39/6.88 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_int )
% 12.39/6.88 = ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1234_signed__take__bit__Suc,axiom,
% 12.39/6.88 ! [N: nat,A: code_integer] :
% 12.39/6.88 ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 12.39/6.88 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1235_signed__take__bit__Suc,axiom,
% 12.39/6.88 ! [N: nat,A: int] :
% 12.39/6.88 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 12.39/6.88 = ( 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 ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1236_power__numeral,axiom,
% 12.39/6.88 ! [K: num,L2: num] :
% 12.39/6.88 ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L2 ) )
% 12.39/6.88 = ( numera6690914467698888265omplex @ ( pow @ K @ L2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1237_power__numeral,axiom,
% 12.39/6.88 ! [K: num,L2: num] :
% 12.39/6.88 ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L2 ) )
% 12.39/6.88 = ( numeral_numeral_real @ ( pow @ K @ L2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1238_power__numeral,axiom,
% 12.39/6.88 ! [K: num,L2: num] :
% 12.39/6.88 ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L2 ) )
% 12.39/6.88 = ( numeral_numeral_rat @ ( pow @ K @ L2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1239_power__numeral,axiom,
% 12.39/6.88 ! [K: num,L2: num] :
% 12.39/6.88 ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L2 ) )
% 12.39/6.88 = ( numeral_numeral_nat @ ( pow @ K @ L2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1240_power__numeral,axiom,
% 12.39/6.88 ! [K: num,L2: num] :
% 12.39/6.88 ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L2 ) )
% 12.39/6.88 = ( numeral_numeral_int @ ( pow @ K @ L2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1241_arith__geo__mean,axiom,
% 12.39/6.88 ! [U: real,X4: real,Y3: real] :
% 12.39/6.88 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.88 = ( times_times_real @ X4 @ Y3 ) )
% 12.39/6.88 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.39/6.88 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.39/6.88 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1242_arith__geo__mean,axiom,
% 12.39/6.88 ! [U: rat,X4: rat,Y3: rat] :
% 12.39/6.88 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.88 = ( times_times_rat @ X4 @ Y3 ) )
% 12.39/6.88 => ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.39/6.88 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.39/6.88 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1243_odd__two__times__div__two__nat,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.39/6.88 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.88 = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1244_less__nat__zero__code,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1245_neq0__conv,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( N != zero_zero_nat )
% 12.39/6.88 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1246_bot__nat__0_Onot__eq__extremum,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( A != zero_zero_nat )
% 12.39/6.88 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1247_le0,axiom,
% 12.39/6.88 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1248_bot__nat__0_Oextremum,axiom,
% 12.39/6.88 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1249_Nat_Oadd__0__right,axiom,
% 12.39/6.88 ! [M: nat] :
% 12.39/6.88 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 12.39/6.88 = M ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1250_add__is__0,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( ( plus_plus_nat @ M @ N )
% 12.39/6.88 = zero_zero_nat )
% 12.39/6.88 = ( ( M = zero_zero_nat )
% 12.39/6.88 & ( N = zero_zero_nat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1251_diff__self__eq__0,axiom,
% 12.39/6.88 ! [M: nat] :
% 12.39/6.88 ( ( minus_minus_nat @ M @ M )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1252_diff__0__eq__0,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1253_mult__cancel2,axiom,
% 12.39/6.88 ! [M: nat,K: nat,N: nat] :
% 12.39/6.88 ( ( ( times_times_nat @ M @ K )
% 12.39/6.88 = ( times_times_nat @ N @ K ) )
% 12.39/6.88 = ( ( M = N )
% 12.39/6.88 | ( K = zero_zero_nat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1254_mult__cancel1,axiom,
% 12.39/6.88 ! [K: nat,M: nat,N: nat] :
% 12.39/6.88 ( ( ( times_times_nat @ K @ M )
% 12.39/6.88 = ( times_times_nat @ K @ N ) )
% 12.39/6.88 = ( ( M = N )
% 12.39/6.88 | ( K = zero_zero_nat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1255_mult__0__right,axiom,
% 12.39/6.88 ! [M: nat] :
% 12.39/6.88 ( ( times_times_nat @ M @ zero_zero_nat )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1256_mult__is__0,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( ( times_times_nat @ M @ N )
% 12.39/6.88 = zero_zero_nat )
% 12.39/6.88 = ( ( M = zero_zero_nat )
% 12.39/6.88 | ( N = zero_zero_nat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1257_div__pos__pos__trivial,axiom,
% 12.39/6.88 ! [K: int,L2: int] :
% 12.39/6.88 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.39/6.88 => ( ( ord_less_int @ K @ L2 )
% 12.39/6.88 => ( ( divide_divide_int @ K @ L2 )
% 12.39/6.88 = zero_zero_int ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1258_div__neg__neg__trivial,axiom,
% 12.39/6.88 ! [K: int,L2: int] :
% 12.39/6.88 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 12.39/6.88 => ( ( ord_less_int @ L2 @ K )
% 12.39/6.88 => ( ( divide_divide_int @ K @ L2 )
% 12.39/6.88 = zero_zero_int ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1259_i0__less,axiom,
% 12.39/6.88 ! [N: extended_enat] :
% 12.39/6.88 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 12.39/6.88 = ( N != zero_z5237406670263579293d_enat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1260_idiff__0,axiom,
% 12.39/6.88 ! [N: extended_enat] :
% 12.39/6.88 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 12.39/6.88 = zero_z5237406670263579293d_enat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1261_idiff__0__right,axiom,
% 12.39/6.88 ! [N: extended_enat] :
% 12.39/6.88 ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 12.39/6.88 = N ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1262_le__zero__eq,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 12.39/6.88 = ( N = zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1263_not__gr__zero,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 12.39/6.88 = ( N = zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1264_mult__cancel__right,axiom,
% 12.39/6.88 ! [A: real,C: real,B2: real] :
% 12.39/6.88 ( ( ( times_times_real @ A @ C )
% 12.39/6.88 = ( times_times_real @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_real )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1265_mult__cancel__right,axiom,
% 12.39/6.88 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.88 ( ( ( times_times_rat @ A @ C )
% 12.39/6.88 = ( times_times_rat @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_rat )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1266_mult__cancel__right,axiom,
% 12.39/6.88 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.88 ( ( ( times_times_nat @ A @ C )
% 12.39/6.88 = ( times_times_nat @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_nat )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1267_mult__cancel__right,axiom,
% 12.39/6.88 ! [A: int,C: int,B2: int] :
% 12.39/6.88 ( ( ( times_times_int @ A @ C )
% 12.39/6.88 = ( times_times_int @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_int )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1268_mult__cancel__left,axiom,
% 12.39/6.88 ! [C: real,A: real,B2: real] :
% 12.39/6.88 ( ( ( times_times_real @ C @ A )
% 12.39/6.88 = ( times_times_real @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_real )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1269_mult__cancel__left,axiom,
% 12.39/6.88 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.88 ( ( ( times_times_rat @ C @ A )
% 12.39/6.88 = ( times_times_rat @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_rat )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1270_mult__cancel__left,axiom,
% 12.39/6.88 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.88 ( ( ( times_times_nat @ C @ A )
% 12.39/6.88 = ( times_times_nat @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_nat )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1271_mult__cancel__left,axiom,
% 12.39/6.88 ! [C: int,A: int,B2: int] :
% 12.39/6.88 ( ( ( times_times_int @ C @ A )
% 12.39/6.88 = ( times_times_int @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_int )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1272_mult__eq__0__iff,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ( times_times_real @ A @ B2 )
% 12.39/6.88 = zero_zero_real )
% 12.39/6.88 = ( ( A = zero_zero_real )
% 12.39/6.88 | ( B2 = zero_zero_real ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1273_mult__eq__0__iff,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ( times_times_rat @ A @ B2 )
% 12.39/6.88 = zero_zero_rat )
% 12.39/6.88 = ( ( A = zero_zero_rat )
% 12.39/6.88 | ( B2 = zero_zero_rat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1274_mult__eq__0__iff,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( ( times_times_nat @ A @ B2 )
% 12.39/6.88 = zero_zero_nat )
% 12.39/6.88 = ( ( A = zero_zero_nat )
% 12.39/6.88 | ( B2 = zero_zero_nat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1275_mult__eq__0__iff,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( ( times_times_int @ A @ B2 )
% 12.39/6.88 = zero_zero_int )
% 12.39/6.88 = ( ( A = zero_zero_int )
% 12.39/6.88 | ( B2 = zero_zero_int ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1276_mult__zero__right,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( times_times_real @ A @ zero_zero_real )
% 12.39/6.88 = zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1277_mult__zero__right,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( times_times_rat @ A @ zero_zero_rat )
% 12.39/6.88 = zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1278_mult__zero__right,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( times_times_nat @ A @ zero_zero_nat )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1279_mult__zero__right,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( times_times_int @ A @ zero_zero_int )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1280_mult__zero__left,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( times_times_real @ zero_zero_real @ A )
% 12.39/6.88 = zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1281_mult__zero__left,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( times_times_rat @ zero_zero_rat @ A )
% 12.39/6.88 = zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1282_mult__zero__left,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( times_times_nat @ zero_zero_nat @ A )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1283_mult__zero__left,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( times_times_int @ zero_zero_int @ A )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1284_add__0,axiom,
% 12.39/6.88 ! [A: literal] :
% 12.39/6.88 ( ( plus_plus_literal @ zero_zero_literal @ A )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1285_add__0,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( plus_plus_real @ zero_zero_real @ A )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1286_add__0,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1287_add__0,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1288_add__0,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( plus_plus_int @ zero_zero_int @ A )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1289_zero__eq__add__iff__both__eq__0,axiom,
% 12.39/6.88 ! [X4: nat,Y3: nat] :
% 12.39/6.88 ( ( zero_zero_nat
% 12.39/6.88 = ( plus_plus_nat @ X4 @ Y3 ) )
% 12.39/6.88 = ( ( X4 = zero_zero_nat )
% 12.39/6.88 & ( Y3 = zero_zero_nat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1290_add__eq__0__iff__both__eq__0,axiom,
% 12.39/6.88 ! [X4: nat,Y3: nat] :
% 12.39/6.88 ( ( ( plus_plus_nat @ X4 @ Y3 )
% 12.39/6.88 = zero_zero_nat )
% 12.39/6.88 = ( ( X4 = zero_zero_nat )
% 12.39/6.88 & ( Y3 = zero_zero_nat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1291_add__cancel__right__right,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( A
% 12.39/6.88 = ( plus_plus_real @ A @ B2 ) )
% 12.39/6.88 = ( B2 = zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1292_add__cancel__right__right,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( A
% 12.39/6.88 = ( plus_plus_rat @ A @ B2 ) )
% 12.39/6.88 = ( B2 = zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1293_add__cancel__right__right,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( A
% 12.39/6.88 = ( plus_plus_nat @ A @ B2 ) )
% 12.39/6.88 = ( B2 = zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1294_add__cancel__right__right,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( A
% 12.39/6.88 = ( plus_plus_int @ A @ B2 ) )
% 12.39/6.88 = ( B2 = zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1295_add__cancel__right__left,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( A
% 12.39/6.88 = ( plus_plus_real @ B2 @ A ) )
% 12.39/6.88 = ( B2 = zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1296_add__cancel__right__left,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( A
% 12.39/6.88 = ( plus_plus_rat @ B2 @ A ) )
% 12.39/6.88 = ( B2 = zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1297_add__cancel__right__left,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( A
% 12.39/6.88 = ( plus_plus_nat @ B2 @ A ) )
% 12.39/6.88 = ( B2 = zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1298_add__cancel__right__left,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( A
% 12.39/6.88 = ( plus_plus_int @ B2 @ A ) )
% 12.39/6.88 = ( B2 = zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1299_add__cancel__left__right,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ( plus_plus_real @ A @ B2 )
% 12.39/6.88 = A )
% 12.39/6.88 = ( B2 = zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1300_add__cancel__left__right,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ( plus_plus_rat @ A @ B2 )
% 12.39/6.88 = A )
% 12.39/6.88 = ( B2 = zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1301_add__cancel__left__right,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( ( plus_plus_nat @ A @ B2 )
% 12.39/6.88 = A )
% 12.39/6.88 = ( B2 = zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1302_add__cancel__left__right,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( ( plus_plus_int @ A @ B2 )
% 12.39/6.88 = A )
% 12.39/6.88 = ( B2 = zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1303_add__cancel__left__left,axiom,
% 12.39/6.88 ! [B2: real,A: real] :
% 12.39/6.88 ( ( ( plus_plus_real @ B2 @ A )
% 12.39/6.88 = A )
% 12.39/6.88 = ( B2 = zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1304_add__cancel__left__left,axiom,
% 12.39/6.88 ! [B2: rat,A: rat] :
% 12.39/6.88 ( ( ( plus_plus_rat @ B2 @ A )
% 12.39/6.88 = A )
% 12.39/6.88 = ( B2 = zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1305_add__cancel__left__left,axiom,
% 12.39/6.88 ! [B2: nat,A: nat] :
% 12.39/6.88 ( ( ( plus_plus_nat @ B2 @ A )
% 12.39/6.88 = A )
% 12.39/6.88 = ( B2 = zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1306_add__cancel__left__left,axiom,
% 12.39/6.88 ! [B2: int,A: int] :
% 12.39/6.88 ( ( ( plus_plus_int @ B2 @ A )
% 12.39/6.88 = A )
% 12.39/6.88 = ( B2 = zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1307_double__zero__sym,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( zero_zero_real
% 12.39/6.88 = ( plus_plus_real @ A @ A ) )
% 12.39/6.88 = ( A = zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1308_double__zero__sym,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( zero_zero_rat
% 12.39/6.88 = ( plus_plus_rat @ A @ A ) )
% 12.39/6.88 = ( A = zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1309_double__zero__sym,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( zero_zero_int
% 12.39/6.88 = ( plus_plus_int @ A @ A ) )
% 12.39/6.88 = ( A = zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1310_add_Oright__neutral,axiom,
% 12.39/6.88 ! [A: literal] :
% 12.39/6.88 ( ( plus_plus_literal @ A @ zero_zero_literal )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1311_add_Oright__neutral,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( plus_plus_real @ A @ zero_zero_real )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1312_add_Oright__neutral,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1313_add_Oright__neutral,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1314_add_Oright__neutral,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( plus_plus_int @ A @ zero_zero_int )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1315_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( minus_minus_real @ A @ A )
% 12.39/6.88 = zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1316_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( minus_minus_rat @ A @ A )
% 12.39/6.88 = zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1317_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( minus_minus_nat @ A @ A )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1318_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( minus_minus_int @ A @ A )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1319_diff__zero,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( minus_minus_real @ A @ zero_zero_real )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1320_diff__zero,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1321_diff__zero,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1322_diff__zero,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( minus_minus_int @ A @ zero_zero_int )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1323_zero__diff,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1324_diff__0__right,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( minus_minus_real @ A @ zero_zero_real )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1325_diff__0__right,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1326_diff__0__right,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( minus_minus_int @ A @ zero_zero_int )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1327_diff__self,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( minus_minus_real @ A @ A )
% 12.39/6.88 = zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1328_diff__self,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( minus_minus_rat @ A @ A )
% 12.39/6.88 = zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1329_diff__self,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( minus_minus_int @ A @ A )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1330_div__0,axiom,
% 12.39/6.88 ! [A: complex] :
% 12.39/6.88 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 12.39/6.88 = zero_zero_complex ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1331_div__0,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( divide_divide_real @ zero_zero_real @ A )
% 12.39/6.88 = zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1332_div__0,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 12.39/6.88 = zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1333_div__0,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1334_div__0,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( divide_divide_int @ zero_zero_int @ A )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1335_div__by__0,axiom,
% 12.39/6.88 ! [A: complex] :
% 12.39/6.88 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 12.39/6.88 = zero_zero_complex ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1336_div__by__0,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( divide_divide_real @ A @ zero_zero_real )
% 12.39/6.88 = zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1337_div__by__0,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 12.39/6.88 = zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1338_div__by__0,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1339_div__by__0,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( divide_divide_int @ A @ zero_zero_int )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1340_bits__div__0,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1341_bits__div__0,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( divide_divide_int @ zero_zero_int @ A )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1342_bits__div__by__0,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1343_bits__div__by__0,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( divide_divide_int @ A @ zero_zero_int )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1344_divide__eq__0__iff,axiom,
% 12.39/6.88 ! [A: complex,B2: complex] :
% 12.39/6.88 ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 12.39/6.88 = zero_zero_complex )
% 12.39/6.88 = ( ( A = zero_zero_complex )
% 12.39/6.88 | ( B2 = zero_zero_complex ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1345_divide__eq__0__iff,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ( divide_divide_real @ A @ B2 )
% 12.39/6.88 = zero_zero_real )
% 12.39/6.88 = ( ( A = zero_zero_real )
% 12.39/6.88 | ( B2 = zero_zero_real ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1346_divide__eq__0__iff,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ( divide_divide_rat @ A @ B2 )
% 12.39/6.88 = zero_zero_rat )
% 12.39/6.88 = ( ( A = zero_zero_rat )
% 12.39/6.88 | ( B2 = zero_zero_rat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1347_divide__cancel__left,axiom,
% 12.39/6.88 ! [C: complex,A: complex,B2: complex] :
% 12.39/6.88 ( ( ( divide1717551699836669952omplex @ C @ A )
% 12.39/6.88 = ( divide1717551699836669952omplex @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_complex )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1348_divide__cancel__left,axiom,
% 12.39/6.88 ! [C: real,A: real,B2: real] :
% 12.39/6.88 ( ( ( divide_divide_real @ C @ A )
% 12.39/6.88 = ( divide_divide_real @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_real )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1349_divide__cancel__left,axiom,
% 12.39/6.88 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.88 ( ( ( divide_divide_rat @ C @ A )
% 12.39/6.88 = ( divide_divide_rat @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_rat )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1350_divide__cancel__right,axiom,
% 12.39/6.88 ! [A: complex,C: complex,B2: complex] :
% 12.39/6.88 ( ( ( divide1717551699836669952omplex @ A @ C )
% 12.39/6.88 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_complex )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1351_divide__cancel__right,axiom,
% 12.39/6.88 ! [A: real,C: real,B2: real] :
% 12.39/6.88 ( ( ( divide_divide_real @ A @ C )
% 12.39/6.88 = ( divide_divide_real @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_real )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1352_divide__cancel__right,axiom,
% 12.39/6.88 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.88 ( ( ( divide_divide_rat @ A @ C )
% 12.39/6.88 = ( divide_divide_rat @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_rat )
% 12.39/6.88 | ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1353_division__ring__divide__zero,axiom,
% 12.39/6.88 ! [A: complex] :
% 12.39/6.88 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 12.39/6.88 = zero_zero_complex ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1354_division__ring__divide__zero,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( divide_divide_real @ A @ zero_zero_real )
% 12.39/6.88 = zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1355_division__ring__divide__zero,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 12.39/6.88 = zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1356_dvd__0__right,axiom,
% 12.39/6.88 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1357_dvd__0__right,axiom,
% 12.39/6.88 ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1358_dvd__0__right,axiom,
% 12.39/6.88 ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1359_dvd__0__right,axiom,
% 12.39/6.88 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1360_dvd__0__right,axiom,
% 12.39/6.88 ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1361_dvd__0__left__iff,axiom,
% 12.39/6.88 ! [A: code_integer] :
% 12.39/6.88 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 12.39/6.88 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1362_dvd__0__left__iff,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 12.39/6.88 = ( A = zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1363_dvd__0__left__iff,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 12.39/6.88 = ( A = zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1364_dvd__0__left__iff,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 12.39/6.88 = ( A = zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1365_dvd__0__left__iff,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 12.39/6.88 = ( A = zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1366_dvd__add__triv__right__iff,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ A ) )
% 12.39/6.88 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1367_dvd__add__triv__right__iff,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 12.39/6.88 = ( dvd_dvd_real @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1368_dvd__add__triv__right__iff,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 12.39/6.88 = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1369_dvd__add__triv__right__iff,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 12.39/6.88 = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1370_dvd__add__triv__right__iff,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 12.39/6.88 = ( dvd_dvd_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1371_dvd__add__triv__left__iff,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 12.39/6.88 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1372_dvd__add__triv__left__iff,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 12.39/6.88 = ( dvd_dvd_real @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1373_dvd__add__triv__left__iff,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 12.39/6.88 = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1374_dvd__add__triv__left__iff,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 12.39/6.88 = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1375_dvd__add__triv__left__iff,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 12.39/6.88 = ( dvd_dvd_int @ A @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1376_power__Suc0__right,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1377_power__Suc0__right,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1378_power__Suc0__right,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1379_power__Suc0__right,axiom,
% 12.39/6.88 ! [A: complex] :
% 12.39/6.88 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1380_mod__0,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1381_mod__0,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1382_mod__0,axiom,
% 12.39/6.88 ! [A: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 12.39/6.88 = zero_z3403309356797280102nteger ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1383_mod__by__0,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1384_mod__by__0,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ A @ zero_zero_int )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1385_mod__by__0,axiom,
% 12.39/6.88 ! [A: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 12.39/6.88 = A ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1386_mod__self,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ A @ A )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1387_mod__self,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ A @ A )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1388_mod__self,axiom,
% 12.39/6.88 ! [A: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ A @ A )
% 12.39/6.88 = zero_z3403309356797280102nteger ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1389_bits__mod__0,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1390_bits__mod__0,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1391_bits__mod__0,axiom,
% 12.39/6.88 ! [A: code_integer] :
% 12.39/6.88 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 12.39/6.88 = zero_z3403309356797280102nteger ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1392_div__dvd__div,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.88 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.88 => ( ( dvd_dvd_Code_integer @ A @ C )
% 12.39/6.88 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B2 @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 12.39/6.88 = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1393_div__dvd__div,axiom,
% 12.39/6.88 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.88 ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.88 => ( ( dvd_dvd_nat @ A @ C )
% 12.39/6.88 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A ) @ ( divide_divide_nat @ C @ A ) )
% 12.39/6.88 = ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1394_div__dvd__div,axiom,
% 12.39/6.88 ! [A: int,B2: int,C: int] :
% 12.39/6.88 ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.88 => ( ( dvd_dvd_int @ A @ C )
% 12.39/6.88 => ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A ) @ ( divide_divide_int @ C @ A ) )
% 12.39/6.88 = ( dvd_dvd_int @ B2 @ C ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1395_zero__less__Suc,axiom,
% 12.39/6.88 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1396_less__Suc0,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.39/6.88 = ( N = zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1397_add__gr__0,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 12.39/6.88 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.39/6.88 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1398_one__eq__mult__iff,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( ( suc @ zero_zero_nat )
% 12.39/6.88 = ( times_times_nat @ M @ N ) )
% 12.39/6.88 = ( ( M
% 12.39/6.88 = ( suc @ zero_zero_nat ) )
% 12.39/6.88 & ( N
% 12.39/6.88 = ( suc @ zero_zero_nat ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1399_mult__eq__1__iff,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( ( times_times_nat @ M @ N )
% 12.39/6.88 = ( suc @ zero_zero_nat ) )
% 12.39/6.88 = ( ( M
% 12.39/6.88 = ( suc @ zero_zero_nat ) )
% 12.39/6.88 & ( N
% 12.39/6.88 = ( suc @ zero_zero_nat ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1400_div__by__Suc__0,axiom,
% 12.39/6.88 ! [M: nat] :
% 12.39/6.88 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 12.39/6.88 = M ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1401_zero__less__diff,axiom,
% 12.39/6.88 ! [N: nat,M: nat] :
% 12.39/6.88 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 12.39/6.88 = ( ord_less_nat @ M @ N ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1402_nat__mult__less__cancel__disj,axiom,
% 12.39/6.88 ! [K: nat,M: nat,N: nat] :
% 12.39/6.88 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.39/6.88 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.39/6.88 & ( ord_less_nat @ M @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1403_nat__0__less__mult__iff,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 12.39/6.88 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.39/6.88 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1404_mult__less__cancel2,axiom,
% 12.39/6.88 ! [M: nat,K: nat,N: nat] :
% 12.39/6.88 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 12.39/6.88 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.39/6.88 & ( ord_less_nat @ M @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1405_div__less,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( ord_less_nat @ M @ N )
% 12.39/6.88 => ( ( divide_divide_nat @ M @ N )
% 12.39/6.88 = zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1406_not__real__square__gt__zero,axiom,
% 12.39/6.88 ! [X4: real] :
% 12.39/6.88 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X4 @ X4 ) ) )
% 12.39/6.88 = ( X4 = zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1407_diff__is__0__eq,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( ( minus_minus_nat @ M @ N )
% 12.39/6.88 = zero_zero_nat )
% 12.39/6.88 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1408_diff__is__0__eq_H,axiom,
% 12.39/6.88 ! [M: nat,N: nat] :
% 12.39/6.88 ( ( ord_less_eq_nat @ M @ N )
% 12.39/6.88 => ( ( minus_minus_nat @ M @ N )
% 12.39/6.88 = zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1409_dvd__1__iff__1,axiom,
% 12.39/6.88 ! [M: nat] :
% 12.39/6.88 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 12.39/6.88 = ( M
% 12.39/6.88 = ( suc @ zero_zero_nat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1410_dvd__1__left,axiom,
% 12.39/6.88 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1411_less__one,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( ord_less_nat @ N @ one_one_nat )
% 12.39/6.88 = ( N = zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1412_power__Suc__0,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.39/6.88 = ( suc @ zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1413_nat__power__eq__Suc__0__iff,axiom,
% 12.39/6.88 ! [X4: nat,M: nat] :
% 12.39/6.88 ( ( ( power_power_nat @ X4 @ M )
% 12.39/6.88 = ( suc @ zero_zero_nat ) )
% 12.39/6.88 = ( ( M = zero_zero_nat )
% 12.39/6.88 | ( X4
% 12.39/6.88 = ( suc @ zero_zero_nat ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1414_nat__zero__less__power__iff,axiom,
% 12.39/6.88 ! [X4: nat,N: nat] :
% 12.39/6.88 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N ) )
% 12.39/6.88 = ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 12.39/6.88 | ( N = zero_zero_nat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1415_nat__mult__div__cancel__disj,axiom,
% 12.39/6.88 ! [K: nat,M: nat,N: nat] :
% 12.39/6.88 ( ( ( K = zero_zero_nat )
% 12.39/6.88 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.39/6.88 = zero_zero_nat ) )
% 12.39/6.88 & ( ( K != zero_zero_nat )
% 12.39/6.88 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.39/6.88 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1416_mod__by__Suc__0,axiom,
% 12.39/6.88 ! [M: nat] :
% 12.39/6.88 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1417_nat__mult__dvd__cancel__disj,axiom,
% 12.39/6.88 ! [K: nat,M: nat,N: nat] :
% 12.39/6.88 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.39/6.88 = ( ( K = zero_zero_nat )
% 12.39/6.88 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1418_nat__dvd__1__iff__1,axiom,
% 12.39/6.88 ! [M: nat] :
% 12.39/6.88 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 12.39/6.88 = ( M = one_one_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1419_signed__take__bit__of__0,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1420_dbl__simps_I2_J,axiom,
% 12.39/6.88 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 12.39/6.88 = zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1421_dbl__simps_I2_J,axiom,
% 12.39/6.88 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 12.39/6.88 = zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1422_dbl__simps_I2_J,axiom,
% 12.39/6.88 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1423_add__le__same__cancel1,axiom,
% 12.39/6.88 ! [B2: real,A: real] :
% 12.39/6.88 ( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
% 12.39/6.88 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1424_add__le__same__cancel1,axiom,
% 12.39/6.88 ! [B2: rat,A: rat] :
% 12.39/6.88 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
% 12.39/6.88 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1425_add__le__same__cancel1,axiom,
% 12.39/6.88 ! [B2: nat,A: nat] :
% 12.39/6.88 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 12.39/6.88 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1426_add__le__same__cancel1,axiom,
% 12.39/6.88 ! [B2: int,A: int] :
% 12.39/6.88 ( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 12.39/6.88 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1427_add__le__same__cancel2,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1428_add__le__same__cancel2,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1429_add__le__same__cancel2,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1430_add__le__same__cancel2,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1431_le__add__same__cancel1,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1432_le__add__same__cancel1,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1433_le__add__same__cancel1,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1434_le__add__same__cancel1,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1435_le__add__same__cancel2,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 12.39/6.88 = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1436_le__add__same__cancel2,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 12.39/6.88 = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1437_le__add__same__cancel2,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 12.39/6.88 = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1438_le__add__same__cancel2,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 12.39/6.88 = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1439_double__add__le__zero__iff__single__add__le__zero,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 12.39/6.88 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1440_double__add__le__zero__iff__single__add__le__zero,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 12.39/6.88 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1441_double__add__le__zero__iff__single__add__le__zero,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 12.39/6.88 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1442_zero__le__double__add__iff__zero__le__single__add,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 12.39/6.88 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1443_zero__le__double__add__iff__zero__le__single__add,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 12.39/6.88 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1444_zero__le__double__add__iff__zero__le__single__add,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 12.39/6.88 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1445_diff__ge__0__iff__ge,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_eq_real @ B2 @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1446_diff__ge__0__iff__ge,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_eq_rat @ B2 @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1447_diff__ge__0__iff__ge,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_eq_int @ B2 @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1448_add__less__same__cancel1,axiom,
% 12.39/6.88 ! [B2: real,A: real] :
% 12.39/6.88 ( ( ord_less_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
% 12.39/6.88 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1449_add__less__same__cancel1,axiom,
% 12.39/6.88 ! [B2: rat,A: rat] :
% 12.39/6.88 ( ( ord_less_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
% 12.39/6.88 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1450_add__less__same__cancel1,axiom,
% 12.39/6.88 ! [B2: nat,A: nat] :
% 12.39/6.88 ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 12.39/6.88 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1451_add__less__same__cancel1,axiom,
% 12.39/6.88 ! [B2: int,A: int] :
% 12.39/6.88 ( ( ord_less_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 12.39/6.88 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1452_add__less__same__cancel2,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1453_add__less__same__cancel2,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1454_add__less__same__cancel2,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1455_add__less__same__cancel2,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 12.39/6.88 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1456_less__add__same__cancel1,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1457_less__add__same__cancel1,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1458_less__add__same__cancel1,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1459_less__add__same__cancel1,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1460_less__add__same__cancel2,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ord_less_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 12.39/6.88 = ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1461_less__add__same__cancel2,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 12.39/6.88 = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1462_less__add__same__cancel2,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 12.39/6.88 = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1463_less__add__same__cancel2,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( ord_less_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 12.39/6.88 = ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1464_double__add__less__zero__iff__single__add__less__zero,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 12.39/6.88 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1465_double__add__less__zero__iff__single__add__less__zero,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 12.39/6.88 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1466_double__add__less__zero__iff__single__add__less__zero,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 12.39/6.88 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1467_zero__less__double__add__iff__zero__less__single__add,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 12.39/6.88 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1468_zero__less__double__add__iff__zero__less__single__add,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 12.39/6.88 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1469_zero__less__double__add__iff__zero__less__single__add,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 12.39/6.88 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1470_diff__gt__0__iff__gt,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_real @ B2 @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1471_diff__gt__0__iff__gt,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_rat @ B2 @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1472_diff__gt__0__iff__gt,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
% 12.39/6.88 = ( ord_less_int @ B2 @ A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1473_mult__cancel__left1,axiom,
% 12.39/6.88 ! [C: complex,B2: complex] :
% 12.39/6.88 ( ( C
% 12.39/6.88 = ( times_times_complex @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_complex )
% 12.39/6.88 | ( B2 = one_one_complex ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1474_mult__cancel__left1,axiom,
% 12.39/6.88 ! [C: real,B2: real] :
% 12.39/6.88 ( ( C
% 12.39/6.88 = ( times_times_real @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_real )
% 12.39/6.88 | ( B2 = one_one_real ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1475_mult__cancel__left1,axiom,
% 12.39/6.88 ! [C: rat,B2: rat] :
% 12.39/6.88 ( ( C
% 12.39/6.88 = ( times_times_rat @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_rat )
% 12.39/6.88 | ( B2 = one_one_rat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1476_mult__cancel__left1,axiom,
% 12.39/6.88 ! [C: int,B2: int] :
% 12.39/6.88 ( ( C
% 12.39/6.88 = ( times_times_int @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_int )
% 12.39/6.88 | ( B2 = one_one_int ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1477_mult__cancel__left2,axiom,
% 12.39/6.88 ! [C: complex,A: complex] :
% 12.39/6.88 ( ( ( times_times_complex @ C @ A )
% 12.39/6.88 = C )
% 12.39/6.88 = ( ( C = zero_zero_complex )
% 12.39/6.88 | ( A = one_one_complex ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1478_mult__cancel__left2,axiom,
% 12.39/6.88 ! [C: real,A: real] :
% 12.39/6.88 ( ( ( times_times_real @ C @ A )
% 12.39/6.88 = C )
% 12.39/6.88 = ( ( C = zero_zero_real )
% 12.39/6.88 | ( A = one_one_real ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1479_mult__cancel__left2,axiom,
% 12.39/6.88 ! [C: rat,A: rat] :
% 12.39/6.88 ( ( ( times_times_rat @ C @ A )
% 12.39/6.88 = C )
% 12.39/6.88 = ( ( C = zero_zero_rat )
% 12.39/6.88 | ( A = one_one_rat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1480_mult__cancel__left2,axiom,
% 12.39/6.88 ! [C: int,A: int] :
% 12.39/6.88 ( ( ( times_times_int @ C @ A )
% 12.39/6.88 = C )
% 12.39/6.88 = ( ( C = zero_zero_int )
% 12.39/6.88 | ( A = one_one_int ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1481_mult__cancel__right1,axiom,
% 12.39/6.88 ! [C: complex,B2: complex] :
% 12.39/6.88 ( ( C
% 12.39/6.88 = ( times_times_complex @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_complex )
% 12.39/6.88 | ( B2 = one_one_complex ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1482_mult__cancel__right1,axiom,
% 12.39/6.88 ! [C: real,B2: real] :
% 12.39/6.88 ( ( C
% 12.39/6.88 = ( times_times_real @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_real )
% 12.39/6.88 | ( B2 = one_one_real ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1483_mult__cancel__right1,axiom,
% 12.39/6.88 ! [C: rat,B2: rat] :
% 12.39/6.88 ( ( C
% 12.39/6.88 = ( times_times_rat @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_rat )
% 12.39/6.88 | ( B2 = one_one_rat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1484_mult__cancel__right1,axiom,
% 12.39/6.88 ! [C: int,B2: int] :
% 12.39/6.88 ( ( C
% 12.39/6.88 = ( times_times_int @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_int )
% 12.39/6.88 | ( B2 = one_one_int ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1485_mult__cancel__right2,axiom,
% 12.39/6.88 ! [A: complex,C: complex] :
% 12.39/6.88 ( ( ( times_times_complex @ A @ C )
% 12.39/6.88 = C )
% 12.39/6.88 = ( ( C = zero_zero_complex )
% 12.39/6.88 | ( A = one_one_complex ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1486_mult__cancel__right2,axiom,
% 12.39/6.88 ! [A: real,C: real] :
% 12.39/6.88 ( ( ( times_times_real @ A @ C )
% 12.39/6.88 = C )
% 12.39/6.88 = ( ( C = zero_zero_real )
% 12.39/6.88 | ( A = one_one_real ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1487_mult__cancel__right2,axiom,
% 12.39/6.88 ! [A: rat,C: rat] :
% 12.39/6.88 ( ( ( times_times_rat @ A @ C )
% 12.39/6.88 = C )
% 12.39/6.88 = ( ( C = zero_zero_rat )
% 12.39/6.88 | ( A = one_one_rat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1488_mult__cancel__right2,axiom,
% 12.39/6.88 ! [A: int,C: int] :
% 12.39/6.88 ( ( ( times_times_int @ A @ C )
% 12.39/6.88 = C )
% 12.39/6.88 = ( ( C = zero_zero_int )
% 12.39/6.88 | ( A = one_one_int ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1489_sum__squares__eq__zero__iff,axiom,
% 12.39/6.88 ! [X4: real,Y3: real] :
% 12.39/6.88 ( ( ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 12.39/6.88 = zero_zero_real )
% 12.39/6.88 = ( ( X4 = zero_zero_real )
% 12.39/6.88 & ( Y3 = zero_zero_real ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1490_sum__squares__eq__zero__iff,axiom,
% 12.39/6.88 ! [X4: rat,Y3: rat] :
% 12.39/6.88 ( ( ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 12.39/6.88 = zero_zero_rat )
% 12.39/6.88 = ( ( X4 = zero_zero_rat )
% 12.39/6.88 & ( Y3 = zero_zero_rat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1491_sum__squares__eq__zero__iff,axiom,
% 12.39/6.88 ! [X4: int,Y3: int] :
% 12.39/6.88 ( ( ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 12.39/6.88 = zero_zero_int )
% 12.39/6.88 = ( ( X4 = zero_zero_int )
% 12.39/6.88 & ( Y3 = zero_zero_int ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1492_div__mult__mult1__if,axiom,
% 12.39/6.88 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.88 ( ( ( C = zero_zero_nat )
% 12.39/6.88 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 12.39/6.88 = zero_zero_nat ) )
% 12.39/6.88 & ( ( C != zero_zero_nat )
% 12.39/6.88 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 12.39/6.88 = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1493_div__mult__mult1__if,axiom,
% 12.39/6.88 ! [C: int,A: int,B2: int] :
% 12.39/6.88 ( ( ( C = zero_zero_int )
% 12.39/6.88 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.39/6.88 = zero_zero_int ) )
% 12.39/6.88 & ( ( C != zero_zero_int )
% 12.39/6.88 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.39/6.88 = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1494_div__mult__mult2,axiom,
% 12.39/6.88 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.88 ( ( C != zero_zero_nat )
% 12.39/6.88 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 12.39/6.88 = ( divide_divide_nat @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1495_div__mult__mult2,axiom,
% 12.39/6.88 ! [C: int,A: int,B2: int] :
% 12.39/6.88 ( ( C != zero_zero_int )
% 12.39/6.88 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 12.39/6.88 = ( divide_divide_int @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1496_div__mult__mult1,axiom,
% 12.39/6.88 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.88 ( ( C != zero_zero_nat )
% 12.39/6.88 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 12.39/6.88 = ( divide_divide_nat @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1497_div__mult__mult1,axiom,
% 12.39/6.88 ! [C: int,A: int,B2: int] :
% 12.39/6.88 ( ( C != zero_zero_int )
% 12.39/6.88 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.39/6.88 = ( divide_divide_int @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1498_nonzero__mult__div__cancel__left,axiom,
% 12.39/6.88 ! [A: complex,B2: complex] :
% 12.39/6.88 ( ( A != zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ A )
% 12.39/6.88 = B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1499_nonzero__mult__div__cancel__left,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( A != zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ A )
% 12.39/6.88 = B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1500_nonzero__mult__div__cancel__left,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( A != zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ A )
% 12.39/6.88 = B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1501_nonzero__mult__div__cancel__left,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( A != zero_zero_nat )
% 12.39/6.88 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ A )
% 12.39/6.88 = B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1502_nonzero__mult__div__cancel__left,axiom,
% 12.39/6.88 ! [A: int,B2: int] :
% 12.39/6.88 ( ( A != zero_zero_int )
% 12.39/6.88 => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ A )
% 12.39/6.88 = B2 ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1503_nonzero__mult__div__cancel__right,axiom,
% 12.39/6.88 ! [B2: complex,A: complex] :
% 12.39/6.88 ( ( B2 != zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ B2 )
% 12.39/6.88 = A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1504_nonzero__mult__div__cancel__right,axiom,
% 12.39/6.88 ! [B2: real,A: real] :
% 12.39/6.88 ( ( B2 != zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ B2 )
% 12.39/6.88 = A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1505_nonzero__mult__div__cancel__right,axiom,
% 12.39/6.88 ! [B2: rat,A: rat] :
% 12.39/6.88 ( ( B2 != zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ B2 )
% 12.39/6.88 = A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1506_nonzero__mult__div__cancel__right,axiom,
% 12.39/6.88 ! [B2: nat,A: nat] :
% 12.39/6.88 ( ( B2 != zero_zero_nat )
% 12.39/6.88 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
% 12.39/6.88 = A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1507_nonzero__mult__div__cancel__right,axiom,
% 12.39/6.88 ! [B2: int,A: int] :
% 12.39/6.88 ( ( B2 != zero_zero_int )
% 12.39/6.88 => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ B2 )
% 12.39/6.88 = A ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1508_mult__divide__mult__cancel__left__if,axiom,
% 12.39/6.88 ! [C: complex,A: complex,B2: complex] :
% 12.39/6.88 ( ( ( C = zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 12.39/6.88 = zero_zero_complex ) )
% 12.39/6.88 & ( ( C != zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 12.39/6.88 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1509_mult__divide__mult__cancel__left__if,axiom,
% 12.39/6.88 ! [C: real,A: real,B2: real] :
% 12.39/6.88 ( ( ( C = zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.39/6.88 = zero_zero_real ) )
% 12.39/6.88 & ( ( C != zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.39/6.88 = ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1510_mult__divide__mult__cancel__left__if,axiom,
% 12.39/6.88 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.88 ( ( ( C = zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.39/6.88 = zero_zero_rat ) )
% 12.39/6.88 & ( ( C != zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.39/6.88 = ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1511_nonzero__mult__divide__mult__cancel__left,axiom,
% 12.39/6.88 ! [C: complex,A: complex,B2: complex] :
% 12.39/6.88 ( ( C != zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 12.39/6.88 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1512_nonzero__mult__divide__mult__cancel__left,axiom,
% 12.39/6.88 ! [C: real,A: real,B2: real] :
% 12.39/6.88 ( ( C != zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.39/6.88 = ( divide_divide_real @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1513_nonzero__mult__divide__mult__cancel__left,axiom,
% 12.39/6.88 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.88 ( ( C != zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.39/6.88 = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1514_nonzero__mult__divide__mult__cancel__left2,axiom,
% 12.39/6.88 ! [C: complex,A: complex,B2: complex] :
% 12.39/6.88 ( ( C != zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B2 @ C ) )
% 12.39/6.88 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1515_nonzero__mult__divide__mult__cancel__left2,axiom,
% 12.39/6.88 ! [C: real,A: real,B2: real] :
% 12.39/6.88 ( ( C != zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B2 @ C ) )
% 12.39/6.88 = ( divide_divide_real @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1516_nonzero__mult__divide__mult__cancel__left2,axiom,
% 12.39/6.88 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.88 ( ( C != zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B2 @ C ) )
% 12.39/6.88 = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1517_nonzero__mult__divide__mult__cancel__right,axiom,
% 12.39/6.88 ! [C: complex,A: complex,B2: complex] :
% 12.39/6.88 ( ( C != zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) )
% 12.39/6.88 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1518_nonzero__mult__divide__mult__cancel__right,axiom,
% 12.39/6.88 ! [C: real,A: real,B2: real] :
% 12.39/6.88 ( ( C != zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 12.39/6.88 = ( divide_divide_real @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1519_nonzero__mult__divide__mult__cancel__right,axiom,
% 12.39/6.88 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.88 ( ( C != zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 12.39/6.88 = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1520_nonzero__mult__divide__mult__cancel__right2,axiom,
% 12.39/6.88 ! [C: complex,A: complex,B2: complex] :
% 12.39/6.88 ( ( C != zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B2 ) )
% 12.39/6.88 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1521_nonzero__mult__divide__mult__cancel__right2,axiom,
% 12.39/6.88 ! [C: real,A: real,B2: real] :
% 12.39/6.88 ( ( C != zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B2 ) )
% 12.39/6.88 = ( divide_divide_real @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1522_nonzero__mult__divide__mult__cancel__right2,axiom,
% 12.39/6.88 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.88 ( ( C != zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B2 ) )
% 12.39/6.88 = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1523_diff__numeral__special_I9_J,axiom,
% 12.39/6.88 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 12.39/6.88 = zero_zero_complex ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1524_diff__numeral__special_I9_J,axiom,
% 12.39/6.88 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 12.39/6.88 = zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1525_diff__numeral__special_I9_J,axiom,
% 12.39/6.88 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 12.39/6.88 = zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1526_diff__numeral__special_I9_J,axiom,
% 12.39/6.88 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1527_diff__add__zero,axiom,
% 12.39/6.88 ! [A: nat,B2: nat] :
% 12.39/6.88 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1528_div__self,axiom,
% 12.39/6.88 ! [A: complex] :
% 12.39/6.88 ( ( A != zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ A @ A )
% 12.39/6.88 = one_one_complex ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1529_div__self,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( A != zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ A @ A )
% 12.39/6.88 = one_one_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1530_div__self,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( A != zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ A @ A )
% 12.39/6.88 = one_one_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1531_div__self,axiom,
% 12.39/6.88 ! [A: nat] :
% 12.39/6.88 ( ( A != zero_zero_nat )
% 12.39/6.88 => ( ( divide_divide_nat @ A @ A )
% 12.39/6.88 = one_one_nat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1532_div__self,axiom,
% 12.39/6.88 ! [A: int] :
% 12.39/6.88 ( ( A != zero_zero_int )
% 12.39/6.88 => ( ( divide_divide_int @ A @ A )
% 12.39/6.88 = one_one_int ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1533_divide__eq__1__iff,axiom,
% 12.39/6.88 ! [A: complex,B2: complex] :
% 12.39/6.88 ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 12.39/6.88 = one_one_complex )
% 12.39/6.88 = ( ( B2 != zero_zero_complex )
% 12.39/6.88 & ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1534_divide__eq__1__iff,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( ( divide_divide_real @ A @ B2 )
% 12.39/6.88 = one_one_real )
% 12.39/6.88 = ( ( B2 != zero_zero_real )
% 12.39/6.88 & ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1535_divide__eq__1__iff,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( ( divide_divide_rat @ A @ B2 )
% 12.39/6.88 = one_one_rat )
% 12.39/6.88 = ( ( B2 != zero_zero_rat )
% 12.39/6.88 & ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1536_one__eq__divide__iff,axiom,
% 12.39/6.88 ! [A: complex,B2: complex] :
% 12.39/6.88 ( ( one_one_complex
% 12.39/6.88 = ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.39/6.88 = ( ( B2 != zero_zero_complex )
% 12.39/6.88 & ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1537_one__eq__divide__iff,axiom,
% 12.39/6.88 ! [A: real,B2: real] :
% 12.39/6.88 ( ( one_one_real
% 12.39/6.88 = ( divide_divide_real @ A @ B2 ) )
% 12.39/6.88 = ( ( B2 != zero_zero_real )
% 12.39/6.88 & ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1538_one__eq__divide__iff,axiom,
% 12.39/6.88 ! [A: rat,B2: rat] :
% 12.39/6.88 ( ( one_one_rat
% 12.39/6.88 = ( divide_divide_rat @ A @ B2 ) )
% 12.39/6.88 = ( ( B2 != zero_zero_rat )
% 12.39/6.88 & ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1539_divide__self,axiom,
% 12.39/6.88 ! [A: complex] :
% 12.39/6.88 ( ( A != zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ A @ A )
% 12.39/6.88 = one_one_complex ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1540_divide__self,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( A != zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ A @ A )
% 12.39/6.88 = one_one_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1541_divide__self,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( A != zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ A @ A )
% 12.39/6.88 = one_one_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1542_divide__self__if,axiom,
% 12.39/6.88 ! [A: complex] :
% 12.39/6.88 ( ( ( A = zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ A @ A )
% 12.39/6.88 = zero_zero_complex ) )
% 12.39/6.88 & ( ( A != zero_zero_complex )
% 12.39/6.88 => ( ( divide1717551699836669952omplex @ A @ A )
% 12.39/6.88 = one_one_complex ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1543_divide__self__if,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( ( A = zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ A @ A )
% 12.39/6.88 = zero_zero_real ) )
% 12.39/6.88 & ( ( A != zero_zero_real )
% 12.39/6.88 => ( ( divide_divide_real @ A @ A )
% 12.39/6.88 = one_one_real ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1544_divide__self__if,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( ( A = zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ A @ A )
% 12.39/6.88 = zero_zero_rat ) )
% 12.39/6.88 & ( ( A != zero_zero_rat )
% 12.39/6.88 => ( ( divide_divide_rat @ A @ A )
% 12.39/6.88 = one_one_rat ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1545_divide__eq__eq__1,axiom,
% 12.39/6.88 ! [B2: real,A: real] :
% 12.39/6.88 ( ( ( divide_divide_real @ B2 @ A )
% 12.39/6.88 = one_one_real )
% 12.39/6.88 = ( ( A != zero_zero_real )
% 12.39/6.88 & ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1546_divide__eq__eq__1,axiom,
% 12.39/6.88 ! [B2: rat,A: rat] :
% 12.39/6.88 ( ( ( divide_divide_rat @ B2 @ A )
% 12.39/6.88 = one_one_rat )
% 12.39/6.88 = ( ( A != zero_zero_rat )
% 12.39/6.88 & ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1547_eq__divide__eq__1,axiom,
% 12.39/6.88 ! [B2: real,A: real] :
% 12.39/6.88 ( ( one_one_real
% 12.39/6.88 = ( divide_divide_real @ B2 @ A ) )
% 12.39/6.88 = ( ( A != zero_zero_real )
% 12.39/6.88 & ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1548_eq__divide__eq__1,axiom,
% 12.39/6.88 ! [B2: rat,A: rat] :
% 12.39/6.88 ( ( one_one_rat
% 12.39/6.88 = ( divide_divide_rat @ B2 @ A ) )
% 12.39/6.88 = ( ( A != zero_zero_rat )
% 12.39/6.88 & ( A = B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1549_one__divide__eq__0__iff,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( ( divide_divide_real @ one_one_real @ A )
% 12.39/6.88 = zero_zero_real )
% 12.39/6.88 = ( A = zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1550_one__divide__eq__0__iff,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 12.39/6.88 = zero_zero_rat )
% 12.39/6.88 = ( A = zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1551_zero__eq__1__divide__iff,axiom,
% 12.39/6.88 ! [A: real] :
% 12.39/6.88 ( ( zero_zero_real
% 12.39/6.88 = ( divide_divide_real @ one_one_real @ A ) )
% 12.39/6.88 = ( A = zero_zero_real ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1552_zero__eq__1__divide__iff,axiom,
% 12.39/6.88 ! [A: rat] :
% 12.39/6.88 ( ( zero_zero_rat
% 12.39/6.88 = ( divide_divide_rat @ one_one_rat @ A ) )
% 12.39/6.88 = ( A = zero_zero_rat ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1553_dvd__times__right__cancel__iff,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.88 ( ( A != zero_z3403309356797280102nteger )
% 12.39/6.88 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 12.39/6.88 = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1554_dvd__times__right__cancel__iff,axiom,
% 12.39/6.88 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.88 ( ( A != zero_zero_nat )
% 12.39/6.88 => ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) )
% 12.39/6.88 = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1555_dvd__times__right__cancel__iff,axiom,
% 12.39/6.88 ! [A: int,B2: int,C: int] :
% 12.39/6.88 ( ( A != zero_zero_int )
% 12.39/6.88 => ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) )
% 12.39/6.88 = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1556_dvd__times__left__cancel__iff,axiom,
% 12.39/6.88 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.88 ( ( A != zero_z3403309356797280102nteger )
% 12.39/6.88 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 12.39/6.88 = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1557_dvd__times__left__cancel__iff,axiom,
% 12.39/6.88 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.88 ( ( A != zero_zero_nat )
% 12.39/6.88 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) )
% 12.39/6.88 = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1558_dvd__times__left__cancel__iff,axiom,
% 12.39/6.88 ! [A: int,B2: int,C: int] :
% 12.39/6.88 ( ( A != zero_zero_int )
% 12.39/6.88 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) )
% 12.39/6.88 = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1559_dvd__mult__cancel__right,axiom,
% 12.39/6.88 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_z3403309356797280102nteger )
% 12.39/6.88 | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1560_dvd__mult__cancel__right,axiom,
% 12.39/6.88 ! [A: real,C: real,B2: real] :
% 12.39/6.88 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_real )
% 12.39/6.88 | ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1561_dvd__mult__cancel__right,axiom,
% 12.39/6.88 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.88 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_rat )
% 12.39/6.88 | ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1562_dvd__mult__cancel__right,axiom,
% 12.39/6.88 ! [A: int,C: int,B2: int] :
% 12.39/6.88 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 12.39/6.88 = ( ( C = zero_zero_int )
% 12.39/6.88 | ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1563_dvd__mult__cancel__left,axiom,
% 12.39/6.88 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 12.39/6.88 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_z3403309356797280102nteger )
% 12.39/6.88 | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1564_dvd__mult__cancel__left,axiom,
% 12.39/6.88 ! [C: real,A: real,B2: real] :
% 12.39/6.88 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_real )
% 12.39/6.88 | ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1565_dvd__mult__cancel__left,axiom,
% 12.39/6.88 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.88 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_rat )
% 12.39/6.88 | ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1566_dvd__mult__cancel__left,axiom,
% 12.39/6.88 ! [C: int,A: int,B2: int] :
% 12.39/6.88 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.39/6.88 = ( ( C = zero_zero_int )
% 12.39/6.88 | ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1567_power__0__Suc,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 12.39/6.88 = zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1568_power__0__Suc,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1569_power__0__Suc,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 12.39/6.88 = zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1570_power__0__Suc,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1571_power__0__Suc,axiom,
% 12.39/6.88 ! [N: nat] :
% 12.39/6.88 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 12.39/6.88 = zero_zero_complex ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1572_power__zero__numeral,axiom,
% 12.39/6.88 ! [K: num] :
% 12.39/6.88 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 12.39/6.88 = zero_zero_rat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1573_power__zero__numeral,axiom,
% 12.39/6.88 ! [K: num] :
% 12.39/6.88 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 12.39/6.88 = zero_zero_nat ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1574_power__zero__numeral,axiom,
% 12.39/6.88 ! [K: num] :
% 12.39/6.88 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 12.39/6.88 = zero_zero_real ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1575_power__zero__numeral,axiom,
% 12.39/6.88 ! [K: num] :
% 12.39/6.88 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 12.39/6.88 = zero_zero_int ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1576_power__zero__numeral,axiom,
% 12.39/6.88 ! [K: num] :
% 12.39/6.88 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 12.39/6.88 = zero_zero_complex ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1577_power__eq__0__iff,axiom,
% 12.39/6.88 ! [A: rat,N: nat] :
% 12.39/6.88 ( ( ( power_power_rat @ A @ N )
% 12.39/6.88 = zero_zero_rat )
% 12.39/6.88 = ( ( A = zero_zero_rat )
% 12.39/6.88 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1578_power__eq__0__iff,axiom,
% 12.39/6.88 ! [A: nat,N: nat] :
% 12.39/6.88 ( ( ( power_power_nat @ A @ N )
% 12.39/6.88 = zero_zero_nat )
% 12.39/6.88 = ( ( A = zero_zero_nat )
% 12.39/6.88 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1579_power__eq__0__iff,axiom,
% 12.39/6.88 ! [A: real,N: nat] :
% 12.39/6.88 ( ( ( power_power_real @ A @ N )
% 12.39/6.88 = zero_zero_real )
% 12.39/6.88 = ( ( A = zero_zero_real )
% 12.39/6.88 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1580_power__eq__0__iff,axiom,
% 12.39/6.88 ! [A: int,N: nat] :
% 12.39/6.88 ( ( ( power_power_int @ A @ N )
% 12.39/6.88 = zero_zero_int )
% 12.39/6.88 = ( ( A = zero_zero_int )
% 12.39/6.88 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.39/6.88
% 12.39/6.88 thf(fact_1581_power__eq__0__iff,axiom,
% 12.39/6.88 ! [A: complex,N: nat] :
% 12.39/6.88 ( ( ( power_power_complex @ A @ N )
% 12.39/6.89 = zero_zero_complex )
% 12.39/6.89 = ( ( A = zero_zero_complex )
% 12.39/6.89 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1582_unit__prod,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1583_unit__prod,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.39/6.89 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.39/6.89 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1584_unit__prod,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.39/6.89 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.39/6.89 => ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1585_dvd__add__times__triv__left__iff,axiom,
% 12.39/6.89 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1586_dvd__add__times__triv__left__iff,axiom,
% 12.39/6.89 ! [A: real,C: real,B2: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_real @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1587_dvd__add__times__triv__left__iff,axiom,
% 12.39/6.89 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1588_dvd__add__times__triv__left__iff,axiom,
% 12.39/6.89 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1589_dvd__add__times__triv__left__iff,axiom,
% 12.39/6.89 ! [A: int,C: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_int @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1590_dvd__add__times__triv__right__iff,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1591_dvd__add__times__triv__right__iff,axiom,
% 12.39/6.89 ! [A: real,B2: real,C: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ ( times_times_real @ C @ A ) ) )
% 12.39/6.89 = ( dvd_dvd_real @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1592_dvd__add__times__triv__right__iff,axiom,
% 12.39/6.89 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ ( times_times_rat @ C @ A ) ) )
% 12.39/6.89 = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1593_dvd__add__times__triv__right__iff,axiom,
% 12.39/6.89 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A ) ) )
% 12.39/6.89 = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1594_dvd__add__times__triv__right__iff,axiom,
% 12.39/6.89 ! [A: int,B2: int,C: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A ) ) )
% 12.39/6.89 = ( dvd_dvd_int @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1595_mod__mult__self1__is__0,axiom,
% 12.39/6.89 ! [B2: nat,A: nat] :
% 12.39/6.89 ( ( modulo_modulo_nat @ ( times_times_nat @ B2 @ A ) @ B2 )
% 12.39/6.89 = zero_zero_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1596_mod__mult__self1__is__0,axiom,
% 12.39/6.89 ! [B2: int,A: int] :
% 12.39/6.89 ( ( modulo_modulo_int @ ( times_times_int @ B2 @ A ) @ B2 )
% 12.39/6.89 = zero_zero_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1597_mod__mult__self1__is__0,axiom,
% 12.39/6.89 ! [B2: code_integer,A: code_integer] :
% 12.39/6.89 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ B2 )
% 12.39/6.89 = zero_z3403309356797280102nteger ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1598_mod__mult__self2__is__0,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
% 12.39/6.89 = zero_zero_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1599_mod__mult__self2__is__0,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ B2 )
% 12.39/6.89 = zero_zero_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1600_mod__mult__self2__is__0,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ B2 )
% 12.39/6.89 = zero_z3403309356797280102nteger ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1601_dvd__div__mult__self,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.89 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
% 12.39/6.89 = B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1602_dvd__div__mult__self,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.89 => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
% 12.39/6.89 = B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1603_dvd__div__mult__self,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.89 => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
% 12.39/6.89 = B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1604_dvd__mult__div__cancel,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.89 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ A ) )
% 12.39/6.89 = B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1605_dvd__mult__div__cancel,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.89 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ A ) )
% 12.39/6.89 = B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1606_dvd__mult__div__cancel,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.89 => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ A ) )
% 12.39/6.89 = B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1607_mod__by__1,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 12.39/6.89 = zero_zero_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1608_mod__by__1,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( modulo_modulo_int @ A @ one_one_int )
% 12.39/6.89 = zero_zero_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1609_mod__by__1,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 12.39/6.89 = zero_z3403309356797280102nteger ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1610_bits__mod__by__1,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 12.39/6.89 = zero_zero_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1611_bits__mod__by__1,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( modulo_modulo_int @ A @ one_one_int )
% 12.39/6.89 = zero_zero_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1612_bits__mod__by__1,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 12.39/6.89 = zero_z3403309356797280102nteger ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1613_unit__div,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1614_unit__div,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.39/6.89 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.39/6.89 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1615_unit__div,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.39/6.89 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.39/6.89 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1616_unit__div__1__unit,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1617_unit__div__1__unit,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.39/6.89 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1618_unit__div__1__unit,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.39/6.89 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1619_unit__div__1__div__1,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.39/6.89 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 12.39/6.89 = A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1620_unit__div__1__div__1,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.39/6.89 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 12.39/6.89 = A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1621_unit__div__1__div__1,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.39/6.89 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 12.39/6.89 = A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1622_div__add,axiom,
% 12.39/6.89 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.39/6.89 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 12.39/6.89 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1623_div__add,axiom,
% 12.39/6.89 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_nat @ C @ B2 )
% 12.39/6.89 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 12.39/6.89 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1624_div__add,axiom,
% 12.39/6.89 ! [C: int,A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_int @ C @ B2 )
% 12.39/6.89 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 12.39/6.89 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1625_div__diff,axiom,
% 12.39/6.89 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.39/6.89 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
% 12.39/6.89 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1626_div__diff,axiom,
% 12.39/6.89 ! [C: int,A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_int @ C @ B2 )
% 12.39/6.89 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 12.39/6.89 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1627_mod__div__trivial,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 12.39/6.89 = zero_zero_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1628_mod__div__trivial,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 12.39/6.89 = zero_zero_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1629_mod__div__trivial,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 12.39/6.89 = zero_z3403309356797280102nteger ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1630_bits__mod__div__trivial,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 12.39/6.89 = zero_zero_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1631_bits__mod__div__trivial,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 12.39/6.89 = zero_zero_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1632_bits__mod__div__trivial,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 12.39/6.89 = zero_z3403309356797280102nteger ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1633_zmod__numeral__Bit0,axiom,
% 12.39/6.89 ! [V: num,W: num] :
% 12.39/6.89 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 12.39/6.89 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1634_dvd__imp__mod__0,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.89 => ( ( modulo_modulo_nat @ B2 @ A )
% 12.39/6.89 = zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1635_dvd__imp__mod__0,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.89 => ( ( modulo_modulo_int @ B2 @ A )
% 12.39/6.89 = zero_zero_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1636_dvd__imp__mod__0,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.89 => ( ( modulo364778990260209775nteger @ B2 @ A )
% 12.39/6.89 = zero_z3403309356797280102nteger ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1637_Suc__pred,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 12.39/6.89 = N ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1638_one__le__mult__iff,axiom,
% 12.39/6.89 ! [M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 12.39/6.89 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 12.39/6.89 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1639_nat__mult__le__cancel__disj,axiom,
% 12.39/6.89 ! [K: nat,M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.39/6.89 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.39/6.89 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1640_mult__le__cancel2,axiom,
% 12.39/6.89 ! [M: nat,K: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 12.39/6.89 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.39/6.89 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1641_div__mult__self1__is__m,axiom,
% 12.39/6.89 ! [N: nat,M: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 12.39/6.89 = M ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1642_div__mult__self__is__m,axiom,
% 12.39/6.89 ! [N: nat,M: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 12.39/6.89 = M ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1643_signed__take__bit__Suc__1,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 12.39/6.89 = one_one_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1644_signed__take__bit__numeral__of__1,axiom,
% 12.39/6.89 ! [K: num] :
% 12.39/6.89 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 12.39/6.89 = one_one_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1645_dbl__simps_I5_J,axiom,
% 12.39/6.89 ! [K: num] :
% 12.39/6.89 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 12.39/6.89 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1646_dbl__simps_I5_J,axiom,
% 12.39/6.89 ! [K: num] :
% 12.39/6.89 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 12.39/6.89 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1647_dbl__simps_I5_J,axiom,
% 12.39/6.89 ! [K: num] :
% 12.39/6.89 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 12.39/6.89 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1648_dbl__simps_I5_J,axiom,
% 12.39/6.89 ! [K: num] :
% 12.39/6.89 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 12.39/6.89 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1649_zero__le__divide__1__iff,axiom,
% 12.39/6.89 ! [A: real] :
% 12.39/6.89 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 12.39/6.89 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1650_zero__le__divide__1__iff,axiom,
% 12.39/6.89 ! [A: rat] :
% 12.39/6.89 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 12.39/6.89 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1651_divide__le__0__1__iff,axiom,
% 12.39/6.89 ! [A: real] :
% 12.39/6.89 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 12.39/6.89 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1652_divide__le__0__1__iff,axiom,
% 12.39/6.89 ! [A: rat] :
% 12.39/6.89 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 12.39/6.89 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1653_zero__less__divide__1__iff,axiom,
% 12.39/6.89 ! [A: real] :
% 12.39/6.89 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 12.39/6.89 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1654_zero__less__divide__1__iff,axiom,
% 12.39/6.89 ! [A: rat] :
% 12.39/6.89 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 12.39/6.89 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1655_less__divide__eq__1__pos,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( ord_less_real @ zero_zero_real @ A )
% 12.39/6.89 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 12.39/6.89 = ( ord_less_real @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1656_less__divide__eq__1__pos,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.39/6.89 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 12.39/6.89 = ( ord_less_rat @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1657_less__divide__eq__1__neg,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( ord_less_real @ A @ zero_zero_real )
% 12.39/6.89 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 12.39/6.89 = ( ord_less_real @ B2 @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1658_less__divide__eq__1__neg,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.39/6.89 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 12.39/6.89 = ( ord_less_rat @ B2 @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1659_divide__less__eq__1__pos,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( ord_less_real @ zero_zero_real @ A )
% 12.39/6.89 => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 12.39/6.89 = ( ord_less_real @ B2 @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1660_divide__less__eq__1__pos,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.39/6.89 => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 12.39/6.89 = ( ord_less_rat @ B2 @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1661_divide__less__eq__1__neg,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( ord_less_real @ A @ zero_zero_real )
% 12.39/6.89 => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 12.39/6.89 = ( ord_less_real @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1662_divide__less__eq__1__neg,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.39/6.89 => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 12.39/6.89 = ( ord_less_rat @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1663_divide__less__0__1__iff,axiom,
% 12.39/6.89 ! [A: real] :
% 12.39/6.89 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 12.39/6.89 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1664_divide__less__0__1__iff,axiom,
% 12.39/6.89 ! [A: rat] :
% 12.39/6.89 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 12.39/6.89 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1665_divide__eq__eq__numeral1_I1_J,axiom,
% 12.39/6.89 ! [B2: complex,W: num,A: complex] :
% 12.39/6.89 ( ( ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) )
% 12.39/6.89 = A )
% 12.39/6.89 = ( ( ( ( numera6690914467698888265omplex @ W )
% 12.39/6.89 != zero_zero_complex )
% 12.39/6.89 => ( B2
% 12.39/6.89 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 12.39/6.89 & ( ( ( numera6690914467698888265omplex @ W )
% 12.39/6.89 = zero_zero_complex )
% 12.39/6.89 => ( A = zero_zero_complex ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1666_divide__eq__eq__numeral1_I1_J,axiom,
% 12.39/6.89 ! [B2: real,W: num,A: real] :
% 12.39/6.89 ( ( ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) )
% 12.39/6.89 = A )
% 12.39/6.89 = ( ( ( ( numeral_numeral_real @ W )
% 12.39/6.89 != zero_zero_real )
% 12.39/6.89 => ( B2
% 12.39/6.89 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 12.39/6.89 & ( ( ( numeral_numeral_real @ W )
% 12.39/6.89 = zero_zero_real )
% 12.39/6.89 => ( A = zero_zero_real ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1667_divide__eq__eq__numeral1_I1_J,axiom,
% 12.39/6.89 ! [B2: rat,W: num,A: rat] :
% 12.39/6.89 ( ( ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) )
% 12.39/6.89 = A )
% 12.39/6.89 = ( ( ( ( numeral_numeral_rat @ W )
% 12.39/6.89 != zero_zero_rat )
% 12.39/6.89 => ( B2
% 12.39/6.89 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 12.39/6.89 & ( ( ( numeral_numeral_rat @ W )
% 12.39/6.89 = zero_zero_rat )
% 12.39/6.89 => ( A = zero_zero_rat ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1668_eq__divide__eq__numeral1_I1_J,axiom,
% 12.39/6.89 ! [A: complex,B2: complex,W: num] :
% 12.39/6.89 ( ( A
% 12.39/6.89 = ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) ) )
% 12.39/6.89 = ( ( ( ( numera6690914467698888265omplex @ W )
% 12.39/6.89 != zero_zero_complex )
% 12.39/6.89 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 12.39/6.89 = B2 ) )
% 12.39/6.89 & ( ( ( numera6690914467698888265omplex @ W )
% 12.39/6.89 = zero_zero_complex )
% 12.39/6.89 => ( A = zero_zero_complex ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1669_eq__divide__eq__numeral1_I1_J,axiom,
% 12.39/6.89 ! [A: real,B2: real,W: num] :
% 12.39/6.89 ( ( A
% 12.39/6.89 = ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 12.39/6.89 = ( ( ( ( numeral_numeral_real @ W )
% 12.39/6.89 != zero_zero_real )
% 12.39/6.89 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 12.39/6.89 = B2 ) )
% 12.39/6.89 & ( ( ( numeral_numeral_real @ W )
% 12.39/6.89 = zero_zero_real )
% 12.39/6.89 => ( A = zero_zero_real ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1670_eq__divide__eq__numeral1_I1_J,axiom,
% 12.39/6.89 ! [A: rat,B2: rat,W: num] :
% 12.39/6.89 ( ( A
% 12.39/6.89 = ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 12.39/6.89 = ( ( ( ( numeral_numeral_rat @ W )
% 12.39/6.89 != zero_zero_rat )
% 12.39/6.89 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 12.39/6.89 = B2 ) )
% 12.39/6.89 & ( ( ( numeral_numeral_rat @ W )
% 12.39/6.89 = zero_zero_rat )
% 12.39/6.89 => ( A = zero_zero_rat ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1671_nonzero__divide__mult__cancel__left,axiom,
% 12.39/6.89 ! [A: complex,B2: complex] :
% 12.39/6.89 ( ( A != zero_zero_complex )
% 12.39/6.89 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B2 ) )
% 12.39/6.89 = ( divide1717551699836669952omplex @ one_one_complex @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1672_nonzero__divide__mult__cancel__left,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( A != zero_zero_real )
% 12.39/6.89 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B2 ) )
% 12.39/6.89 = ( divide_divide_real @ one_one_real @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1673_nonzero__divide__mult__cancel__left,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( A != zero_zero_rat )
% 12.39/6.89 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B2 ) )
% 12.39/6.89 = ( divide_divide_rat @ one_one_rat @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1674_nonzero__divide__mult__cancel__right,axiom,
% 12.39/6.89 ! [B2: complex,A: complex] :
% 12.39/6.89 ( ( B2 != zero_zero_complex )
% 12.39/6.89 => ( ( divide1717551699836669952omplex @ B2 @ ( times_times_complex @ A @ B2 ) )
% 12.39/6.89 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1675_nonzero__divide__mult__cancel__right,axiom,
% 12.39/6.89 ! [B2: real,A: real] :
% 12.39/6.89 ( ( B2 != zero_zero_real )
% 12.39/6.89 => ( ( divide_divide_real @ B2 @ ( times_times_real @ A @ B2 ) )
% 12.39/6.89 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1676_nonzero__divide__mult__cancel__right,axiom,
% 12.39/6.89 ! [B2: rat,A: rat] :
% 12.39/6.89 ( ( B2 != zero_zero_rat )
% 12.39/6.89 => ( ( divide_divide_rat @ B2 @ ( times_times_rat @ A @ B2 ) )
% 12.39/6.89 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1677_div__mult__self1,axiom,
% 12.39/6.89 ! [B2: nat,A: nat,C: nat] :
% 12.39/6.89 ( ( B2 != zero_zero_nat )
% 12.39/6.89 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
% 12.39/6.89 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1678_div__mult__self1,axiom,
% 12.39/6.89 ! [B2: int,A: int,C: int] :
% 12.39/6.89 ( ( B2 != zero_zero_int )
% 12.39/6.89 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
% 12.39/6.89 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1679_div__mult__self2,axiom,
% 12.39/6.89 ! [B2: nat,A: nat,C: nat] :
% 12.39/6.89 ( ( B2 != zero_zero_nat )
% 12.39/6.89 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
% 12.39/6.89 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1680_div__mult__self2,axiom,
% 12.39/6.89 ! [B2: int,A: int,C: int] :
% 12.39/6.89 ( ( B2 != zero_zero_int )
% 12.39/6.89 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
% 12.39/6.89 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1681_div__mult__self3,axiom,
% 12.39/6.89 ! [B2: nat,C: nat,A: nat] :
% 12.39/6.89 ( ( B2 != zero_zero_nat )
% 12.39/6.89 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
% 12.39/6.89 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1682_div__mult__self3,axiom,
% 12.39/6.89 ! [B2: int,C: int,A: int] :
% 12.39/6.89 ( ( B2 != zero_zero_int )
% 12.39/6.89 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
% 12.39/6.89 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1683_div__mult__self4,axiom,
% 12.39/6.89 ! [B2: nat,C: nat,A: nat] :
% 12.39/6.89 ( ( B2 != zero_zero_nat )
% 12.39/6.89 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
% 12.39/6.89 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1684_div__mult__self4,axiom,
% 12.39/6.89 ! [B2: int,C: int,A: int] :
% 12.39/6.89 ( ( B2 != zero_zero_int )
% 12.39/6.89 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
% 12.39/6.89 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1685_power__mono__iff,axiom,
% 12.39/6.89 ! [A: real,B2: real,N: nat] :
% 12.39/6.89 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.39/6.89 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_eq_real @ A @ B2 ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1686_power__mono__iff,axiom,
% 12.39/6.89 ! [A: rat,B2: rat,N: nat] :
% 12.39/6.89 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.39/6.89 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_eq_rat @ A @ B2 ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1687_power__mono__iff,axiom,
% 12.39/6.89 ! [A: nat,B2: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.39/6.89 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_eq_nat @ A @ B2 ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1688_power__mono__iff,axiom,
% 12.39/6.89 ! [A: int,B2: int,N: nat] :
% 12.39/6.89 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.39/6.89 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_eq_int @ A @ B2 ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1689_unit__mult__div__div,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.39/6.89 => ( ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 12.39/6.89 = ( divide6298287555418463151nteger @ B2 @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1690_unit__mult__div__div,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.39/6.89 => ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A ) )
% 12.39/6.89 = ( divide_divide_nat @ B2 @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1691_unit__mult__div__div,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.39/6.89 => ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A ) )
% 12.39/6.89 = ( divide_divide_int @ B2 @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1692_unit__div__mult__self,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.39/6.89 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
% 12.39/6.89 = B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1693_unit__div__mult__self,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.39/6.89 => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
% 12.39/6.89 = B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1694_unit__div__mult__self,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.39/6.89 => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
% 12.39/6.89 = B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1695_signed__take__bit__Suc__bit0,axiom,
% 12.39/6.89 ! [N: nat,K: num] :
% 12.39/6.89 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 12.39/6.89 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1696_Suc__diff__1,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 12.39/6.89 = N ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1697_half__negative__int__iff,axiom,
% 12.39/6.89 ! [K: int] :
% 12.39/6.89 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 12.39/6.89 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1698_half__nonnegative__int__iff,axiom,
% 12.39/6.89 ! [K: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1699_le__divide__eq__1__pos,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( ord_less_real @ zero_zero_real @ A )
% 12.39/6.89 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 12.39/6.89 = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1700_le__divide__eq__1__pos,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.39/6.89 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 12.39/6.89 = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1701_le__divide__eq__1__neg,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( ord_less_real @ A @ zero_zero_real )
% 12.39/6.89 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 12.39/6.89 = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1702_le__divide__eq__1__neg,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.39/6.89 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 12.39/6.89 = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1703_divide__le__eq__1__pos,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( ord_less_real @ zero_zero_real @ A )
% 12.39/6.89 => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 12.39/6.89 = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1704_divide__le__eq__1__pos,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.39/6.89 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 12.39/6.89 = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1705_divide__le__eq__1__neg,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( ord_less_real @ A @ zero_zero_real )
% 12.39/6.89 => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 12.39/6.89 = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1706_divide__le__eq__1__neg,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.39/6.89 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 12.39/6.89 = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1707_power__strict__decreasing__iff,axiom,
% 12.39/6.89 ! [B2: real,M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.39/6.89 => ( ( ord_less_real @ B2 @ one_one_real )
% 12.39/6.89 => ( ( ord_less_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1708_power__strict__decreasing__iff,axiom,
% 12.39/6.89 ! [B2: rat,M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 12.39/6.89 => ( ( ord_less_rat @ B2 @ one_one_rat )
% 12.39/6.89 => ( ( ord_less_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1709_power__strict__decreasing__iff,axiom,
% 12.39/6.89 ! [B2: nat,M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.39/6.89 => ( ( ord_less_nat @ B2 @ one_one_nat )
% 12.39/6.89 => ( ( ord_less_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1710_power__strict__decreasing__iff,axiom,
% 12.39/6.89 ! [B2: int,M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.39/6.89 => ( ( ord_less_int @ B2 @ one_one_int )
% 12.39/6.89 => ( ( ord_less_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1711_even__mult__iff,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 12.39/6.89 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1712_even__mult__iff,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B2 ) )
% 12.39/6.89 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1713_even__mult__iff,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B2 ) )
% 12.39/6.89 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1714_zero__eq__power2,axiom,
% 12.39/6.89 ! [A: rat] :
% 12.39/6.89 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_rat )
% 12.39/6.89 = ( A = zero_zero_rat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1715_zero__eq__power2,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_nat )
% 12.39/6.89 = ( A = zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1716_zero__eq__power2,axiom,
% 12.39/6.89 ! [A: real] :
% 12.39/6.89 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_real )
% 12.39/6.89 = ( A = zero_zero_real ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1717_zero__eq__power2,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_int )
% 12.39/6.89 = ( A = zero_zero_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1718_zero__eq__power2,axiom,
% 12.39/6.89 ! [A: complex] :
% 12.39/6.89 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_complex )
% 12.39/6.89 = ( A = zero_zero_complex ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1719_odd__add,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) )
% 12.39/6.89 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 12.39/6.89 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1720_odd__add,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) )
% 12.39/6.89 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 12.39/6.89 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1721_odd__add,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) )
% 12.39/6.89 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.39/6.89 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1722_even__add,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 12.39/6.89 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1723_even__add,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) )
% 12.39/6.89 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1724_even__add,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) )
% 12.39/6.89 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1725_even__mod__2__iff,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1726_even__mod__2__iff,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1727_even__mod__2__iff,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1728_even__Suc,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 12.39/6.89 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1729_even__Suc__Suc__iff,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 12.39/6.89 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1730_not__mod2__eq__Suc__0__eq__0,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 != ( suc @ zero_zero_nat ) )
% 12.39/6.89 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1731_add__self__mod__2,axiom,
% 12.39/6.89 ! [M: nat] :
% 12.39/6.89 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1732_one__div__two__eq__zero,axiom,
% 12.39/6.89 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1733_one__div__two__eq__zero,axiom,
% 12.39/6.89 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1734_bits__1__div__2,axiom,
% 12.39/6.89 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1735_bits__1__div__2,axiom,
% 12.39/6.89 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1736_power__decreasing__iff,axiom,
% 12.39/6.89 ! [B2: real,M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.39/6.89 => ( ( ord_less_real @ B2 @ one_one_real )
% 12.39/6.89 => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1737_power__decreasing__iff,axiom,
% 12.39/6.89 ! [B2: rat,M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 12.39/6.89 => ( ( ord_less_rat @ B2 @ one_one_rat )
% 12.39/6.89 => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1738_power__decreasing__iff,axiom,
% 12.39/6.89 ! [B2: nat,M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.39/6.89 => ( ( ord_less_nat @ B2 @ one_one_nat )
% 12.39/6.89 => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1739_power__decreasing__iff,axiom,
% 12.39/6.89 ! [B2: int,M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.39/6.89 => ( ( ord_less_int @ B2 @ one_one_int )
% 12.39/6.89 => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
% 12.39/6.89 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1740_power2__less__eq__zero__iff,axiom,
% 12.39/6.89 ! [A: real] :
% 12.39/6.89 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 12.39/6.89 = ( A = zero_zero_real ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1741_power2__less__eq__zero__iff,axiom,
% 12.39/6.89 ! [A: rat] :
% 12.39/6.89 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 12.39/6.89 = ( A = zero_zero_rat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1742_power2__less__eq__zero__iff,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 12.39/6.89 = ( A = zero_zero_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1743_power2__eq__iff__nonneg,axiom,
% 12.39/6.89 ! [X4: real,Y3: real] :
% 12.39/6.89 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.39/6.89 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.39/6.89 => ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( X4 = Y3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1744_power2__eq__iff__nonneg,axiom,
% 12.39/6.89 ! [X4: rat,Y3: rat] :
% 12.39/6.89 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.39/6.89 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.39/6.89 => ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( X4 = Y3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1745_power2__eq__iff__nonneg,axiom,
% 12.39/6.89 ! [X4: nat,Y3: nat] :
% 12.39/6.89 ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
% 12.39/6.89 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 12.39/6.89 => ( ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( X4 = Y3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1746_power2__eq__iff__nonneg,axiom,
% 12.39/6.89 ! [X4: int,Y3: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.39/6.89 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.39/6.89 => ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( X4 = Y3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1747_zero__less__power2,axiom,
% 12.39/6.89 ! [A: real] :
% 12.39/6.89 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( A != zero_zero_real ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1748_zero__less__power2,axiom,
% 12.39/6.89 ! [A: rat] :
% 12.39/6.89 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( A != zero_zero_rat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1749_zero__less__power2,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( A != zero_zero_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1750_sum__power2__eq__zero__iff,axiom,
% 12.39/6.89 ! [X4: rat,Y3: rat] :
% 12.39/6.89 ( ( ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = zero_zero_rat )
% 12.39/6.89 = ( ( X4 = zero_zero_rat )
% 12.39/6.89 & ( Y3 = zero_zero_rat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1751_sum__power2__eq__zero__iff,axiom,
% 12.39/6.89 ! [X4: real,Y3: real] :
% 12.39/6.89 ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = zero_zero_real )
% 12.39/6.89 = ( ( X4 = zero_zero_real )
% 12.39/6.89 & ( Y3 = zero_zero_real ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1752_sum__power2__eq__zero__iff,axiom,
% 12.39/6.89 ! [X4: int,Y3: int] :
% 12.39/6.89 ( ( ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = zero_zero_int )
% 12.39/6.89 = ( ( X4 = zero_zero_int )
% 12.39/6.89 & ( Y3 = zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1753_even__plus__one__iff,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 12.39/6.89 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1754_even__plus__one__iff,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 12.39/6.89 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1755_even__plus__one__iff,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 12.39/6.89 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1756_even__diff,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1757_even__diff,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1758_not__mod__2__eq__0__eq__1,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 != zero_zero_nat )
% 12.39/6.89 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = one_one_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1759_not__mod__2__eq__0__eq__1,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.89 != zero_zero_int )
% 12.39/6.89 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.89 = one_one_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1760_not__mod__2__eq__0__eq__1,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.39/6.89 != zero_z3403309356797280102nteger )
% 12.39/6.89 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.39/6.89 = one_one_Code_integer ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1761_not__mod__2__eq__1__eq__0,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 != one_one_nat )
% 12.39/6.89 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1762_not__mod__2__eq__1__eq__0,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.89 != one_one_int )
% 12.39/6.89 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_zero_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1763_not__mod__2__eq__1__eq__0,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.39/6.89 != one_one_Code_integer )
% 12.39/6.89 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.39/6.89 = zero_z3403309356797280102nteger ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1764_even__power,axiom,
% 12.39/6.89 ! [A: code_integer,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 12.39/6.89 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1765_even__power,axiom,
% 12.39/6.89 ! [A: nat,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 12.39/6.89 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1766_even__power,axiom,
% 12.39/6.89 ! [A: int,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 12.39/6.89 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1767_odd__Suc__minus__one,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.39/6.89 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 12.39/6.89 = N ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1768_odd__Suc__div__two,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.39/6.89 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1769_even__Suc__div__two,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.39/6.89 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1770_mod2__gr__0,axiom,
% 12.39/6.89 ! [M: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.39/6.89 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = one_one_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1771_unset__bit__0,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 12.39/6.89 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1772_unset__bit__0,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 12.39/6.89 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1773_odd__succ__div__two,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1774_odd__succ__div__two,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1775_odd__succ__div__two,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1776_even__succ__div__two,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1777_even__succ__div__two,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1778_even__succ__div__two,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1779_even__succ__div__2,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1780_even__succ__div__2,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1781_even__succ__div__2,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.39/6.89 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1782_semiring__parity__class_Oeven__mask__iff,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 12.39/6.89 = ( N = zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1783_semiring__parity__class_Oeven__mask__iff,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 12.39/6.89 = ( N = zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1784_semiring__parity__class_Oeven__mask__iff,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 12.39/6.89 = ( N = zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1785_zero__le__power__eq__numeral,axiom,
% 12.39/6.89 ! [A: real,W: num] :
% 12.39/6.89 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.39/6.89 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1786_zero__le__power__eq__numeral,axiom,
% 12.39/6.89 ! [A: rat,W: num] :
% 12.39/6.89 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.39/6.89 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1787_zero__le__power__eq__numeral,axiom,
% 12.39/6.89 ! [A: int,W: num] :
% 12.39/6.89 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.39/6.89 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1788_zero__less__power__eq__numeral,axiom,
% 12.39/6.89 ! [A: real,W: num] :
% 12.39/6.89 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.39/6.89 = ( ( ( numeral_numeral_nat @ W )
% 12.39/6.89 = zero_zero_nat )
% 12.39/6.89 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( A != zero_zero_real ) )
% 12.39/6.89 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1789_zero__less__power__eq__numeral,axiom,
% 12.39/6.89 ! [A: rat,W: num] :
% 12.39/6.89 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.39/6.89 = ( ( ( numeral_numeral_nat @ W )
% 12.39/6.89 = zero_zero_nat )
% 12.39/6.89 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( A != zero_zero_rat ) )
% 12.39/6.89 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1790_zero__less__power__eq__numeral,axiom,
% 12.39/6.89 ! [A: int,W: num] :
% 12.39/6.89 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.39/6.89 = ( ( ( numeral_numeral_nat @ W )
% 12.39/6.89 = zero_zero_nat )
% 12.39/6.89 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( A != zero_zero_int ) )
% 12.39/6.89 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1791_power__less__zero__eq,axiom,
% 12.39/6.89 ! [A: real,N: nat] :
% 12.39/6.89 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 12.39/6.89 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.39/6.89 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1792_power__less__zero__eq,axiom,
% 12.39/6.89 ! [A: rat,N: nat] :
% 12.39/6.89 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 12.39/6.89 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.39/6.89 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1793_power__less__zero__eq,axiom,
% 12.39/6.89 ! [A: int,N: nat] :
% 12.39/6.89 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 12.39/6.89 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.39/6.89 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1794_power__less__zero__eq__numeral,axiom,
% 12.39/6.89 ! [A: real,W: num] :
% 12.39/6.89 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 12.39/6.89 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1795_power__less__zero__eq__numeral,axiom,
% 12.39/6.89 ! [A: rat,W: num] :
% 12.39/6.89 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 12.39/6.89 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1796_power__less__zero__eq__numeral,axiom,
% 12.39/6.89 ! [A: int,W: num] :
% 12.39/6.89 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 12.39/6.89 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1797_even__diff__nat,axiom,
% 12.39/6.89 ! [M: nat,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 12.39/6.89 = ( ( ord_less_nat @ M @ N )
% 12.39/6.89 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1798_odd__two__times__div__two__succ,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 12.39/6.89 = A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1799_odd__two__times__div__two__succ,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 12.39/6.89 = A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1800_odd__two__times__div__two__succ,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 12.39/6.89 = A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1801_power__le__zero__eq__numeral,axiom,
% 12.39/6.89 ! [A: real,W: num] :
% 12.39/6.89 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 12.39/6.89 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 12.39/6.89 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( A = zero_zero_real ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1802_power__le__zero__eq__numeral,axiom,
% 12.39/6.89 ! [A: rat,W: num] :
% 12.39/6.89 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 12.39/6.89 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 12.39/6.89 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( A = zero_zero_rat ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1803_power__le__zero__eq__numeral,axiom,
% 12.39/6.89 ! [A: int,W: num] :
% 12.39/6.89 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 12.39/6.89 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 12.39/6.89 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.39/6.89 & ( A = zero_zero_int ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1804_set__bit__0,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 12.39/6.89 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1805_set__bit__0,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 12.39/6.89 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1806_even__succ__div__exp,axiom,
% 12.39/6.89 ! [A: code_integer,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.89 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1807_even__succ__div__exp,axiom,
% 12.39/6.89 ! [A: nat,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.89 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1808_even__succ__div__exp,axiom,
% 12.39/6.89 ! [A: int,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.89 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1809_even__succ__mod__exp,axiom,
% 12.39/6.89 ! [A: nat,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.89 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1810_even__succ__mod__exp,axiom,
% 12.39/6.89 ! [A: int,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.89 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1811_even__succ__mod__exp,axiom,
% 12.39/6.89 ! [A: code_integer,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.39/6.89 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1812_split__zdiv,axiom,
% 12.39/6.89 ! [P: int > $o,N: int,K: int] :
% 12.39/6.89 ( ( P @ ( divide_divide_int @ N @ K ) )
% 12.39/6.89 = ( ( ( K = zero_zero_int )
% 12.39/6.89 => ( P @ zero_zero_int ) )
% 12.39/6.89 & ( ( ord_less_int @ zero_zero_int @ K )
% 12.39/6.89 => ! [I2: int,J3: int] :
% 12.39/6.89 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 12.39/6.89 & ( ord_less_int @ J3 @ K )
% 12.39/6.89 & ( N
% 12.39/6.89 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.39/6.89 => ( P @ I2 ) ) )
% 12.39/6.89 & ( ( ord_less_int @ K @ zero_zero_int )
% 12.39/6.89 => ! [I2: int,J3: int] :
% 12.39/6.89 ( ( ( ord_less_int @ K @ J3 )
% 12.39/6.89 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 12.39/6.89 & ( N
% 12.39/6.89 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.39/6.89 => ( P @ I2 ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1813_div__pos__geq,axiom,
% 12.39/6.89 ! [L2: int,K: int] :
% 12.39/6.89 ( ( ord_less_int @ zero_zero_int @ L2 )
% 12.39/6.89 => ( ( ord_less_eq_int @ L2 @ K )
% 12.39/6.89 => ( ( divide_divide_int @ K @ L2 )
% 12.39/6.89 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1814_int__div__neg__eq,axiom,
% 12.39/6.89 ! [A: int,B2: int,Q2: int,R2: int] :
% 12.39/6.89 ( ( A
% 12.39/6.89 = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 12.39/6.89 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 12.39/6.89 => ( ( ord_less_int @ B2 @ R2 )
% 12.39/6.89 => ( ( divide_divide_int @ A @ B2 )
% 12.39/6.89 = Q2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1815_int__div__pos__eq,axiom,
% 12.39/6.89 ! [A: int,B2: int,Q2: int,R2: int] :
% 12.39/6.89 ( ( A
% 12.39/6.89 = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 12.39/6.89 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 12.39/6.89 => ( ( ord_less_int @ R2 @ B2 )
% 12.39/6.89 => ( ( divide_divide_int @ A @ B2 )
% 12.39/6.89 = Q2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1816_div__mod__decomp__int,axiom,
% 12.39/6.89 ! [A3: int,N: int] :
% 12.39/6.89 ( A3
% 12.39/6.89 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ N ) @ N ) @ ( modulo_modulo_int @ A3 @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1817_split__neg__lemma,axiom,
% 12.39/6.89 ! [K: int,P: int > int > $o,N: int] :
% 12.39/6.89 ( ( ord_less_int @ K @ zero_zero_int )
% 12.39/6.89 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 12.39/6.89 = ( ! [I2: int,J3: int] :
% 12.39/6.89 ( ( ( ord_less_int @ K @ J3 )
% 12.39/6.89 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 12.39/6.89 & ( N
% 12.39/6.89 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.39/6.89 => ( P @ I2 @ J3 ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1818_split__pos__lemma,axiom,
% 12.39/6.89 ! [K: int,P: int > int > $o,N: int] :
% 12.39/6.89 ( ( ord_less_int @ zero_zero_int @ K )
% 12.39/6.89 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 12.39/6.89 = ( ! [I2: int,J3: int] :
% 12.39/6.89 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 12.39/6.89 & ( ord_less_int @ J3 @ K )
% 12.39/6.89 & ( N
% 12.39/6.89 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.39/6.89 => ( P @ I2 @ J3 ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1819_int__div__less__self,axiom,
% 12.39/6.89 ! [X4: int,K: int] :
% 12.39/6.89 ( ( ord_less_int @ zero_zero_int @ X4 )
% 12.39/6.89 => ( ( ord_less_int @ one_one_int @ K )
% 12.39/6.89 => ( ord_less_int @ ( divide_divide_int @ X4 @ K ) @ X4 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1820_verit__le__mono__div__int,axiom,
% 12.39/6.89 ! [A3: int,B4: int,N: int] :
% 12.39/6.89 ( ( ord_less_int @ A3 @ B4 )
% 12.39/6.89 => ( ( ord_less_int @ zero_zero_int @ N )
% 12.39/6.89 => ( ord_less_eq_int
% 12.39/6.89 @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N )
% 12.39/6.89 @ ( if_int
% 12.39/6.89 @ ( ( modulo_modulo_int @ B4 @ N )
% 12.39/6.89 = zero_zero_int )
% 12.39/6.89 @ one_one_int
% 12.39/6.89 @ zero_zero_int ) )
% 12.39/6.89 @ ( divide_divide_int @ B4 @ N ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1821_zdiv__zmult2__eq,axiom,
% 12.39/6.89 ! [C: int,A: int,B2: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.39/6.89 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 12.39/6.89 = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1822_zmod__zmult2__eq,axiom,
% 12.39/6.89 ! [C: int,A: int,B2: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.39/6.89 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
% 12.39/6.89 = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1823_nonneg1__imp__zdiv__pos__iff,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.39/6.89 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 12.39/6.89 = ( ( ord_less_eq_int @ B2 @ A )
% 12.39/6.89 & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1824_pos__imp__zdiv__nonneg__iff,axiom,
% 12.39/6.89 ! [B2: int,A: int] :
% 12.39/6.89 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.39/6.89 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 12.39/6.89 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1825_neg__imp__zdiv__nonneg__iff,axiom,
% 12.39/6.89 ! [B2: int,A: int] :
% 12.39/6.89 ( ( ord_less_int @ B2 @ zero_zero_int )
% 12.39/6.89 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 12.39/6.89 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1826_pos__imp__zdiv__pos__iff,axiom,
% 12.39/6.89 ! [K: int,I: int] :
% 12.39/6.89 ( ( ord_less_int @ zero_zero_int @ K )
% 12.39/6.89 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 12.39/6.89 = ( ord_less_eq_int @ K @ I ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1827_pos__imp__zdiv__neg__iff,axiom,
% 12.39/6.89 ! [B2: int,A: int] :
% 12.39/6.89 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.39/6.89 => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
% 12.39/6.89 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1828_neg__imp__zdiv__neg__iff,axiom,
% 12.39/6.89 ! [B2: int,A: int] :
% 12.39/6.89 ( ( ord_less_int @ B2 @ zero_zero_int )
% 12.39/6.89 => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
% 12.39/6.89 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1829_div__nonpos__pos__le0,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.39/6.89 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.39/6.89 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1830_div__nonneg__neg__le0,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.39/6.89 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 12.39/6.89 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1831_div__neg__pos__less0,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( ord_less_int @ A @ zero_zero_int )
% 12.39/6.89 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.39/6.89 => ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1832_div__positive__int,axiom,
% 12.39/6.89 ! [L2: int,K: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ L2 @ K )
% 12.39/6.89 => ( ( ord_less_int @ zero_zero_int @ L2 )
% 12.39/6.89 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1833_div__int__pos__iff,axiom,
% 12.39/6.89 ! [K: int,L2: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
% 12.39/6.89 = ( ( K = zero_zero_int )
% 12.39/6.89 | ( L2 = zero_zero_int )
% 12.39/6.89 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.39/6.89 & ( ord_less_eq_int @ zero_zero_int @ L2 ) )
% 12.39/6.89 | ( ( ord_less_int @ K @ zero_zero_int )
% 12.39/6.89 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1834_zdiv__mono2__neg,axiom,
% 12.39/6.89 ! [A: int,B: int,B2: int] :
% 12.39/6.89 ( ( ord_less_int @ A @ zero_zero_int )
% 12.39/6.89 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.39/6.89 => ( ( ord_less_eq_int @ B @ B2 )
% 12.39/6.89 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1835_zdiv__mono1__neg,axiom,
% 12.39/6.89 ! [A: int,A2: int,B2: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ A @ A2 )
% 12.39/6.89 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 12.39/6.89 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1836_zdiv__eq__0__iff,axiom,
% 12.39/6.89 ! [I: int,K: int] :
% 12.39/6.89 ( ( ( divide_divide_int @ I @ K )
% 12.39/6.89 = zero_zero_int )
% 12.39/6.89 = ( ( K = zero_zero_int )
% 12.39/6.89 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 12.39/6.89 & ( ord_less_int @ I @ K ) )
% 12.39/6.89 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 12.39/6.89 & ( ord_less_int @ K @ I ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1837_zdiv__mono__strict,axiom,
% 12.39/6.89 ! [A3: int,B4: int,N: int] :
% 12.39/6.89 ( ( ord_less_int @ A3 @ B4 )
% 12.39/6.89 => ( ( ord_less_int @ zero_zero_int @ N )
% 12.39/6.89 => ( ( ( modulo_modulo_int @ A3 @ N )
% 12.39/6.89 = zero_zero_int )
% 12.39/6.89 => ( ( ( modulo_modulo_int @ B4 @ N )
% 12.39/6.89 = zero_zero_int )
% 12.39/6.89 => ( ord_less_int @ ( divide_divide_int @ A3 @ N ) @ ( divide_divide_int @ B4 @ N ) ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1838_zdiv__mono2,axiom,
% 12.39/6.89 ! [A: int,B: int,B2: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.39/6.89 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.39/6.89 => ( ( ord_less_eq_int @ B @ B2 )
% 12.39/6.89 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1839_zdiv__mono1,axiom,
% 12.39/6.89 ! [A: int,A2: int,B2: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ A @ A2 )
% 12.39/6.89 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.39/6.89 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1840_dvd__field__iff,axiom,
% 12.39/6.89 ( dvd_dvd_real
% 12.39/6.89 = ( ^ [A4: real,B3: real] :
% 12.39/6.89 ( ( A4 = zero_zero_real )
% 12.39/6.89 => ( B3 = zero_zero_real ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1841_dvd__field__iff,axiom,
% 12.39/6.89 ( dvd_dvd_rat
% 12.39/6.89 = ( ^ [A4: rat,B3: rat] :
% 12.39/6.89 ( ( A4 = zero_zero_rat )
% 12.39/6.89 => ( B3 = zero_zero_rat ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1842_dvd__refl,axiom,
% 12.39/6.89 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1843_dvd__refl,axiom,
% 12.39/6.89 ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1844_dvd__refl,axiom,
% 12.39/6.89 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1845_dvd__trans,axiom,
% 12.39/6.89 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_nat @ B2 @ C )
% 12.39/6.89 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1846_dvd__trans,axiom,
% 12.39/6.89 ! [A: int,B2: int,C: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_int @ B2 @ C )
% 12.39/6.89 => ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1847_dvd__trans,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ B2 @ C )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1848_dvd__0__left,axiom,
% 12.39/6.89 ! [A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 12.39/6.89 => ( A = zero_z3403309356797280102nteger ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1849_dvd__0__left,axiom,
% 12.39/6.89 ! [A: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 12.39/6.89 => ( A = zero_zero_real ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1850_dvd__0__left,axiom,
% 12.39/6.89 ! [A: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 12.39/6.89 => ( A = zero_zero_rat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1851_dvd__0__left,axiom,
% 12.39/6.89 ! [A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 12.39/6.89 => ( A = zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1852_dvd__0__left,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 12.39/6.89 => ( A = zero_zero_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1853_dvd__antisym,axiom,
% 12.39/6.89 ! [M: nat,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ M @ N )
% 12.39/6.89 => ( ( dvd_dvd_nat @ N @ M )
% 12.39/6.89 => ( M = N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1854_zero__reorient,axiom,
% 12.39/6.89 ! [X4: literal] :
% 12.39/6.89 ( ( zero_zero_literal = X4 )
% 12.39/6.89 = ( X4 = zero_zero_literal ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1855_zero__reorient,axiom,
% 12.39/6.89 ! [X4: real] :
% 12.39/6.89 ( ( zero_zero_real = X4 )
% 12.39/6.89 = ( X4 = zero_zero_real ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1856_zero__reorient,axiom,
% 12.39/6.89 ! [X4: rat] :
% 12.39/6.89 ( ( zero_zero_rat = X4 )
% 12.39/6.89 = ( X4 = zero_zero_rat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1857_zero__reorient,axiom,
% 12.39/6.89 ! [X4: nat] :
% 12.39/6.89 ( ( zero_zero_nat = X4 )
% 12.39/6.89 = ( X4 = zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1858_zero__reorient,axiom,
% 12.39/6.89 ! [X4: int] :
% 12.39/6.89 ( ( zero_zero_int = X4 )
% 12.39/6.89 = ( X4 = zero_zero_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1859_not__is__unit__0,axiom,
% 12.39/6.89 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1860_not__is__unit__0,axiom,
% 12.39/6.89 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1861_not__is__unit__0,axiom,
% 12.39/6.89 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1862_dvd__div__eq__0__iff,axiom,
% 12.39/6.89 ! [B2: code_integer,A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 12.39/6.89 => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 12.39/6.89 = zero_z3403309356797280102nteger )
% 12.39/6.89 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1863_dvd__div__eq__0__iff,axiom,
% 12.39/6.89 ! [B2: complex,A: complex] :
% 12.39/6.89 ( ( dvd_dvd_complex @ B2 @ A )
% 12.39/6.89 => ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 12.39/6.89 = zero_zero_complex )
% 12.39/6.89 = ( A = zero_zero_complex ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1864_dvd__div__eq__0__iff,axiom,
% 12.39/6.89 ! [B2: real,A: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ B2 @ A )
% 12.39/6.89 => ( ( ( divide_divide_real @ A @ B2 )
% 12.39/6.89 = zero_zero_real )
% 12.39/6.89 = ( A = zero_zero_real ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1865_dvd__div__eq__0__iff,axiom,
% 12.39/6.89 ! [B2: rat,A: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ B2 @ A )
% 12.39/6.89 => ( ( ( divide_divide_rat @ A @ B2 )
% 12.39/6.89 = zero_zero_rat )
% 12.39/6.89 = ( A = zero_zero_rat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1866_dvd__div__eq__0__iff,axiom,
% 12.39/6.89 ! [B2: nat,A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ B2 @ A )
% 12.39/6.89 => ( ( ( divide_divide_nat @ A @ B2 )
% 12.39/6.89 = zero_zero_nat )
% 12.39/6.89 = ( A = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1867_dvd__div__eq__0__iff,axiom,
% 12.39/6.89 ! [B2: int,A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ B2 @ A )
% 12.39/6.89 => ( ( ( divide_divide_int @ A @ B2 )
% 12.39/6.89 = zero_zero_int )
% 12.39/6.89 = ( A = zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1868_nat__dvd__not__less,axiom,
% 12.39/6.89 ! [M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.39/6.89 => ( ( ord_less_nat @ M @ N )
% 12.39/6.89 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1869_mod__0__imp__dvd,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( ( modulo_modulo_nat @ A @ B2 )
% 12.39/6.89 = zero_zero_nat )
% 12.39/6.89 => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1870_mod__0__imp__dvd,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( ( modulo_modulo_int @ A @ B2 )
% 12.39/6.89 = zero_zero_int )
% 12.39/6.89 => ( dvd_dvd_int @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1871_mod__0__imp__dvd,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 12.39/6.89 = zero_z3403309356797280102nteger )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1872_dvd__eq__mod__eq__0,axiom,
% 12.39/6.89 ( dvd_dvd_nat
% 12.39/6.89 = ( ^ [A4: nat,B3: nat] :
% 12.39/6.89 ( ( modulo_modulo_nat @ B3 @ A4 )
% 12.39/6.89 = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1873_dvd__eq__mod__eq__0,axiom,
% 12.39/6.89 ( dvd_dvd_int
% 12.39/6.89 = ( ^ [A4: int,B3: int] :
% 12.39/6.89 ( ( modulo_modulo_int @ B3 @ A4 )
% 12.39/6.89 = zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1874_dvd__eq__mod__eq__0,axiom,
% 12.39/6.89 ( dvd_dvd_Code_integer
% 12.39/6.89 = ( ^ [A4: code_integer,B3: code_integer] :
% 12.39/6.89 ( ( modulo364778990260209775nteger @ B3 @ A4 )
% 12.39/6.89 = zero_z3403309356797280102nteger ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1875_mod__eq__0__iff__dvd,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( ( modulo_modulo_nat @ A @ B2 )
% 12.39/6.89 = zero_zero_nat )
% 12.39/6.89 = ( dvd_dvd_nat @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1876_mod__eq__0__iff__dvd,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( ( modulo_modulo_int @ A @ B2 )
% 12.39/6.89 = zero_zero_int )
% 12.39/6.89 = ( dvd_dvd_int @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1877_mod__eq__0__iff__dvd,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 12.39/6.89 = zero_z3403309356797280102nteger )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1878_list__decode_Ocases,axiom,
% 12.39/6.89 ! [X4: nat] :
% 12.39/6.89 ( ( X4 != zero_zero_nat )
% 12.39/6.89 => ~ ! [N4: nat] :
% 12.39/6.89 ( X4
% 12.39/6.89 != ( suc @ N4 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1879_pos__zmod__mult__2,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.39/6.89 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.39/6.89 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ A ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1880_neg__zmod__mult__2,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.39/6.89 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.39/6.89 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1881_unit__dvdE,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.39/6.89 => ~ ( ( A != zero_z3403309356797280102nteger )
% 12.39/6.89 => ! [C3: code_integer] :
% 12.39/6.89 ( B2
% 12.39/6.89 != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1882_unit__dvdE,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.39/6.89 => ~ ( ( A != zero_zero_nat )
% 12.39/6.89 => ! [C3: nat] :
% 12.39/6.89 ( B2
% 12.39/6.89 != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1883_unit__dvdE,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.39/6.89 => ~ ( ( A != zero_zero_int )
% 12.39/6.89 => ! [C3: int] :
% 12.39/6.89 ( B2
% 12.39/6.89 != ( times_times_int @ A @ C3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1884_dvd__div__eq__mult,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.89 ( ( A != zero_z3403309356797280102nteger )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.89 => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 12.39/6.89 = C )
% 12.39/6.89 = ( B2
% 12.39/6.89 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1885_dvd__div__eq__mult,axiom,
% 12.39/6.89 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.89 ( ( A != zero_zero_nat )
% 12.39/6.89 => ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.89 => ( ( ( divide_divide_nat @ B2 @ A )
% 12.39/6.89 = C )
% 12.39/6.89 = ( B2
% 12.39/6.89 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1886_dvd__div__eq__mult,axiom,
% 12.39/6.89 ! [A: int,B2: int,C: int] :
% 12.39/6.89 ( ( A != zero_zero_int )
% 12.39/6.89 => ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.89 => ( ( ( divide_divide_int @ B2 @ A )
% 12.39/6.89 = C )
% 12.39/6.89 = ( B2
% 12.39/6.89 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1887_div__dvd__iff__mult,axiom,
% 12.39/6.89 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 12.39/6.89 ( ( B2 != zero_z3403309356797280102nteger )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1888_div__dvd__iff__mult,axiom,
% 12.39/6.89 ! [B2: nat,A: nat,C: nat] :
% 12.39/6.89 ( ( B2 != zero_zero_nat )
% 12.39/6.89 => ( ( dvd_dvd_nat @ B2 @ A )
% 12.39/6.89 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 12.39/6.89 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1889_div__dvd__iff__mult,axiom,
% 12.39/6.89 ! [B2: int,A: int,C: int] :
% 12.39/6.89 ( ( B2 != zero_zero_int )
% 12.39/6.89 => ( ( dvd_dvd_int @ B2 @ A )
% 12.39/6.89 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 12.39/6.89 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1890_dvd__div__iff__mult,axiom,
% 12.39/6.89 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 12.39/6.89 ( ( C != zero_z3403309356797280102nteger )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1891_dvd__div__iff__mult,axiom,
% 12.39/6.89 ! [C: nat,B2: nat,A: nat] :
% 12.39/6.89 ( ( C != zero_zero_nat )
% 12.39/6.89 => ( ( dvd_dvd_nat @ C @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1892_dvd__div__iff__mult,axiom,
% 12.39/6.89 ! [C: int,B2: int,A: int] :
% 12.39/6.89 ( ( C != zero_zero_int )
% 12.39/6.89 => ( ( dvd_dvd_int @ C @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1893_dvd__div__div__eq__mult,axiom,
% 12.39/6.89 ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
% 12.39/6.89 ( ( A != zero_z3403309356797280102nteger )
% 12.39/6.89 => ( ( C != zero_z3403309356797280102nteger )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ C @ D )
% 12.39/6.89 => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 12.39/6.89 = ( divide6298287555418463151nteger @ D @ C ) )
% 12.39/6.89 = ( ( times_3573771949741848930nteger @ B2 @ C )
% 12.39/6.89 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1894_dvd__div__div__eq__mult,axiom,
% 12.39/6.89 ! [A: nat,C: nat,B2: nat,D: nat] :
% 12.39/6.89 ( ( A != zero_zero_nat )
% 12.39/6.89 => ( ( C != zero_zero_nat )
% 12.39/6.89 => ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_nat @ C @ D )
% 12.39/6.89 => ( ( ( divide_divide_nat @ B2 @ A )
% 12.39/6.89 = ( divide_divide_nat @ D @ C ) )
% 12.39/6.89 = ( ( times_times_nat @ B2 @ C )
% 12.39/6.89 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1895_dvd__div__div__eq__mult,axiom,
% 12.39/6.89 ! [A: int,C: int,B2: int,D: int] :
% 12.39/6.89 ( ( A != zero_zero_int )
% 12.39/6.89 => ( ( C != zero_zero_int )
% 12.39/6.89 => ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_int @ C @ D )
% 12.39/6.89 => ( ( ( divide_divide_int @ B2 @ A )
% 12.39/6.89 = ( divide_divide_int @ D @ C ) )
% 12.39/6.89 = ( ( times_times_int @ B2 @ C )
% 12.39/6.89 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1896_unit__div__eq__0__iff,axiom,
% 12.39/6.89 ! [B2: code_integer,A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.39/6.89 => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 12.39/6.89 = zero_z3403309356797280102nteger )
% 12.39/6.89 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1897_unit__div__eq__0__iff,axiom,
% 12.39/6.89 ! [B2: nat,A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.39/6.89 => ( ( ( divide_divide_nat @ A @ B2 )
% 12.39/6.89 = zero_zero_nat )
% 12.39/6.89 = ( A = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1898_unit__div__eq__0__iff,axiom,
% 12.39/6.89 ! [B2: int,A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.39/6.89 => ( ( ( divide_divide_int @ A @ B2 )
% 12.39/6.89 = zero_zero_int )
% 12.39/6.89 = ( A = zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1899_is__unit__power__iff,axiom,
% 12.39/6.89 ! [A: code_integer,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 12.39/6.89 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.39/6.89 | ( N = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1900_is__unit__power__iff,axiom,
% 12.39/6.89 ! [A: nat,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 12.39/6.89 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.39/6.89 | ( N = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1901_is__unit__power__iff,axiom,
% 12.39/6.89 ! [A: int,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 12.39/6.89 = ( ( dvd_dvd_int @ A @ one_one_int )
% 12.39/6.89 | ( N = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1902_unit__imp__mod__eq__0,axiom,
% 12.39/6.89 ! [B2: nat,A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.39/6.89 => ( ( modulo_modulo_nat @ A @ B2 )
% 12.39/6.89 = zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1903_unit__imp__mod__eq__0,axiom,
% 12.39/6.89 ! [B2: int,A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.39/6.89 => ( ( modulo_modulo_int @ A @ B2 )
% 12.39/6.89 = zero_zero_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1904_unit__imp__mod__eq__0,axiom,
% 12.39/6.89 ! [B2: code_integer,A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.39/6.89 => ( ( modulo364778990260209775nteger @ A @ B2 )
% 12.39/6.89 = zero_z3403309356797280102nteger ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1905_dvd__imp__le,axiom,
% 12.39/6.89 ! [K: nat,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ K @ N )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1906_nat__mult__dvd__cancel1,axiom,
% 12.39/6.89 ! [K: nat,M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.39/6.89 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.39/6.89 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1907_dvd__mult__cancel,axiom,
% 12.39/6.89 ! [K: nat,M: nat,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.39/6.89 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.39/6.89 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1908_power__0__left,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ( N = zero_zero_nat )
% 12.39/6.89 => ( ( power_power_rat @ zero_zero_rat @ N )
% 12.39/6.89 = one_one_rat ) )
% 12.39/6.89 & ( ( N != zero_zero_nat )
% 12.39/6.89 => ( ( power_power_rat @ zero_zero_rat @ N )
% 12.39/6.89 = zero_zero_rat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1909_power__0__left,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ( N = zero_zero_nat )
% 12.39/6.89 => ( ( power_power_nat @ zero_zero_nat @ N )
% 12.39/6.89 = one_one_nat ) )
% 12.39/6.89 & ( ( N != zero_zero_nat )
% 12.39/6.89 => ( ( power_power_nat @ zero_zero_nat @ N )
% 12.39/6.89 = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1910_power__0__left,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ( N = zero_zero_nat )
% 12.39/6.89 => ( ( power_power_real @ zero_zero_real @ N )
% 12.39/6.89 = one_one_real ) )
% 12.39/6.89 & ( ( N != zero_zero_nat )
% 12.39/6.89 => ( ( power_power_real @ zero_zero_real @ N )
% 12.39/6.89 = zero_zero_real ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1911_power__0__left,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ( N = zero_zero_nat )
% 12.39/6.89 => ( ( power_power_int @ zero_zero_int @ N )
% 12.39/6.89 = one_one_int ) )
% 12.39/6.89 & ( ( N != zero_zero_nat )
% 12.39/6.89 => ( ( power_power_int @ zero_zero_int @ N )
% 12.39/6.89 = zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1912_power__0__left,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ( N = zero_zero_nat )
% 12.39/6.89 => ( ( power_power_complex @ zero_zero_complex @ N )
% 12.39/6.89 = one_one_complex ) )
% 12.39/6.89 & ( ( N != zero_zero_nat )
% 12.39/6.89 => ( ( power_power_complex @ zero_zero_complex @ N )
% 12.39/6.89 = zero_zero_complex ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1913_zero__power,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( power_power_rat @ zero_zero_rat @ N )
% 12.39/6.89 = zero_zero_rat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1914_zero__power,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( power_power_nat @ zero_zero_nat @ N )
% 12.39/6.89 = zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1915_zero__power,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( power_power_real @ zero_zero_real @ N )
% 12.39/6.89 = zero_zero_real ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1916_zero__power,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( power_power_int @ zero_zero_int @ N )
% 12.39/6.89 = zero_zero_int ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1917_zero__power,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 => ( ( power_power_complex @ zero_zero_complex @ N )
% 12.39/6.89 = zero_zero_complex ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1918_mod__greater__zero__iff__not__dvd,axiom,
% 12.39/6.89 ! [M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 12.39/6.89 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1919_even__set__bit__iff,axiom,
% 12.39/6.89 ! [M: nat,A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 12.39/6.89 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 & ( M != zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1920_even__set__bit__iff,axiom,
% 12.39/6.89 ! [M: nat,A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 12.39/6.89 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 & ( M != zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1921_even__set__bit__iff,axiom,
% 12.39/6.89 ! [M: nat,A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 12.39/6.89 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 & ( M != zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1922_even__flip__bit__iff,axiom,
% 12.39/6.89 ! [M: nat,A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 12.39/6.89 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 != ( M = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1923_even__flip__bit__iff,axiom,
% 12.39/6.89 ! [M: nat,A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 12.39/6.89 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 != ( M = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1924_even__flip__bit__iff,axiom,
% 12.39/6.89 ! [M: nat,A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 12.39/6.89 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 != ( M = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1925_even__unset__bit__iff,axiom,
% 12.39/6.89 ! [M: nat,A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 12.39/6.89 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 | ( M = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1926_even__unset__bit__iff,axiom,
% 12.39/6.89 ! [M: nat,A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 12.39/6.89 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 | ( M = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1927_even__unset__bit__iff,axiom,
% 12.39/6.89 ! [M: nat,A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 12.39/6.89 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.39/6.89 | ( M = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1928_dvd__triv__right,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1929_dvd__triv__right,axiom,
% 12.39/6.89 ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1930_dvd__triv__right,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1931_dvd__triv__right,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1932_dvd__triv__right,axiom,
% 12.39/6.89 ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1933_dvd__mult__right,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ B2 @ C ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1934_dvd__mult__right,axiom,
% 12.39/6.89 ! [A: real,B2: real,C: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
% 12.39/6.89 => ( dvd_dvd_real @ B2 @ C ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1935_dvd__mult__right,axiom,
% 12.39/6.89 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 12.39/6.89 => ( dvd_dvd_rat @ B2 @ C ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1936_dvd__mult__right,axiom,
% 12.39/6.89 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 12.39/6.89 => ( dvd_dvd_nat @ B2 @ C ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1937_dvd__mult__right,axiom,
% 12.39/6.89 ! [A: int,B2: int,C: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 12.39/6.89 => ( dvd_dvd_int @ B2 @ C ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1938_mult__dvd__mono,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ C @ D )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1939_mult__dvd__mono,axiom,
% 12.39/6.89 ! [A: real,B2: real,C: real,D: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_real @ C @ D )
% 12.39/6.89 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1940_mult__dvd__mono,axiom,
% 12.39/6.89 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_rat @ C @ D )
% 12.39/6.89 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1941_mult__dvd__mono,axiom,
% 12.39/6.89 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_nat @ C @ D )
% 12.39/6.89 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1942_mult__dvd__mono,axiom,
% 12.39/6.89 ! [A: int,B2: int,C: int,D: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_int @ C @ D )
% 12.39/6.89 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1943_dvd__triv__left,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1944_dvd__triv__left,axiom,
% 12.39/6.89 ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1945_dvd__triv__left,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1946_dvd__triv__left,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1947_dvd__triv__left,axiom,
% 12.39/6.89 ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B2 ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1948_dvd__mult__left,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1949_dvd__mult__left,axiom,
% 12.39/6.89 ! [A: real,B2: real,C: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
% 12.39/6.89 => ( dvd_dvd_real @ A @ C ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1950_dvd__mult__left,axiom,
% 12.39/6.89 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 12.39/6.89 => ( dvd_dvd_rat @ A @ C ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1951_dvd__mult__left,axiom,
% 12.39/6.89 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 12.39/6.89 => ( dvd_dvd_nat @ A @ C ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1952_dvd__mult__left,axiom,
% 12.39/6.89 ! [A: int,B2: int,C: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 12.39/6.89 => ( dvd_dvd_int @ A @ C ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1953_dvd__mult2,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1954_dvd__mult2,axiom,
% 12.39/6.89 ! [A: real,B2: real,C: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ A @ B2 )
% 12.39/6.89 => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1955_dvd__mult2,axiom,
% 12.39/6.89 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ A @ B2 )
% 12.39/6.89 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1956_dvd__mult2,axiom,
% 12.39/6.89 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.89 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1957_dvd__mult2,axiom,
% 12.39/6.89 ! [A: int,B2: int,C: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.89 => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1958_dvd__mult,axiom,
% 12.39/6.89 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ C )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1959_dvd__mult,axiom,
% 12.39/6.89 ! [A: real,C: real,B2: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ A @ C )
% 12.39/6.89 => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1960_dvd__mult,axiom,
% 12.39/6.89 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ A @ C )
% 12.39/6.89 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1961_dvd__mult,axiom,
% 12.39/6.89 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ C )
% 12.39/6.89 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1962_dvd__mult,axiom,
% 12.39/6.89 ! [A: int,C: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ C )
% 12.39/6.89 => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1963_dvd__def,axiom,
% 12.39/6.89 ( dvd_dvd_Code_integer
% 12.39/6.89 = ( ^ [B3: code_integer,A4: code_integer] :
% 12.39/6.89 ? [K3: code_integer] :
% 12.39/6.89 ( A4
% 12.39/6.89 = ( times_3573771949741848930nteger @ B3 @ K3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1964_dvd__def,axiom,
% 12.39/6.89 ( dvd_dvd_real
% 12.39/6.89 = ( ^ [B3: real,A4: real] :
% 12.39/6.89 ? [K3: real] :
% 12.39/6.89 ( A4
% 12.39/6.89 = ( times_times_real @ B3 @ K3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1965_dvd__def,axiom,
% 12.39/6.89 ( dvd_dvd_rat
% 12.39/6.89 = ( ^ [B3: rat,A4: rat] :
% 12.39/6.89 ? [K3: rat] :
% 12.39/6.89 ( A4
% 12.39/6.89 = ( times_times_rat @ B3 @ K3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1966_dvd__def,axiom,
% 12.39/6.89 ( dvd_dvd_nat
% 12.39/6.89 = ( ^ [B3: nat,A4: nat] :
% 12.39/6.89 ? [K3: nat] :
% 12.39/6.89 ( A4
% 12.39/6.89 = ( times_times_nat @ B3 @ K3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1967_dvd__def,axiom,
% 12.39/6.89 ( dvd_dvd_int
% 12.39/6.89 = ( ^ [B3: int,A4: int] :
% 12.39/6.89 ? [K3: int] :
% 12.39/6.89 ( A4
% 12.39/6.89 = ( times_times_int @ B3 @ K3 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1968_dvdI,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer,K: code_integer] :
% 12.39/6.89 ( ( A
% 12.39/6.89 = ( times_3573771949741848930nteger @ B2 @ K ) )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1969_dvdI,axiom,
% 12.39/6.89 ! [A: real,B2: real,K: real] :
% 12.39/6.89 ( ( A
% 12.39/6.89 = ( times_times_real @ B2 @ K ) )
% 12.39/6.89 => ( dvd_dvd_real @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1970_dvdI,axiom,
% 12.39/6.89 ! [A: rat,B2: rat,K: rat] :
% 12.39/6.89 ( ( A
% 12.39/6.89 = ( times_times_rat @ B2 @ K ) )
% 12.39/6.89 => ( dvd_dvd_rat @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1971_dvdI,axiom,
% 12.39/6.89 ! [A: nat,B2: nat,K: nat] :
% 12.39/6.89 ( ( A
% 12.39/6.89 = ( times_times_nat @ B2 @ K ) )
% 12.39/6.89 => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1972_dvdI,axiom,
% 12.39/6.89 ! [A: int,B2: int,K: int] :
% 12.39/6.89 ( ( A
% 12.39/6.89 = ( times_times_int @ B2 @ K ) )
% 12.39/6.89 => ( dvd_dvd_int @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1973_dvdE,axiom,
% 12.39/6.89 ! [B2: code_integer,A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 12.39/6.89 => ~ ! [K2: code_integer] :
% 12.39/6.89 ( A
% 12.39/6.89 != ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1974_dvdE,axiom,
% 12.39/6.89 ! [B2: real,A: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ B2 @ A )
% 12.39/6.89 => ~ ! [K2: real] :
% 12.39/6.89 ( A
% 12.39/6.89 != ( times_times_real @ B2 @ K2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1975_dvdE,axiom,
% 12.39/6.89 ! [B2: rat,A: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ B2 @ A )
% 12.39/6.89 => ~ ! [K2: rat] :
% 12.39/6.89 ( A
% 12.39/6.89 != ( times_times_rat @ B2 @ K2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1976_dvdE,axiom,
% 12.39/6.89 ! [B2: nat,A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ B2 @ A )
% 12.39/6.89 => ~ ! [K2: nat] :
% 12.39/6.89 ( A
% 12.39/6.89 != ( times_times_nat @ B2 @ K2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1977_dvdE,axiom,
% 12.39/6.89 ! [B2: int,A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ B2 @ A )
% 12.39/6.89 => ~ ! [K2: int] :
% 12.39/6.89 ( A
% 12.39/6.89 != ( times_times_int @ B2 @ K2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1978_dvd__unit__imp__unit,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1979_dvd__unit__imp__unit,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.39/6.89 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1980_dvd__unit__imp__unit,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.39/6.89 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1981_unit__imp__dvd,axiom,
% 12.39/6.89 ! [B2: code_integer,A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1982_unit__imp__dvd,axiom,
% 12.39/6.89 ! [B2: nat,A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.39/6.89 => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1983_unit__imp__dvd,axiom,
% 12.39/6.89 ! [B2: int,A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.39/6.89 => ( dvd_dvd_int @ B2 @ A ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1984_one__dvd,axiom,
% 12.39/6.89 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1985_one__dvd,axiom,
% 12.39/6.89 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1986_one__dvd,axiom,
% 12.39/6.89 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1987_one__dvd,axiom,
% 12.39/6.89 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1988_one__dvd,axiom,
% 12.39/6.89 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1989_one__dvd,axiom,
% 12.39/6.89 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1990_dvd__add__right__iff,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1991_dvd__add__right__iff,axiom,
% 12.39/6.89 ! [A: real,B2: real,C: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1992_dvd__add__right__iff,axiom,
% 12.39/6.89 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1993_dvd__add__right__iff,axiom,
% 12.39/6.89 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1994_dvd__add__right__iff,axiom,
% 12.39/6.89 ! [A: int,B2: int,C: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1995_dvd__add__left__iff,axiom,
% 12.39/6.89 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ C )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1996_dvd__add__left__iff,axiom,
% 12.39/6.89 ! [A: real,C: real,B2: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ A @ C )
% 12.39/6.89 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1997_dvd__add__left__iff,axiom,
% 12.39/6.89 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ A @ C )
% 12.39/6.89 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1998_dvd__add__left__iff,axiom,
% 12.39/6.89 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ C )
% 12.39/6.89 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_nat @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_1999_dvd__add__left__iff,axiom,
% 12.39/6.89 ! [A: int,C: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ C )
% 12.39/6.89 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 12.39/6.89 = ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2000_dvd__add,axiom,
% 12.39/6.89 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ A @ C )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2001_dvd__add,axiom,
% 12.39/6.89 ! [A: real,B2: real,C: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_real @ A @ C )
% 12.39/6.89 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2002_dvd__add,axiom,
% 12.39/6.89 ! [A: rat,B2: rat,C: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_rat @ A @ C )
% 12.39/6.89 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2003_dvd__add,axiom,
% 12.39/6.89 ! [A: nat,B2: nat,C: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_nat @ A @ C )
% 12.39/6.89 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2004_dvd__add,axiom,
% 12.39/6.89 ! [A: int,B2: int,C: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_int @ A @ C )
% 12.39/6.89 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2005_dvd__diff__commute,axiom,
% 12.39/6.89 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2006_dvd__diff__commute,axiom,
% 12.39/6.89 ! [A: int,C: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2007_dvd__diff,axiom,
% 12.39/6.89 ! [X4: code_integer,Y3: code_integer,Z: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ X4 @ Y3 )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ X4 @ Z )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ X4 @ ( minus_8373710615458151222nteger @ Y3 @ Z ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2008_dvd__diff,axiom,
% 12.39/6.89 ! [X4: real,Y3: real,Z: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ X4 @ Y3 )
% 12.39/6.89 => ( ( dvd_dvd_real @ X4 @ Z )
% 12.39/6.89 => ( dvd_dvd_real @ X4 @ ( minus_minus_real @ Y3 @ Z ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2009_dvd__diff,axiom,
% 12.39/6.89 ! [X4: rat,Y3: rat,Z: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ X4 @ Y3 )
% 12.39/6.89 => ( ( dvd_dvd_rat @ X4 @ Z )
% 12.39/6.89 => ( dvd_dvd_rat @ X4 @ ( minus_minus_rat @ Y3 @ Z ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2010_dvd__diff,axiom,
% 12.39/6.89 ! [X4: int,Y3: int,Z: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ X4 @ Y3 )
% 12.39/6.89 => ( ( dvd_dvd_int @ X4 @ Z )
% 12.39/6.89 => ( dvd_dvd_int @ X4 @ ( minus_minus_int @ Y3 @ Z ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2011_dvd__div__eq__iff,axiom,
% 12.39/6.89 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.39/6.89 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 12.39/6.89 = ( divide6298287555418463151nteger @ B2 @ C ) )
% 12.39/6.89 = ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2012_dvd__div__eq__iff,axiom,
% 12.39/6.89 ! [C: complex,A: complex,B2: complex] :
% 12.39/6.89 ( ( dvd_dvd_complex @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_complex @ C @ B2 )
% 12.39/6.89 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 12.39/6.89 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 12.39/6.89 = ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2013_dvd__div__eq__iff,axiom,
% 12.39/6.89 ! [C: real,A: real,B2: real] :
% 12.39/6.89 ( ( dvd_dvd_real @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_real @ C @ B2 )
% 12.39/6.89 => ( ( ( divide_divide_real @ A @ C )
% 12.39/6.89 = ( divide_divide_real @ B2 @ C ) )
% 12.39/6.89 = ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2014_dvd__div__eq__iff,axiom,
% 12.39/6.89 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.89 ( ( dvd_dvd_rat @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_rat @ C @ B2 )
% 12.39/6.89 => ( ( ( divide_divide_rat @ A @ C )
% 12.39/6.89 = ( divide_divide_rat @ B2 @ C ) )
% 12.39/6.89 = ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2015_dvd__div__eq__iff,axiom,
% 12.39/6.89 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_nat @ C @ B2 )
% 12.39/6.89 => ( ( ( divide_divide_nat @ A @ C )
% 12.39/6.89 = ( divide_divide_nat @ B2 @ C ) )
% 12.39/6.89 = ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2016_dvd__div__eq__iff,axiom,
% 12.39/6.89 ! [C: int,A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_int @ C @ B2 )
% 12.39/6.89 => ( ( ( divide_divide_int @ A @ C )
% 12.39/6.89 = ( divide_divide_int @ B2 @ C ) )
% 12.39/6.89 = ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2017_dvd__div__eq__cancel,axiom,
% 12.39/6.89 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( ( divide6298287555418463151nteger @ A @ C )
% 12.39/6.89 = ( divide6298287555418463151nteger @ B2 @ C ) )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.39/6.89 => ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2018_dvd__div__eq__cancel,axiom,
% 12.39/6.89 ! [A: complex,C: complex,B2: complex] :
% 12.39/6.89 ( ( ( divide1717551699836669952omplex @ A @ C )
% 12.39/6.89 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 12.39/6.89 => ( ( dvd_dvd_complex @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_complex @ C @ B2 )
% 12.39/6.89 => ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2019_dvd__div__eq__cancel,axiom,
% 12.39/6.89 ! [A: real,C: real,B2: real] :
% 12.39/6.89 ( ( ( divide_divide_real @ A @ C )
% 12.39/6.89 = ( divide_divide_real @ B2 @ C ) )
% 12.39/6.89 => ( ( dvd_dvd_real @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_real @ C @ B2 )
% 12.39/6.89 => ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2020_dvd__div__eq__cancel,axiom,
% 12.39/6.89 ! [A: rat,C: rat,B2: rat] :
% 12.39/6.89 ( ( ( divide_divide_rat @ A @ C )
% 12.39/6.89 = ( divide_divide_rat @ B2 @ C ) )
% 12.39/6.89 => ( ( dvd_dvd_rat @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_rat @ C @ B2 )
% 12.39/6.89 => ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2021_dvd__div__eq__cancel,axiom,
% 12.39/6.89 ! [A: nat,C: nat,B2: nat] :
% 12.39/6.89 ( ( ( divide_divide_nat @ A @ C )
% 12.39/6.89 = ( divide_divide_nat @ B2 @ C ) )
% 12.39/6.89 => ( ( dvd_dvd_nat @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_nat @ C @ B2 )
% 12.39/6.89 => ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2022_dvd__div__eq__cancel,axiom,
% 12.39/6.89 ! [A: int,C: int,B2: int] :
% 12.39/6.89 ( ( ( divide_divide_int @ A @ C )
% 12.39/6.89 = ( divide_divide_int @ B2 @ C ) )
% 12.39/6.89 => ( ( dvd_dvd_int @ C @ A )
% 12.39/6.89 => ( ( dvd_dvd_int @ C @ B2 )
% 12.39/6.89 => ( A = B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2023_div__div__div__same,axiom,
% 12.39/6.89 ! [D: code_integer,B2: code_integer,A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ D @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 12.39/6.89 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B2 @ D ) )
% 12.39/6.89 = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2024_div__div__div__same,axiom,
% 12.39/6.89 ! [D: nat,B2: nat,A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ D @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_nat @ B2 @ A )
% 12.39/6.89 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B2 @ D ) )
% 12.39/6.89 = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2025_div__div__div__same,axiom,
% 12.39/6.89 ! [D: int,B2: int,A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ D @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_int @ B2 @ A )
% 12.39/6.89 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B2 @ D ) )
% 12.39/6.89 = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2026_dvd__power__same,axiom,
% 12.39/6.89 ! [X4: code_integer,Y3: code_integer,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ X4 @ Y3 )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2027_dvd__power__same,axiom,
% 12.39/6.89 ! [X4: nat,Y3: nat,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ X4 @ Y3 )
% 12.39/6.89 => ( dvd_dvd_nat @ ( power_power_nat @ X4 @ N ) @ ( power_power_nat @ Y3 @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2028_dvd__power__same,axiom,
% 12.39/6.89 ! [X4: real,Y3: real,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_real @ X4 @ Y3 )
% 12.39/6.89 => ( dvd_dvd_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ Y3 @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2029_dvd__power__same,axiom,
% 12.39/6.89 ! [X4: int,Y3: int,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_int @ X4 @ Y3 )
% 12.39/6.89 => ( dvd_dvd_int @ ( power_power_int @ X4 @ N ) @ ( power_power_int @ Y3 @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2030_dvd__power__same,axiom,
% 12.39/6.89 ! [X4: complex,Y3: complex,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_complex @ X4 @ Y3 )
% 12.39/6.89 => ( dvd_dvd_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ Y3 @ N ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2031_dvd__mod__iff,axiom,
% 12.39/6.89 ! [C: nat,B2: nat,A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ C @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2032_dvd__mod__iff,axiom,
% 12.39/6.89 ! [C: int,B2: int,A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ C @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_int @ C @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2033_dvd__mod__iff,axiom,
% 12.39/6.89 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 12.39/6.89 = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2034_dvd__mod__imp__dvd,axiom,
% 12.39/6.89 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 12.39/6.89 => ( ( dvd_dvd_nat @ C @ B2 )
% 12.39/6.89 => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2035_dvd__mod__imp__dvd,axiom,
% 12.39/6.89 ! [C: int,A: int,B2: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 12.39/6.89 => ( ( dvd_dvd_int @ C @ B2 )
% 12.39/6.89 => ( dvd_dvd_int @ C @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2036_dvd__mod__imp__dvd,axiom,
% 12.39/6.89 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2037_dvd__mod,axiom,
% 12.39/6.89 ! [K: nat,M: nat,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ K @ M )
% 12.39/6.89 => ( ( dvd_dvd_nat @ K @ N )
% 12.39/6.89 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2038_dvd__mod,axiom,
% 12.39/6.89 ! [K: int,M: int,N: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ K @ M )
% 12.39/6.89 => ( ( dvd_dvd_int @ K @ N )
% 12.39/6.89 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2039_dvd__mod,axiom,
% 12.39/6.89 ! [K: code_integer,M: code_integer,N: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ K @ M )
% 12.39/6.89 => ( ( dvd_dvd_Code_integer @ K @ N )
% 12.39/6.89 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2040_mod__mod__cancel,axiom,
% 12.39/6.89 ! [C: nat,B2: nat,A: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ C @ B2 )
% 12.39/6.89 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C )
% 12.39/6.89 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2041_mod__mod__cancel,axiom,
% 12.39/6.89 ! [C: int,B2: int,A: int] :
% 12.39/6.89 ( ( dvd_dvd_int @ C @ B2 )
% 12.39/6.89 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ C )
% 12.39/6.89 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2042_mod__mod__cancel,axiom,
% 12.39/6.89 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 12.39/6.89 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.39/6.89 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C )
% 12.39/6.89 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2043_dvd__diff__nat,axiom,
% 12.39/6.89 ! [K: nat,M: nat,N: nat] :
% 12.39/6.89 ( ( dvd_dvd_nat @ K @ M )
% 12.39/6.89 => ( ( dvd_dvd_nat @ K @ N )
% 12.39/6.89 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2044_zero__le,axiom,
% 12.39/6.89 ! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2045_le__numeral__extra_I3_J,axiom,
% 12.39/6.89 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2046_le__numeral__extra_I3_J,axiom,
% 12.39/6.89 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2047_le__numeral__extra_I3_J,axiom,
% 12.39/6.89 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2048_le__numeral__extra_I3_J,axiom,
% 12.39/6.89 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2049_zero__less__iff__neq__zero,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.39/6.89 = ( N != zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2050_gr__implies__not__zero,axiom,
% 12.39/6.89 ! [M: nat,N: nat] :
% 12.39/6.89 ( ( ord_less_nat @ M @ N )
% 12.39/6.89 => ( N != zero_zero_nat ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2051_not__less__zero,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2052_gr__zeroI,axiom,
% 12.39/6.89 ! [N: nat] :
% 12.39/6.89 ( ( N != zero_zero_nat )
% 12.39/6.89 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2053_less__numeral__extra_I3_J,axiom,
% 12.39/6.89 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2054_less__numeral__extra_I3_J,axiom,
% 12.39/6.89 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2055_less__numeral__extra_I3_J,axiom,
% 12.39/6.89 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2056_less__numeral__extra_I3_J,axiom,
% 12.39/6.89 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2057_field__lbound__gt__zero,axiom,
% 12.39/6.89 ! [D1: real,D22: real] :
% 12.39/6.89 ( ( ord_less_real @ zero_zero_real @ D1 )
% 12.39/6.89 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 12.39/6.89 => ? [E2: real] :
% 12.39/6.89 ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.39/6.89 & ( ord_less_real @ E2 @ D1 )
% 12.39/6.89 & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2058_field__lbound__gt__zero,axiom,
% 12.39/6.89 ! [D1: rat,D22: rat] :
% 12.39/6.89 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 12.39/6.89 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 12.39/6.89 => ? [E2: rat] :
% 12.39/6.89 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 12.39/6.89 & ( ord_less_rat @ E2 @ D1 )
% 12.39/6.89 & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2059_zero__neq__numeral,axiom,
% 12.39/6.89 ! [N: num] :
% 12.39/6.89 ( zero_zero_complex
% 12.39/6.89 != ( numera6690914467698888265omplex @ N ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2060_zero__neq__numeral,axiom,
% 12.39/6.89 ! [N: num] :
% 12.39/6.89 ( zero_zero_real
% 12.39/6.89 != ( numeral_numeral_real @ N ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2061_zero__neq__numeral,axiom,
% 12.39/6.89 ! [N: num] :
% 12.39/6.89 ( zero_zero_rat
% 12.39/6.89 != ( numeral_numeral_rat @ N ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2062_zero__neq__numeral,axiom,
% 12.39/6.89 ! [N: num] :
% 12.39/6.89 ( zero_zero_nat
% 12.39/6.89 != ( numeral_numeral_nat @ N ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2063_zero__neq__numeral,axiom,
% 12.39/6.89 ! [N: num] :
% 12.39/6.89 ( zero_zero_int
% 12.39/6.89 != ( numeral_numeral_int @ N ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2064_mult__right__cancel,axiom,
% 12.39/6.89 ! [C: real,A: real,B2: real] :
% 12.39/6.89 ( ( C != zero_zero_real )
% 12.39/6.89 => ( ( ( times_times_real @ A @ C )
% 12.39/6.89 = ( times_times_real @ B2 @ C ) )
% 12.39/6.89 = ( A = B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2065_mult__right__cancel,axiom,
% 12.39/6.89 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.89 ( ( C != zero_zero_rat )
% 12.39/6.89 => ( ( ( times_times_rat @ A @ C )
% 12.39/6.89 = ( times_times_rat @ B2 @ C ) )
% 12.39/6.89 = ( A = B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2066_mult__right__cancel,axiom,
% 12.39/6.89 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.89 ( ( C != zero_zero_nat )
% 12.39/6.89 => ( ( ( times_times_nat @ A @ C )
% 12.39/6.89 = ( times_times_nat @ B2 @ C ) )
% 12.39/6.89 = ( A = B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2067_mult__right__cancel,axiom,
% 12.39/6.89 ! [C: int,A: int,B2: int] :
% 12.39/6.89 ( ( C != zero_zero_int )
% 12.39/6.89 => ( ( ( times_times_int @ A @ C )
% 12.39/6.89 = ( times_times_int @ B2 @ C ) )
% 12.39/6.89 = ( A = B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2068_mult__left__cancel,axiom,
% 12.39/6.89 ! [C: real,A: real,B2: real] :
% 12.39/6.89 ( ( C != zero_zero_real )
% 12.39/6.89 => ( ( ( times_times_real @ C @ A )
% 12.39/6.89 = ( times_times_real @ C @ B2 ) )
% 12.39/6.89 = ( A = B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2069_mult__left__cancel,axiom,
% 12.39/6.89 ! [C: rat,A: rat,B2: rat] :
% 12.39/6.89 ( ( C != zero_zero_rat )
% 12.39/6.89 => ( ( ( times_times_rat @ C @ A )
% 12.39/6.89 = ( times_times_rat @ C @ B2 ) )
% 12.39/6.89 = ( A = B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2070_mult__left__cancel,axiom,
% 12.39/6.89 ! [C: nat,A: nat,B2: nat] :
% 12.39/6.89 ( ( C != zero_zero_nat )
% 12.39/6.89 => ( ( ( times_times_nat @ C @ A )
% 12.39/6.89 = ( times_times_nat @ C @ B2 ) )
% 12.39/6.89 = ( A = B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2071_mult__left__cancel,axiom,
% 12.39/6.89 ! [C: int,A: int,B2: int] :
% 12.39/6.89 ( ( C != zero_zero_int )
% 12.39/6.89 => ( ( ( times_times_int @ C @ A )
% 12.39/6.89 = ( times_times_int @ C @ B2 ) )
% 12.39/6.89 = ( A = B2 ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2072_no__zero__divisors,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( A != zero_zero_real )
% 12.39/6.89 => ( ( B2 != zero_zero_real )
% 12.39/6.89 => ( ( times_times_real @ A @ B2 )
% 12.39/6.89 != zero_zero_real ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2073_no__zero__divisors,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( A != zero_zero_rat )
% 12.39/6.89 => ( ( B2 != zero_zero_rat )
% 12.39/6.89 => ( ( times_times_rat @ A @ B2 )
% 12.39/6.89 != zero_zero_rat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2074_no__zero__divisors,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( A != zero_zero_nat )
% 12.39/6.89 => ( ( B2 != zero_zero_nat )
% 12.39/6.89 => ( ( times_times_nat @ A @ B2 )
% 12.39/6.89 != zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2075_no__zero__divisors,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( A != zero_zero_int )
% 12.39/6.89 => ( ( B2 != zero_zero_int )
% 12.39/6.89 => ( ( times_times_int @ A @ B2 )
% 12.39/6.89 != zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2076_divisors__zero,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( ( times_times_real @ A @ B2 )
% 12.39/6.89 = zero_zero_real )
% 12.39/6.89 => ( ( A = zero_zero_real )
% 12.39/6.89 | ( B2 = zero_zero_real ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2077_divisors__zero,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( ( times_times_rat @ A @ B2 )
% 12.39/6.89 = zero_zero_rat )
% 12.39/6.89 => ( ( A = zero_zero_rat )
% 12.39/6.89 | ( B2 = zero_zero_rat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2078_divisors__zero,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( ( times_times_nat @ A @ B2 )
% 12.39/6.89 = zero_zero_nat )
% 12.39/6.89 => ( ( A = zero_zero_nat )
% 12.39/6.89 | ( B2 = zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2079_divisors__zero,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( ( times_times_int @ A @ B2 )
% 12.39/6.89 = zero_zero_int )
% 12.39/6.89 => ( ( A = zero_zero_int )
% 12.39/6.89 | ( B2 = zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2080_mult__not__zero,axiom,
% 12.39/6.89 ! [A: real,B2: real] :
% 12.39/6.89 ( ( ( times_times_real @ A @ B2 )
% 12.39/6.89 != zero_zero_real )
% 12.39/6.89 => ( ( A != zero_zero_real )
% 12.39/6.89 & ( B2 != zero_zero_real ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2081_mult__not__zero,axiom,
% 12.39/6.89 ! [A: rat,B2: rat] :
% 12.39/6.89 ( ( ( times_times_rat @ A @ B2 )
% 12.39/6.89 != zero_zero_rat )
% 12.39/6.89 => ( ( A != zero_zero_rat )
% 12.39/6.89 & ( B2 != zero_zero_rat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2082_mult__not__zero,axiom,
% 12.39/6.89 ! [A: nat,B2: nat] :
% 12.39/6.89 ( ( ( times_times_nat @ A @ B2 )
% 12.39/6.89 != zero_zero_nat )
% 12.39/6.89 => ( ( A != zero_zero_nat )
% 12.39/6.89 & ( B2 != zero_zero_nat ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2083_mult__not__zero,axiom,
% 12.39/6.89 ! [A: int,B2: int] :
% 12.39/6.89 ( ( ( times_times_int @ A @ B2 )
% 12.39/6.89 != zero_zero_int )
% 12.39/6.89 => ( ( A != zero_zero_int )
% 12.39/6.89 & ( B2 != zero_zero_int ) ) ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2084_zero__neq__one,axiom,
% 12.39/6.89 zero_zero_complex != one_one_complex ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2085_zero__neq__one,axiom,
% 12.39/6.89 zero_zero_real != one_one_real ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2086_zero__neq__one,axiom,
% 12.39/6.89 zero_zero_rat != one_one_rat ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2087_zero__neq__one,axiom,
% 12.39/6.89 zero_zero_nat != one_one_nat ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2088_zero__neq__one,axiom,
% 12.39/6.89 zero_zero_int != one_one_int ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2089_add_Ogroup__left__neutral,axiom,
% 12.39/6.89 ! [A: real] :
% 12.39/6.89 ( ( plus_plus_real @ zero_zero_real @ A )
% 12.39/6.89 = A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2090_add_Ogroup__left__neutral,axiom,
% 12.39/6.89 ! [A: rat] :
% 12.39/6.89 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 12.39/6.89 = A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2091_add_Ogroup__left__neutral,axiom,
% 12.39/6.89 ! [A: int] :
% 12.39/6.89 ( ( plus_plus_int @ zero_zero_int @ A )
% 12.39/6.89 = A ) ).
% 12.39/6.89
% 12.39/6.89 thf(fact_2092_add_Ocomm__neutral,axiom,
% 12.39/6.89 ! [A: real] :
% 12.39/6.89 ( ( plus_plus_real @ A @ zero_zero_real )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2093_add_Ocomm__neutral,axiom,
% 12.39/6.90 ! [A: rat] :
% 12.39/6.90 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2094_add_Ocomm__neutral,axiom,
% 12.39/6.90 ! [A: nat] :
% 12.39/6.90 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2095_add_Ocomm__neutral,axiom,
% 12.39/6.90 ! [A: int] :
% 12.39/6.90 ( ( plus_plus_int @ A @ zero_zero_int )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2096_comm__monoid__add__class_Oadd__0,axiom,
% 12.39/6.90 ! [A: real] :
% 12.39/6.90 ( ( plus_plus_real @ zero_zero_real @ A )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2097_comm__monoid__add__class_Oadd__0,axiom,
% 12.39/6.90 ! [A: rat] :
% 12.39/6.90 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2098_comm__monoid__add__class_Oadd__0,axiom,
% 12.39/6.90 ! [A: nat] :
% 12.39/6.90 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2099_comm__monoid__add__class_Oadd__0,axiom,
% 12.39/6.90 ! [A: int] :
% 12.39/6.90 ( ( plus_plus_int @ zero_zero_int @ A )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2100_verit__sum__simplify,axiom,
% 12.39/6.90 ! [A: real] :
% 12.39/6.90 ( ( plus_plus_real @ A @ zero_zero_real )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2101_verit__sum__simplify,axiom,
% 12.39/6.90 ! [A: rat] :
% 12.39/6.90 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2102_verit__sum__simplify,axiom,
% 12.39/6.90 ! [A: nat] :
% 12.39/6.90 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2103_verit__sum__simplify,axiom,
% 12.39/6.90 ! [A: int] :
% 12.39/6.90 ( ( plus_plus_int @ A @ zero_zero_int )
% 12.39/6.90 = A ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2104_eq__iff__diff__eq__0,axiom,
% 12.39/6.90 ( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
% 12.39/6.90 = ( ^ [A4: real,B3: real] :
% 12.39/6.90 ( ( minus_minus_real @ A4 @ B3 )
% 12.39/6.90 = zero_zero_real ) ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2105_eq__iff__diff__eq__0,axiom,
% 12.39/6.90 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 12.39/6.90 = ( ^ [A4: rat,B3: rat] :
% 12.39/6.90 ( ( minus_minus_rat @ A4 @ B3 )
% 12.39/6.90 = zero_zero_rat ) ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2106_eq__iff__diff__eq__0,axiom,
% 12.39/6.90 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 12.39/6.90 = ( ^ [A4: int,B3: int] :
% 12.39/6.90 ( ( minus_minus_int @ A4 @ B3 )
% 12.39/6.90 = zero_zero_int ) ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2107_power__not__zero,axiom,
% 12.39/6.90 ! [A: rat,N: nat] :
% 12.39/6.90 ( ( A != zero_zero_rat )
% 12.39/6.90 => ( ( power_power_rat @ A @ N )
% 12.39/6.90 != zero_zero_rat ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2108_power__not__zero,axiom,
% 12.39/6.90 ! [A: nat,N: nat] :
% 12.39/6.90 ( ( A != zero_zero_nat )
% 12.39/6.90 => ( ( power_power_nat @ A @ N )
% 12.39/6.90 != zero_zero_nat ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2109_power__not__zero,axiom,
% 12.39/6.90 ! [A: real,N: nat] :
% 12.39/6.90 ( ( A != zero_zero_real )
% 12.39/6.90 => ( ( power_power_real @ A @ N )
% 12.39/6.90 != zero_zero_real ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2110_power__not__zero,axiom,
% 12.39/6.90 ! [A: int,N: nat] :
% 12.39/6.90 ( ( A != zero_zero_int )
% 12.39/6.90 => ( ( power_power_int @ A @ N )
% 12.39/6.90 != zero_zero_int ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2111_power__not__zero,axiom,
% 12.39/6.90 ! [A: complex,N: nat] :
% 12.39/6.90 ( ( A != zero_zero_complex )
% 12.39/6.90 => ( ( power_power_complex @ A @ N )
% 12.39/6.90 != zero_zero_complex ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2112_not0__implies__Suc,axiom,
% 12.39/6.90 ! [N: nat] :
% 12.39/6.90 ( ( N != zero_zero_nat )
% 12.39/6.90 => ? [M4: nat] :
% 12.39/6.90 ( N
% 12.39/6.90 = ( suc @ M4 ) ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2113_Zero__not__Suc,axiom,
% 12.39/6.90 ! [M: nat] :
% 12.39/6.90 ( zero_zero_nat
% 12.39/6.90 != ( suc @ M ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2114_Zero__neq__Suc,axiom,
% 12.39/6.90 ! [M: nat] :
% 12.39/6.90 ( zero_zero_nat
% 12.39/6.90 != ( suc @ M ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2115_Suc__neq__Zero,axiom,
% 12.39/6.90 ! [M: nat] :
% 12.39/6.90 ( ( suc @ M )
% 12.39/6.90 != zero_zero_nat ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2116_zero__induct,axiom,
% 12.39/6.90 ! [P: nat > $o,K: nat] :
% 12.39/6.90 ( ( P @ K )
% 12.39/6.90 => ( ! [N4: nat] :
% 12.39/6.90 ( ( P @ ( suc @ N4 ) )
% 12.39/6.90 => ( P @ N4 ) )
% 12.39/6.90 => ( P @ zero_zero_nat ) ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2117_diff__induct,axiom,
% 12.39/6.90 ! [P: nat > nat > $o,M: nat,N: nat] :
% 12.39/6.90 ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 12.39/6.90 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 12.39/6.90 => ( ! [X3: nat,Y4: nat] :
% 12.39/6.90 ( ( P @ X3 @ Y4 )
% 12.39/6.90 => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 12.39/6.90 => ( P @ M @ N ) ) ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2118_nat__induct,axiom,
% 12.39/6.90 ! [P: nat > $o,N: nat] :
% 12.39/6.90 ( ( P @ zero_zero_nat )
% 12.39/6.90 => ( ! [N4: nat] :
% 12.39/6.90 ( ( P @ N4 )
% 12.39/6.90 => ( P @ ( suc @ N4 ) ) )
% 12.39/6.90 => ( P @ N ) ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2119_old_Onat_Oexhaust,axiom,
% 12.39/6.90 ! [Y3: nat] :
% 12.39/6.90 ( ( Y3 != zero_zero_nat )
% 12.39/6.90 => ~ ! [Nat3: nat] :
% 12.39/6.90 ( Y3
% 12.39/6.90 != ( suc @ Nat3 ) ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2120_nat_OdiscI,axiom,
% 12.39/6.90 ! [Nat: nat,X2: nat] :
% 12.39/6.90 ( ( Nat
% 12.39/6.90 = ( suc @ X2 ) )
% 12.39/6.90 => ( Nat != zero_zero_nat ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2121_old_Onat_Odistinct_I1_J,axiom,
% 12.39/6.90 ! [Nat2: nat] :
% 12.39/6.90 ( zero_zero_nat
% 12.39/6.90 != ( suc @ Nat2 ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2122_old_Onat_Odistinct_I2_J,axiom,
% 12.39/6.90 ! [Nat2: nat] :
% 12.39/6.90 ( ( suc @ Nat2 )
% 12.39/6.90 != zero_zero_nat ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2123_nat_Odistinct_I1_J,axiom,
% 12.39/6.90 ! [X2: nat] :
% 12.39/6.90 ( zero_zero_nat
% 12.39/6.90 != ( suc @ X2 ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2124_infinite__descent0,axiom,
% 12.39/6.90 ! [P: nat > $o,N: nat] :
% 12.39/6.90 ( ( P @ zero_zero_nat )
% 12.39/6.90 => ( ! [N4: nat] :
% 12.39/6.90 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 12.39/6.90 => ( ~ ( P @ N4 )
% 12.39/6.90 => ? [M5: nat] :
% 12.39/6.90 ( ( ord_less_nat @ M5 @ N4 )
% 12.39/6.90 & ~ ( P @ M5 ) ) ) )
% 12.39/6.90 => ( P @ N ) ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2125_gr__implies__not0,axiom,
% 12.39/6.90 ! [M: nat,N: nat] :
% 12.39/6.90 ( ( ord_less_nat @ M @ N )
% 12.39/6.90 => ( N != zero_zero_nat ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2126_less__zeroE,axiom,
% 12.39/6.90 ! [N: nat] :
% 12.39/6.90 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2127_not__less0,axiom,
% 12.39/6.90 ! [N: nat] :
% 12.39/6.90 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2128_not__gr0,axiom,
% 12.39/6.90 ! [N: nat] :
% 12.39/6.90 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 12.39/6.90 = ( N = zero_zero_nat ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2129_gr0I,axiom,
% 12.39/6.90 ! [N: nat] :
% 12.39/6.90 ( ( N != zero_zero_nat )
% 12.39/6.90 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2130_bot__nat__0_Oextremum__strict,axiom,
% 12.39/6.90 ! [A: nat] :
% 12.39/6.90 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2131_le__0__eq,axiom,
% 12.39/6.90 ! [N: nat] :
% 12.39/6.90 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 12.39/6.90 = ( N = zero_zero_nat ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2132_bot__nat__0_Oextremum__uniqueI,axiom,
% 12.39/6.90 ! [A: nat] :
% 12.39/6.90 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.39/6.90 => ( A = zero_zero_nat ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2133_bot__nat__0_Oextremum__unique,axiom,
% 12.39/6.90 ! [A: nat] :
% 12.39/6.90 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.39/6.90 = ( A = zero_zero_nat ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2134_less__eq__nat_Osimps_I1_J,axiom,
% 12.39/6.90 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2135_add__eq__self__zero,axiom,
% 12.39/6.90 ! [M: nat,N: nat] :
% 12.39/6.90 ( ( ( plus_plus_nat @ M @ N )
% 12.39/6.90 = M )
% 12.39/6.90 => ( N = zero_zero_nat ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2136_plus__nat_Oadd__0,axiom,
% 12.39/6.90 ! [N: nat] :
% 12.39/6.90 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 12.39/6.90 = N ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2137_diffs0__imp__equal,axiom,
% 12.39/6.90 ! [M: nat,N: nat] :
% 12.39/6.90 ( ( ( minus_minus_nat @ M @ N )
% 12.39/6.90 = zero_zero_nat )
% 12.39/6.90 => ( ( ( minus_minus_nat @ N @ M )
% 12.39/6.90 = zero_zero_nat )
% 12.39/6.90 => ( M = N ) ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2138_minus__nat_Odiff__0,axiom,
% 12.39/6.90 ! [M: nat] :
% 12.39/6.90 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 12.39/6.90 = M ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2139_nat__mult__eq__cancel__disj,axiom,
% 12.39/6.90 ! [K: nat,M: nat,N: nat] :
% 12.39/6.90 ( ( ( times_times_nat @ K @ M )
% 12.39/6.90 = ( times_times_nat @ K @ N ) )
% 12.39/6.90 = ( ( K = zero_zero_nat )
% 12.39/6.90 | ( M = N ) ) ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2140_mult__0,axiom,
% 12.39/6.90 ! [N: nat] :
% 12.39/6.90 ( ( times_times_nat @ zero_zero_nat @ N )
% 12.39/6.90 = zero_zero_nat ) ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2141_even__zero,axiom,
% 12.39/6.90 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2142_even__zero,axiom,
% 12.39/6.90 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2143_even__zero,axiom,
% 12.39/6.90 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 12.39/6.90
% 12.39/6.90 thf(fact_2144_is__unitE,axiom,
% 12.39/6.90 ! [A: code_integer,C: code_integer] :
% 12.39/6.90 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.39/6.90 => ~ ( ( A != zero_z3403309356797280102nteger )
% 12.39/6.90 => ! [B5: code_integer] :
% 12.39/6.90 ( ( B5 != zero_z3403309356797280102nteger )
% 12.39/6.90 => ( ( dvd_dvd_Code_integer @ B5 @ one_one_Code_integer )
% 12.39/6.90 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 12.39/6.90 = B5 )
% 12.39/6.90 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B5 )
% 12.39/6.90 = A )
% 12.45/6.90 => ( ( ( times_3573771949741848930nteger @ A @ B5 )
% 12.45/6.90 = one_one_Code_integer )
% 12.45/6.90 => ( ( divide6298287555418463151nteger @ C @ A )
% 12.45/6.90 != ( times_3573771949741848930nteger @ C @ B5 ) ) ) ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2145_is__unitE,axiom,
% 12.45/6.90 ! [A: nat,C: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.45/6.90 => ~ ( ( A != zero_zero_nat )
% 12.45/6.90 => ! [B5: nat] :
% 12.45/6.90 ( ( B5 != zero_zero_nat )
% 12.45/6.90 => ( ( dvd_dvd_nat @ B5 @ one_one_nat )
% 12.45/6.90 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 12.45/6.90 = B5 )
% 12.45/6.90 => ( ( ( divide_divide_nat @ one_one_nat @ B5 )
% 12.45/6.90 = A )
% 12.45/6.90 => ( ( ( times_times_nat @ A @ B5 )
% 12.45/6.90 = one_one_nat )
% 12.45/6.90 => ( ( divide_divide_nat @ C @ A )
% 12.45/6.90 != ( times_times_nat @ C @ B5 ) ) ) ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2146_is__unitE,axiom,
% 12.45/6.90 ! [A: int,C: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.45/6.90 => ~ ( ( A != zero_zero_int )
% 12.45/6.90 => ! [B5: int] :
% 12.45/6.90 ( ( B5 != zero_zero_int )
% 12.45/6.90 => ( ( dvd_dvd_int @ B5 @ one_one_int )
% 12.45/6.90 => ( ( ( divide_divide_int @ one_one_int @ A )
% 12.45/6.90 = B5 )
% 12.45/6.90 => ( ( ( divide_divide_int @ one_one_int @ B5 )
% 12.45/6.90 = A )
% 12.45/6.90 => ( ( ( times_times_int @ A @ B5 )
% 12.45/6.90 = one_one_int )
% 12.45/6.90 => ( ( divide_divide_int @ C @ A )
% 12.45/6.90 != ( times_times_int @ C @ B5 ) ) ) ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2147_is__unit__div__mult__cancel__left,axiom,
% 12.45/6.90 ! [A: code_integer,B2: code_integer] :
% 12.45/6.90 ( ( A != zero_z3403309356797280102nteger )
% 12.45/6.90 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.45/6.90 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 12.45/6.90 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2148_is__unit__div__mult__cancel__left,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( A != zero_zero_nat )
% 12.45/6.90 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.45/6.90 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B2 ) )
% 12.45/6.90 = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2149_is__unit__div__mult__cancel__left,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( A != zero_zero_int )
% 12.45/6.90 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.45/6.90 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B2 ) )
% 12.45/6.90 = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2150_is__unit__div__mult__cancel__right,axiom,
% 12.45/6.90 ! [A: code_integer,B2: code_integer] :
% 12.45/6.90 ( ( A != zero_z3403309356797280102nteger )
% 12.45/6.90 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.45/6.90 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ A ) )
% 12.45/6.90 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2151_is__unit__div__mult__cancel__right,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( A != zero_zero_nat )
% 12.45/6.90 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.45/6.90 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ A ) )
% 12.45/6.90 = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2152_is__unit__div__mult__cancel__right,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( A != zero_zero_int )
% 12.45/6.90 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.45/6.90 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ A ) )
% 12.45/6.90 = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2153_dvd__power__iff,axiom,
% 12.45/6.90 ! [X4: code_integer,M: nat,N: nat] :
% 12.45/6.90 ( ( X4 != zero_z3403309356797280102nteger )
% 12.45/6.90 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ M ) @ ( power_8256067586552552935nteger @ X4 @ N ) )
% 12.45/6.90 = ( ( dvd_dvd_Code_integer @ X4 @ one_one_Code_integer )
% 12.45/6.90 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2154_dvd__power__iff,axiom,
% 12.45/6.90 ! [X4: nat,M: nat,N: nat] :
% 12.45/6.90 ( ( X4 != zero_zero_nat )
% 12.45/6.90 => ( ( dvd_dvd_nat @ ( power_power_nat @ X4 @ M ) @ ( power_power_nat @ X4 @ N ) )
% 12.45/6.90 = ( ( dvd_dvd_nat @ X4 @ one_one_nat )
% 12.45/6.90 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2155_dvd__power__iff,axiom,
% 12.45/6.90 ! [X4: int,M: nat,N: nat] :
% 12.45/6.90 ( ( X4 != zero_zero_int )
% 12.45/6.90 => ( ( dvd_dvd_int @ ( power_power_int @ X4 @ M ) @ ( power_power_int @ X4 @ N ) )
% 12.45/6.90 = ( ( dvd_dvd_int @ X4 @ one_one_int )
% 12.45/6.90 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2156_dvd__power,axiom,
% 12.45/6.90 ! [N: nat,X4: code_integer] :
% 12.45/6.90 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 | ( X4 = one_one_Code_integer ) )
% 12.45/6.90 => ( dvd_dvd_Code_integer @ X4 @ ( power_8256067586552552935nteger @ X4 @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2157_dvd__power,axiom,
% 12.45/6.90 ! [N: nat,X4: rat] :
% 12.45/6.90 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 | ( X4 = one_one_rat ) )
% 12.45/6.90 => ( dvd_dvd_rat @ X4 @ ( power_power_rat @ X4 @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2158_dvd__power,axiom,
% 12.45/6.90 ! [N: nat,X4: nat] :
% 12.45/6.90 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 | ( X4 = one_one_nat ) )
% 12.45/6.90 => ( dvd_dvd_nat @ X4 @ ( power_power_nat @ X4 @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2159_dvd__power,axiom,
% 12.45/6.90 ! [N: nat,X4: real] :
% 12.45/6.90 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 | ( X4 = one_one_real ) )
% 12.45/6.90 => ( dvd_dvd_real @ X4 @ ( power_power_real @ X4 @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2160_dvd__power,axiom,
% 12.45/6.90 ! [N: nat,X4: int] :
% 12.45/6.90 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 | ( X4 = one_one_int ) )
% 12.45/6.90 => ( dvd_dvd_int @ X4 @ ( power_power_int @ X4 @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2161_dvd__power,axiom,
% 12.45/6.90 ! [N: nat,X4: complex] :
% 12.45/6.90 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 | ( X4 = one_one_complex ) )
% 12.45/6.90 => ( dvd_dvd_complex @ X4 @ ( power_power_complex @ X4 @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2162_enat__0__less__mult__iff,axiom,
% 12.45/6.90 ! [M: extended_enat,N: extended_enat] :
% 12.45/6.90 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 12.45/6.90 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 12.45/6.90 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2163_not__iless0,axiom,
% 12.45/6.90 ! [N: extended_enat] :
% 12.45/6.90 ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2164_iadd__is__0,axiom,
% 12.45/6.90 ! [M: extended_enat,N: extended_enat] :
% 12.45/6.90 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 12.45/6.90 = zero_z5237406670263579293d_enat )
% 12.45/6.90 = ( ( M = zero_z5237406670263579293d_enat )
% 12.45/6.90 & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2165_ile0__eq,axiom,
% 12.45/6.90 ! [N: extended_enat] :
% 12.45/6.90 ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 12.45/6.90 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2166_i0__lb,axiom,
% 12.45/6.90 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2167_dvd__mult__cancel1,axiom,
% 12.45/6.90 ! [M: nat,N: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.90 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 12.45/6.90 = ( N = one_one_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2168_dvd__mult__cancel2,axiom,
% 12.45/6.90 ! [M: nat,N: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.90 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 12.45/6.90 = ( N = one_one_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2169_power__eq__imp__eq__base,axiom,
% 12.45/6.90 ! [A: real,N: nat,B2: real] :
% 12.45/6.90 ( ( ( power_power_real @ A @ N )
% 12.45/6.90 = ( power_power_real @ B2 @ N ) )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( A = B2 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2170_power__eq__imp__eq__base,axiom,
% 12.45/6.90 ! [A: rat,N: nat,B2: rat] :
% 12.45/6.90 ( ( ( power_power_rat @ A @ N )
% 12.45/6.90 = ( power_power_rat @ B2 @ N ) )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( A = B2 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2171_power__eq__imp__eq__base,axiom,
% 12.45/6.90 ! [A: nat,N: nat,B2: nat] :
% 12.45/6.90 ( ( ( power_power_nat @ A @ N )
% 12.45/6.90 = ( power_power_nat @ B2 @ N ) )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( A = B2 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2172_power__eq__imp__eq__base,axiom,
% 12.45/6.90 ! [A: int,N: nat,B2: int] :
% 12.45/6.90 ( ( ( power_power_int @ A @ N )
% 12.45/6.90 = ( power_power_int @ B2 @ N ) )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( A = B2 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2173_power__eq__iff__eq__base,axiom,
% 12.45/6.90 ! [N: nat,A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.90 => ( ( ( power_power_real @ A @ N )
% 12.45/6.90 = ( power_power_real @ B2 @ N ) )
% 12.45/6.90 = ( A = B2 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2174_power__eq__iff__eq__base,axiom,
% 12.45/6.90 ! [N: nat,A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.90 => ( ( ( power_power_rat @ A @ N )
% 12.45/6.90 = ( power_power_rat @ B2 @ N ) )
% 12.45/6.90 = ( A = B2 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2175_power__eq__iff__eq__base,axiom,
% 12.45/6.90 ! [N: nat,A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.45/6.90 => ( ( ( power_power_nat @ A @ N )
% 12.45/6.90 = ( power_power_nat @ B2 @ N ) )
% 12.45/6.90 = ( A = B2 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2176_power__eq__iff__eq__base,axiom,
% 12.45/6.90 ! [N: nat,A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.90 => ( ( ( power_power_int @ A @ N )
% 12.45/6.90 = ( power_power_int @ B2 @ N ) )
% 12.45/6.90 = ( A = B2 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2177_int__power__div__base,axiom,
% 12.45/6.90 ! [M: nat,K: int] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.90 => ( ( ord_less_int @ zero_zero_int @ K )
% 12.45/6.90 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 12.45/6.90 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2178_zero__less__power__eq,axiom,
% 12.45/6.90 ! [A: real,N: nat] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 12.45/6.90 = ( ( N = zero_zero_nat )
% 12.45/6.90 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( A != zero_zero_real ) )
% 12.45/6.90 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2179_zero__less__power__eq,axiom,
% 12.45/6.90 ! [A: rat,N: nat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 12.45/6.90 = ( ( N = zero_zero_nat )
% 12.45/6.90 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( A != zero_zero_rat ) )
% 12.45/6.90 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2180_zero__less__power__eq,axiom,
% 12.45/6.90 ! [A: int,N: nat] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 12.45/6.90 = ( ( N = zero_zero_nat )
% 12.45/6.90 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( A != zero_zero_int ) )
% 12.45/6.90 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2181_even__signed__take__bit__iff,axiom,
% 12.45/6.90 ! [M: nat,A: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 12.45/6.90 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2182_even__signed__take__bit__iff,axiom,
% 12.45/6.90 ! [M: nat,A: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 12.45/6.90 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2183_realpow__pos__nth,axiom,
% 12.45/6.90 ! [N: nat,A: real] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 => ? [R3: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ R3 )
% 12.45/6.90 & ( ( power_power_real @ R3 @ N )
% 12.45/6.90 = A ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2184_realpow__pos__nth__unique,axiom,
% 12.45/6.90 ! [N: nat,A: real] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 => ? [X3: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ X3 )
% 12.45/6.90 & ( ( power_power_real @ X3 @ N )
% 12.45/6.90 = A )
% 12.45/6.90 & ! [Y5: real] :
% 12.45/6.90 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 12.45/6.90 & ( ( power_power_real @ Y5 @ N )
% 12.45/6.90 = A ) )
% 12.45/6.90 => ( Y5 = X3 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2185_power__le__zero__eq,axiom,
% 12.45/6.90 ! [A: real,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 12.45/6.90 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 12.45/6.90 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( A = zero_zero_real ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2186_power__le__zero__eq,axiom,
% 12.45/6.90 ! [A: rat,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 12.45/6.90 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 12.45/6.90 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( A = zero_zero_rat ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2187_power__le__zero__eq,axiom,
% 12.45/6.90 ! [A: int,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 12.45/6.90 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 12.45/6.90 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 & ( A = zero_zero_int ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2188_even__iff__mod__2__eq__zero,axiom,
% 12.45/6.90 ! [A: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.90 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.90 = zero_zero_nat ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2189_even__iff__mod__2__eq__zero,axiom,
% 12.45/6.90 ! [A: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.90 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.90 = zero_zero_int ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2190_even__iff__mod__2__eq__zero,axiom,
% 12.45/6.90 ! [A: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.90 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.90 = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2191_odd__pos,axiom,
% 12.45/6.90 ! [N: nat] :
% 12.45/6.90 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.90 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2192_signed__take__bit__int__less__self__iff,axiom,
% 12.45/6.90 ! [N: nat,K: int] :
% 12.45/6.90 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 12.45/6.90 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2193_signed__take__bit__int__greater__eq__self__iff,axiom,
% 12.45/6.90 ! [K: int,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 12.45/6.90 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2194_signed__take__bit__int__less__exp,axiom,
% 12.45/6.90 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2195_is__unit__mult__iff,axiom,
% 12.45/6.90 ! [A: code_integer,B2: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer )
% 12.45/6.90 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.45/6.90 & ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2196_is__unit__mult__iff,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat )
% 12.45/6.90 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.45/6.90 & ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2197_is__unit__mult__iff,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int )
% 12.45/6.90 = ( ( dvd_dvd_int @ A @ one_one_int )
% 12.45/6.90 & ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2198_dvd__mult__unit__iff,axiom,
% 12.45/6.90 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.45/6.90 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) )
% 12.45/6.90 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2199_dvd__mult__unit__iff,axiom,
% 12.45/6.90 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.45/6.90 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) )
% 12.45/6.90 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2200_dvd__mult__unit__iff,axiom,
% 12.45/6.90 ! [B2: int,A: int,C: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.45/6.90 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) )
% 12.45/6.90 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2201_mult__unit__dvd__iff,axiom,
% 12.45/6.90 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.45/6.90 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 12.45/6.90 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2202_mult__unit__dvd__iff,axiom,
% 12.45/6.90 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.45/6.90 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 12.45/6.90 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2203_mult__unit__dvd__iff,axiom,
% 12.45/6.90 ! [B2: int,A: int,C: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.45/6.90 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 12.45/6.90 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2204_dvd__mult__unit__iff_H,axiom,
% 12.45/6.90 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.45/6.90 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 12.45/6.90 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2205_dvd__mult__unit__iff_H,axiom,
% 12.45/6.90 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.45/6.90 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 12.45/6.90 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2206_dvd__mult__unit__iff_H,axiom,
% 12.45/6.90 ! [B2: int,A: int,C: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.45/6.90 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
% 12.45/6.90 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2207_mult__unit__dvd__iff_H,axiom,
% 12.45/6.90 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.45/6.90 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 12.45/6.90 = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2208_mult__unit__dvd__iff_H,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.45/6.90 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 12.45/6.90 = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2209_mult__unit__dvd__iff_H,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.45/6.90 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 12.45/6.90 = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2210_unit__mult__left__cancel,axiom,
% 12.45/6.90 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.45/6.90 => ( ( ( times_3573771949741848930nteger @ A @ B2 )
% 12.45/6.90 = ( times_3573771949741848930nteger @ A @ C ) )
% 12.45/6.90 = ( B2 = C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2211_unit__mult__left__cancel,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.45/6.90 => ( ( ( times_times_nat @ A @ B2 )
% 12.45/6.90 = ( times_times_nat @ A @ C ) )
% 12.45/6.90 = ( B2 = C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2212_unit__mult__left__cancel,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.45/6.90 => ( ( ( times_times_int @ A @ B2 )
% 12.45/6.90 = ( times_times_int @ A @ C ) )
% 12.45/6.90 = ( B2 = C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2213_unit__mult__right__cancel,axiom,
% 12.45/6.90 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.45/6.90 => ( ( ( times_3573771949741848930nteger @ B2 @ A )
% 12.45/6.90 = ( times_3573771949741848930nteger @ C @ A ) )
% 12.45/6.90 = ( B2 = C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2214_unit__mult__right__cancel,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.45/6.90 => ( ( ( times_times_nat @ B2 @ A )
% 12.45/6.90 = ( times_times_nat @ C @ A ) )
% 12.45/6.90 = ( B2 = C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2215_unit__mult__right__cancel,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.45/6.90 => ( ( ( times_times_int @ B2 @ A )
% 12.45/6.90 = ( times_times_int @ C @ A ) )
% 12.45/6.90 = ( B2 = C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2216_dvd__div__mult,axiom,
% 12.45/6.90 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.45/6.90 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ C ) @ A )
% 12.45/6.90 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2217_dvd__div__mult,axiom,
% 12.45/6.90 ! [C: nat,B2: nat,A: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ C @ B2 )
% 12.45/6.90 => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A )
% 12.45/6.90 = ( divide_divide_nat @ ( times_times_nat @ B2 @ A ) @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2218_dvd__div__mult,axiom,
% 12.45/6.90 ! [C: int,B2: int,A: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ C @ B2 )
% 12.45/6.90 => ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A )
% 12.45/6.90 = ( divide_divide_int @ ( times_times_int @ B2 @ A ) @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2219_div__mult__swap,axiom,
% 12.45/6.90 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.45/6.90 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 12.45/6.90 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2220_div__mult__swap,axiom,
% 12.45/6.90 ! [C: nat,B2: nat,A: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ C @ B2 )
% 12.45/6.90 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 12.45/6.90 = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2221_div__mult__swap,axiom,
% 12.45/6.90 ! [C: int,B2: int,A: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ C @ B2 )
% 12.45/6.90 => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 12.45/6.90 = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2222_div__div__eq__right,axiom,
% 12.45/6.90 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.45/6.90 => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 12.45/6.90 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 12.45/6.90 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2223_div__div__eq__right,axiom,
% 12.45/6.90 ! [C: nat,B2: nat,A: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ C @ B2 )
% 12.45/6.90 => ( ( dvd_dvd_nat @ B2 @ A )
% 12.45/6.90 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 12.45/6.90 = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2224_div__div__eq__right,axiom,
% 12.45/6.90 ! [C: int,B2: int,A: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ C @ B2 )
% 12.45/6.90 => ( ( dvd_dvd_int @ B2 @ A )
% 12.45/6.90 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 12.45/6.90 = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2225_dvd__div__mult2__eq,axiom,
% 12.45/6.90 ! [B2: code_integer,C: code_integer,A: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ C ) @ A )
% 12.45/6.90 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 12.45/6.90 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2226_dvd__div__mult2__eq,axiom,
% 12.45/6.90 ! [B2: nat,C: nat,A: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A )
% 12.45/6.90 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 12.45/6.90 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2227_dvd__div__mult2__eq,axiom,
% 12.45/6.90 ! [B2: int,C: int,A: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A )
% 12.45/6.90 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 12.45/6.90 = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2228_dvd__mult__imp__div,axiom,
% 12.45/6.90 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 )
% 12.45/6.90 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2229_dvd__mult__imp__div,axiom,
% 12.45/6.90 ! [A: nat,C: nat,B2: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 )
% 12.45/6.90 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2230_dvd__mult__imp__div,axiom,
% 12.45/6.90 ! [A: int,C: int,B2: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 )
% 12.45/6.90 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2231_div__mult__div__if__dvd,axiom,
% 12.45/6.90 ! [B2: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 12.45/6.90 => ( ( dvd_dvd_Code_integer @ D @ C )
% 12.45/6.90 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 12.45/6.90 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2232_div__mult__div__if__dvd,axiom,
% 12.45/6.90 ! [B2: nat,A: nat,D: nat,C: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ B2 @ A )
% 12.45/6.90 => ( ( dvd_dvd_nat @ D @ C )
% 12.45/6.90 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ ( divide_divide_nat @ C @ D ) )
% 12.45/6.90 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2233_div__mult__div__if__dvd,axiom,
% 12.45/6.90 ! [B2: int,A: int,D: int,C: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ B2 @ A )
% 12.45/6.90 => ( ( dvd_dvd_int @ D @ C )
% 12.45/6.90 => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ C @ D ) )
% 12.45/6.90 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2234_unit__div__cancel,axiom,
% 12.45/6.90 ! [A: code_integer,B2: code_integer,C: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.45/6.90 => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 12.45/6.90 = ( divide6298287555418463151nteger @ C @ A ) )
% 12.45/6.90 = ( B2 = C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2235_unit__div__cancel,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.45/6.90 => ( ( ( divide_divide_nat @ B2 @ A )
% 12.45/6.90 = ( divide_divide_nat @ C @ A ) )
% 12.45/6.90 = ( B2 = C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2236_unit__div__cancel,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.45/6.90 => ( ( ( divide_divide_int @ B2 @ A )
% 12.45/6.90 = ( divide_divide_int @ C @ A ) )
% 12.45/6.90 = ( B2 = C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2237_div__unit__dvd__iff,axiom,
% 12.45/6.90 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.45/6.90 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 12.45/6.90 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2238_div__unit__dvd__iff,axiom,
% 12.45/6.90 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.45/6.90 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 12.45/6.90 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2239_div__unit__dvd__iff,axiom,
% 12.45/6.90 ! [B2: int,A: int,C: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.45/6.90 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 12.45/6.90 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2240_dvd__div__unit__iff,axiom,
% 12.45/6.90 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.45/6.90 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B2 ) )
% 12.45/6.90 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2241_dvd__div__unit__iff,axiom,
% 12.45/6.90 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.45/6.90 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B2 ) )
% 12.45/6.90 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2242_dvd__div__unit__iff,axiom,
% 12.45/6.90 ! [B2: int,A: int,C: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.45/6.90 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B2 ) )
% 12.45/6.90 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2243_div__plus__div__distrib__dvd__right,axiom,
% 12.45/6.90 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ C @ B2 )
% 12.45/6.90 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 12.45/6.90 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2244_div__plus__div__distrib__dvd__right,axiom,
% 12.45/6.90 ! [C: nat,B2: nat,A: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ C @ B2 )
% 12.45/6.90 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 12.45/6.90 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2245_div__plus__div__distrib__dvd__right,axiom,
% 12.45/6.90 ! [C: int,B2: int,A: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ C @ B2 )
% 12.45/6.90 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 12.45/6.90 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2246_div__plus__div__distrib__dvd__left,axiom,
% 12.45/6.90 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ C @ A )
% 12.45/6.90 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 12.45/6.90 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2247_div__plus__div__distrib__dvd__left,axiom,
% 12.45/6.90 ! [C: nat,A: nat,B2: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ C @ A )
% 12.45/6.90 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 12.45/6.90 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2248_div__plus__div__distrib__dvd__left,axiom,
% 12.45/6.90 ! [C: int,A: int,B2: int] :
% 12.45/6.90 ( ( dvd_dvd_int @ C @ A )
% 12.45/6.90 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 12.45/6.90 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2249_div__power,axiom,
% 12.45/6.90 ! [B2: code_integer,A: code_integer,N: nat] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 12.45/6.90 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ N )
% 12.45/6.90 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2250_div__power,axiom,
% 12.45/6.90 ! [B2: nat,A: nat,N: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ B2 @ A )
% 12.45/6.90 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B2 ) @ N )
% 12.45/6.90 = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2251_div__power,axiom,
% 12.45/6.90 ! [B2: int,A: int,N: nat] :
% 12.45/6.90 ( ( dvd_dvd_int @ B2 @ A )
% 12.45/6.90 => ( ( power_power_int @ ( divide_divide_int @ A @ B2 ) @ N )
% 12.45/6.90 = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2252_power__strict__mono,axiom,
% 12.45/6.90 ! [A: real,B2: real,N: nat] :
% 12.45/6.90 ( ( ord_less_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2253_power__strict__mono,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,N: nat] :
% 12.45/6.90 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2254_power__strict__mono,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,N: nat] :
% 12.45/6.90 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2255_power__strict__mono,axiom,
% 12.45/6.90 ! [A: int,B2: int,N: nat] :
% 12.45/6.90 ( ( ord_less_int @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2256_le__imp__power__dvd,axiom,
% 12.45/6.90 ! [M: nat,N: nat,A: code_integer] :
% 12.45/6.90 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.90 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2257_le__imp__power__dvd,axiom,
% 12.45/6.90 ! [M: nat,N: nat,A: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.90 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2258_le__imp__power__dvd,axiom,
% 12.45/6.90 ! [M: nat,N: nat,A: real] :
% 12.45/6.90 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.90 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2259_le__imp__power__dvd,axiom,
% 12.45/6.90 ! [M: nat,N: nat,A: int] :
% 12.45/6.90 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.90 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2260_le__imp__power__dvd,axiom,
% 12.45/6.90 ! [M: nat,N: nat,A: complex] :
% 12.45/6.90 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.90 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2261_power__le__dvd,axiom,
% 12.45/6.90 ! [A: code_integer,N: nat,B2: code_integer,M: nat] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.90 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2262_power__le__dvd,axiom,
% 12.45/6.90 ! [A: nat,N: nat,B2: nat,M: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.90 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2263_power__le__dvd,axiom,
% 12.45/6.90 ! [A: real,N: nat,B2: real,M: nat] :
% 12.45/6.90 ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.90 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2264_power__le__dvd,axiom,
% 12.45/6.90 ! [A: int,N: nat,B2: int,M: nat] :
% 12.45/6.90 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.90 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2265_power__le__dvd,axiom,
% 12.45/6.90 ! [A: complex,N: nat,B2: complex,M: nat] :
% 12.45/6.90 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.90 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2266_dvd__power__le,axiom,
% 12.45/6.90 ! [X4: code_integer,Y3: code_integer,N: nat,M: nat] :
% 12.45/6.90 ( ( dvd_dvd_Code_integer @ X4 @ Y3 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.90 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2267_dvd__power__le,axiom,
% 12.45/6.90 ! [X4: nat,Y3: nat,N: nat,M: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ X4 @ Y3 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.90 => ( dvd_dvd_nat @ ( power_power_nat @ X4 @ N ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2268_dvd__power__le,axiom,
% 12.45/6.90 ! [X4: real,Y3: real,N: nat,M: nat] :
% 12.45/6.90 ( ( dvd_dvd_real @ X4 @ Y3 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.90 => ( dvd_dvd_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2269_dvd__power__le,axiom,
% 12.45/6.90 ! [X4: int,Y3: int,N: nat,M: nat] :
% 12.45/6.90 ( ( dvd_dvd_int @ X4 @ Y3 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.90 => ( dvd_dvd_int @ ( power_power_int @ X4 @ N ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2270_dvd__power__le,axiom,
% 12.45/6.90 ! [X4: complex,Y3: complex,N: nat,M: nat] :
% 12.45/6.90 ( ( dvd_dvd_complex @ X4 @ Y3 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.90 => ( dvd_dvd_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2271_mod__eq__dvd__iff,axiom,
% 12.45/6.90 ! [A: int,C: int,B2: int] :
% 12.45/6.90 ( ( ( modulo_modulo_int @ A @ C )
% 12.45/6.90 = ( modulo_modulo_int @ B2 @ C ) )
% 12.45/6.90 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2272_mod__eq__dvd__iff,axiom,
% 12.45/6.90 ! [A: code_integer,C: code_integer,B2: code_integer] :
% 12.45/6.90 ( ( ( modulo364778990260209775nteger @ A @ C )
% 12.45/6.90 = ( modulo364778990260209775nteger @ B2 @ C ) )
% 12.45/6.90 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2273_dvd__minus__mod,axiom,
% 12.45/6.90 ! [B2: nat,A: nat] : ( dvd_dvd_nat @ B2 @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2274_dvd__minus__mod,axiom,
% 12.45/6.90 ! [B2: int,A: int] : ( dvd_dvd_int @ B2 @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2275_dvd__minus__mod,axiom,
% 12.45/6.90 ! [B2: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B2 @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2276_dvd__minus__self,axiom,
% 12.45/6.90 ! [M: nat,N: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 12.45/6.90 = ( ( ord_less_nat @ N @ M )
% 12.45/6.90 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2277_dvd__diffD,axiom,
% 12.45/6.90 ! [K: nat,M: nat,N: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.90 => ( ( dvd_dvd_nat @ K @ N )
% 12.45/6.90 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.90 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2278_dvd__diffD1,axiom,
% 12.45/6.90 ! [K: nat,M: nat,N: nat] :
% 12.45/6.90 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.90 => ( ( dvd_dvd_nat @ K @ M )
% 12.45/6.90 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.90 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2279_less__eq__dvd__minus,axiom,
% 12.45/6.90 ! [M: nat,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.90 => ( ( dvd_dvd_nat @ M @ N )
% 12.45/6.90 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2280_dbl__def,axiom,
% 12.45/6.90 ( neg_numeral_dbl_real
% 12.45/6.90 = ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2281_dbl__def,axiom,
% 12.45/6.90 ( neg_numeral_dbl_rat
% 12.45/6.90 = ( ^ [X: rat] : ( plus_plus_rat @ X @ X ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2282_dbl__def,axiom,
% 12.45/6.90 ( neg_numeral_dbl_int
% 12.45/6.90 = ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2283_zero__le__numeral,axiom,
% 12.45/6.90 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2284_zero__le__numeral,axiom,
% 12.45/6.90 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2285_zero__le__numeral,axiom,
% 12.45/6.90 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2286_zero__le__numeral,axiom,
% 12.45/6.90 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2287_not__numeral__le__zero,axiom,
% 12.45/6.90 ! [N: num] :
% 12.45/6.90 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2288_not__numeral__le__zero,axiom,
% 12.45/6.90 ! [N: num] :
% 12.45/6.90 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2289_not__numeral__le__zero,axiom,
% 12.45/6.90 ! [N: num] :
% 12.45/6.90 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2290_not__numeral__le__zero,axiom,
% 12.45/6.90 ! [N: num] :
% 12.45/6.90 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2291_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2292_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2293_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2294_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.90 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2295_zero__le__mult__iff,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 12.45/6.90 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.90 & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2296_zero__le__mult__iff,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 12.45/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2297_zero__le__mult__iff,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
% 12.45/6.90 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.45/6.90 & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2298_mult__nonneg__nonpos2,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_eq_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2299_mult__nonneg__nonpos2,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2300_mult__nonneg__nonpos2,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 12.45/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2301_mult__nonneg__nonpos2,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 12.45/6.90 => ( ord_less_eq_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2302_mult__nonpos__nonneg,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2303_mult__nonpos__nonneg,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2304_mult__nonpos__nonneg,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.45/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2305_mult__nonpos__nonneg,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2306_mult__nonneg__nonpos,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2307_mult__nonneg__nonpos,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2308_mult__nonneg__nonpos,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 12.45/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2309_mult__nonneg__nonpos,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 12.45/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2310_mult__nonneg__nonneg,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.90 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2311_mult__nonneg__nonneg,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.90 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2312_mult__nonneg__nonneg,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.45/6.90 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2313_mult__nonneg__nonneg,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.90 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2314_split__mult__neg__le,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 12.45/6.90 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.90 & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
% 12.45/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2315_split__mult__neg__le,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 12.45/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) )
% 12.45/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2316_split__mult__neg__le,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
% 12.45/6.90 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.45/6.90 & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
% 12.45/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2317_split__mult__neg__le,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
% 12.45/6.90 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.45/6.90 & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
% 12.45/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2318_mult__le__0__iff,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 12.45/6.90 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 12.45/6.90 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.90 & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2319_mult__le__0__iff,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 12.45/6.90 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 12.45/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2320_mult__le__0__iff,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
% 12.45/6.90 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
% 12.45/6.90 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.45/6.90 & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2321_mult__right__mono,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2322_mult__right__mono,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2323_mult__right__mono,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2324_mult__right__mono,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2325_mult__right__mono__neg,axiom,
% 12.45/6.90 ! [B2: real,A: real,C: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.45/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2326_mult__right__mono__neg,axiom,
% 12.45/6.90 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2327_mult__right__mono__neg,axiom,
% 12.45/6.90 ! [B2: int,A: int,C: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.45/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2328_mult__left__mono,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2329_mult__left__mono,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2330_mult__left__mono,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2331_mult__left__mono,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.90 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2332_mult__nonpos__nonpos,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2333_mult__nonpos__nonpos,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2334_mult__nonpos__nonpos,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.45/6.90 => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 12.45/6.90 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2335_mult__left__mono__neg,axiom,
% 12.45/6.90 ! [B2: real,A: real,C: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.45/6.90 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2336_mult__left__mono__neg,axiom,
% 12.45/6.90 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2337_mult__left__mono__neg,axiom,
% 12.45/6.90 ! [B2: int,A: int,C: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.45/6.90 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2338_split__mult__pos__le,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 12.45/6.90 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.90 & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
% 12.45/6.90 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2339_split__mult__pos__le,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 12.45/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
% 12.45/6.90 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2340_split__mult__pos__le,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
% 12.45/6.90 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.45/6.90 & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
% 12.45/6.90 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2341_zero__le__square,axiom,
% 12.45/6.90 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2342_zero__le__square,axiom,
% 12.45/6.90 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2343_zero__le__square,axiom,
% 12.45/6.90 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2344_mult__mono_H,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_real @ C @ D )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2345_mult__mono_H,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ C @ D )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2346_mult__mono_H,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ C @ D )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2347_mult__mono_H,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int,D: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_int @ C @ D )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2348_mult__mono,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_real @ C @ D )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2349_mult__mono,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ C @ D )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2350_mult__mono,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ C @ D )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2351_mult__mono,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int,D: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_int @ C @ D )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2352_not__numeral__less__zero,axiom,
% 12.45/6.90 ! [N: num] :
% 12.45/6.90 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2353_not__numeral__less__zero,axiom,
% 12.45/6.90 ! [N: num] :
% 12.45/6.90 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2354_not__numeral__less__zero,axiom,
% 12.45/6.90 ! [N: num] :
% 12.45/6.90 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2355_not__numeral__less__zero,axiom,
% 12.45/6.90 ! [N: num] :
% 12.45/6.90 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2356_zero__less__numeral,axiom,
% 12.45/6.90 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2357_zero__less__numeral,axiom,
% 12.45/6.90 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2358_zero__less__numeral,axiom,
% 12.45/6.90 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2359_zero__less__numeral,axiom,
% 12.45/6.90 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2360_zero__less__one__class_Ozero__le__one,axiom,
% 12.45/6.90 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2361_zero__less__one__class_Ozero__le__one,axiom,
% 12.45/6.90 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2362_zero__less__one__class_Ozero__le__one,axiom,
% 12.45/6.90 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2363_zero__less__one__class_Ozero__le__one,axiom,
% 12.45/6.90 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2364_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 12.45/6.90 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2365_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 12.45/6.90 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2366_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 12.45/6.90 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2367_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 12.45/6.90 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2368_not__one__le__zero,axiom,
% 12.45/6.90 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2369_not__one__le__zero,axiom,
% 12.45/6.90 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2370_not__one__le__zero,axiom,
% 12.45/6.90 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2371_not__one__le__zero,axiom,
% 12.45/6.90 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2372_add__decreasing,axiom,
% 12.45/6.90 ! [A: real,C: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_eq_real @ C @ B2 )
% 12.45/6.90 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2373_add__decreasing,axiom,
% 12.45/6.90 ! [A: rat,C: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_eq_rat @ C @ B2 )
% 12.45/6.90 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2374_add__decreasing,axiom,
% 12.45/6.90 ! [A: nat,C: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.45/6.90 => ( ( ord_less_eq_nat @ C @ B2 )
% 12.45/6.90 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2375_add__decreasing,axiom,
% 12.45/6.90 ! [A: int,C: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.45/6.90 => ( ( ord_less_eq_int @ C @ B2 )
% 12.45/6.90 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2376_add__increasing,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_eq_real @ B2 @ C )
% 12.45/6.90 => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2377_add__increasing,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.90 => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2378_add__increasing,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_eq_nat @ B2 @ C )
% 12.45/6.90 => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2379_add__increasing,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_eq_int @ B2 @ C )
% 12.45/6.90 => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2380_add__decreasing2,axiom,
% 12.45/6.90 ! [C: real,A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.90 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2381_add__decreasing2,axiom,
% 12.45/6.90 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.90 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2382_add__decreasing2,axiom,
% 12.45/6.90 ! [C: nat,A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 12.45/6.90 => ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.90 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2383_add__decreasing2,axiom,
% 12.45/6.90 ! [C: int,A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.45/6.90 => ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.90 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2384_add__increasing2,axiom,
% 12.45/6.90 ! [C: real,B2: real,A: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ( ord_less_eq_real @ B2 @ A )
% 12.45/6.90 => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2385_add__increasing2,axiom,
% 12.45/6.90 ! [C: rat,B2: rat,A: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ( ord_less_eq_rat @ B2 @ A )
% 12.45/6.90 => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2386_add__increasing2,axiom,
% 12.45/6.90 ! [C: nat,B2: nat,A: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.90 => ( ( ord_less_eq_nat @ B2 @ A )
% 12.45/6.90 => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2387_add__increasing2,axiom,
% 12.45/6.90 ! [C: int,B2: int,A: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.90 => ( ( ord_less_eq_int @ B2 @ A )
% 12.45/6.90 => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2388_add__nonneg__nonneg,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.90 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2389_add__nonneg__nonneg,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.90 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2390_add__nonneg__nonneg,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.45/6.90 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2391_add__nonneg__nonneg,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.90 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2392_add__nonpos__nonpos,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2393_add__nonpos__nonpos,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2394_add__nonpos__nonpos,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.45/6.90 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 12.45/6.90 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2395_add__nonpos__nonpos,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.45/6.90 => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 12.45/6.90 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2396_add__nonneg__eq__0__iff,axiom,
% 12.45/6.90 ! [X4: real,Y3: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.90 => ( ( ( plus_plus_real @ X4 @ Y3 )
% 12.45/6.90 = zero_zero_real )
% 12.45/6.90 = ( ( X4 = zero_zero_real )
% 12.45/6.90 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2397_add__nonneg__eq__0__iff,axiom,
% 12.45/6.90 ! [X4: rat,Y3: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.45/6.90 => ( ( ( plus_plus_rat @ X4 @ Y3 )
% 12.45/6.90 = zero_zero_rat )
% 12.45/6.90 = ( ( X4 = zero_zero_rat )
% 12.45/6.90 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2398_add__nonneg__eq__0__iff,axiom,
% 12.45/6.90 ! [X4: nat,Y3: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 12.45/6.90 => ( ( ( plus_plus_nat @ X4 @ Y3 )
% 12.45/6.90 = zero_zero_nat )
% 12.45/6.90 = ( ( X4 = zero_zero_nat )
% 12.45/6.90 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2399_add__nonneg__eq__0__iff,axiom,
% 12.45/6.90 ! [X4: int,Y3: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.90 => ( ( ( plus_plus_int @ X4 @ Y3 )
% 12.45/6.90 = zero_zero_int )
% 12.45/6.90 = ( ( X4 = zero_zero_int )
% 12.45/6.90 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2400_add__nonpos__eq__0__iff,axiom,
% 12.45/6.90 ! [X4: real,Y3: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 12.45/6.90 => ( ( ( plus_plus_real @ X4 @ Y3 )
% 12.45/6.90 = zero_zero_real )
% 12.45/6.90 = ( ( X4 = zero_zero_real )
% 12.45/6.90 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2401_add__nonpos__eq__0__iff,axiom,
% 12.45/6.90 ! [X4: rat,Y3: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 12.45/6.90 => ( ( ( plus_plus_rat @ X4 @ Y3 )
% 12.45/6.90 = zero_zero_rat )
% 12.45/6.90 = ( ( X4 = zero_zero_rat )
% 12.45/6.90 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2402_add__nonpos__eq__0__iff,axiom,
% 12.45/6.90 ! [X4: nat,Y3: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
% 12.45/6.90 => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 12.45/6.90 => ( ( ( plus_plus_nat @ X4 @ Y3 )
% 12.45/6.90 = zero_zero_nat )
% 12.45/6.90 = ( ( X4 = zero_zero_nat )
% 12.45/6.90 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2403_add__nonpos__eq__0__iff,axiom,
% 12.45/6.90 ! [X4: int,Y3: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ X4 @ zero_zero_int )
% 12.45/6.90 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 12.45/6.90 => ( ( ( plus_plus_int @ X4 @ Y3 )
% 12.45/6.90 = zero_zero_int )
% 12.45/6.90 = ( ( X4 = zero_zero_int )
% 12.45/6.90 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2404_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( ord_less_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2405_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2406_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.45/6.90 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2407_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( ord_less_int @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.90 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2408_mult__less__cancel__right__disj,axiom,
% 12.45/6.90 ! [A: real,C: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 12.45/6.90 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.90 & ( ord_less_real @ A @ B2 ) )
% 12.45/6.90 | ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.90 & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2409_mult__less__cancel__right__disj,axiom,
% 12.45/6.90 ! [A: rat,C: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 12.45/6.90 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.90 & ( ord_less_rat @ A @ B2 ) )
% 12.45/6.90 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2410_mult__less__cancel__right__disj,axiom,
% 12.45/6.90 ! [A: int,C: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 12.45/6.90 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.90 & ( ord_less_int @ A @ B2 ) )
% 12.45/6.90 | ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.90 & ( ord_less_int @ B2 @ A ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2411_mult__strict__right__mono,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( ord_less_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2412_mult__strict__right__mono,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2413_mult__strict__right__mono,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.45/6.90 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2414_mult__strict__right__mono,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( ord_less_int @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.90 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2415_mult__strict__right__mono__neg,axiom,
% 12.45/6.90 ! [B2: real,A: real,C: real] :
% 12.45/6.90 ( ( ord_less_real @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.90 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2416_mult__strict__right__mono__neg,axiom,
% 12.45/6.90 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_rat @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2417_mult__strict__right__mono__neg,axiom,
% 12.45/6.90 ! [B2: int,A: int,C: int] :
% 12.45/6.90 ( ( ord_less_int @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.90 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2418_mult__less__cancel__left__disj,axiom,
% 12.45/6.90 ! [C: real,A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.90 & ( ord_less_real @ A @ B2 ) )
% 12.45/6.90 | ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.90 & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2419_mult__less__cancel__left__disj,axiom,
% 12.45/6.90 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.90 & ( ord_less_rat @ A @ B2 ) )
% 12.45/6.90 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2420_mult__less__cancel__left__disj,axiom,
% 12.45/6.90 ! [C: int,A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.90 & ( ord_less_int @ A @ B2 ) )
% 12.45/6.90 | ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.90 & ( ord_less_int @ B2 @ A ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2421_mult__strict__left__mono,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( ord_less_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2422_mult__strict__left__mono,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2423_mult__strict__left__mono,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.45/6.90 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2424_mult__strict__left__mono,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( ord_less_int @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.90 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2425_mult__strict__left__mono__neg,axiom,
% 12.45/6.90 ! [B2: real,A: real,C: real] :
% 12.45/6.90 ( ( ord_less_real @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.90 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2426_mult__strict__left__mono__neg,axiom,
% 12.45/6.90 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_rat @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2427_mult__strict__left__mono__neg,axiom,
% 12.45/6.90 ! [B2: int,A: int,C: int] :
% 12.45/6.90 ( ( ord_less_int @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.90 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2428_mult__less__cancel__left__pos,axiom,
% 12.45/6.90 ! [C: real,A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.45/6.90 = ( ord_less_real @ A @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2429_mult__less__cancel__left__pos,axiom,
% 12.45/6.90 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.45/6.90 = ( ord_less_rat @ A @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2430_mult__less__cancel__left__pos,axiom,
% 12.45/6.90 ! [C: int,A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.90 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.45/6.90 = ( ord_less_int @ A @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2431_mult__less__cancel__left__neg,axiom,
% 12.45/6.90 ! [C: real,A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.45/6.90 = ( ord_less_real @ B2 @ A ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2432_mult__less__cancel__left__neg,axiom,
% 12.45/6.90 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.45/6.90 = ( ord_less_rat @ B2 @ A ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2433_mult__less__cancel__left__neg,axiom,
% 12.45/6.90 ! [C: int,A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.90 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.45/6.90 = ( ord_less_int @ B2 @ A ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2434_zero__less__mult__pos2,axiom,
% 12.45/6.90 ! [B2: real,A: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A ) )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2435_zero__less__mult__pos2,axiom,
% 12.45/6.90 ! [B2: rat,A: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B2 @ A ) )
% 12.45/6.90 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2436_zero__less__mult__pos2,axiom,
% 12.45/6.90 ! [B2: nat,A: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A ) )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2437_zero__less__mult__pos2,axiom,
% 12.45/6.90 ! [B2: int,A: int] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A ) )
% 12.45/6.90 => ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2438_zero__less__mult__pos,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2439_zero__less__mult__pos,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.90 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2440_zero__less__mult__pos,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2441_zero__less__mult__pos,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 12.45/6.90 => ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2442_zero__less__mult__iff,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 & ( ord_less_real @ zero_zero_real @ B2 ) )
% 12.45/6.90 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.90 & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2443_zero__less__mult__iff,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 & ( ord_less_rat @ zero_zero_rat @ B2 ) )
% 12.45/6.90 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2444_zero__less__mult__iff,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.90 & ( ord_less_int @ zero_zero_int @ B2 ) )
% 12.45/6.90 | ( ( ord_less_int @ A @ zero_zero_int )
% 12.45/6.90 & ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2445_mult__pos__neg2,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2446_mult__pos__neg2,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2447_mult__pos__neg2,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 12.45/6.90 => ( ord_less_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2448_mult__pos__neg2,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 12.45/6.90 => ( ord_less_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2449_mult__pos__pos,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.90 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2450_mult__pos__pos,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 12.45/6.90 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2451_mult__pos__pos,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.45/6.90 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2452_mult__pos__pos,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.90 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2453_mult__pos__neg,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2454_mult__pos__neg,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2455_mult__pos__neg,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 12.45/6.90 => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2456_mult__pos__neg,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 12.45/6.90 => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2457_mult__neg__pos,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.90 => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2458_mult__neg__pos,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 12.45/6.90 => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2459_mult__neg__pos,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_nat @ A @ zero_zero_nat )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.45/6.90 => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2460_mult__neg__pos,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ A @ zero_zero_int )
% 12.45/6.90 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.90 => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2461_mult__less__0__iff,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 12.45/6.90 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 & ( ord_less_real @ B2 @ zero_zero_real ) )
% 12.45/6.90 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.90 & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2462_mult__less__0__iff,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 12.45/6.90 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 & ( ord_less_rat @ B2 @ zero_zero_rat ) )
% 12.45/6.90 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2463_mult__less__0__iff,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
% 12.45/6.90 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.90 & ( ord_less_int @ B2 @ zero_zero_int ) )
% 12.45/6.90 | ( ( ord_less_int @ A @ zero_zero_int )
% 12.45/6.90 & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2464_not__square__less__zero,axiom,
% 12.45/6.90 ! [A: real] :
% 12.45/6.90 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2465_not__square__less__zero,axiom,
% 12.45/6.90 ! [A: rat] :
% 12.45/6.90 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2466_not__square__less__zero,axiom,
% 12.45/6.90 ! [A: int] :
% 12.45/6.90 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2467_mult__neg__neg,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2468_mult__neg__neg,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2469_mult__neg__neg,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ A @ zero_zero_int )
% 12.45/6.90 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 12.45/6.90 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2470_le__iff__diff__le__0,axiom,
% 12.45/6.90 ( ord_less_eq_real
% 12.45/6.90 = ( ^ [A4: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2471_le__iff__diff__le__0,axiom,
% 12.45/6.90 ( ord_less_eq_rat
% 12.45/6.90 = ( ^ [A4: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B3 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2472_le__iff__diff__le__0,axiom,
% 12.45/6.90 ( ord_less_eq_int
% 12.45/6.90 = ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2473_less__numeral__extra_I1_J,axiom,
% 12.45/6.90 ord_less_real @ zero_zero_real @ one_one_real ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2474_less__numeral__extra_I1_J,axiom,
% 12.45/6.90 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2475_less__numeral__extra_I1_J,axiom,
% 12.45/6.90 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2476_less__numeral__extra_I1_J,axiom,
% 12.45/6.90 ord_less_int @ zero_zero_int @ one_one_int ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2477_zero__less__one,axiom,
% 12.45/6.90 ord_less_real @ zero_zero_real @ one_one_real ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2478_zero__less__one,axiom,
% 12.45/6.90 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2479_zero__less__one,axiom,
% 12.45/6.90 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2480_zero__less__one,axiom,
% 12.45/6.90 ord_less_int @ zero_zero_int @ one_one_int ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2481_not__one__less__zero,axiom,
% 12.45/6.90 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2482_not__one__less__zero,axiom,
% 12.45/6.90 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2483_not__one__less__zero,axiom,
% 12.45/6.90 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2484_not__one__less__zero,axiom,
% 12.45/6.90 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2485_add__neg__neg,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2486_add__neg__neg,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2487_add__neg__neg,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_nat @ A @ zero_zero_nat )
% 12.45/6.90 => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 12.45/6.90 => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2488_add__neg__neg,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ A @ zero_zero_int )
% 12.45/6.90 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 12.45/6.90 => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2489_add__pos__pos,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.90 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2490_add__pos__pos,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 12.45/6.90 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2491_add__pos__pos,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.45/6.90 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2492_add__pos__pos,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.90 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2493_canonically__ordered__monoid__add__class_OlessE,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.90 => ~ ! [C3: nat] :
% 12.45/6.90 ( ( B2
% 12.45/6.90 = ( plus_plus_nat @ A @ C3 ) )
% 12.45/6.90 => ( C3 = zero_zero_nat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2494_pos__add__strict,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.90 => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2495_pos__add__strict,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.90 => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2496_pos__add__strict,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ( ord_less_nat @ B2 @ C )
% 12.45/6.90 => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2497_pos__add__strict,axiom,
% 12.45/6.90 ! [A: int,B2: int,C: int] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( ord_less_int @ B2 @ C )
% 12.45/6.90 => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2498_add__less__zeroD,axiom,
% 12.45/6.90 ! [X4: real,Y3: real] :
% 12.45/6.90 ( ( ord_less_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 12.45/6.90 | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2499_add__less__zeroD,axiom,
% 12.45/6.90 ! [X4: rat,Y3: rat] :
% 12.45/6.90 ( ( ord_less_rat @ ( plus_plus_rat @ X4 @ Y3 ) @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_rat @ X4 @ zero_zero_rat )
% 12.45/6.90 | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2500_add__less__zeroD,axiom,
% 12.45/6.90 ! [X4: int,Y3: int] :
% 12.45/6.90 ( ( ord_less_int @ ( plus_plus_int @ X4 @ Y3 ) @ zero_zero_int )
% 12.45/6.90 => ( ( ord_less_int @ X4 @ zero_zero_int )
% 12.45/6.90 | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2501_divide__le__0__iff,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
% 12.45/6.90 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 12.45/6.90 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.90 & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2502_divide__le__0__iff,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
% 12.45/6.90 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 12.45/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2503_divide__right__mono,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2504_divide__right__mono,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2505_zero__le__divide__iff,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 12.45/6.90 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.90 & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2506_zero__le__divide__iff,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 12.45/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2507_divide__nonneg__nonneg,axiom,
% 12.45/6.90 ! [X4: real,Y3: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.90 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2508_divide__nonneg__nonneg,axiom,
% 12.45/6.90 ! [X4: rat,Y3: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.45/6.90 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2509_divide__nonneg__nonpos,axiom,
% 12.45/6.90 ! [X4: real,Y3: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.90 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2510_divide__nonneg__nonpos,axiom,
% 12.45/6.90 ! [X4: rat,Y3: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2511_divide__nonpos__nonneg,axiom,
% 12.45/6.90 ! [X4: real,Y3: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.90 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2512_divide__nonpos__nonneg,axiom,
% 12.45/6.90 ! [X4: rat,Y3: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.45/6.90 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2513_divide__nonpos__nonpos,axiom,
% 12.45/6.90 ! [X4: real,Y3: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2514_divide__nonpos__nonpos,axiom,
% 12.45/6.90 ! [X4: rat,Y3: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2515_divide__right__mono__neg,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.45/6.90 => ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2516_divide__right__mono__neg,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2517_less__iff__diff__less__0,axiom,
% 12.45/6.90 ( ord_less_real
% 12.45/6.90 = ( ^ [A4: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2518_less__iff__diff__less__0,axiom,
% 12.45/6.90 ( ord_less_rat
% 12.45/6.90 = ( ^ [A4: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B3 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2519_less__iff__diff__less__0,axiom,
% 12.45/6.90 ( ord_less_int
% 12.45/6.90 = ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2520_divide__strict__right__mono__neg,axiom,
% 12.45/6.90 ! [B2: real,A: real,C: real] :
% 12.45/6.90 ( ( ord_less_real @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.90 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2521_divide__strict__right__mono__neg,axiom,
% 12.45/6.90 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_rat @ B2 @ A )
% 12.45/6.90 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2522_divide__strict__right__mono,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( ord_less_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2523_divide__strict__right__mono,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2524_zero__less__divide__iff,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 & ( ord_less_real @ zero_zero_real @ B2 ) )
% 12.45/6.90 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.90 & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2525_zero__less__divide__iff,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
% 12.45/6.90 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 & ( ord_less_rat @ zero_zero_rat @ B2 ) )
% 12.45/6.90 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2526_divide__less__cancel,axiom,
% 12.45/6.90 ! [A: real,C: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.90 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.90 => ( ord_less_real @ A @ B2 ) )
% 12.45/6.90 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.90 => ( ord_less_real @ B2 @ A ) )
% 12.45/6.90 & ( C != zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2527_divide__less__cancel,axiom,
% 12.45/6.90 ! [A: rat,C: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.90 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.90 => ( ord_less_rat @ A @ B2 ) )
% 12.45/6.90 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_rat @ B2 @ A ) )
% 12.45/6.90 & ( C != zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2528_divide__less__0__iff,axiom,
% 12.45/6.90 ! [A: real,B2: real] :
% 12.45/6.90 ( ( ord_less_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
% 12.45/6.90 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 & ( ord_less_real @ B2 @ zero_zero_real ) )
% 12.45/6.90 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.90 & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2529_divide__less__0__iff,axiom,
% 12.45/6.90 ! [A: rat,B2: rat] :
% 12.45/6.90 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
% 12.45/6.90 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 & ( ord_less_rat @ B2 @ zero_zero_rat ) )
% 12.45/6.90 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.90 & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2530_divide__pos__pos,axiom,
% 12.45/6.90 ! [X4: real,Y3: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.90 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2531_divide__pos__pos,axiom,
% 12.45/6.90 ! [X4: rat,Y3: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 12.45/6.90 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 12.45/6.90 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2532_divide__pos__neg,axiom,
% 12.45/6.90 ! [X4: real,Y3: real] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.90 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2533_divide__pos__neg,axiom,
% 12.45/6.90 ! [X4: rat,Y3: rat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 12.45/6.90 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2534_divide__neg__pos,axiom,
% 12.45/6.90 ! [X4: real,Y3: real] :
% 12.45/6.90 ( ( ord_less_real @ X4 @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.90 => ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2535_divide__neg__pos,axiom,
% 12.45/6.90 ! [X4: rat,Y3: rat] :
% 12.45/6.90 ( ( ord_less_rat @ X4 @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 12.45/6.90 => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2536_divide__neg__neg,axiom,
% 12.45/6.90 ! [X4: real,Y3: real] :
% 12.45/6.90 ( ( ord_less_real @ X4 @ zero_zero_real )
% 12.45/6.90 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 12.45/6.90 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2537_divide__neg__neg,axiom,
% 12.45/6.90 ! [X4: rat,Y3: rat] :
% 12.45/6.90 ( ( ord_less_rat @ X4 @ zero_zero_rat )
% 12.45/6.90 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 12.45/6.90 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2538_zero__le__power,axiom,
% 12.45/6.90 ! [A: real,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2539_zero__le__power,axiom,
% 12.45/6.90 ! [A: rat,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2540_zero__le__power,axiom,
% 12.45/6.90 ! [A: nat,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2541_zero__le__power,axiom,
% 12.45/6.90 ! [A: int,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2542_power__mono,axiom,
% 12.45/6.90 ! [A: real,B2: real,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2543_power__mono,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2544_power__mono,axiom,
% 12.45/6.90 ! [A: nat,B2: nat,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2545_power__mono,axiom,
% 12.45/6.90 ! [A: int,B2: int,N: nat] :
% 12.45/6.90 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2546_zero__less__power,axiom,
% 12.45/6.90 ! [A: real,N: nat] :
% 12.45/6.90 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.90 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2547_zero__less__power,axiom,
% 12.45/6.90 ! [A: rat,N: nat] :
% 12.45/6.90 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.90 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2548_zero__less__power,axiom,
% 12.45/6.90 ! [A: nat,N: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2549_zero__less__power,axiom,
% 12.45/6.90 ! [A: int,N: nat] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2550_frac__eq__eq,axiom,
% 12.45/6.90 ! [Y3: complex,Z: complex,X4: complex,W: complex] :
% 12.45/6.90 ( ( Y3 != zero_zero_complex )
% 12.45/6.90 => ( ( Z != zero_zero_complex )
% 12.45/6.90 => ( ( ( divide1717551699836669952omplex @ X4 @ Y3 )
% 12.45/6.90 = ( divide1717551699836669952omplex @ W @ Z ) )
% 12.45/6.90 = ( ( times_times_complex @ X4 @ Z )
% 12.45/6.90 = ( times_times_complex @ W @ Y3 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2551_frac__eq__eq,axiom,
% 12.45/6.90 ! [Y3: real,Z: real,X4: real,W: real] :
% 12.45/6.90 ( ( Y3 != zero_zero_real )
% 12.45/6.90 => ( ( Z != zero_zero_real )
% 12.45/6.90 => ( ( ( divide_divide_real @ X4 @ Y3 )
% 12.45/6.90 = ( divide_divide_real @ W @ Z ) )
% 12.45/6.90 = ( ( times_times_real @ X4 @ Z )
% 12.45/6.90 = ( times_times_real @ W @ Y3 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2552_frac__eq__eq,axiom,
% 12.45/6.90 ! [Y3: rat,Z: rat,X4: rat,W: rat] :
% 12.45/6.90 ( ( Y3 != zero_zero_rat )
% 12.45/6.90 => ( ( Z != zero_zero_rat )
% 12.45/6.90 => ( ( ( divide_divide_rat @ X4 @ Y3 )
% 12.45/6.90 = ( divide_divide_rat @ W @ Z ) )
% 12.45/6.90 = ( ( times_times_rat @ X4 @ Z )
% 12.45/6.90 = ( times_times_rat @ W @ Y3 ) ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2553_divide__eq__eq,axiom,
% 12.45/6.90 ! [B2: complex,C: complex,A: complex] :
% 12.45/6.90 ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 12.45/6.90 = A )
% 12.45/6.90 = ( ( ( C != zero_zero_complex )
% 12.45/6.90 => ( B2
% 12.45/6.90 = ( times_times_complex @ A @ C ) ) )
% 12.45/6.90 & ( ( C = zero_zero_complex )
% 12.45/6.90 => ( A = zero_zero_complex ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2554_divide__eq__eq,axiom,
% 12.45/6.90 ! [B2: real,C: real,A: real] :
% 12.45/6.90 ( ( ( divide_divide_real @ B2 @ C )
% 12.45/6.90 = A )
% 12.45/6.90 = ( ( ( C != zero_zero_real )
% 12.45/6.90 => ( B2
% 12.45/6.90 = ( times_times_real @ A @ C ) ) )
% 12.45/6.90 & ( ( C = zero_zero_real )
% 12.45/6.90 => ( A = zero_zero_real ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2555_divide__eq__eq,axiom,
% 12.45/6.90 ! [B2: rat,C: rat,A: rat] :
% 12.45/6.90 ( ( ( divide_divide_rat @ B2 @ C )
% 12.45/6.90 = A )
% 12.45/6.90 = ( ( ( C != zero_zero_rat )
% 12.45/6.90 => ( B2
% 12.45/6.90 = ( times_times_rat @ A @ C ) ) )
% 12.45/6.90 & ( ( C = zero_zero_rat )
% 12.45/6.90 => ( A = zero_zero_rat ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2556_eq__divide__eq,axiom,
% 12.45/6.90 ! [A: complex,B2: complex,C: complex] :
% 12.45/6.90 ( ( A
% 12.45/6.90 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 12.45/6.90 = ( ( ( C != zero_zero_complex )
% 12.45/6.90 => ( ( times_times_complex @ A @ C )
% 12.45/6.90 = B2 ) )
% 12.45/6.90 & ( ( C = zero_zero_complex )
% 12.45/6.90 => ( A = zero_zero_complex ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2557_eq__divide__eq,axiom,
% 12.45/6.90 ! [A: real,B2: real,C: real] :
% 12.45/6.90 ( ( A
% 12.45/6.90 = ( divide_divide_real @ B2 @ C ) )
% 12.45/6.90 = ( ( ( C != zero_zero_real )
% 12.45/6.90 => ( ( times_times_real @ A @ C )
% 12.45/6.90 = B2 ) )
% 12.45/6.90 & ( ( C = zero_zero_real )
% 12.45/6.90 => ( A = zero_zero_real ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2558_eq__divide__eq,axiom,
% 12.45/6.90 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.90 ( ( A
% 12.45/6.90 = ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.90 = ( ( ( C != zero_zero_rat )
% 12.45/6.90 => ( ( times_times_rat @ A @ C )
% 12.45/6.90 = B2 ) )
% 12.45/6.90 & ( ( C = zero_zero_rat )
% 12.45/6.90 => ( A = zero_zero_rat ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2559_divide__eq__imp,axiom,
% 12.45/6.90 ! [C: complex,B2: complex,A: complex] :
% 12.45/6.90 ( ( C != zero_zero_complex )
% 12.45/6.90 => ( ( B2
% 12.45/6.90 = ( times_times_complex @ A @ C ) )
% 12.45/6.90 => ( ( divide1717551699836669952omplex @ B2 @ C )
% 12.45/6.90 = A ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2560_divide__eq__imp,axiom,
% 12.45/6.90 ! [C: real,B2: real,A: real] :
% 12.45/6.90 ( ( C != zero_zero_real )
% 12.45/6.90 => ( ( B2
% 12.45/6.90 = ( times_times_real @ A @ C ) )
% 12.45/6.90 => ( ( divide_divide_real @ B2 @ C )
% 12.45/6.90 = A ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2561_divide__eq__imp,axiom,
% 12.45/6.90 ! [C: rat,B2: rat,A: rat] :
% 12.45/6.90 ( ( C != zero_zero_rat )
% 12.45/6.90 => ( ( B2
% 12.45/6.90 = ( times_times_rat @ A @ C ) )
% 12.45/6.90 => ( ( divide_divide_rat @ B2 @ C )
% 12.45/6.90 = A ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2562_eq__divide__imp,axiom,
% 12.45/6.90 ! [C: complex,A: complex,B2: complex] :
% 12.45/6.90 ( ( C != zero_zero_complex )
% 12.45/6.90 => ( ( ( times_times_complex @ A @ C )
% 12.45/6.90 = B2 )
% 12.45/6.90 => ( A
% 12.45/6.90 = ( divide1717551699836669952omplex @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2563_eq__divide__imp,axiom,
% 12.45/6.90 ! [C: real,A: real,B2: real] :
% 12.45/6.90 ( ( C != zero_zero_real )
% 12.45/6.90 => ( ( ( times_times_real @ A @ C )
% 12.45/6.90 = B2 )
% 12.45/6.90 => ( A
% 12.45/6.90 = ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2564_eq__divide__imp,axiom,
% 12.45/6.90 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.90 ( ( C != zero_zero_rat )
% 12.45/6.90 => ( ( ( times_times_rat @ A @ C )
% 12.45/6.90 = B2 )
% 12.45/6.90 => ( A
% 12.45/6.90 = ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2565_nonzero__divide__eq__eq,axiom,
% 12.45/6.90 ! [C: complex,B2: complex,A: complex] :
% 12.45/6.90 ( ( C != zero_zero_complex )
% 12.45/6.90 => ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 12.45/6.90 = A )
% 12.45/6.90 = ( B2
% 12.45/6.90 = ( times_times_complex @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2566_nonzero__divide__eq__eq,axiom,
% 12.45/6.90 ! [C: real,B2: real,A: real] :
% 12.45/6.90 ( ( C != zero_zero_real )
% 12.45/6.90 => ( ( ( divide_divide_real @ B2 @ C )
% 12.45/6.90 = A )
% 12.45/6.90 = ( B2
% 12.45/6.90 = ( times_times_real @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2567_nonzero__divide__eq__eq,axiom,
% 12.45/6.90 ! [C: rat,B2: rat,A: rat] :
% 12.45/6.90 ( ( C != zero_zero_rat )
% 12.45/6.90 => ( ( ( divide_divide_rat @ B2 @ C )
% 12.45/6.90 = A )
% 12.45/6.90 = ( B2
% 12.45/6.90 = ( times_times_rat @ A @ C ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2568_nonzero__eq__divide__eq,axiom,
% 12.45/6.90 ! [C: complex,A: complex,B2: complex] :
% 12.45/6.90 ( ( C != zero_zero_complex )
% 12.45/6.90 => ( ( A
% 12.45/6.90 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 12.45/6.90 = ( ( times_times_complex @ A @ C )
% 12.45/6.90 = B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2569_nonzero__eq__divide__eq,axiom,
% 12.45/6.90 ! [C: real,A: real,B2: real] :
% 12.45/6.90 ( ( C != zero_zero_real )
% 12.45/6.90 => ( ( A
% 12.45/6.90 = ( divide_divide_real @ B2 @ C ) )
% 12.45/6.90 = ( ( times_times_real @ A @ C )
% 12.45/6.90 = B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2570_nonzero__eq__divide__eq,axiom,
% 12.45/6.90 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.90 ( ( C != zero_zero_rat )
% 12.45/6.90 => ( ( A
% 12.45/6.90 = ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.90 = ( ( times_times_rat @ A @ C )
% 12.45/6.90 = B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2571_right__inverse__eq,axiom,
% 12.45/6.90 ! [B2: complex,A: complex] :
% 12.45/6.90 ( ( B2 != zero_zero_complex )
% 12.45/6.90 => ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 12.45/6.90 = one_one_complex )
% 12.45/6.90 = ( A = B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2572_right__inverse__eq,axiom,
% 12.45/6.90 ! [B2: real,A: real] :
% 12.45/6.90 ( ( B2 != zero_zero_real )
% 12.45/6.90 => ( ( ( divide_divide_real @ A @ B2 )
% 12.45/6.90 = one_one_real )
% 12.45/6.90 = ( A = B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2573_right__inverse__eq,axiom,
% 12.45/6.90 ! [B2: rat,A: rat] :
% 12.45/6.90 ( ( B2 != zero_zero_rat )
% 12.45/6.90 => ( ( ( divide_divide_rat @ A @ B2 )
% 12.45/6.90 = one_one_rat )
% 12.45/6.90 = ( A = B2 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2574_not__exp__less__eq__0__int,axiom,
% 12.45/6.90 ! [N: nat] :
% 12.45/6.90 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2575_pos__zdiv__mult__2,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.45/6.90 = ( divide_divide_int @ B2 @ A ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2576_neg__zdiv__mult__2,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.45/6.90 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.45/6.90 = ( divide_divide_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2577_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 12.45/6.90 ! [A: code_integer,B2: code_integer] :
% 12.45/6.90 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.45/6.90 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ A ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2578_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.90 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B2 ) @ A ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2579_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.90 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ A ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2580_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 12.45/6.90 ! [B2: nat,A: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.45/6.90 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2581_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 12.45/6.90 ! [B2: int,A: int] :
% 12.45/6.90 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.90 => ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2582_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 12.45/6.90 ! [B2: code_integer,A: code_integer] :
% 12.45/6.90 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 12.45/6.90 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2583_power__0,axiom,
% 12.45/6.90 ! [A: rat] :
% 12.45/6.90 ( ( power_power_rat @ A @ zero_zero_nat )
% 12.45/6.90 = one_one_rat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2584_power__0,axiom,
% 12.45/6.90 ! [A: nat] :
% 12.45/6.90 ( ( power_power_nat @ A @ zero_zero_nat )
% 12.45/6.90 = one_one_nat ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2585_power__0,axiom,
% 12.45/6.90 ! [A: real] :
% 12.45/6.90 ( ( power_power_real @ A @ zero_zero_nat )
% 12.45/6.90 = one_one_real ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2586_power__0,axiom,
% 12.45/6.90 ! [A: int] :
% 12.45/6.90 ( ( power_power_int @ A @ zero_zero_nat )
% 12.45/6.90 = one_one_int ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2587_power__0,axiom,
% 12.45/6.90 ! [A: complex] :
% 12.45/6.90 ( ( power_power_complex @ A @ zero_zero_nat )
% 12.45/6.90 = one_one_complex ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2588_mod__eq__self__iff__div__eq__0,axiom,
% 12.45/6.90 ! [A: nat,B2: nat] :
% 12.45/6.90 ( ( ( modulo_modulo_nat @ A @ B2 )
% 12.45/6.90 = A )
% 12.45/6.90 = ( ( divide_divide_nat @ A @ B2 )
% 12.45/6.90 = zero_zero_nat ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2589_mod__eq__self__iff__div__eq__0,axiom,
% 12.45/6.90 ! [A: int,B2: int] :
% 12.45/6.90 ( ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.90 = A )
% 12.45/6.90 = ( ( divide_divide_int @ A @ B2 )
% 12.45/6.90 = zero_zero_int ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2590_mod__eq__self__iff__div__eq__0,axiom,
% 12.45/6.90 ! [A: code_integer,B2: code_integer] :
% 12.45/6.90 ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 12.45/6.90 = A )
% 12.45/6.90 = ( ( divide6298287555418463151nteger @ A @ B2 )
% 12.45/6.90 = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2591_less__Suc__eq__0__disj,axiom,
% 12.45/6.90 ! [M: nat,N: nat] :
% 12.45/6.90 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 12.45/6.90 = ( ( M = zero_zero_nat )
% 12.45/6.90 | ? [J3: nat] :
% 12.45/6.90 ( ( M
% 12.45/6.90 = ( suc @ J3 ) )
% 12.45/6.90 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2592_gr0__implies__Suc,axiom,
% 12.45/6.90 ! [N: nat] :
% 12.45/6.90 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.90 => ? [M4: nat] :
% 12.45/6.90 ( N
% 12.45/6.90 = ( suc @ M4 ) ) ) ).
% 12.45/6.90
% 12.45/6.90 thf(fact_2593_All__less__Suc2,axiom,
% 12.45/6.90 ! [N: nat,P: nat > $o] :
% 12.45/6.90 ( ( ! [I2: nat] :
% 12.45/6.90 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 12.45/6.90 => ( P @ I2 ) ) )
% 12.45/6.90 = ( ( P @ zero_zero_nat )
% 12.45/6.90 & ! [I2: nat] :
% 12.45/6.90 ( ( ord_less_nat @ I2 @ N )
% 12.45/6.91 => ( P @ ( suc @ I2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2594_gr0__conv__Suc,axiom,
% 12.45/6.91 ! [N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 = ( ? [M6: nat] :
% 12.45/6.91 ( N
% 12.45/6.91 = ( suc @ M6 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2595_Ex__less__Suc2,axiom,
% 12.45/6.91 ! [N: nat,P: nat > $o] :
% 12.45/6.91 ( ( ? [I2: nat] :
% 12.45/6.91 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 12.45/6.91 & ( P @ I2 ) ) )
% 12.45/6.91 = ( ( P @ zero_zero_nat )
% 12.45/6.91 | ? [I2: nat] :
% 12.45/6.91 ( ( ord_less_nat @ I2 @ N )
% 12.45/6.91 & ( P @ ( suc @ I2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2596_add__is__1,axiom,
% 12.45/6.91 ! [M: nat,N: nat] :
% 12.45/6.91 ( ( ( plus_plus_nat @ M @ N )
% 12.45/6.91 = ( suc @ zero_zero_nat ) )
% 12.45/6.91 = ( ( ( M
% 12.45/6.91 = ( suc @ zero_zero_nat ) )
% 12.45/6.91 & ( N = zero_zero_nat ) )
% 12.45/6.91 | ( ( M = zero_zero_nat )
% 12.45/6.91 & ( N
% 12.45/6.91 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2597_one__is__add,axiom,
% 12.45/6.91 ! [M: nat,N: nat] :
% 12.45/6.91 ( ( ( suc @ zero_zero_nat )
% 12.45/6.91 = ( plus_plus_nat @ M @ N ) )
% 12.45/6.91 = ( ( ( M
% 12.45/6.91 = ( suc @ zero_zero_nat ) )
% 12.45/6.91 & ( N = zero_zero_nat ) )
% 12.45/6.91 | ( ( M = zero_zero_nat )
% 12.45/6.91 & ( N
% 12.45/6.91 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2598_ex__least__nat__le,axiom,
% 12.45/6.91 ! [P: nat > $o,N: nat] :
% 12.45/6.91 ( ( P @ N )
% 12.45/6.91 => ( ~ ( P @ zero_zero_nat )
% 12.45/6.91 => ? [K2: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ K2 @ N )
% 12.45/6.91 & ! [I4: nat] :
% 12.45/6.91 ( ( ord_less_nat @ I4 @ K2 )
% 12.45/6.91 => ~ ( P @ I4 ) )
% 12.45/6.91 & ( P @ K2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2599_less__imp__add__positive,axiom,
% 12.45/6.91 ! [I: nat,J: nat] :
% 12.45/6.91 ( ( ord_less_nat @ I @ J )
% 12.45/6.91 => ? [K2: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 12.45/6.91 & ( ( plus_plus_nat @ I @ K2 )
% 12.45/6.91 = J ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2600_diff__less,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.91 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2601_nat__mult__less__cancel1,axiom,
% 12.45/6.91 ! [K: nat,M: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.91 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.45/6.91 = ( ord_less_nat @ M @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2602_nat__mult__eq__cancel1,axiom,
% 12.45/6.91 ! [K: nat,M: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.91 => ( ( ( times_times_nat @ K @ M )
% 12.45/6.91 = ( times_times_nat @ K @ N ) )
% 12.45/6.91 = ( M = N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2603_mult__less__mono2,axiom,
% 12.45/6.91 ! [I: nat,J: nat,K: nat] :
% 12.45/6.91 ( ( ord_less_nat @ I @ J )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.91 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2604_mult__less__mono1,axiom,
% 12.45/6.91 ! [I: nat,J: nat,K: nat] :
% 12.45/6.91 ( ( ord_less_nat @ I @ J )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.91 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2605_One__nat__def,axiom,
% 12.45/6.91 ( one_one_nat
% 12.45/6.91 = ( suc @ zero_zero_nat ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2606_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 12.45/6.91 ! [M: nat,N: nat] :
% 12.45/6.91 ( ( ( divide_divide_nat @ M @ N )
% 12.45/6.91 = zero_zero_nat )
% 12.45/6.91 = ( ( ord_less_nat @ M @ N )
% 12.45/6.91 | ( N = zero_zero_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2607_diff__add__0,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 12.45/6.91 = zero_zero_nat ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2608_nat__power__less__imp__less,axiom,
% 12.45/6.91 ! [I: nat,M: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ I )
% 12.45/6.91 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 12.45/6.91 => ( ord_less_nat @ M @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2609_mult__eq__self__implies__10,axiom,
% 12.45/6.91 ! [M: nat,N: nat] :
% 12.45/6.91 ( ( M
% 12.45/6.91 = ( times_times_nat @ M @ N ) )
% 12.45/6.91 => ( ( N = one_one_nat )
% 12.45/6.91 | ( M = zero_zero_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2610_mod__Suc,axiom,
% 12.45/6.91 ! [M: nat,N: nat] :
% 12.45/6.91 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 12.45/6.91 = N )
% 12.45/6.91 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 12.45/6.91 = zero_zero_nat ) )
% 12.45/6.91 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 12.45/6.91 != N )
% 12.45/6.91 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 12.45/6.91 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2611_mod__less__divisor,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2612_mod2__eq__if,axiom,
% 12.45/6.91 ! [A: nat] :
% 12.45/6.91 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 = zero_zero_nat ) )
% 12.45/6.91 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 = one_one_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2613_mod2__eq__if,axiom,
% 12.45/6.91 ! [A: int] :
% 12.45/6.91 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.91 = zero_zero_int ) )
% 12.45/6.91 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.91 = one_one_int ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2614_mod2__eq__if,axiom,
% 12.45/6.91 ! [A: code_integer] :
% 12.45/6.91 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.91 = zero_z3403309356797280102nteger ) )
% 12.45/6.91 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.91 = one_one_Code_integer ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2615_parity__cases,axiom,
% 12.45/6.91 ! [A: nat] :
% 12.45/6.91 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 != zero_zero_nat ) )
% 12.45/6.91 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 != one_one_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2616_parity__cases,axiom,
% 12.45/6.91 ! [A: int] :
% 12.45/6.91 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.91 != zero_zero_int ) )
% 12.45/6.91 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.91 != one_one_int ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2617_parity__cases,axiom,
% 12.45/6.91 ! [A: code_integer] :
% 12.45/6.91 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.91 != zero_z3403309356797280102nteger ) )
% 12.45/6.91 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.91 != one_one_Code_integer ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2618_pow_Osimps_I1_J,axiom,
% 12.45/6.91 ! [X4: num] :
% 12.45/6.91 ( ( pow @ X4 @ one )
% 12.45/6.91 = X4 ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2619_zero__le__power__eq,axiom,
% 12.45/6.91 ! [A: real,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 12.45/6.91 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2620_zero__le__power__eq,axiom,
% 12.45/6.91 ! [A: rat,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 12.45/6.91 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2621_zero__le__power__eq,axiom,
% 12.45/6.91 ! [A: int,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 12.45/6.91 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2622_zero__le__odd__power,axiom,
% 12.45/6.91 ! [N: nat,A: real] :
% 12.45/6.91 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 12.45/6.91 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2623_zero__le__odd__power,axiom,
% 12.45/6.91 ! [N: nat,A: rat] :
% 12.45/6.91 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 12.45/6.91 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2624_zero__le__odd__power,axiom,
% 12.45/6.91 ! [N: nat,A: int] :
% 12.45/6.91 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 12.45/6.91 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2625_zero__le__even__power,axiom,
% 12.45/6.91 ! [N: nat,A: real] :
% 12.45/6.91 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2626_zero__le__even__power,axiom,
% 12.45/6.91 ! [N: nat,A: rat] :
% 12.45/6.91 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2627_zero__le__even__power,axiom,
% 12.45/6.91 ! [N: nat,A: int] :
% 12.45/6.91 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2628_mod__eq__0D,axiom,
% 12.45/6.91 ! [M: nat,D: nat] :
% 12.45/6.91 ( ( ( modulo_modulo_nat @ M @ D )
% 12.45/6.91 = zero_zero_nat )
% 12.45/6.91 => ? [Q3: nat] :
% 12.45/6.91 ( M
% 12.45/6.91 = ( times_times_nat @ D @ Q3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2629_div2__even__ext__nat,axiom,
% 12.45/6.91 ! [X4: nat,Y3: nat] :
% 12.45/6.91 ( ( ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 = ( divide_divide_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.91 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 )
% 12.45/6.91 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y3 ) )
% 12.45/6.91 => ( X4 = Y3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2630_even__numeral,axiom,
% 12.45/6.91 ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2631_even__numeral,axiom,
% 12.45/6.91 ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2632_even__numeral,axiom,
% 12.45/6.91 ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2633_signed__take__bit__int__less__eq,axiom,
% 12.45/6.91 ! [N: nat,K: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 12.45/6.91 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2634_unit__eq__div1,axiom,
% 12.45/6.91 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 12.45/6.91 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.45/6.91 => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 12.45/6.91 = C )
% 12.45/6.91 = ( A
% 12.45/6.91 = ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2635_unit__eq__div1,axiom,
% 12.45/6.91 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.91 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.45/6.91 => ( ( ( divide_divide_nat @ A @ B2 )
% 12.45/6.91 = C )
% 12.45/6.91 = ( A
% 12.45/6.91 = ( times_times_nat @ C @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2636_unit__eq__div1,axiom,
% 12.45/6.91 ! [B2: int,A: int,C: int] :
% 12.45/6.91 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.45/6.91 => ( ( ( divide_divide_int @ A @ B2 )
% 12.45/6.91 = C )
% 12.45/6.91 = ( A
% 12.45/6.91 = ( times_times_int @ C @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2637_unit__eq__div2,axiom,
% 12.45/6.91 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 12.45/6.91 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.45/6.91 => ( ( A
% 12.45/6.91 = ( divide6298287555418463151nteger @ C @ B2 ) )
% 12.45/6.91 = ( ( times_3573771949741848930nteger @ A @ B2 )
% 12.45/6.91 = C ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2638_unit__eq__div2,axiom,
% 12.45/6.91 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.91 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.45/6.91 => ( ( A
% 12.45/6.91 = ( divide_divide_nat @ C @ B2 ) )
% 12.45/6.91 = ( ( times_times_nat @ A @ B2 )
% 12.45/6.91 = C ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2639_unit__eq__div2,axiom,
% 12.45/6.91 ! [B2: int,A: int,C: int] :
% 12.45/6.91 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.45/6.91 => ( ( A
% 12.45/6.91 = ( divide_divide_int @ C @ B2 ) )
% 12.45/6.91 = ( ( times_times_int @ A @ B2 )
% 12.45/6.91 = C ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2640_div__mult__unit2,axiom,
% 12.45/6.91 ! [C: code_integer,B2: code_integer,A: code_integer] :
% 12.45/6.91 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 12.45/6.91 => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 12.45/6.91 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 12.45/6.91 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2641_div__mult__unit2,axiom,
% 12.45/6.91 ! [C: nat,B2: nat,A: nat] :
% 12.45/6.91 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 12.45/6.91 => ( ( dvd_dvd_nat @ B2 @ A )
% 12.45/6.91 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 12.45/6.91 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2642_div__mult__unit2,axiom,
% 12.45/6.91 ! [C: int,B2: int,A: int] :
% 12.45/6.91 ( ( dvd_dvd_int @ C @ one_one_int )
% 12.45/6.91 => ( ( dvd_dvd_int @ B2 @ A )
% 12.45/6.91 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 12.45/6.91 = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2643_unit__div__commute,axiom,
% 12.45/6.91 ! [B2: code_integer,A: code_integer,C: code_integer] :
% 12.45/6.91 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.45/6.91 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 12.45/6.91 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2644_unit__div__commute,axiom,
% 12.45/6.91 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.91 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.45/6.91 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 12.45/6.91 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2645_unit__div__commute,axiom,
% 12.45/6.91 ! [B2: int,A: int,C: int] :
% 12.45/6.91 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.45/6.91 => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 12.45/6.91 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2646_unit__div__mult__swap,axiom,
% 12.45/6.91 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 12.45/6.91 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 12.45/6.91 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 12.45/6.91 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2647_unit__div__mult__swap,axiom,
% 12.45/6.91 ! [C: nat,A: nat,B2: nat] :
% 12.45/6.91 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 12.45/6.91 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 12.45/6.91 = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2648_unit__div__mult__swap,axiom,
% 12.45/6.91 ! [C: int,A: int,B2: int] :
% 12.45/6.91 ( ( dvd_dvd_int @ C @ one_one_int )
% 12.45/6.91 => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 12.45/6.91 = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2649_is__unit__div__mult2__eq,axiom,
% 12.45/6.91 ! [B2: code_integer,C: code_integer,A: code_integer] :
% 12.45/6.91 ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 12.45/6.91 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 12.45/6.91 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 12.45/6.91 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2650_is__unit__div__mult2__eq,axiom,
% 12.45/6.91 ! [B2: nat,C: nat,A: nat] :
% 12.45/6.91 ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 12.45/6.91 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 12.45/6.91 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 12.45/6.91 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2651_is__unit__div__mult2__eq,axiom,
% 12.45/6.91 ! [B2: int,C: int,A: int] :
% 12.45/6.91 ( ( dvd_dvd_int @ B2 @ one_one_int )
% 12.45/6.91 => ( ( dvd_dvd_int @ C @ one_one_int )
% 12.45/6.91 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 12.45/6.91 = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2652_mod__eq__dvd__iff__nat,axiom,
% 12.45/6.91 ! [N: nat,M: nat,Q2: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.91 => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 12.45/6.91 = ( modulo_modulo_nat @ N @ Q2 ) )
% 12.45/6.91 = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2653_mult__less__le__imp__less,axiom,
% 12.45/6.91 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.91 ( ( ord_less_real @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_real @ C @ D )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2654_mult__less__le__imp__less,axiom,
% 12.45/6.91 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.45/6.91 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ C @ D )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2655_mult__less__le__imp__less,axiom,
% 12.45/6.91 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.45/6.91 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_nat @ C @ D )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.45/6.91 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2656_mult__less__le__imp__less,axiom,
% 12.45/6.91 ! [A: int,B2: int,C: int,D: int] :
% 12.45/6.91 ( ( ord_less_int @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_int @ C @ D )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2657_mult__le__less__imp__less,axiom,
% 12.45/6.91 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_real @ C @ D )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2658_mult__le__less__imp__less,axiom,
% 12.45/6.91 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_rat @ C @ D )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2659_mult__le__less__imp__less,axiom,
% 12.45/6.91 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_nat @ C @ D )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.91 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2660_mult__le__less__imp__less,axiom,
% 12.45/6.91 ! [A: int,B2: int,C: int,D: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_int @ C @ D )
% 12.45/6.91 => ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2661_mult__right__le__imp__le,axiom,
% 12.45/6.91 ! [A: real,C: real,B2: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2662_mult__right__le__imp__le,axiom,
% 12.45/6.91 ! [A: rat,C: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2663_mult__right__le__imp__le,axiom,
% 12.45/6.91 ! [A: nat,C: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.45/6.91 => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2664_mult__right__le__imp__le,axiom,
% 12.45/6.91 ! [A: int,C: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 12.45/6.91 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2665_mult__left__le__imp__le,axiom,
% 12.45/6.91 ! [C: real,A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2666_mult__left__le__imp__le,axiom,
% 12.45/6.91 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2667_mult__left__le__imp__le,axiom,
% 12.45/6.91 ! [C: nat,A: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.45/6.91 => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2668_mult__left__le__imp__le,axiom,
% 12.45/6.91 ! [C: int,A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.45/6.91 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2669_mult__le__cancel__left__pos,axiom,
% 12.45/6.91 ! [C: real,A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.45/6.91 = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2670_mult__le__cancel__left__pos,axiom,
% 12.45/6.91 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.45/6.91 = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2671_mult__le__cancel__left__pos,axiom,
% 12.45/6.91 ! [C: int,A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.45/6.91 = ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2672_mult__le__cancel__left__neg,axiom,
% 12.45/6.91 ! [C: real,A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.45/6.91 = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2673_mult__le__cancel__left__neg,axiom,
% 12.45/6.91 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.45/6.91 = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2674_mult__le__cancel__left__neg,axiom,
% 12.45/6.91 ! [C: int,A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.45/6.91 = ( ord_less_eq_int @ B2 @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2675_mult__less__cancel__right,axiom,
% 12.45/6.91 ! [A: real,C: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2676_mult__less__cancel__right,axiom,
% 12.45/6.91 ! [A: rat,C: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2677_mult__less__cancel__right,axiom,
% 12.45/6.91 ! [A: int,C: int,B2: int] :
% 12.45/6.91 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_int @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2678_mult__strict__mono_H,axiom,
% 12.45/6.91 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.91 ( ( ord_less_real @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_real @ C @ D )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2679_mult__strict__mono_H,axiom,
% 12.45/6.91 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.45/6.91 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_rat @ C @ D )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2680_mult__strict__mono_H,axiom,
% 12.45/6.91 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.45/6.91 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_nat @ C @ D )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.91 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2681_mult__strict__mono_H,axiom,
% 12.45/6.91 ! [A: int,B2: int,C: int,D: int] :
% 12.45/6.91 ( ( ord_less_int @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_int @ C @ D )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2682_mult__right__less__imp__less,axiom,
% 12.45/6.91 ! [A: real,C: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2683_mult__right__less__imp__less,axiom,
% 12.45/6.91 ! [A: rat,C: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2684_mult__right__less__imp__less,axiom,
% 12.45/6.91 ! [A: nat,C: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.91 => ( ord_less_nat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2685_mult__right__less__imp__less,axiom,
% 12.45/6.91 ! [A: int,C: int,B2: int] :
% 12.45/6.91 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2686_mult__less__cancel__left,axiom,
% 12.45/6.91 ! [C: real,A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2687_mult__less__cancel__left,axiom,
% 12.45/6.91 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2688_mult__less__cancel__left,axiom,
% 12.45/6.91 ! [C: int,A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_int @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2689_mult__strict__mono,axiom,
% 12.45/6.91 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.91 ( ( ord_less_real @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_real @ C @ D )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2690_mult__strict__mono,axiom,
% 12.45/6.91 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.45/6.91 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_rat @ C @ D )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2691_mult__strict__mono,axiom,
% 12.45/6.91 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.45/6.91 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_nat @ C @ D )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.91 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2692_mult__strict__mono,axiom,
% 12.45/6.91 ! [A: int,B2: int,C: int,D: int] :
% 12.45/6.91 ( ( ord_less_int @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_int @ C @ D )
% 12.45/6.91 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2693_mult__left__less__imp__less,axiom,
% 12.45/6.91 ! [C: real,A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2694_mult__left__less__imp__less,axiom,
% 12.45/6.91 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2695_mult__left__less__imp__less,axiom,
% 12.45/6.91 ! [C: nat,A: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.91 => ( ord_less_nat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2696_mult__left__less__imp__less,axiom,
% 12.45/6.91 ! [C: int,A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2697_mult__le__cancel__right,axiom,
% 12.45/6.91 ! [A: real,C: real,B2: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_eq_real @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2698_mult__le__cancel__right,axiom,
% 12.45/6.91 ! [A: rat,C: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_eq_rat @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2699_mult__le__cancel__right,axiom,
% 12.45/6.91 ! [A: int,C: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_eq_int @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2700_mult__le__cancel__left,axiom,
% 12.45/6.91 ! [C: real,A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_eq_real @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2701_mult__le__cancel__left,axiom,
% 12.45/6.91 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_eq_rat @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2702_mult__le__cancel__left,axiom,
% 12.45/6.91 ! [C: int,A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_eq_int @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2703_add__strict__increasing2,axiom,
% 12.45/6.91 ! [A: real,B2: real,C: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.91 => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2704_add__strict__increasing2,axiom,
% 12.45/6.91 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.91 => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2705_add__strict__increasing2,axiom,
% 12.45/6.91 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ B2 @ C )
% 12.45/6.91 => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2706_add__strict__increasing2,axiom,
% 12.45/6.91 ! [A: int,B2: int,C: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_int @ B2 @ C )
% 12.45/6.91 => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2707_add__strict__increasing,axiom,
% 12.45/6.91 ! [A: real,B2: real,C: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_eq_real @ B2 @ C )
% 12.45/6.91 => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2708_add__strict__increasing,axiom,
% 12.45/6.91 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.91 => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2709_add__strict__increasing,axiom,
% 12.45/6.91 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_eq_nat @ B2 @ C )
% 12.45/6.91 => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2710_add__strict__increasing,axiom,
% 12.45/6.91 ! [A: int,B2: int,C: int] :
% 12.45/6.91 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_eq_int @ B2 @ C )
% 12.45/6.91 => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2711_add__pos__nonneg,axiom,
% 12.45/6.91 ! [A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.91 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2712_add__pos__nonneg,axiom,
% 12.45/6.91 ! [A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.91 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2713_add__pos__nonneg,axiom,
% 12.45/6.91 ! [A: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.45/6.91 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2714_add__pos__nonneg,axiom,
% 12.45/6.91 ! [A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.91 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2715_add__nonpos__neg,axiom,
% 12.45/6.91 ! [A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.91 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2716_add__nonpos__neg,axiom,
% 12.45/6.91 ! [A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2717_add__nonpos__neg,axiom,
% 12.45/6.91 ! [A: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.45/6.91 => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 12.45/6.91 => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2718_add__nonpos__neg,axiom,
% 12.45/6.91 ! [A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.45/6.91 => ( ( ord_less_int @ B2 @ zero_zero_int )
% 12.45/6.91 => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2719_add__nonneg__pos,axiom,
% 12.45/6.91 ! [A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.91 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2720_add__nonneg__pos,axiom,
% 12.45/6.91 ! [A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 12.45/6.91 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2721_add__nonneg__pos,axiom,
% 12.45/6.91 ! [A: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.45/6.91 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2722_add__nonneg__pos,axiom,
% 12.45/6.91 ! [A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.91 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2723_add__neg__nonpos,axiom,
% 12.45/6.91 ! [A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.91 => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2724_add__neg__nonpos,axiom,
% 12.45/6.91 ! [A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2725_add__neg__nonpos,axiom,
% 12.45/6.91 ! [A: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_nat @ A @ zero_zero_nat )
% 12.45/6.91 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 12.45/6.91 => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2726_add__neg__nonpos,axiom,
% 12.45/6.91 ! [A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_int @ A @ zero_zero_int )
% 12.45/6.91 => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 12.45/6.91 => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2727_field__le__epsilon,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] :
% 12.45/6.91 ( ! [E2: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.45/6.91 => ( ord_less_eq_real @ X4 @ ( plus_plus_real @ Y3 @ E2 ) ) )
% 12.45/6.91 => ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2728_field__le__epsilon,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] :
% 12.45/6.91 ( ! [E2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 12.45/6.91 => ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ Y3 @ E2 ) ) )
% 12.45/6.91 => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2729_div__positive,axiom,
% 12.45/6.91 ! [B2: nat,A: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_nat @ B2 @ A )
% 12.45/6.91 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2730_div__positive,axiom,
% 12.45/6.91 ! [B2: int,A: int] :
% 12.45/6.91 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_int @ B2 @ A )
% 12.45/6.91 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2731_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 12.45/6.91 ! [A: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ A @ B2 )
% 12.45/6.91 => ( ( divide_divide_nat @ A @ B2 )
% 12.45/6.91 = zero_zero_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2732_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 12.45/6.91 ! [A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_int @ A @ B2 )
% 12.45/6.91 => ( ( divide_divide_int @ A @ B2 )
% 12.45/6.91 = zero_zero_int ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2733_divide__nonpos__pos,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.91 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2734_divide__nonpos__pos,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 12.45/6.91 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2735_divide__nonpos__neg,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.91 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2736_divide__nonpos__neg,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2737_divide__nonneg__pos,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.91 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2738_divide__nonneg__pos,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 12.45/6.91 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2739_divide__nonneg__neg,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.91 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ zero_zero_real ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2740_divide__nonneg__neg,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.45/6.91 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2741_divide__le__cancel,axiom,
% 12.45/6.91 ! [A: real,C: real,B2: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_eq_real @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2742_divide__le__cancel,axiom,
% 12.45/6.91 ! [A: rat,C: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_eq_rat @ A @ B2 ) )
% 12.45/6.91 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2743_frac__less2,axiom,
% 12.45/6.91 ! [X4: real,Y3: real,W: real,Z: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.91 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ W )
% 12.45/6.91 => ( ( ord_less_real @ W @ Z )
% 12.45/6.91 => ( ord_less_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2744_frac__less2,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat,W: rat,Z: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 12.45/6.91 => ( ( ord_less_rat @ W @ Z )
% 12.45/6.91 => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2745_frac__less,axiom,
% 12.45/6.91 ! [X4: real,Y3: real,W: real,Z: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.91 => ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ W )
% 12.45/6.91 => ( ( ord_less_eq_real @ W @ Z )
% 12.45/6.91 => ( ord_less_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2746_frac__less,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat,W: rat,Z: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.45/6.91 => ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 12.45/6.91 => ( ( ord_less_eq_rat @ W @ Z )
% 12.45/6.91 => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2747_frac__le,axiom,
% 12.45/6.91 ! [Y3: real,X4: real,W: real,Z: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ W )
% 12.45/6.91 => ( ( ord_less_eq_real @ W @ Z )
% 12.45/6.91 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2748_frac__le,axiom,
% 12.45/6.91 ! [Y3: rat,X4: rat,W: rat,Z: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 12.45/6.91 => ( ( ord_less_eq_rat @ W @ Z )
% 12.45/6.91 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2749_mult__left__le__one__le,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.91 => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X4 ) @ X4 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2750_mult__left__le__one__le,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X4 ) @ X4 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2751_mult__left__le__one__le,axiom,
% 12.45/6.91 ! [X4: int,Y3: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 12.45/6.91 => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X4 ) @ X4 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2752_mult__right__le__one__le,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.91 => ( ord_less_eq_real @ ( times_times_real @ X4 @ Y3 ) @ X4 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2753_mult__right__le__one__le,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ ( times_times_rat @ X4 @ Y3 ) @ X4 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2754_mult__right__le__one__le,axiom,
% 12.45/6.91 ! [X4: int,Y3: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 12.45/6.91 => ( ord_less_eq_int @ ( times_times_int @ X4 @ Y3 ) @ X4 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2755_mult__le__one,axiom,
% 12.45/6.91 ! [A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ A @ one_one_real )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_real @ B2 @ one_one_real )
% 12.45/6.91 => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ one_one_real ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2756_mult__le__one,axiom,
% 12.45/6.91 ! [A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ B2 @ one_one_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ one_one_rat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2757_mult__le__one,axiom,
% 12.45/6.91 ! [A: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_nat @ B2 @ one_one_nat )
% 12.45/6.91 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2758_mult__le__one,axiom,
% 12.45/6.91 ! [A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ A @ one_one_int )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_int @ B2 @ one_one_int )
% 12.45/6.91 => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2759_mult__left__le,axiom,
% 12.45/6.91 ! [C: real,A: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ C @ one_one_real )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2760_mult__left__le,axiom,
% 12.45/6.91 ! [C: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2761_mult__left__le,axiom,
% 12.45/6.91 ! [C: nat,A: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2762_mult__left__le,axiom,
% 12.45/6.91 ! [C: int,A: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ C @ one_one_int )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2763_sum__squares__le__zero__iff,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 12.45/6.91 = ( ( X4 = zero_zero_real )
% 12.45/6.91 & ( Y3 = zero_zero_real ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2764_sum__squares__le__zero__iff,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 12.45/6.91 = ( ( X4 = zero_zero_rat )
% 12.45/6.91 & ( Y3 = zero_zero_rat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2765_sum__squares__le__zero__iff,axiom,
% 12.45/6.91 ! [X4: int,Y3: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 12.45/6.91 = ( ( X4 = zero_zero_int )
% 12.45/6.91 & ( Y3 = zero_zero_int ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2766_sum__squares__ge__zero,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2767_sum__squares__ge__zero,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2768_sum__squares__ge__zero,axiom,
% 12.45/6.91 ! [X4: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2769_power__less__imp__less__base,axiom,
% 12.45/6.91 ! [A: real,N: nat,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.91 => ( ord_less_real @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2770_power__less__imp__less__base,axiom,
% 12.45/6.91 ! [A: rat,N: nat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.91 => ( ord_less_rat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2771_power__less__imp__less__base,axiom,
% 12.45/6.91 ! [A: nat,N: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.45/6.91 => ( ord_less_nat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2772_power__less__imp__less__base,axiom,
% 12.45/6.91 ! [A: int,N: nat,B2: int] :
% 12.45/6.91 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.91 => ( ord_less_int @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2773_sum__squares__gt__zero__iff,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 12.45/6.91 = ( ( X4 != zero_zero_real )
% 12.45/6.91 | ( Y3 != zero_zero_real ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2774_sum__squares__gt__zero__iff,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 12.45/6.91 = ( ( X4 != zero_zero_rat )
% 12.45/6.91 | ( Y3 != zero_zero_rat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2775_sum__squares__gt__zero__iff,axiom,
% 12.45/6.91 ! [X4: int,Y3: int] :
% 12.45/6.91 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 12.45/6.91 = ( ( X4 != zero_zero_int )
% 12.45/6.91 | ( Y3 != zero_zero_int ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2776_not__sum__squares__lt__zero,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] :
% 12.45/6.91 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2777_not__sum__squares__lt__zero,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] :
% 12.45/6.91 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2778_not__sum__squares__lt__zero,axiom,
% 12.45/6.91 ! [X4: int,Y3: int] :
% 12.45/6.91 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2779_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 12.45/6.91 ! [C: nat,A: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.91 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 12.45/6.91 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2780_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 12.45/6.91 ! [C: int,A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 12.45/6.91 = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2781_zero__less__two,axiom,
% 12.45/6.91 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2782_zero__less__two,axiom,
% 12.45/6.91 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2783_zero__less__two,axiom,
% 12.45/6.91 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2784_zero__less__two,axiom,
% 12.45/6.91 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2785_divide__strict__left__mono__neg,axiom,
% 12.45/6.91 ! [A: real,B2: real,C: real] :
% 12.45/6.91 ( ( ord_less_real @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.91 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2786_divide__strict__left__mono__neg,axiom,
% 12.45/6.91 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.91 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.91 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2787_divide__strict__left__mono,axiom,
% 12.45/6.91 ! [B2: real,A: real,C: real] :
% 12.45/6.91 ( ( ord_less_real @ B2 @ A )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.91 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2788_divide__strict__left__mono,axiom,
% 12.45/6.91 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.91 ( ( ord_less_rat @ B2 @ A )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.91 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2789_mult__imp__less__div__pos,axiom,
% 12.45/6.91 ! [Y3: real,Z: real,X4: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.91 => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X4 )
% 12.45/6.91 => ( ord_less_real @ Z @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2790_mult__imp__less__div__pos,axiom,
% 12.45/6.91 ! [Y3: rat,Z: rat,X4: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 12.45/6.91 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X4 )
% 12.45/6.91 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2791_mult__imp__div__pos__less,axiom,
% 12.45/6.91 ! [Y3: real,X4: real,Z: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.91 => ( ( ord_less_real @ X4 @ ( times_times_real @ Z @ Y3 ) )
% 12.45/6.91 => ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2792_mult__imp__div__pos__less,axiom,
% 12.45/6.91 ! [Y3: rat,X4: rat,Z: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 12.45/6.91 => ( ( ord_less_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) )
% 12.45/6.91 => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2793_pos__less__divide__eq,axiom,
% 12.45/6.91 ! [C: real,A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.91 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2794_pos__less__divide__eq,axiom,
% 12.45/6.91 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.91 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2795_pos__divide__less__eq,axiom,
% 12.45/6.91 ! [C: real,B2: real,A: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 12.45/6.91 = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2796_pos__divide__less__eq,axiom,
% 12.45/6.91 ! [C: rat,B2: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 12.45/6.91 = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2797_neg__less__divide__eq,axiom,
% 12.45/6.91 ! [C: real,A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.91 = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2798_neg__less__divide__eq,axiom,
% 12.45/6.91 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.91 = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2799_neg__divide__less__eq,axiom,
% 12.45/6.91 ! [C: real,B2: real,A: real] :
% 12.45/6.91 ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 12.45/6.91 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2800_neg__divide__less__eq,axiom,
% 12.45/6.91 ! [C: rat,B2: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 12.45/6.91 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2801_less__divide__eq,axiom,
% 12.45/6.91 ! [A: real,B2: real,C: real] :
% 12.45/6.91 ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2802_less__divide__eq,axiom,
% 12.45/6.91 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.91 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2803_divide__less__eq,axiom,
% 12.45/6.91 ! [B2: real,C: real,A: real] :
% 12.45/6.91 ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2804_divide__less__eq,axiom,
% 12.45/6.91 ! [B2: rat,C: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2805_less__divide__eq__1,axiom,
% 12.45/6.91 ! [B2: real,A: real] :
% 12.45/6.91 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.91 & ( ord_less_real @ A @ B2 ) )
% 12.45/6.91 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.91 & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2806_less__divide__eq__1,axiom,
% 12.45/6.91 ! [B2: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.91 & ( ord_less_rat @ A @ B2 ) )
% 12.45/6.91 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.91 & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2807_divide__less__eq__1,axiom,
% 12.45/6.91 ! [B2: real,A: real] :
% 12.45/6.91 ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.91 & ( ord_less_real @ B2 @ A ) )
% 12.45/6.91 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.91 & ( ord_less_real @ A @ B2 ) )
% 12.45/6.91 | ( A = zero_zero_real ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2808_divide__less__eq__1,axiom,
% 12.45/6.91 ! [B2: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.91 & ( ord_less_rat @ B2 @ A ) )
% 12.45/6.91 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.91 & ( ord_less_rat @ A @ B2 ) )
% 12.45/6.91 | ( A = zero_zero_rat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2809_power__le__one,axiom,
% 12.45/6.91 ! [A: real,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_eq_real @ A @ one_one_real )
% 12.45/6.91 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2810_power__le__one,axiom,
% 12.45/6.91 ! [A: rat,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2811_power__le__one,axiom,
% 12.45/6.91 ! [A: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 12.45/6.91 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2812_power__le__one,axiom,
% 12.45/6.91 ! [A: int,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_eq_int @ A @ one_one_int )
% 12.45/6.91 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2813_divide__eq__eq__numeral_I1_J,axiom,
% 12.45/6.91 ! [B2: complex,C: complex,W: num] :
% 12.45/6.91 ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 12.45/6.91 = ( numera6690914467698888265omplex @ W ) )
% 12.45/6.91 = ( ( ( C != zero_zero_complex )
% 12.45/6.91 => ( B2
% 12.45/6.91 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 12.45/6.91 & ( ( C = zero_zero_complex )
% 12.45/6.91 => ( ( numera6690914467698888265omplex @ W )
% 12.45/6.91 = zero_zero_complex ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2814_divide__eq__eq__numeral_I1_J,axiom,
% 12.45/6.91 ! [B2: real,C: real,W: num] :
% 12.45/6.91 ( ( ( divide_divide_real @ B2 @ C )
% 12.45/6.91 = ( numeral_numeral_real @ W ) )
% 12.45/6.91 = ( ( ( C != zero_zero_real )
% 12.45/6.91 => ( B2
% 12.45/6.91 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 12.45/6.91 & ( ( C = zero_zero_real )
% 12.45/6.91 => ( ( numeral_numeral_real @ W )
% 12.45/6.91 = zero_zero_real ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2815_divide__eq__eq__numeral_I1_J,axiom,
% 12.45/6.91 ! [B2: rat,C: rat,W: num] :
% 12.45/6.91 ( ( ( divide_divide_rat @ B2 @ C )
% 12.45/6.91 = ( numeral_numeral_rat @ W ) )
% 12.45/6.91 = ( ( ( C != zero_zero_rat )
% 12.45/6.91 => ( B2
% 12.45/6.91 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 12.45/6.91 & ( ( C = zero_zero_rat )
% 12.45/6.91 => ( ( numeral_numeral_rat @ W )
% 12.45/6.91 = zero_zero_rat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2816_eq__divide__eq__numeral_I1_J,axiom,
% 12.45/6.91 ! [W: num,B2: complex,C: complex] :
% 12.45/6.91 ( ( ( numera6690914467698888265omplex @ W )
% 12.45/6.91 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 12.45/6.91 = ( ( ( C != zero_zero_complex )
% 12.45/6.91 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 12.45/6.91 = B2 ) )
% 12.45/6.91 & ( ( C = zero_zero_complex )
% 12.45/6.91 => ( ( numera6690914467698888265omplex @ W )
% 12.45/6.91 = zero_zero_complex ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2817_eq__divide__eq__numeral_I1_J,axiom,
% 12.45/6.91 ! [W: num,B2: real,C: real] :
% 12.45/6.91 ( ( ( numeral_numeral_real @ W )
% 12.45/6.91 = ( divide_divide_real @ B2 @ C ) )
% 12.45/6.91 = ( ( ( C != zero_zero_real )
% 12.45/6.91 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 12.45/6.91 = B2 ) )
% 12.45/6.91 & ( ( C = zero_zero_real )
% 12.45/6.91 => ( ( numeral_numeral_real @ W )
% 12.45/6.91 = zero_zero_real ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2818_eq__divide__eq__numeral_I1_J,axiom,
% 12.45/6.91 ! [W: num,B2: rat,C: rat] :
% 12.45/6.91 ( ( ( numeral_numeral_rat @ W )
% 12.45/6.91 = ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.91 = ( ( ( C != zero_zero_rat )
% 12.45/6.91 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 12.45/6.91 = B2 ) )
% 12.45/6.91 & ( ( C = zero_zero_rat )
% 12.45/6.91 => ( ( numeral_numeral_rat @ W )
% 12.45/6.91 = zero_zero_rat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2819_add__divide__eq__if__simps_I2_J,axiom,
% 12.45/6.91 ! [Z: complex,A: complex,B2: complex] :
% 12.45/6.91 ( ( ( Z = zero_zero_complex )
% 12.45/6.91 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 12.45/6.91 = B2 ) )
% 12.45/6.91 & ( ( Z != zero_zero_complex )
% 12.45/6.91 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2820_add__divide__eq__if__simps_I2_J,axiom,
% 12.45/6.91 ! [Z: real,A: real,B2: real] :
% 12.45/6.91 ( ( ( Z = zero_zero_real )
% 12.45/6.91 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 12.45/6.91 = B2 ) )
% 12.45/6.91 & ( ( Z != zero_zero_real )
% 12.45/6.91 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 12.45/6.91 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2821_add__divide__eq__if__simps_I2_J,axiom,
% 12.45/6.91 ! [Z: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ( Z = zero_zero_rat )
% 12.45/6.91 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 12.45/6.91 = B2 ) )
% 12.45/6.91 & ( ( Z != zero_zero_rat )
% 12.45/6.91 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 12.45/6.91 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2822_add__divide__eq__if__simps_I1_J,axiom,
% 12.45/6.91 ! [Z: complex,A: complex,B2: complex] :
% 12.45/6.91 ( ( ( Z = zero_zero_complex )
% 12.45/6.91 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 12.45/6.91 = A ) )
% 12.45/6.91 & ( ( Z != zero_zero_complex )
% 12.45/6.91 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2823_add__divide__eq__if__simps_I1_J,axiom,
% 12.45/6.91 ! [Z: real,A: real,B2: real] :
% 12.45/6.91 ( ( ( Z = zero_zero_real )
% 12.45/6.91 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 12.45/6.91 = A ) )
% 12.45/6.91 & ( ( Z != zero_zero_real )
% 12.45/6.91 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 12.45/6.91 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2824_add__divide__eq__if__simps_I1_J,axiom,
% 12.45/6.91 ! [Z: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ( Z = zero_zero_rat )
% 12.45/6.91 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 12.45/6.91 = A ) )
% 12.45/6.91 & ( ( Z != zero_zero_rat )
% 12.45/6.91 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 12.45/6.91 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2825_add__frac__eq,axiom,
% 12.45/6.91 ! [Y3: complex,Z: complex,X4: complex,W: complex] :
% 12.45/6.91 ( ( Y3 != zero_zero_complex )
% 12.45/6.91 => ( ( Z != zero_zero_complex )
% 12.45/6.91 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2826_add__frac__eq,axiom,
% 12.45/6.91 ! [Y3: real,Z: real,X4: real,W: real] :
% 12.45/6.91 ( ( Y3 != zero_zero_real )
% 12.45/6.91 => ( ( Z != zero_zero_real )
% 12.45/6.91 => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 12.45/6.91 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2827_add__frac__eq,axiom,
% 12.45/6.91 ! [Y3: rat,Z: rat,X4: rat,W: rat] :
% 12.45/6.91 ( ( Y3 != zero_zero_rat )
% 12.45/6.91 => ( ( Z != zero_zero_rat )
% 12.45/6.91 => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 12.45/6.91 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2828_add__frac__num,axiom,
% 12.45/6.91 ! [Y3: complex,X4: complex,Z: complex] :
% 12.45/6.91 ( ( Y3 != zero_zero_complex )
% 12.45/6.91 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ Z )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2829_add__frac__num,axiom,
% 12.45/6.91 ! [Y3: real,X4: real,Z: real] :
% 12.45/6.91 ( ( Y3 != zero_zero_real )
% 12.45/6.91 => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Y3 ) @ Z )
% 12.45/6.91 = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2830_add__frac__num,axiom,
% 12.45/6.91 ! [Y3: rat,X4: rat,Z: rat] :
% 12.45/6.91 ( ( Y3 != zero_zero_rat )
% 12.45/6.91 => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ Z )
% 12.45/6.91 = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2831_add__num__frac,axiom,
% 12.45/6.91 ! [Y3: complex,Z: complex,X4: complex] :
% 12.45/6.91 ( ( Y3 != zero_zero_complex )
% 12.45/6.91 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2832_add__num__frac,axiom,
% 12.45/6.91 ! [Y3: real,Z: real,X4: real] :
% 12.45/6.91 ( ( Y3 != zero_zero_real )
% 12.45/6.91 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X4 @ Y3 ) )
% 12.45/6.91 = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2833_add__num__frac,axiom,
% 12.45/6.91 ! [Y3: rat,Z: rat,X4: rat] :
% 12.45/6.91 ( ( Y3 != zero_zero_rat )
% 12.45/6.91 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X4 @ Y3 ) )
% 12.45/6.91 = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2834_add__divide__eq__iff,axiom,
% 12.45/6.91 ! [Z: complex,X4: complex,Y3: complex] :
% 12.45/6.91 ( ( Z != zero_zero_complex )
% 12.45/6.91 => ( ( plus_plus_complex @ X4 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2835_add__divide__eq__iff,axiom,
% 12.45/6.91 ! [Z: real,X4: real,Y3: real] :
% 12.45/6.91 ( ( Z != zero_zero_real )
% 12.45/6.91 => ( ( plus_plus_real @ X4 @ ( divide_divide_real @ Y3 @ Z ) )
% 12.45/6.91 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2836_add__divide__eq__iff,axiom,
% 12.45/6.91 ! [Z: rat,X4: rat,Y3: rat] :
% 12.45/6.91 ( ( Z != zero_zero_rat )
% 12.45/6.91 => ( ( plus_plus_rat @ X4 @ ( divide_divide_rat @ Y3 @ Z ) )
% 12.45/6.91 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2837_divide__add__eq__iff,axiom,
% 12.45/6.91 ! [Z: complex,X4: complex,Y3: complex] :
% 12.45/6.91 ( ( Z != zero_zero_complex )
% 12.45/6.91 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Z ) @ Y3 )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2838_divide__add__eq__iff,axiom,
% 12.45/6.91 ! [Z: real,X4: real,Y3: real] :
% 12.45/6.91 ( ( Z != zero_zero_real )
% 12.45/6.91 => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Z ) @ Y3 )
% 12.45/6.91 = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2839_divide__add__eq__iff,axiom,
% 12.45/6.91 ! [Z: rat,X4: rat,Y3: rat] :
% 12.45/6.91 ( ( Z != zero_zero_rat )
% 12.45/6.91 => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Z ) @ Y3 )
% 12.45/6.91 = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2840_power__le__imp__le__base,axiom,
% 12.45/6.91 ! [A: real,N: nat,B2: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B2 @ ( suc @ N ) ) )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.91 => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2841_power__le__imp__le__base,axiom,
% 12.45/6.91 ! [A: rat,N: nat,B2: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B2 @ ( suc @ N ) ) )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.91 => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2842_power__le__imp__le__base,axiom,
% 12.45/6.91 ! [A: nat,N: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B2 @ ( suc @ N ) ) )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.45/6.91 => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2843_power__le__imp__le__base,axiom,
% 12.45/6.91 ! [A: int,N: nat,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B2 @ ( suc @ N ) ) )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.91 => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2844_power__inject__base,axiom,
% 12.45/6.91 ! [A: real,N: nat,B2: real] :
% 12.45/6.91 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 12.45/6.91 = ( power_power_real @ B2 @ ( suc @ N ) ) )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.91 => ( A = B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2845_power__inject__base,axiom,
% 12.45/6.91 ! [A: rat,N: nat,B2: rat] :
% 12.45/6.91 ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 12.45/6.91 = ( power_power_rat @ B2 @ ( suc @ N ) ) )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.91 => ( A = B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2846_power__inject__base,axiom,
% 12.45/6.91 ! [A: nat,N: nat,B2: nat] :
% 12.45/6.91 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 12.45/6.91 = ( power_power_nat @ B2 @ ( suc @ N ) ) )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 12.45/6.91 => ( A = B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2847_power__inject__base,axiom,
% 12.45/6.91 ! [A: int,N: nat,B2: int] :
% 12.45/6.91 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 12.45/6.91 = ( power_power_int @ B2 @ ( suc @ N ) ) )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.91 => ( A = B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2848_div__add__self1,axiom,
% 12.45/6.91 ! [B2: nat,A: nat] :
% 12.45/6.91 ( ( B2 != zero_zero_nat )
% 12.45/6.91 => ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 12.45/6.91 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2849_div__add__self1,axiom,
% 12.45/6.91 ! [B2: int,A: int] :
% 12.45/6.91 ( ( B2 != zero_zero_int )
% 12.45/6.91 => ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 12.45/6.91 = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2850_div__add__self2,axiom,
% 12.45/6.91 ! [B2: nat,A: nat] :
% 12.45/6.91 ( ( B2 != zero_zero_nat )
% 12.45/6.91 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 12.45/6.91 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2851_div__add__self2,axiom,
% 12.45/6.91 ! [B2: int,A: int] :
% 12.45/6.91 ( ( B2 != zero_zero_int )
% 12.45/6.91 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 12.45/6.91 = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2852_add__divide__eq__if__simps_I4_J,axiom,
% 12.45/6.91 ! [Z: complex,A: complex,B2: complex] :
% 12.45/6.91 ( ( ( Z = zero_zero_complex )
% 12.45/6.91 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 12.45/6.91 = A ) )
% 12.45/6.91 & ( ( Z != zero_zero_complex )
% 12.45/6.91 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2853_add__divide__eq__if__simps_I4_J,axiom,
% 12.45/6.91 ! [Z: real,A: real,B2: real] :
% 12.45/6.91 ( ( ( Z = zero_zero_real )
% 12.45/6.91 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 12.45/6.91 = A ) )
% 12.45/6.91 & ( ( Z != zero_zero_real )
% 12.45/6.91 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 12.45/6.91 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2854_add__divide__eq__if__simps_I4_J,axiom,
% 12.45/6.91 ! [Z: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ( Z = zero_zero_rat )
% 12.45/6.91 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 12.45/6.91 = A ) )
% 12.45/6.91 & ( ( Z != zero_zero_rat )
% 12.45/6.91 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 12.45/6.91 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2855_diff__frac__eq,axiom,
% 12.45/6.91 ! [Y3: complex,Z: complex,X4: complex,W: complex] :
% 12.45/6.91 ( ( Y3 != zero_zero_complex )
% 12.45/6.91 => ( ( Z != zero_zero_complex )
% 12.45/6.91 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X4 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2856_diff__frac__eq,axiom,
% 12.45/6.91 ! [Y3: real,Z: real,X4: real,W: real] :
% 12.45/6.91 ( ( Y3 != zero_zero_real )
% 12.45/6.91 => ( ( Z != zero_zero_real )
% 12.45/6.91 => ( ( minus_minus_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 12.45/6.91 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2857_diff__frac__eq,axiom,
% 12.45/6.91 ! [Y3: rat,Z: rat,X4: rat,W: rat] :
% 12.45/6.91 ( ( Y3 != zero_zero_rat )
% 12.45/6.91 => ( ( Z != zero_zero_rat )
% 12.45/6.91 => ( ( minus_minus_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 12.45/6.91 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2858_diff__divide__eq__iff,axiom,
% 12.45/6.91 ! [Z: complex,X4: complex,Y3: complex] :
% 12.45/6.91 ( ( Z != zero_zero_complex )
% 12.45/6.91 => ( ( minus_minus_complex @ X4 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2859_diff__divide__eq__iff,axiom,
% 12.45/6.91 ! [Z: real,X4: real,Y3: real] :
% 12.45/6.91 ( ( Z != zero_zero_real )
% 12.45/6.91 => ( ( minus_minus_real @ X4 @ ( divide_divide_real @ Y3 @ Z ) )
% 12.45/6.91 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2860_diff__divide__eq__iff,axiom,
% 12.45/6.91 ! [Z: rat,X4: rat,Y3: rat] :
% 12.45/6.91 ( ( Z != zero_zero_rat )
% 12.45/6.91 => ( ( minus_minus_rat @ X4 @ ( divide_divide_rat @ Y3 @ Z ) )
% 12.45/6.91 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2861_divide__diff__eq__iff,axiom,
% 12.45/6.91 ! [Z: complex,X4: complex,Y3: complex] :
% 12.45/6.91 ( ( Z != zero_zero_complex )
% 12.45/6.91 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X4 @ Z ) @ Y3 )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X4 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2862_divide__diff__eq__iff,axiom,
% 12.45/6.91 ! [Z: real,X4: real,Y3: real] :
% 12.45/6.91 ( ( Z != zero_zero_real )
% 12.45/6.91 => ( ( minus_minus_real @ ( divide_divide_real @ X4 @ Z ) @ Y3 )
% 12.45/6.91 = ( divide_divide_real @ ( minus_minus_real @ X4 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2863_divide__diff__eq__iff,axiom,
% 12.45/6.91 ! [Z: rat,X4: rat,Y3: rat] :
% 12.45/6.91 ( ( Z != zero_zero_rat )
% 12.45/6.91 => ( ( minus_minus_rat @ ( divide_divide_rat @ X4 @ Z ) @ Y3 )
% 12.45/6.91 = ( divide_divide_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2864_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 12.45/6.91 ! [B2: code_integer,A: code_integer] :
% 12.45/6.91 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 12.45/6.91 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2865_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 12.45/6.91 ! [B2: nat,A: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.45/6.91 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2866_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 12.45/6.91 ! [B2: int,A: int] :
% 12.45/6.91 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.91 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2867_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 12.45/6.91 ! [A: code_integer,B2: code_integer] :
% 12.45/6.91 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.45/6.91 => ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 12.45/6.91 => ( ( modulo364778990260209775nteger @ A @ B2 )
% 12.45/6.91 = A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2868_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 12.45/6.91 ! [A: nat,B2: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ A @ B2 )
% 12.45/6.91 => ( ( modulo_modulo_nat @ A @ B2 )
% 12.45/6.91 = A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2869_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 12.45/6.91 ! [A: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_int @ A @ B2 )
% 12.45/6.91 => ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.91 = A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2870_cong__exp__iff__simps_I2_J,axiom,
% 12.45/6.91 ! [N: num,Q2: num] :
% 12.45/6.91 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.45/6.91 = zero_zero_nat )
% 12.45/6.91 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 12.45/6.91 = zero_zero_nat ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2871_cong__exp__iff__simps_I2_J,axiom,
% 12.45/6.91 ! [N: num,Q2: num] :
% 12.45/6.91 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.45/6.91 = zero_zero_int )
% 12.45/6.91 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 12.45/6.91 = zero_zero_int ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2872_cong__exp__iff__simps_I2_J,axiom,
% 12.45/6.91 ! [N: num,Q2: num] :
% 12.45/6.91 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.45/6.91 = zero_z3403309356797280102nteger )
% 12.45/6.91 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 12.45/6.91 = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2873_cong__exp__iff__simps_I1_J,axiom,
% 12.45/6.91 ! [N: num] :
% 12.45/6.91 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 12.45/6.91 = zero_zero_nat ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2874_cong__exp__iff__simps_I1_J,axiom,
% 12.45/6.91 ! [N: num] :
% 12.45/6.91 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 12.45/6.91 = zero_zero_int ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2875_cong__exp__iff__simps_I1_J,axiom,
% 12.45/6.91 ! [N: num] :
% 12.45/6.91 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 12.45/6.91 = zero_z3403309356797280102nteger ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2876_numeral__1__eq__Suc__0,axiom,
% 12.45/6.91 ( ( numeral_numeral_nat @ one )
% 12.45/6.91 = ( suc @ zero_zero_nat ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2877_ex__least__nat__less,axiom,
% 12.45/6.91 ! [P: nat > $o,N: nat] :
% 12.45/6.91 ( ( P @ N )
% 12.45/6.91 => ( ~ ( P @ zero_zero_nat )
% 12.45/6.91 => ? [K2: nat] :
% 12.45/6.91 ( ( ord_less_nat @ K2 @ N )
% 12.45/6.91 & ! [I4: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ I4 @ K2 )
% 12.45/6.91 => ~ ( P @ I4 ) )
% 12.45/6.91 & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2878_diff__Suc__less,axiom,
% 12.45/6.91 ! [N: nat,I: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2879_n__less__n__mult__m,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 12.45/6.91 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2880_n__less__m__mult__n,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 12.45/6.91 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2881_one__less__mult,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.45/6.91 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 12.45/6.91 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2882_realpow__pos__nth2,axiom,
% 12.45/6.91 ! [A: real,N: nat] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.91 => ? [R3: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ R3 )
% 12.45/6.91 & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 12.45/6.91 = A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2883_nat__induct__non__zero,axiom,
% 12.45/6.91 ! [N: nat,P: nat > $o] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ( P @ one_one_nat )
% 12.45/6.91 => ( ! [N4: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 12.45/6.91 => ( ( P @ N4 )
% 12.45/6.91 => ( P @ ( suc @ N4 ) ) ) )
% 12.45/6.91 => ( P @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2884_nat__mult__le__cancel1,axiom,
% 12.45/6.91 ! [K: nat,M: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.91 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.45/6.91 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2885_div__greater__zero__iff,axiom,
% 12.45/6.91 ! [M: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 12.45/6.91 = ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.91 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2886_div__le__mono2,axiom,
% 12.45/6.91 ! [M: nat,N: nat,K: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.91 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.91 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2887_power__gt__expt,axiom,
% 12.45/6.91 ! [N: nat,K: nat] :
% 12.45/6.91 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.45/6.91 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2888_nat__diff__split,axiom,
% 12.45/6.91 ! [P: nat > $o,A: nat,B2: nat] :
% 12.45/6.91 ( ( P @ ( minus_minus_nat @ A @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_nat @ A @ B2 )
% 12.45/6.91 => ( P @ zero_zero_nat ) )
% 12.45/6.91 & ! [D2: nat] :
% 12.45/6.91 ( ( A
% 12.45/6.91 = ( plus_plus_nat @ B2 @ D2 ) )
% 12.45/6.91 => ( P @ D2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2889_nat__diff__split__asm,axiom,
% 12.45/6.91 ! [P: nat > $o,A: nat,B2: nat] :
% 12.45/6.91 ( ( P @ ( minus_minus_nat @ A @ B2 ) )
% 12.45/6.91 = ( ~ ( ( ( ord_less_nat @ A @ B2 )
% 12.45/6.91 & ~ ( P @ zero_zero_nat ) )
% 12.45/6.91 | ? [D2: nat] :
% 12.45/6.91 ( ( A
% 12.45/6.91 = ( plus_plus_nat @ B2 @ D2 ) )
% 12.45/6.91 & ~ ( P @ D2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2890_nat__one__le__power,axiom,
% 12.45/6.91 ! [I: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 12.45/6.91 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2891_real__arch__pow__inv,axiom,
% 12.45/6.91 ! [Y3: real,X4: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.91 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/6.91 => ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X4 @ N4 ) @ Y3 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2892_div__less__iff__less__mult,axiom,
% 12.45/6.91 ! [Q2: nat,M: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 12.45/6.91 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
% 12.45/6.91 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2893_nat__mult__div__cancel1,axiom,
% 12.45/6.91 ! [K: nat,M: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.91 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.45/6.91 = ( divide_divide_nat @ M @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2894_div__eq__dividend__iff,axiom,
% 12.45/6.91 ! [M: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.91 => ( ( ( divide_divide_nat @ M @ N )
% 12.45/6.91 = M )
% 12.45/6.91 = ( N = one_one_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2895_div__less__dividend,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ one_one_nat @ N )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.91 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2896_mod__le__divisor,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2897_div__less__mono,axiom,
% 12.45/6.91 ! [A3: nat,B4: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ A3 @ B4 )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ( ( modulo_modulo_nat @ A3 @ N )
% 12.45/6.91 = zero_zero_nat )
% 12.45/6.91 => ( ( ( modulo_modulo_nat @ B4 @ N )
% 12.45/6.91 = zero_zero_nat )
% 12.45/6.91 => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N ) @ ( divide_divide_nat @ B4 @ N ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2898_evenE,axiom,
% 12.45/6.91 ! [A: code_integer] :
% 12.45/6.91 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ~ ! [B5: code_integer] :
% 12.45/6.91 ( A
% 12.45/6.91 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2899_evenE,axiom,
% 12.45/6.91 ! [A: nat] :
% 12.45/6.91 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ~ ! [B5: nat] :
% 12.45/6.91 ( A
% 12.45/6.91 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2900_evenE,axiom,
% 12.45/6.91 ! [A: int] :
% 12.45/6.91 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ~ ! [B5: int] :
% 12.45/6.91 ( A
% 12.45/6.91 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2901_odd__one,axiom,
% 12.45/6.91 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2902_odd__one,axiom,
% 12.45/6.91 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2903_odd__one,axiom,
% 12.45/6.91 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2904_odd__even__add,axiom,
% 12.45/6.91 ! [A: code_integer,B2: code_integer] :
% 12.45/6.91 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 )
% 12.45/6.91 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2905_odd__even__add,axiom,
% 12.45/6.91 ! [A: nat,B2: nat] :
% 12.45/6.91 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 12.45/6.91 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2906_odd__even__add,axiom,
% 12.45/6.91 ! [A: int,B2: int] :
% 12.45/6.91 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 )
% 12.45/6.91 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2907_bit__eq__rec,axiom,
% 12.45/6.91 ( ( ^ [Y6: code_integer,Z4: code_integer] : ( Y6 = Z4 ) )
% 12.45/6.91 = ( ^ [A4: code_integer,B3: code_integer] :
% 12.45/6.91 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 )
% 12.45/6.91 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) )
% 12.45/6.91 & ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.91 = ( divide6298287555418463151nteger @ B3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2908_bit__eq__rec,axiom,
% 12.45/6.91 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 12.45/6.91 = ( ^ [A4: nat,B3: nat] :
% 12.45/6.91 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 )
% 12.45/6.91 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) )
% 12.45/6.91 & ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 = ( divide_divide_nat @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2909_bit__eq__rec,axiom,
% 12.45/6.91 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 12.45/6.91 = ( ^ [A4: int,B3: int] :
% 12.45/6.91 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 )
% 12.45/6.91 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) )
% 12.45/6.91 & ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.91 = ( divide_divide_int @ B3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2910_even__mask__div__iff,axiom,
% 12.45/6.91 ! [M: nat,N: nat] :
% 12.45/6.91 ( ( 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 ) ) )
% 12.45/6.91 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 = zero_z3403309356797280102nteger )
% 12.45/6.91 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2911_even__mask__div__iff,axiom,
% 12.45/6.91 ! [M: nat,N: nat] :
% 12.45/6.91 ( ( 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 ) ) )
% 12.45/6.91 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 = zero_zero_nat )
% 12.45/6.91 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2912_even__mask__div__iff,axiom,
% 12.45/6.91 ! [M: nat,N: nat] :
% 12.45/6.91 ( ( 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 ) ) )
% 12.45/6.91 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 = zero_zero_int )
% 12.45/6.91 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2913_dvd__minus__add,axiom,
% 12.45/6.91 ! [Q2: nat,N: nat,R2: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ Q2 @ N )
% 12.45/6.91 => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M ) )
% 12.45/6.91 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
% 12.45/6.91 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2914_power__dvd__imp__le,axiom,
% 12.45/6.91 ! [I: nat,M: nat,N: nat] :
% 12.45/6.91 ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 12.45/6.91 => ( ( ord_less_nat @ one_one_nat @ I )
% 12.45/6.91 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2915_mod__nat__eqI,axiom,
% 12.45/6.91 ! [R2: nat,N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ R2 @ N )
% 12.45/6.91 => ( ( ord_less_eq_nat @ R2 @ M )
% 12.45/6.91 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 12.45/6.91 => ( ( modulo_modulo_nat @ M @ N )
% 12.45/6.91 = R2 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2916_mult__less__cancel__right2,axiom,
% 12.45/6.91 ! [A: real,C: real] :
% 12.45/6.91 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 12.45/6.91 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ A @ one_one_real ) )
% 12.45/6.91 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2917_mult__less__cancel__right2,axiom,
% 12.45/6.91 ! [A: rat,C: rat] :
% 12.45/6.91 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 12.45/6.91 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ A @ one_one_rat ) )
% 12.45/6.91 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2918_mult__less__cancel__right2,axiom,
% 12.45/6.91 ! [A: int,C: int] :
% 12.45/6.91 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 12.45/6.91 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ A @ one_one_int ) )
% 12.45/6.91 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2919_mult__less__cancel__right1,axiom,
% 12.45/6.91 ! [C: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ C @ ( times_times_real @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ one_one_real @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2920_mult__less__cancel__right1,axiom,
% 12.45/6.91 ! [C: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ C @ ( times_times_rat @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ one_one_rat @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2921_mult__less__cancel__right1,axiom,
% 12.45/6.91 ! [C: int,B2: int] :
% 12.45/6.91 ( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ one_one_int @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2922_mult__less__cancel__left2,axiom,
% 12.45/6.91 ! [C: real,A: real] :
% 12.45/6.91 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 12.45/6.91 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ A @ one_one_real ) )
% 12.45/6.91 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2923_mult__less__cancel__left2,axiom,
% 12.45/6.91 ! [C: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 12.45/6.91 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ A @ one_one_rat ) )
% 12.45/6.91 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2924_mult__less__cancel__left2,axiom,
% 12.45/6.91 ! [C: int,A: int] :
% 12.45/6.91 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 12.45/6.91 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ A @ one_one_int ) )
% 12.45/6.91 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2925_mult__less__cancel__left1,axiom,
% 12.45/6.91 ! [C: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ C @ ( times_times_real @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ one_one_real @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2926_mult__less__cancel__left1,axiom,
% 12.45/6.91 ! [C: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ one_one_rat @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2927_mult__less__cancel__left1,axiom,
% 12.45/6.91 ! [C: int,B2: int] :
% 12.45/6.91 ( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_int @ one_one_int @ B2 ) )
% 12.45/6.91 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2928_mult__le__cancel__right2,axiom,
% 12.45/6.91 ! [A: real,C: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_eq_real @ A @ one_one_real ) )
% 12.45/6.91 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2929_mult__le__cancel__right2,axiom,
% 12.45/6.91 ! [A: rat,C: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 12.45/6.91 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2930_mult__le__cancel__right2,axiom,
% 12.45/6.91 ! [A: int,C: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 12.45/6.91 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_eq_int @ A @ one_one_int ) )
% 12.45/6.91 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2931_mult__le__cancel__right1,axiom,
% 12.45/6.91 ! [C: real,B2: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ C @ ( times_times_real @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_eq_real @ one_one_real @ B2 ) )
% 12.45/6.91 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2932_mult__le__cancel__right1,axiom,
% 12.45/6.91 ! [C: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
% 12.45/6.91 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2933_mult__le__cancel__right1,axiom,
% 12.45/6.91 ! [C: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_eq_int @ one_one_int @ B2 ) )
% 12.45/6.91 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2934_mult__le__cancel__left2,axiom,
% 12.45/6.91 ! [C: real,A: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_eq_real @ A @ one_one_real ) )
% 12.45/6.91 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2935_mult__le__cancel__left2,axiom,
% 12.45/6.91 ! [C: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 12.45/6.91 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2936_mult__le__cancel__left2,axiom,
% 12.45/6.91 ! [C: int,A: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 12.45/6.91 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_eq_int @ A @ one_one_int ) )
% 12.45/6.91 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2937_mult__le__cancel__left1,axiom,
% 12.45/6.91 ! [C: real,B2: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_eq_real @ one_one_real @ B2 ) )
% 12.45/6.91 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2938_mult__le__cancel__left1,axiom,
% 12.45/6.91 ! [C: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
% 12.45/6.91 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2939_mult__le__cancel__left1,axiom,
% 12.45/6.91 ! [C: int,B2: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
% 12.45/6.91 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.45/6.91 => ( ord_less_eq_int @ one_one_int @ B2 ) )
% 12.45/6.91 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.45/6.91 => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2940_field__le__mult__one__interval,axiom,
% 12.45/6.91 ! [X4: real,Y3: real] :
% 12.45/6.91 ( ! [Z2: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 12.45/6.91 => ( ( ord_less_real @ Z2 @ one_one_real )
% 12.45/6.91 => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X4 ) @ Y3 ) ) )
% 12.45/6.91 => ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2941_field__le__mult__one__interval,axiom,
% 12.45/6.91 ! [X4: rat,Y3: rat] :
% 12.45/6.91 ( ! [Z2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 12.45/6.91 => ( ( ord_less_rat @ Z2 @ one_one_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X4 ) @ Y3 ) ) )
% 12.45/6.91 => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2942_divide__left__mono__neg,axiom,
% 12.45/6.91 ! [A: real,B2: real,C: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.91 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2943_divide__left__mono__neg,axiom,
% 12.45/6.91 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.91 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2944_mult__imp__le__div__pos,axiom,
% 12.45/6.91 ! [Y3: real,Z: real,X4: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X4 )
% 12.45/6.91 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2945_mult__imp__le__div__pos,axiom,
% 12.45/6.91 ! [Y3: rat,Z: rat,X4: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X4 )
% 12.45/6.91 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2946_mult__imp__div__pos__le,axiom,
% 12.45/6.91 ! [Y3: real,X4: real,Z: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_real @ X4 @ ( times_times_real @ Z @ Y3 ) )
% 12.45/6.91 => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2947_mult__imp__div__pos__le,axiom,
% 12.45/6.91 ! [Y3: rat,X4: rat,Z: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ X4 @ ( times_times_rat @ Z @ Y3 ) )
% 12.45/6.91 => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ Z ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2948_pos__le__divide__eq,axiom,
% 12.45/6.91 ! [C: real,A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.91 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2949_pos__le__divide__eq,axiom,
% 12.45/6.91 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.91 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2950_pos__divide__le__eq,axiom,
% 12.45/6.91 ! [C: real,B2: real,A: real] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 12.45/6.91 = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2951_pos__divide__le__eq,axiom,
% 12.45/6.91 ! [C: rat,B2: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 12.45/6.91 = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2952_neg__le__divide__eq,axiom,
% 12.45/6.91 ! [C: real,A: real,B2: real] :
% 12.45/6.91 ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.91 = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2953_neg__le__divide__eq,axiom,
% 12.45/6.91 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.91 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.91 = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2954_neg__divide__le__eq,axiom,
% 12.45/6.91 ! [C: real,B2: real,A: real] :
% 12.45/6.91 ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 12.45/6.91 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2955_neg__divide__le__eq,axiom,
% 12.45/6.91 ! [C: rat,B2: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 12.45/6.91 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2956_divide__left__mono,axiom,
% 12.45/6.91 ! [B2: real,A: real,C: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ B2 @ A )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.91 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2957_divide__left__mono,axiom,
% 12.45/6.91 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ B2 @ A )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.91 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2958_le__divide__eq,axiom,
% 12.45/6.91 ! [A: real,B2: real,C: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2959_le__divide__eq,axiom,
% 12.45/6.91 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2960_divide__le__eq,axiom,
% 12.45/6.91 ! [B2: real,C: real,A: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2961_divide__le__eq,axiom,
% 12.45/6.91 ! [B2: rat,C: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2962_le__divide__eq__1,axiom,
% 12.45/6.91 ! [B2: real,A: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.91 & ( ord_less_eq_real @ A @ B2 ) )
% 12.45/6.91 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.91 & ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2963_le__divide__eq__1,axiom,
% 12.45/6.91 ! [B2: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.91 & ( ord_less_eq_rat @ A @ B2 ) )
% 12.45/6.91 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.91 & ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2964_divide__le__eq__1,axiom,
% 12.45/6.91 ! [B2: real,A: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.91 & ( ord_less_eq_real @ B2 @ A ) )
% 12.45/6.91 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.91 & ( ord_less_eq_real @ A @ B2 ) )
% 12.45/6.91 | ( A = zero_zero_real ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2965_divide__le__eq__1,axiom,
% 12.45/6.91 ! [B2: rat,A: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.91 & ( ord_less_eq_rat @ B2 @ A ) )
% 12.45/6.91 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.91 & ( ord_less_eq_rat @ A @ B2 ) )
% 12.45/6.91 | ( A = zero_zero_rat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2966_convex__bound__le,axiom,
% 12.45/6.91 ! [X4: real,A: real,Y3: real,U: real,V: real] :
% 12.45/6.91 ( ( ord_less_eq_real @ X4 @ A )
% 12.45/6.91 => ( ( ord_less_eq_real @ Y3 @ A )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 12.45/6.91 => ( ( ( plus_plus_real @ U @ V )
% 12.45/6.91 = one_one_real )
% 12.45/6.91 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2967_convex__bound__le,axiom,
% 12.45/6.91 ! [X4: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ X4 @ A )
% 12.45/6.91 => ( ( ord_less_eq_rat @ Y3 @ A )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 12.45/6.91 => ( ( ( plus_plus_rat @ U @ V )
% 12.45/6.91 = one_one_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X4 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2968_convex__bound__le,axiom,
% 12.45/6.91 ! [X4: int,A: int,Y3: int,U: int,V: int] :
% 12.45/6.91 ( ( ord_less_eq_int @ X4 @ A )
% 12.45/6.91 => ( ( ord_less_eq_int @ Y3 @ A )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 12.45/6.91 => ( ( ( plus_plus_int @ U @ V )
% 12.45/6.91 = one_one_int )
% 12.45/6.91 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2969_divide__less__eq__numeral_I1_J,axiom,
% 12.45/6.91 ! [B2: real,C: real,W: num] :
% 12.45/6.91 ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2970_divide__less__eq__numeral_I1_J,axiom,
% 12.45/6.91 ! [B2: rat,C: rat,W: num] :
% 12.45/6.91 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2971_less__divide__eq__numeral_I1_J,axiom,
% 12.45/6.91 ! [W: num,B2: real,C: real] :
% 12.45/6.91 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.91 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.91 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2972_less__divide__eq__numeral_I1_J,axiom,
% 12.45/6.91 ! [W: num,B2: rat,C: rat] :
% 12.45/6.91 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.91 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.91 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 12.45/6.91 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.91 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2973_frac__le__eq,axiom,
% 12.45/6.91 ! [Y3: real,Z: real,X4: real,W: real] :
% 12.45/6.91 ( ( Y3 != zero_zero_real )
% 12.45/6.91 => ( ( Z != zero_zero_real )
% 12.45/6.91 => ( ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 12.45/6.91 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2974_frac__le__eq,axiom,
% 12.45/6.91 ! [Y3: rat,Z: rat,X4: rat,W: rat] :
% 12.45/6.91 ( ( Y3 != zero_zero_rat )
% 12.45/6.91 => ( ( Z != zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 12.45/6.91 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2975_frac__less__eq,axiom,
% 12.45/6.91 ! [Y3: real,Z: real,X4: real,W: real] :
% 12.45/6.91 ( ( Y3 != zero_zero_real )
% 12.45/6.91 => ( ( Z != zero_zero_real )
% 12.45/6.91 => ( ( ord_less_real @ ( divide_divide_real @ X4 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 12.45/6.91 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2976_frac__less__eq,axiom,
% 12.45/6.91 ! [Y3: rat,Z: rat,X4: rat,W: rat] :
% 12.45/6.91 ( ( Y3 != zero_zero_rat )
% 12.45/6.91 => ( ( Z != zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 12.45/6.91 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2977_power__Suc__less,axiom,
% 12.45/6.91 ! [A: real,N: nat] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_real @ A @ one_one_real )
% 12.45/6.91 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2978_power__Suc__less,axiom,
% 12.45/6.91 ! [A: rat,N: nat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_rat @ A @ one_one_rat )
% 12.45/6.91 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2979_power__Suc__less,axiom,
% 12.45/6.91 ! [A: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ A @ one_one_nat )
% 12.45/6.91 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2980_power__Suc__less,axiom,
% 12.45/6.91 ! [A: int,N: nat] :
% 12.45/6.91 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_int @ A @ one_one_int )
% 12.45/6.91 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2981_power__Suc__le__self,axiom,
% 12.45/6.91 ! [A: real,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_eq_real @ A @ one_one_real )
% 12.45/6.91 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2982_power__Suc__le__self,axiom,
% 12.45/6.91 ! [A: rat,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2983_power__Suc__le__self,axiom,
% 12.45/6.91 ! [A: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 12.45/6.91 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2984_power__Suc__le__self,axiom,
% 12.45/6.91 ! [A: int,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_eq_int @ A @ one_one_int )
% 12.45/6.91 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2985_power__Suc__less__one,axiom,
% 12.45/6.91 ! [A: real,N: nat] :
% 12.45/6.91 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_real @ A @ one_one_real )
% 12.45/6.91 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2986_power__Suc__less__one,axiom,
% 12.45/6.91 ! [A: rat,N: nat] :
% 12.45/6.91 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_rat @ A @ one_one_rat )
% 12.45/6.91 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2987_power__Suc__less__one,axiom,
% 12.45/6.91 ! [A: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ A @ one_one_nat )
% 12.45/6.91 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2988_power__Suc__less__one,axiom,
% 12.45/6.91 ! [A: int,N: nat] :
% 12.45/6.91 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_int @ A @ one_one_int )
% 12.45/6.91 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2989_power__strict__decreasing,axiom,
% 12.45/6.91 ! [N: nat,N3: nat,A: real] :
% 12.45/6.91 ( ( ord_less_nat @ N @ N3 )
% 12.45/6.91 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_real @ A @ one_one_real )
% 12.45/6.91 => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2990_power__strict__decreasing,axiom,
% 12.45/6.91 ! [N: nat,N3: nat,A: rat] :
% 12.45/6.91 ( ( ord_less_nat @ N @ N3 )
% 12.45/6.91 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_rat @ A @ one_one_rat )
% 12.45/6.91 => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2991_power__strict__decreasing,axiom,
% 12.45/6.91 ! [N: nat,N3: nat,A: nat] :
% 12.45/6.91 ( ( ord_less_nat @ N @ N3 )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ A @ one_one_nat )
% 12.45/6.91 => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2992_power__strict__decreasing,axiom,
% 12.45/6.91 ! [N: nat,N3: nat,A: int] :
% 12.45/6.91 ( ( ord_less_nat @ N @ N3 )
% 12.45/6.91 => ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_int @ A @ one_one_int )
% 12.45/6.91 => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2993_power__decreasing,axiom,
% 12.45/6.91 ! [N: nat,N3: nat,A: real] :
% 12.45/6.91 ( ( ord_less_eq_nat @ N @ N3 )
% 12.45/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.91 => ( ( ord_less_eq_real @ A @ one_one_real )
% 12.45/6.91 => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2994_power__decreasing,axiom,
% 12.45/6.91 ! [N: nat,N3: nat,A: rat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ N @ N3 )
% 12.45/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.91 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 12.45/6.91 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2995_power__decreasing,axiom,
% 12.45/6.91 ! [N: nat,N3: nat,A: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ N @ N3 )
% 12.45/6.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.91 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 12.45/6.91 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2996_power__decreasing,axiom,
% 12.45/6.91 ! [N: nat,N3: nat,A: int] :
% 12.45/6.91 ( ( ord_less_eq_nat @ N @ N3 )
% 12.45/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.91 => ( ( ord_less_eq_int @ A @ one_one_int )
% 12.45/6.91 => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2997_zero__power2,axiom,
% 12.45/6.91 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 = zero_zero_rat ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2998_zero__power2,axiom,
% 12.45/6.91 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 = zero_zero_nat ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_2999_zero__power2,axiom,
% 12.45/6.91 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 = zero_zero_real ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3000_zero__power2,axiom,
% 12.45/6.91 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 = zero_zero_int ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3001_zero__power2,axiom,
% 12.45/6.91 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 = zero_zero_complex ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3002_self__le__power,axiom,
% 12.45/6.91 ! [A: real,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_real @ one_one_real @ A )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3003_self__le__power,axiom,
% 12.45/6.91 ! [A: rat,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3004_self__le__power,axiom,
% 12.45/6.91 ! [A: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3005_self__le__power,axiom,
% 12.45/6.91 ! [A: int,N: nat] :
% 12.45/6.91 ( ( ord_less_eq_int @ one_one_int @ A )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3006_one__less__power,axiom,
% 12.45/6.91 ! [A: real,N: nat] :
% 12.45/6.91 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3007_one__less__power,axiom,
% 12.45/6.91 ! [A: rat,N: nat] :
% 12.45/6.91 ( ( ord_less_rat @ one_one_rat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3008_one__less__power,axiom,
% 12.45/6.91 ! [A: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ one_one_nat @ A )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3009_one__less__power,axiom,
% 12.45/6.91 ! [A: int,N: nat] :
% 12.45/6.91 ( ( ord_less_int @ one_one_int @ A )
% 12.45/6.91 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3010_numeral__2__eq__2,axiom,
% 12.45/6.91 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 12.45/6.91 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3011_pos2,axiom,
% 12.45/6.91 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3012_power__diff,axiom,
% 12.45/6.91 ! [A: complex,N: nat,M: nat] :
% 12.45/6.91 ( ( A != zero_zero_complex )
% 12.45/6.91 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.91 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.91 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3013_power__diff,axiom,
% 12.45/6.91 ! [A: real,N: nat,M: nat] :
% 12.45/6.91 ( ( A != zero_zero_real )
% 12.45/6.91 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.91 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.91 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3014_power__diff,axiom,
% 12.45/6.91 ! [A: rat,N: nat,M: nat] :
% 12.45/6.91 ( ( A != zero_zero_rat )
% 12.45/6.91 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.91 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.91 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3015_power__diff,axiom,
% 12.45/6.91 ! [A: nat,N: nat,M: nat] :
% 12.45/6.91 ( ( A != zero_zero_nat )
% 12.45/6.91 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.91 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.91 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3016_power__diff,axiom,
% 12.45/6.91 ! [A: int,N: nat,M: nat] :
% 12.45/6.91 ( ( A != zero_zero_int )
% 12.45/6.91 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.91 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.91 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3017_even__mult__exp__div__exp__iff,axiom,
% 12.45/6.91 ! [A: code_integer,M: nat,N: nat] :
% 12.45/6.91 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/6.91 = ( ( ord_less_nat @ N @ M )
% 12.45/6.91 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 = zero_z3403309356797280102nteger )
% 12.45/6.91 | ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.91 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3018_even__mult__exp__div__exp__iff,axiom,
% 12.45/6.91 ! [A: nat,M: nat,N: nat] :
% 12.45/6.91 ( ( 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 ) ) )
% 12.45/6.91 = ( ( ord_less_nat @ N @ M )
% 12.45/6.91 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 = zero_zero_nat )
% 12.45/6.91 | ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.91 & ( 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 ) ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3019_even__mult__exp__div__exp__iff,axiom,
% 12.45/6.91 ! [A: int,M: nat,N: nat] :
% 12.45/6.91 ( ( 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 ) ) )
% 12.45/6.91 = ( ( ord_less_nat @ N @ M )
% 12.45/6.91 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 = zero_zero_int )
% 12.45/6.91 | ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.91 & ( 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 ) ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3020_div__if,axiom,
% 12.45/6.91 ( divide_divide_nat
% 12.45/6.91 = ( ^ [M6: nat,N2: nat] :
% 12.45/6.91 ( if_nat
% 12.45/6.91 @ ( ( ord_less_nat @ M6 @ N2 )
% 12.45/6.91 | ( N2 = zero_zero_nat ) )
% 12.45/6.91 @ zero_zero_nat
% 12.45/6.91 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3021_div__geq,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ~ ( ord_less_nat @ M @ N )
% 12.45/6.91 => ( ( divide_divide_nat @ M @ N )
% 12.45/6.91 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3022_Suc__pred_H,axiom,
% 12.45/6.91 ! [N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( N
% 12.45/6.91 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3023_Suc__diff__eq__diff__pred,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 12.45/6.91 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3024_less__eq__div__iff__mult__less__eq,axiom,
% 12.45/6.91 ! [Q2: nat,M: nat,N: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 12.45/6.91 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
% 12.45/6.91 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3025_add__eq__if,axiom,
% 12.45/6.91 ( plus_plus_nat
% 12.45/6.91 = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3026_dividend__less__times__div,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3027_dividend__less__div__times,axiom,
% 12.45/6.91 ! [N: nat,M: nat] :
% 12.45/6.91 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.91 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3028_split__div,axiom,
% 12.45/6.91 ! [P: nat > $o,M: nat,N: nat] :
% 12.45/6.91 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 12.45/6.91 = ( ( ( N = zero_zero_nat )
% 12.45/6.91 => ( P @ zero_zero_nat ) )
% 12.45/6.91 & ( ( N != zero_zero_nat )
% 12.45/6.91 => ! [I2: nat,J3: nat] :
% 12.45/6.91 ( ( ord_less_nat @ J3 @ N )
% 12.45/6.91 => ( ( M
% 12.45/6.91 = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
% 12.45/6.91 => ( P @ I2 ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3029_mult__eq__if,axiom,
% 12.45/6.91 ( times_times_nat
% 12.45/6.91 = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3030_split__mod,axiom,
% 12.45/6.91 ! [P: nat > $o,M: nat,N: nat] :
% 12.45/6.91 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 12.45/6.91 = ( ( ( N = zero_zero_nat )
% 12.45/6.91 => ( P @ M ) )
% 12.45/6.91 & ( ( N != zero_zero_nat )
% 12.45/6.91 => ! [I2: nat,J3: nat] :
% 12.45/6.91 ( ( ord_less_nat @ J3 @ N )
% 12.45/6.91 => ( ( M
% 12.45/6.91 = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
% 12.45/6.91 => ( P @ J3 ) ) ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3031_even__two__times__div__two,axiom,
% 12.45/6.91 ! [A: code_integer] :
% 12.45/6.91 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 12.45/6.91 = A ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3032_even__two__times__div__two,axiom,
% 12.45/6.91 ! [A: nat] :
% 12.45/6.91 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.91 = A ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3033_even__two__times__div__two,axiom,
% 12.45/6.91 ! [A: int] :
% 12.45/6.91 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.91 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 12.45/6.91 = A ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3034_odd__iff__mod__2__eq__one,axiom,
% 12.45/6.91 ! [A: nat] :
% 12.45/6.91 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 12.45/6.91 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.91 = one_one_nat ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3035_odd__iff__mod__2__eq__one,axiom,
% 12.45/6.91 ! [A: int] :
% 12.45/6.91 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.45/6.91 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.91 = one_one_int ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3036_odd__iff__mod__2__eq__one,axiom,
% 12.45/6.91 ! [A: code_integer] :
% 12.45/6.91 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 12.45/6.91 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.91 = one_one_Code_integer ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3037_power__mono__odd,axiom,
% 12.45/6.91 ! [N: nat,A: real,B2: real] :
% 12.45/6.91 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 => ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.91 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3038_power__mono__odd,axiom,
% 12.45/6.91 ! [N: nat,A: rat,B2: rat] :
% 12.45/6.91 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 => ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.91 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 12.45/6.91
% 12.45/6.91 thf(fact_3039_power__mono__odd,axiom,
% 12.45/6.91 ! [N: nat,A: int,B2: int] :
% 12.45/6.91 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.91 => ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.91 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3040_dvd__power__iff__le,axiom,
% 12.45/6.92 ! [K: nat,M: nat,N: nat] :
% 12.45/6.92 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 12.45/6.92 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 12.45/6.92 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3041_convex__bound__lt,axiom,
% 12.45/6.92 ! [X4: real,A: real,Y3: real,U: real,V: real] :
% 12.45/6.92 ( ( ord_less_real @ X4 @ A )
% 12.45/6.92 => ( ( ord_less_real @ Y3 @ A )
% 12.45/6.92 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 12.45/6.92 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 12.45/6.92 => ( ( ( plus_plus_real @ U @ V )
% 12.45/6.92 = one_one_real )
% 12.45/6.92 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3042_convex__bound__lt,axiom,
% 12.45/6.92 ! [X4: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X4 @ A )
% 12.45/6.92 => ( ( ord_less_rat @ Y3 @ A )
% 12.45/6.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 12.45/6.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 12.45/6.92 => ( ( ( plus_plus_rat @ U @ V )
% 12.45/6.92 = one_one_rat )
% 12.45/6.92 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X4 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3043_convex__bound__lt,axiom,
% 12.45/6.92 ! [X4: int,A: int,Y3: int,U: int,V: int] :
% 12.45/6.92 ( ( ord_less_int @ X4 @ A )
% 12.45/6.92 => ( ( ord_less_int @ Y3 @ A )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 12.45/6.92 => ( ( ( plus_plus_int @ U @ V )
% 12.45/6.92 = one_one_int )
% 12.45/6.92 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3044_le__divide__eq__numeral_I1_J,axiom,
% 12.45/6.92 ! [W: num,B2: real,C: real] :
% 12.45/6.92 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.92 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.92 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 12.45/6.92 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.92 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.92 => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 12.45/6.92 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.92 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3045_le__divide__eq__numeral_I1_J,axiom,
% 12.45/6.92 ! [W: num,B2: rat,C: rat] :
% 12.45/6.92 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.92 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.92 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 12.45/6.92 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.92 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.92 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 12.45/6.92 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.92 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3046_divide__le__eq__numeral_I1_J,axiom,
% 12.45/6.92 ! [B2: real,C: real,W: num] :
% 12.45/6.92 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
% 12.45/6.92 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.92 => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 12.45/6.92 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.92 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.92 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 12.45/6.92 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.92 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3047_divide__le__eq__numeral_I1_J,axiom,
% 12.45/6.92 ! [B2: rat,C: rat,W: num] :
% 12.45/6.92 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
% 12.45/6.92 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.92 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 12.45/6.92 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.92 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.92 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 12.45/6.92 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.92 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3048_half__gt__zero,axiom,
% 12.45/6.92 ! [A: real] :
% 12.45/6.92 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.92 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3049_half__gt__zero,axiom,
% 12.45/6.92 ! [A: rat] :
% 12.45/6.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.92 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3050_half__gt__zero__iff,axiom,
% 12.45/6.92 ! [A: real] :
% 12.45/6.92 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.92 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3051_half__gt__zero__iff,axiom,
% 12.45/6.92 ! [A: rat] :
% 12.45/6.92 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3052_scaling__mono,axiom,
% 12.45/6.92 ! [U: real,V: real,R2: real,S: real] :
% 12.45/6.92 ( ( ord_less_eq_real @ U @ V )
% 12.45/6.92 => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 12.45/6.92 => ( ( ord_less_eq_real @ R2 @ S )
% 12.45/6.92 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3053_scaling__mono,axiom,
% 12.45/6.92 ! [U: rat,V: rat,R2: rat,S: rat] :
% 12.45/6.92 ( ( ord_less_eq_rat @ U @ V )
% 12.45/6.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 12.45/6.92 => ( ( ord_less_eq_rat @ R2 @ S )
% 12.45/6.92 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3054_zero__le__power2,axiom,
% 12.45/6.92 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3055_zero__le__power2,axiom,
% 12.45/6.92 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3056_zero__le__power2,axiom,
% 12.45/6.92 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3057_power2__eq__imp__eq,axiom,
% 12.45/6.92 ! [X4: real,Y3: real] :
% 12.45/6.92 ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.92 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.92 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.92 => ( X4 = Y3 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3058_power2__eq__imp__eq,axiom,
% 12.45/6.92 ! [X4: rat,Y3: rat] :
% 12.45/6.92 ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.92 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.45/6.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.45/6.92 => ( X4 = Y3 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3059_power2__eq__imp__eq,axiom,
% 12.45/6.92 ! [X4: nat,Y3: nat] :
% 12.45/6.92 ( ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.92 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
% 12.45/6.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 12.45/6.92 => ( X4 = Y3 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3060_power2__eq__imp__eq,axiom,
% 12.45/6.92 ! [X4: int,Y3: int] :
% 12.45/6.92 ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.92 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.92 => ( X4 = Y3 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3061_power2__le__imp__le,axiom,
% 12.45/6.92 ! [X4: real,Y3: real] :
% 12.45/6.92 ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.92 => ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3062_power2__le__imp__le,axiom,
% 12.45/6.92 ! [X4: rat,Y3: rat] :
% 12.45/6.92 ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.45/6.92 => ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3063_power2__le__imp__le,axiom,
% 12.45/6.92 ! [X4: nat,Y3: nat] :
% 12.45/6.92 ( ( ord_less_eq_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 12.45/6.92 => ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3064_power2__le__imp__le,axiom,
% 12.45/6.92 ! [X4: int,Y3: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.92 => ( ord_less_eq_int @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3065_power2__less__0,axiom,
% 12.45/6.92 ! [A: real] :
% 12.45/6.92 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3066_power2__less__0,axiom,
% 12.45/6.92 ! [A: rat] :
% 12.45/6.92 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3067_power2__less__0,axiom,
% 12.45/6.92 ! [A: int] :
% 12.45/6.92 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3068_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 12.45/6.92 ! [C: code_integer,A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 12.45/6.92 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 12.45/6.92 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3069_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 12.45/6.92 ! [C: nat,A: nat,B2: nat] :
% 12.45/6.92 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.45/6.92 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 12.45/6.92 = ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3070_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 12.45/6.92 ! [C: int,A: int,B2: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.45/6.92 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
% 12.45/6.92 = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3071_exp__add__not__zero__imp__left,axiom,
% 12.45/6.92 ! [M: nat,N: nat] :
% 12.45/6.92 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 12.45/6.92 != zero_zero_nat )
% 12.45/6.92 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 12.45/6.92 != zero_zero_nat ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3072_exp__add__not__zero__imp__left,axiom,
% 12.45/6.92 ! [M: nat,N: nat] :
% 12.45/6.92 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 12.45/6.92 != zero_zero_int )
% 12.45/6.92 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 12.45/6.92 != zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3073_exp__add__not__zero__imp__right,axiom,
% 12.45/6.92 ! [M: nat,N: nat] :
% 12.45/6.92 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 12.45/6.92 != zero_zero_nat )
% 12.45/6.92 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.92 != zero_zero_nat ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3074_exp__add__not__zero__imp__right,axiom,
% 12.45/6.92 ! [M: nat,N: nat] :
% 12.45/6.92 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 12.45/6.92 != zero_zero_int )
% 12.45/6.92 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.45/6.92 != zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3075_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 12.45/6.92 ! [N: nat,M: nat] :
% 12.45/6.92 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.92 != zero_zero_nat )
% 12.45/6.92 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 12.45/6.92 != zero_zero_nat ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3076_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 12.45/6.92 ! [N: nat,M: nat] :
% 12.45/6.92 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.45/6.92 != zero_zero_int )
% 12.45/6.92 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 12.45/6.92 != zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3077_power__diff__power__eq,axiom,
% 12.45/6.92 ! [A: nat,N: nat,M: nat] :
% 12.45/6.92 ( ( A != zero_zero_nat )
% 12.45/6.92 => ( ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.92 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 12.45/6.92 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 12.45/6.92 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.45/6.92 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 12.45/6.92 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3078_power__diff__power__eq,axiom,
% 12.45/6.92 ! [A: int,N: nat,M: nat] :
% 12.45/6.92 ( ( A != zero_zero_int )
% 12.45/6.92 => ( ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.92 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 12.45/6.92 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 12.45/6.92 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.45/6.92 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 12.45/6.92 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3079_less__2__cases,axiom,
% 12.45/6.92 ! [N: nat] :
% 12.45/6.92 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.92 => ( ( N = zero_zero_nat )
% 12.45/6.92 | ( N
% 12.45/6.92 = ( suc @ zero_zero_nat ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3080_less__2__cases__iff,axiom,
% 12.45/6.92 ! [N: nat] :
% 12.45/6.92 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.92 = ( ( N = zero_zero_nat )
% 12.45/6.92 | ( N
% 12.45/6.92 = ( suc @ zero_zero_nat ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3081_nat__induct2,axiom,
% 12.45/6.92 ! [P: nat > $o,N: nat] :
% 12.45/6.92 ( ( P @ zero_zero_nat )
% 12.45/6.92 => ( ( P @ one_one_nat )
% 12.45/6.92 => ( ! [N4: nat] :
% 12.45/6.92 ( ( P @ N4 )
% 12.45/6.92 => ( P @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.92 => ( P @ N ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3082_power__eq__if,axiom,
% 12.45/6.92 ( power_power_complex
% 12.45/6.92 = ( ^ [P3: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P3 @ ( power_power_complex @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3083_power__eq__if,axiom,
% 12.45/6.92 ( power_power_real
% 12.45/6.92 = ( ^ [P3: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P3 @ ( power_power_real @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3084_power__eq__if,axiom,
% 12.45/6.92 ( power_power_rat
% 12.45/6.92 = ( ^ [P3: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P3 @ ( power_power_rat @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3085_power__eq__if,axiom,
% 12.45/6.92 ( power_power_nat
% 12.45/6.92 = ( ^ [P3: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3086_power__eq__if,axiom,
% 12.45/6.92 ( power_power_int
% 12.45/6.92 = ( ^ [P3: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3087_power__minus__mult,axiom,
% 12.45/6.92 ! [N: nat,A: complex] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 12.45/6.92 = ( power_power_complex @ A @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3088_power__minus__mult,axiom,
% 12.45/6.92 ! [N: nat,A: real] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 12.45/6.92 = ( power_power_real @ A @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3089_power__minus__mult,axiom,
% 12.45/6.92 ! [N: nat,A: rat] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 12.45/6.92 = ( power_power_rat @ A @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3090_power__minus__mult,axiom,
% 12.45/6.92 ! [N: nat,A: nat] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 12.45/6.92 = ( power_power_nat @ A @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3091_power__minus__mult,axiom,
% 12.45/6.92 ! [N: nat,A: int] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 12.45/6.92 = ( power_power_int @ A @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3092_le__div__geq,axiom,
% 12.45/6.92 ! [N: nat,M: nat] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.92 => ( ( divide_divide_nat @ M @ N )
% 12.45/6.92 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3093_split__div_H,axiom,
% 12.45/6.92 ! [P: nat > $o,M: nat,N: nat] :
% 12.45/6.92 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 12.45/6.92 = ( ( ( N = zero_zero_nat )
% 12.45/6.92 & ( P @ zero_zero_nat ) )
% 12.45/6.92 | ? [Q4: nat] :
% 12.45/6.92 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
% 12.45/6.92 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
% 12.45/6.92 & ( P @ Q4 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3094_Suc__times__mod__eq,axiom,
% 12.45/6.92 ! [M: nat,N: nat] :
% 12.45/6.92 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 12.45/6.92 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 12.45/6.92 = one_one_nat ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3095_oddE,axiom,
% 12.45/6.92 ! [A: code_integer] :
% 12.45/6.92 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.92 => ~ ! [B5: code_integer] :
% 12.45/6.92 ( A
% 12.45/6.92 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) @ one_one_Code_integer ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3096_oddE,axiom,
% 12.45/6.92 ! [A: nat] :
% 12.45/6.92 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.92 => ~ ! [B5: nat] :
% 12.45/6.92 ( A
% 12.45/6.92 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) @ one_one_nat ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3097_oddE,axiom,
% 12.45/6.92 ! [A: int] :
% 12.45/6.92 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.92 => ~ ! [B5: int] :
% 12.45/6.92 ( A
% 12.45/6.92 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) @ one_one_int ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3098_power2__less__imp__less,axiom,
% 12.45/6.92 ! [X4: real,Y3: real] :
% 12.45/6.92 ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.92 => ( ord_less_real @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3099_power2__less__imp__less,axiom,
% 12.45/6.92 ! [X4: rat,Y3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.45/6.92 => ( ord_less_rat @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3100_power2__less__imp__less,axiom,
% 12.45/6.92 ! [X4: nat,Y3: nat] :
% 12.45/6.92 ( ( ord_less_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 12.45/6.92 => ( ord_less_nat @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3101_power2__less__imp__less,axiom,
% 12.45/6.92 ! [X4: int,Y3: int] :
% 12.45/6.92 ( ( ord_less_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.92 => ( ord_less_int @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3102_sum__power2__ge__zero,axiom,
% 12.45/6.92 ! [X4: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3103_sum__power2__ge__zero,axiom,
% 12.45/6.92 ! [X4: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3104_sum__power2__ge__zero,axiom,
% 12.45/6.92 ! [X4: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3105_sum__power2__le__zero__iff,axiom,
% 12.45/6.92 ! [X4: real,Y3: real] :
% 12.45/6.92 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 12.45/6.92 = ( ( X4 = zero_zero_real )
% 12.45/6.92 & ( Y3 = zero_zero_real ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3106_sum__power2__le__zero__iff,axiom,
% 12.45/6.92 ! [X4: rat,Y3: rat] :
% 12.45/6.92 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 12.45/6.92 = ( ( X4 = zero_zero_rat )
% 12.45/6.92 & ( Y3 = zero_zero_rat ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3107_sum__power2__le__zero__iff,axiom,
% 12.45/6.92 ! [X4: int,Y3: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 12.45/6.92 = ( ( X4 = zero_zero_int )
% 12.45/6.92 & ( Y3 = zero_zero_int ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3108_not__sum__power2__lt__zero,axiom,
% 12.45/6.92 ! [X4: real,Y3: real] :
% 12.45/6.92 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3109_not__sum__power2__lt__zero,axiom,
% 12.45/6.92 ! [X4: rat,Y3: rat] :
% 12.45/6.92 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3110_not__sum__power2__lt__zero,axiom,
% 12.45/6.92 ! [X4: int,Y3: int] :
% 12.45/6.92 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3111_sum__power2__gt__zero__iff,axiom,
% 12.45/6.92 ! [X4: real,Y3: real] :
% 12.45/6.92 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.92 = ( ( X4 != zero_zero_real )
% 12.45/6.92 | ( Y3 != zero_zero_real ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3112_sum__power2__gt__zero__iff,axiom,
% 12.45/6.92 ! [X4: rat,Y3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.92 = ( ( X4 != zero_zero_rat )
% 12.45/6.92 | ( Y3 != zero_zero_rat ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3113_sum__power2__gt__zero__iff,axiom,
% 12.45/6.92 ! [X4: int,Y3: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.92 = ( ( X4 != zero_zero_int )
% 12.45/6.92 | ( Y3 != zero_zero_int ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3114_divmod__digit__0_I2_J,axiom,
% 12.45/6.92 ! [B2: nat,A: nat] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.45/6.92 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 12.45/6.92 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
% 12.45/6.92 = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3115_divmod__digit__0_I2_J,axiom,
% 12.45/6.92 ! [B2: int,A: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.92 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 12.45/6.92 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
% 12.45/6.92 = ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3116_divmod__digit__0_I2_J,axiom,
% 12.45/6.92 ! [B2: code_integer,A: code_integer] :
% 12.45/6.92 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 12.45/6.92 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 12.45/6.92 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
% 12.45/6.92 = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3117_bits__stable__imp__add__self,axiom,
% 12.45/6.92 ! [A: nat] :
% 12.45/6.92 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.92 = A )
% 12.45/6.92 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.92 = zero_zero_nat ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3118_bits__stable__imp__add__self,axiom,
% 12.45/6.92 ! [A: int] :
% 12.45/6.92 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.92 = A )
% 12.45/6.92 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 12.45/6.92 = zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3119_bits__stable__imp__add__self,axiom,
% 12.45/6.92 ! [A: code_integer] :
% 12.45/6.92 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.92 = A )
% 12.45/6.92 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 12.45/6.92 = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3120_zero__le__even__power_H,axiom,
% 12.45/6.92 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3121_zero__le__even__power_H,axiom,
% 12.45/6.92 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3122_zero__le__even__power_H,axiom,
% 12.45/6.92 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3123_nat__bit__induct,axiom,
% 12.45/6.92 ! [P: nat > $o,N: nat] :
% 12.45/6.92 ( ( P @ zero_zero_nat )
% 12.45/6.92 => ( ! [N4: nat] :
% 12.45/6.92 ( ( P @ N4 )
% 12.45/6.92 => ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 12.45/6.92 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 12.45/6.92 => ( ! [N4: nat] :
% 12.45/6.92 ( ( P @ N4 )
% 12.45/6.92 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 12.45/6.92 => ( P @ N ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3124_div__2__gt__zero,axiom,
% 12.45/6.92 ! [N: nat] :
% 12.45/6.92 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.45/6.92 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3125_Suc__n__div__2__gt__zero,axiom,
% 12.45/6.92 ! [N: nat] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3126_verit__le__mono__div,axiom,
% 12.45/6.92 ! [A3: nat,B4: nat,N: nat] :
% 12.45/6.92 ( ( ord_less_nat @ A3 @ B4 )
% 12.45/6.92 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ord_less_eq_nat
% 12.45/6.92 @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N )
% 12.45/6.92 @ ( if_nat
% 12.45/6.92 @ ( ( modulo_modulo_nat @ B4 @ N )
% 12.45/6.92 = zero_zero_nat )
% 12.45/6.92 @ one_one_nat
% 12.45/6.92 @ zero_zero_nat ) )
% 12.45/6.92 @ ( divide_divide_nat @ B4 @ N ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3127_divmod__digit__0_I1_J,axiom,
% 12.45/6.92 ! [B2: nat,A: nat] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.45/6.92 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 12.45/6.92 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 12.45/6.92 = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3128_divmod__digit__0_I1_J,axiom,
% 12.45/6.92 ! [B2: int,A: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.92 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 12.45/6.92 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 12.45/6.92 = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3129_divmod__digit__0_I1_J,axiom,
% 12.45/6.92 ! [B2: code_integer,A: code_integer] :
% 12.45/6.92 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 12.45/6.92 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 12.45/6.92 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 12.45/6.92 = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3130_odd__0__le__power__imp__0__le,axiom,
% 12.45/6.92 ! [A: real,N: nat] :
% 12.45/6.92 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 12.45/6.92 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3131_odd__0__le__power__imp__0__le,axiom,
% 12.45/6.92 ! [A: rat,N: nat] :
% 12.45/6.92 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 12.45/6.92 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3132_odd__0__le__power__imp__0__le,axiom,
% 12.45/6.92 ! [A: int,N: nat] :
% 12.45/6.92 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 12.45/6.92 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3133_odd__power__less__zero,axiom,
% 12.45/6.92 ! [A: real,N: nat] :
% 12.45/6.92 ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.92 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3134_odd__power__less__zero,axiom,
% 12.45/6.92 ! [A: rat,N: nat] :
% 12.45/6.92 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.92 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3135_odd__power__less__zero,axiom,
% 12.45/6.92 ! [A: int,N: nat] :
% 12.45/6.92 ( ( ord_less_int @ A @ zero_zero_int )
% 12.45/6.92 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3136_even__mask__div__iff_H,axiom,
% 12.45/6.92 ! [M: nat,N: nat] :
% 12.45/6.92 ( ( 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 ) ) )
% 12.45/6.92 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3137_even__mask__div__iff_H,axiom,
% 12.45/6.92 ! [M: nat,N: nat] :
% 12.45/6.92 ( ( 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 ) ) )
% 12.45/6.92 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3138_even__mask__div__iff_H,axiom,
% 12.45/6.92 ! [M: nat,N: nat] :
% 12.45/6.92 ( ( 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 ) ) )
% 12.45/6.92 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3139_mod__double__modulus,axiom,
% 12.45/6.92 ! [M: code_integer,X4: code_integer] :
% 12.45/6.92 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 12.45/6.92 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 12.45/6.92 => ( ( ( modulo364778990260209775nteger @ X4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.92 = ( modulo364778990260209775nteger @ X4 @ M ) )
% 12.45/6.92 | ( ( modulo364778990260209775nteger @ X4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.92 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X4 @ M ) @ M ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3140_mod__double__modulus,axiom,
% 12.45/6.92 ! [M: nat,X4: nat] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
% 12.45/6.92 => ( ( ( modulo_modulo_nat @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.92 = ( modulo_modulo_nat @ X4 @ M ) )
% 12.45/6.92 | ( ( modulo_modulo_nat @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.92 = ( plus_plus_nat @ ( modulo_modulo_nat @ X4 @ M ) @ M ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3141_mod__double__modulus,axiom,
% 12.45/6.92 ! [M: int,X4: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ M )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.92 => ( ( ( modulo_modulo_int @ X4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.92 = ( modulo_modulo_int @ X4 @ M ) )
% 12.45/6.92 | ( ( modulo_modulo_int @ X4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.92 = ( plus_plus_int @ ( modulo_modulo_int @ X4 @ M ) @ M ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3142_divmod__digit__1_I2_J,axiom,
% 12.45/6.92 ! [A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.45/6.92 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 12.45/6.92 => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 12.45/6.92 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 12.45/6.92 = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3143_divmod__digit__1_I2_J,axiom,
% 12.45/6.92 ! [A: nat,B2: nat] :
% 12.45/6.92 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.45/6.92 => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 12.45/6.92 => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 12.45/6.92 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 12.45/6.92 = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3144_divmod__digit__1_I2_J,axiom,
% 12.45/6.92 ! [A: int,B2: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.92 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.92 => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 12.45/6.92 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 12.45/6.92 = ( modulo_modulo_int @ A @ B2 ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3145_even__mod__4__div__2,axiom,
% 12.45/6.92 ! [N: nat] :
% 12.45/6.92 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.45/6.92 = ( suc @ zero_zero_nat ) )
% 12.45/6.92 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3146_pow__divides__pow__iff,axiom,
% 12.45/6.92 ! [N: nat,A: nat,B2: nat] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 12.45/6.92 = ( dvd_dvd_nat @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3147_pow__divides__pow__iff,axiom,
% 12.45/6.92 ! [N: nat,A: int,B2: int] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 12.45/6.92 = ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3148_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 12.45/6.92 ! [X4: nat,N: nat,M: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 12.45/6.92 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.92 => ( ord_less_nat @ ( vEBT_VEBT_low @ X4 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3149_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 12.45/6.92 ! [X4: nat,N: nat,M: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 12.45/6.92 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.92 => ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3150_even__even__mod__4__iff,axiom,
% 12.45/6.92 ! [N: nat] :
% 12.45/6.92 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.92 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3151_double__eq__0__iff,axiom,
% 12.45/6.92 ! [A: real] :
% 12.45/6.92 ( ( ( plus_plus_real @ A @ A )
% 12.45/6.92 = zero_zero_real )
% 12.45/6.92 = ( A = zero_zero_real ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3152_double__eq__0__iff,axiom,
% 12.45/6.92 ! [A: rat] :
% 12.45/6.92 ( ( ( plus_plus_rat @ A @ A )
% 12.45/6.92 = zero_zero_rat )
% 12.45/6.92 = ( A = zero_zero_rat ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3153_double__eq__0__iff,axiom,
% 12.45/6.92 ! [A: int] :
% 12.45/6.92 ( ( ( plus_plus_int @ A @ A )
% 12.45/6.92 = zero_zero_int )
% 12.45/6.92 = ( A = zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3154_inf__period_I3_J,axiom,
% 12.45/6.92 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 12.45/6.92 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 12.45/6.92 => ! [X5: code_integer,K4: code_integer] :
% 12.45/6.92 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
% 12.45/6.92 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3155_inf__period_I3_J,axiom,
% 12.45/6.92 ! [D: real,D4: real,T: real] :
% 12.45/6.92 ( ( dvd_dvd_real @ D @ D4 )
% 12.45/6.92 => ! [X5: real,K4: real] :
% 12.45/6.92 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
% 12.45/6.92 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3156_inf__period_I3_J,axiom,
% 12.45/6.92 ! [D: rat,D4: rat,T: rat] :
% 12.45/6.92 ( ( dvd_dvd_rat @ D @ D4 )
% 12.45/6.92 => ! [X5: rat,K4: rat] :
% 12.45/6.92 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
% 12.45/6.92 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3157_inf__period_I3_J,axiom,
% 12.45/6.92 ! [D: int,D4: int,T: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ D @ D4 )
% 12.45/6.92 => ! [X5: int,K4: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 12.45/6.92 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3158_inf__period_I4_J,axiom,
% 12.45/6.92 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 12.45/6.92 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 12.45/6.92 => ! [X5: code_integer,K4: code_integer] :
% 12.45/6.92 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3159_inf__period_I4_J,axiom,
% 12.45/6.92 ! [D: real,D4: real,T: real] :
% 12.45/6.92 ( ( dvd_dvd_real @ D @ D4 )
% 12.45/6.92 => ! [X5: real,K4: real] :
% 12.45/6.92 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3160_inf__period_I4_J,axiom,
% 12.45/6.92 ! [D: rat,D4: rat,T: rat] :
% 12.45/6.92 ( ( dvd_dvd_rat @ D @ D4 )
% 12.45/6.92 => ! [X5: rat,K4: rat] :
% 12.45/6.92 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3161_inf__period_I4_J,axiom,
% 12.45/6.92 ! [D: int,D4: int,T: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ D @ D4 )
% 12.45/6.92 => ! [X5: int,K4: int] :
% 12.45/6.92 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3162_unity__coeff__ex,axiom,
% 12.45/6.92 ! [P: code_integer > $o,L2: code_integer] :
% 12.45/6.92 ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X ) ) )
% 12.45/6.92 = ( ? [X: code_integer] :
% 12.45/6.92 ( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
% 12.45/6.92 & ( P @ X ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3163_unity__coeff__ex,axiom,
% 12.45/6.92 ! [P: real > $o,L2: real] :
% 12.45/6.92 ( ( ? [X: real] : ( P @ ( times_times_real @ L2 @ X ) ) )
% 12.45/6.92 = ( ? [X: real] :
% 12.45/6.92 ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X @ zero_zero_real ) )
% 12.45/6.92 & ( P @ X ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3164_unity__coeff__ex,axiom,
% 12.45/6.92 ! [P: rat > $o,L2: rat] :
% 12.45/6.92 ( ( ? [X: rat] : ( P @ ( times_times_rat @ L2 @ X ) ) )
% 12.45/6.92 = ( ? [X: rat] :
% 12.45/6.92 ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X @ zero_zero_rat ) )
% 12.45/6.92 & ( P @ X ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3165_unity__coeff__ex,axiom,
% 12.45/6.92 ! [P: nat > $o,L2: nat] :
% 12.45/6.92 ( ( ? [X: nat] : ( P @ ( times_times_nat @ L2 @ X ) ) )
% 12.45/6.92 = ( ? [X: nat] :
% 12.45/6.92 ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X @ zero_zero_nat ) )
% 12.45/6.92 & ( P @ X ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3166_unity__coeff__ex,axiom,
% 12.45/6.92 ! [P: int > $o,L2: int] :
% 12.45/6.92 ( ( ? [X: int] : ( P @ ( times_times_int @ L2 @ X ) ) )
% 12.45/6.92 = ( ? [X: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X @ zero_zero_int ) )
% 12.45/6.92 & ( P @ X ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3167_unset__bit__nonnegative__int__iff,axiom,
% 12.45/6.92 ! [N: nat,K: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 12.45/6.92 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3168_flip__bit__nonnegative__int__iff,axiom,
% 12.45/6.92 ! [N: nat,K: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 12.45/6.92 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3169_set__bit__nonnegative__int__iff,axiom,
% 12.45/6.92 ! [N: nat,K: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 12.45/6.92 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3170_set__bit__negative__int__iff,axiom,
% 12.45/6.92 ! [N: nat,K: int] :
% 12.45/6.92 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 12.45/6.92 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3171_flip__bit__negative__int__iff,axiom,
% 12.45/6.92 ! [N: nat,K: int] :
% 12.45/6.92 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 12.45/6.92 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3172_unset__bit__negative__int__iff,axiom,
% 12.45/6.92 ! [N: nat,K: int] :
% 12.45/6.92 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 12.45/6.92 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3173_zle__add1__eq__le,axiom,
% 12.45/6.92 ! [W: int,Z: int] :
% 12.45/6.92 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 12.45/6.92 = ( ord_less_eq_int @ W @ Z ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3174_zle__diff1__eq,axiom,
% 12.45/6.92 ! [W: int,Z: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 12.45/6.92 = ( ord_less_int @ W @ Z ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3175_mod__pos__pos__trivial,axiom,
% 12.45/6.92 ! [K: int,L2: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.92 => ( ( ord_less_int @ K @ L2 )
% 12.45/6.92 => ( ( modulo_modulo_int @ K @ L2 )
% 12.45/6.92 = K ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3176_mod__neg__neg__trivial,axiom,
% 12.45/6.92 ! [K: int,L2: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 12.45/6.92 => ( ( ord_less_int @ L2 @ K )
% 12.45/6.92 => ( ( modulo_modulo_int @ K @ L2 )
% 12.45/6.92 = K ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3177_le__imp__0__less,axiom,
% 12.45/6.92 ! [Z: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.45/6.92 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3178_split__zmod,axiom,
% 12.45/6.92 ! [P: int > $o,N: int,K: int] :
% 12.45/6.92 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 12.45/6.92 = ( ( ( K = zero_zero_int )
% 12.45/6.92 => ( P @ N ) )
% 12.45/6.92 & ( ( ord_less_int @ zero_zero_int @ K )
% 12.45/6.92 => ! [I2: int,J3: int] :
% 12.45/6.92 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 12.45/6.92 & ( ord_less_int @ J3 @ K )
% 12.45/6.92 & ( N
% 12.45/6.92 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.45/6.92 => ( P @ J3 ) ) )
% 12.45/6.92 & ( ( ord_less_int @ K @ zero_zero_int )
% 12.45/6.92 => ! [I2: int,J3: int] :
% 12.45/6.92 ( ( ( ord_less_int @ K @ J3 )
% 12.45/6.92 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 12.45/6.92 & ( N
% 12.45/6.92 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.45/6.92 => ( P @ J3 ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3179_odd__less__0__iff,axiom,
% 12.45/6.92 ! [Z: int] :
% 12.45/6.92 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 12.45/6.92 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3180_mod__pos__geq,axiom,
% 12.45/6.92 ! [L2: int,K: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ L2 )
% 12.45/6.92 => ( ( ord_less_eq_int @ L2 @ K )
% 12.45/6.92 => ( ( modulo_modulo_int @ K @ L2 )
% 12.45/6.92 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3181_q__pos__lemma,axiom,
% 12.45/6.92 ! [B: int,Q5: int,R4: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 12.45/6.92 => ( ( ord_less_int @ R4 @ B )
% 12.45/6.92 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.45/6.92 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3182_zmult__zless__mono2,axiom,
% 12.45/6.92 ! [I: int,J: int,K: int] :
% 12.45/6.92 ( ( ord_less_int @ I @ J )
% 12.45/6.92 => ( ( ord_less_int @ zero_zero_int @ K )
% 12.45/6.92 => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3183_int__mod__neg__eq,axiom,
% 12.45/6.92 ! [A: int,B2: int,Q2: int,R2: int] :
% 12.45/6.92 ( ( A
% 12.45/6.92 = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 12.45/6.92 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 12.45/6.92 => ( ( ord_less_int @ B2 @ R2 )
% 12.45/6.92 => ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.92 = R2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3184_int__mod__pos__eq,axiom,
% 12.45/6.92 ! [A: int,B2: int,Q2: int,R2: int] :
% 12.45/6.92 ( ( A
% 12.45/6.92 = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 12.45/6.92 => ( ( ord_less_int @ R2 @ B2 )
% 12.45/6.92 => ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.92 = R2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3185_pos__zmult__eq__1__iff,axiom,
% 12.45/6.92 ! [M: int,N: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ M )
% 12.45/6.92 => ( ( ( times_times_int @ M @ N )
% 12.45/6.92 = one_one_int )
% 12.45/6.92 = ( ( M = one_one_int )
% 12.45/6.92 & ( N = one_one_int ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3186_plusinfinity,axiom,
% 12.45/6.92 ! [D: int,P4: int > $o,P: int > $o] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ D )
% 12.45/6.92 => ( ! [X3: int,K2: int] :
% 12.45/6.92 ( ( P4 @ X3 )
% 12.45/6.92 = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
% 12.45/6.92 => ( ? [Z3: int] :
% 12.45/6.92 ! [X3: int] :
% 12.45/6.92 ( ( ord_less_int @ Z3 @ X3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [X_12: int] : ( P4 @ X_12 )
% 12.45/6.92 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3187_zdiv__mono2__lemma,axiom,
% 12.45/6.92 ! [B2: int,Q2: int,R2: int,B: int,Q5: int,R4: int] :
% 12.45/6.92 ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 )
% 12.45/6.92 = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 12.45/6.92 => ( ( ord_less_int @ R4 @ B )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 12.45/6.92 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.45/6.92 => ( ( ord_less_eq_int @ B @ B2 )
% 12.45/6.92 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3188_minusinfinity,axiom,
% 12.45/6.92 ! [D: int,P1: int > $o,P: int > $o] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ D )
% 12.45/6.92 => ( ! [X3: int,K2: int] :
% 12.45/6.92 ( ( P1 @ X3 )
% 12.45/6.92 = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
% 12.45/6.92 => ( ? [Z3: int] :
% 12.45/6.92 ! [X3: int] :
% 12.45/6.92 ( ( ord_less_int @ X3 @ Z3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P1 @ X3 ) ) )
% 12.45/6.92 => ( ? [X_12: int] : ( P1 @ X_12 )
% 12.45/6.92 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3189_decr__mult__lemma,axiom,
% 12.45/6.92 ! [D: int,P: int > $o,K: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ D )
% 12.45/6.92 => ( ! [X3: int] :
% 12.45/6.92 ( ( P @ X3 )
% 12.45/6.92 => ( P @ ( minus_minus_int @ X3 @ D ) ) )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.92 => ! [X5: int] :
% 12.45/6.92 ( ( P @ X5 )
% 12.45/6.92 => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3190_incr__mult__lemma,axiom,
% 12.45/6.92 ! [D: int,P: int > $o,K: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ D )
% 12.45/6.92 => ( ! [X3: int] :
% 12.45/6.92 ( ( P @ X3 )
% 12.45/6.92 => ( P @ ( plus_plus_int @ X3 @ D ) ) )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.92 => ! [X5: int] :
% 12.45/6.92 ( ( P @ X5 )
% 12.45/6.92 => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3191_zdiv__mono2__neg__lemma,axiom,
% 12.45/6.92 ! [B2: int,Q2: int,R2: int,B: int,Q5: int,R4: int] :
% 12.45/6.92 ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 )
% 12.45/6.92 = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 12.45/6.92 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ zero_zero_int )
% 12.45/6.92 => ( ( ord_less_int @ R2 @ B2 )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 12.45/6.92 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.45/6.92 => ( ( ord_less_eq_int @ B @ B2 )
% 12.45/6.92 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3192_int__one__le__iff__zero__less,axiom,
% 12.45/6.92 ! [Z: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ one_one_int @ Z )
% 12.45/6.92 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3193_unique__quotient__lemma,axiom,
% 12.45/6.92 ! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 12.45/6.92 => ( ( ord_less_int @ R4 @ B2 )
% 12.45/6.92 => ( ( ord_less_int @ R2 @ B2 )
% 12.45/6.92 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3194_unique__quotient__lemma__neg,axiom,
% 12.45/6.92 ! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R2 ) )
% 12.45/6.92 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 12.45/6.92 => ( ( ord_less_int @ B2 @ R2 )
% 12.45/6.92 => ( ( ord_less_int @ B2 @ R4 )
% 12.45/6.92 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3195_mod__pos__neg__trivial,axiom,
% 12.45/6.92 ! [K: int,L2: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ K )
% 12.45/6.92 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 12.45/6.92 => ( ( modulo_modulo_int @ K @ L2 )
% 12.45/6.92 = ( plus_plus_int @ K @ L2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3196_zdvd__reduce,axiom,
% 12.45/6.92 ! [K: int,N: int,M: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 12.45/6.92 = ( dvd_dvd_int @ K @ N ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3197_zdvd__period,axiom,
% 12.45/6.92 ! [A: int,D: int,X4: int,T: int,C: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ A @ D )
% 12.45/6.92 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X4 @ T ) )
% 12.45/6.92 = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X4 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3198_odd__nonzero,axiom,
% 12.45/6.92 ! [Z: int] :
% 12.45/6.92 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 12.45/6.92 != zero_zero_int ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3199_times__int__code_I2_J,axiom,
% 12.45/6.92 ! [L2: int] :
% 12.45/6.92 ( ( times_times_int @ zero_zero_int @ L2 )
% 12.45/6.92 = zero_zero_int ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3200_times__int__code_I1_J,axiom,
% 12.45/6.92 ! [K: int] :
% 12.45/6.92 ( ( times_times_int @ K @ zero_zero_int )
% 12.45/6.92 = zero_zero_int ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3201_zmod__eq__0D,axiom,
% 12.45/6.92 ! [M: int,D: int] :
% 12.45/6.92 ( ( ( modulo_modulo_int @ M @ D )
% 12.45/6.92 = zero_zero_int )
% 12.45/6.92 => ? [Q3: int] :
% 12.45/6.92 ( M
% 12.45/6.92 = ( times_times_int @ D @ Q3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3202_zdvd__mult__cancel,axiom,
% 12.45/6.92 ! [K: int,M: int,N: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 12.45/6.92 => ( ( K != zero_zero_int )
% 12.45/6.92 => ( dvd_dvd_int @ M @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3203_zdvd__mono,axiom,
% 12.45/6.92 ! [K: int,M: int,T: int] :
% 12.45/6.92 ( ( K != zero_zero_int )
% 12.45/6.92 => ( ( dvd_dvd_int @ M @ T )
% 12.45/6.92 = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3204_zmod__eq__0__iff,axiom,
% 12.45/6.92 ! [M: int,D: int] :
% 12.45/6.92 ( ( ( modulo_modulo_int @ M @ D )
% 12.45/6.92 = zero_zero_int )
% 12.45/6.92 = ( ? [Q4: int] :
% 12.45/6.92 ( M
% 12.45/6.92 = ( times_times_int @ D @ Q4 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3205_imult__is__0,axiom,
% 12.45/6.92 ! [M: extended_enat,N: extended_enat] :
% 12.45/6.92 ( ( ( times_7803423173614009249d_enat @ M @ N )
% 12.45/6.92 = zero_z5237406670263579293d_enat )
% 12.45/6.92 = ( ( M = zero_z5237406670263579293d_enat )
% 12.45/6.92 | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3206_signed__take__bit__mult,axiom,
% 12.45/6.92 ! [N: nat,K: int,L2: int] :
% 12.45/6.92 ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 12.45/6.92 = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3207_Euclidean__Division_Opos__mod__bound,axiom,
% 12.45/6.92 ! [L2: int,K: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ L2 )
% 12.45/6.92 => ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3208_neg__mod__bound,axiom,
% 12.45/6.92 ! [L2: int,K: int] :
% 12.45/6.92 ( ( ord_less_int @ L2 @ zero_zero_int )
% 12.45/6.92 => ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3209_Euclidean__Division_Opos__mod__sign,axiom,
% 12.45/6.92 ! [L2: int,K: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ L2 )
% 12.45/6.92 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3210_neg__mod__sign,axiom,
% 12.45/6.92 ! [L2: int,K: int] :
% 12.45/6.92 ( ( ord_less_int @ L2 @ zero_zero_int )
% 12.45/6.92 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3211_zmod__le__nonneg__dividend,axiom,
% 12.45/6.92 ! [M: int,K: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 12.45/6.92 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3212_zmod__trivial__iff,axiom,
% 12.45/6.92 ! [I: int,K: int] :
% 12.45/6.92 ( ( ( modulo_modulo_int @ I @ K )
% 12.45/6.92 = I )
% 12.45/6.92 = ( ( K = zero_zero_int )
% 12.45/6.92 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 12.45/6.92 & ( ord_less_int @ I @ K ) )
% 12.45/6.92 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 12.45/6.92 & ( ord_less_int @ K @ I ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3213_mod__int__pos__iff,axiom,
% 12.45/6.92 ! [K: int,L2: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) )
% 12.45/6.92 = ( ( dvd_dvd_int @ L2 @ K )
% 12.45/6.92 | ( ( L2 = zero_zero_int )
% 12.45/6.92 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 12.45/6.92 | ( ord_less_int @ zero_zero_int @ L2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3214_pos__mod__conj,axiom,
% 12.45/6.92 ! [B2: int,A: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.92 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) )
% 12.45/6.92 & ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3215_neg__mod__conj,axiom,
% 12.45/6.92 ! [B2: int,A: int] :
% 12.45/6.92 ( ( ord_less_int @ B2 @ zero_zero_int )
% 12.45/6.92 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ zero_zero_int )
% 12.45/6.92 & ( ord_less_int @ B2 @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3216_zdvd__not__zless,axiom,
% 12.45/6.92 ! [M: int,N: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ M )
% 12.45/6.92 => ( ( ord_less_int @ M @ N )
% 12.45/6.92 => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3217_zdvd__imp__le,axiom,
% 12.45/6.92 ! [Z: int,N: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ Z @ N )
% 12.45/6.92 => ( ( ord_less_int @ zero_zero_int @ N )
% 12.45/6.92 => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3218_less__int__code_I1_J,axiom,
% 12.45/6.92 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3219_plus__int__code_I2_J,axiom,
% 12.45/6.92 ! [L2: int] :
% 12.45/6.92 ( ( plus_plus_int @ zero_zero_int @ L2 )
% 12.45/6.92 = L2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3220_plus__int__code_I1_J,axiom,
% 12.45/6.92 ! [K: int] :
% 12.45/6.92 ( ( plus_plus_int @ K @ zero_zero_int )
% 12.45/6.92 = K ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3221_minus__int__code_I1_J,axiom,
% 12.45/6.92 ! [K: int] :
% 12.45/6.92 ( ( minus_minus_int @ K @ zero_zero_int )
% 12.45/6.92 = K ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3222_zdvd__zdiffD,axiom,
% 12.45/6.92 ! [K: int,M: int,N: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 12.45/6.92 => ( ( dvd_dvd_int @ K @ N )
% 12.45/6.92 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3223_signed__take__bit__add,axiom,
% 12.45/6.92 ! [N: nat,K: int,L2: int] :
% 12.45/6.92 ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 12.45/6.92 = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3224_signed__take__bit__diff,axiom,
% 12.45/6.92 ! [N: nat,K: int,L2: int] :
% 12.45/6.92 ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 12.45/6.92 = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3225_zero__one__enat__neq_I1_J,axiom,
% 12.45/6.92 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3226_set__bit__greater__eq,axiom,
% 12.45/6.92 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3227_unset__bit__less__eq,axiom,
% 12.45/6.92 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3228_int__distrib_I4_J,axiom,
% 12.45/6.92 ! [W: int,Z1: int,Z22: int] :
% 12.45/6.92 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 12.45/6.92 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3229_int__distrib_I3_J,axiom,
% 12.45/6.92 ! [Z1: int,Z22: int,W: int] :
% 12.45/6.92 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 12.45/6.92 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3230_int__distrib_I2_J,axiom,
% 12.45/6.92 ! [W: int,Z1: int,Z22: int] :
% 12.45/6.92 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 12.45/6.92 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3231_int__distrib_I1_J,axiom,
% 12.45/6.92 ! [Z1: int,Z22: int,W: int] :
% 12.45/6.92 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 12.45/6.92 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3232_zless__imp__add1__zle,axiom,
% 12.45/6.92 ! [W: int,Z: int] :
% 12.45/6.92 ( ( ord_less_int @ W @ Z )
% 12.45/6.92 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3233_int__less__induct,axiom,
% 12.45/6.92 ! [I: int,K: int,P: int > $o] :
% 12.45/6.92 ( ( ord_less_int @ I @ K )
% 12.45/6.92 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 12.45/6.92 => ( ! [I3: int] :
% 12.45/6.92 ( ( ord_less_int @ I3 @ K )
% 12.45/6.92 => ( ( P @ I3 )
% 12.45/6.92 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 12.45/6.92 => ( P @ I ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3234_zless__add1__eq,axiom,
% 12.45/6.92 ! [W: int,Z: int] :
% 12.45/6.92 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 12.45/6.92 = ( ( ord_less_int @ W @ Z )
% 12.45/6.92 | ( W = Z ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3235_int__le__induct,axiom,
% 12.45/6.92 ! [I: int,K: int,P: int > $o] :
% 12.45/6.92 ( ( ord_less_eq_int @ I @ K )
% 12.45/6.92 => ( ( P @ K )
% 12.45/6.92 => ( ! [I3: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ I3 @ K )
% 12.45/6.92 => ( ( P @ I3 )
% 12.45/6.92 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 12.45/6.92 => ( P @ I ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3236_int__gr__induct,axiom,
% 12.45/6.92 ! [K: int,I: int,P: int > $o] :
% 12.45/6.92 ( ( ord_less_int @ K @ I )
% 12.45/6.92 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 12.45/6.92 => ( ! [I3: int] :
% 12.45/6.92 ( ( ord_less_int @ K @ I3 )
% 12.45/6.92 => ( ( P @ I3 )
% 12.45/6.92 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 12.45/6.92 => ( P @ I ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3237_int__ge__induct,axiom,
% 12.45/6.92 ! [K: int,I: int,P: int > $o] :
% 12.45/6.92 ( ( ord_less_eq_int @ K @ I )
% 12.45/6.92 => ( ( P @ K )
% 12.45/6.92 => ( ! [I3: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ K @ I3 )
% 12.45/6.92 => ( ( P @ I3 )
% 12.45/6.92 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 12.45/6.92 => ( P @ I ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3238_add1__zle__eq,axiom,
% 12.45/6.92 ! [W: int,Z: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 12.45/6.92 = ( ord_less_int @ W @ Z ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3239_int__induct,axiom,
% 12.45/6.92 ! [P: int > $o,K: int,I: int] :
% 12.45/6.92 ( ( P @ K )
% 12.45/6.92 => ( ! [I3: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ K @ I3 )
% 12.45/6.92 => ( ( P @ I3 )
% 12.45/6.92 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 12.45/6.92 => ( ! [I3: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ I3 @ K )
% 12.45/6.92 => ( ( P @ I3 )
% 12.45/6.92 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 12.45/6.92 => ( P @ I ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3240_even__diff__iff,axiom,
% 12.45/6.92 ! [K: int,L2: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
% 12.45/6.92 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3241_minf_I7_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.45/6.92 => ~ ( ord_less_real @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3242_minf_I7_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.45/6.92 => ~ ( ord_less_rat @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3243_minf_I7_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ X5 @ Z2 )
% 12.45/6.92 => ~ ( ord_less_num @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3244_minf_I7_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.45/6.92 => ~ ( ord_less_nat @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3245_minf_I7_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.45/6.92 => ~ ( ord_less_int @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3246_minf_I5_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.45/6.92 => ( ord_less_real @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3247_minf_I5_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.45/6.92 => ( ord_less_rat @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3248_minf_I5_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ X5 @ Z2 )
% 12.45/6.92 => ( ord_less_num @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3249_minf_I5_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.45/6.92 => ( ord_less_nat @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3250_minf_I5_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.45/6.92 => ( ord_less_int @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3251_minf_I4_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3252_minf_I4_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3253_minf_I4_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ X5 @ Z2 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3254_minf_I4_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3255_minf_I4_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3256_minf_I3_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3257_minf_I3_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3258_minf_I3_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ X5 @ Z2 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3259_minf_I3_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3260_minf_I3_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3261_minf_I2_J,axiom,
% 12.45/6.92 ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
% 12.45/6.92 ( ? [Z3: real] :
% 12.45/6.92 ! [X3: real] :
% 12.45/6.92 ( ( ord_less_real @ X3 @ Z3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: real] :
% 12.45/6.92 ! [X3: real] :
% 12.45/6.92 ( ( ord_less_real @ X3 @ Z3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3262_minf_I2_J,axiom,
% 12.45/6.92 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 12.45/6.92 ( ? [Z3: rat] :
% 12.45/6.92 ! [X3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X3 @ Z3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: rat] :
% 12.45/6.92 ! [X3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X3 @ Z3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3263_minf_I2_J,axiom,
% 12.45/6.92 ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
% 12.45/6.92 ( ? [Z3: num] :
% 12.45/6.92 ! [X3: num] :
% 12.45/6.92 ( ( ord_less_num @ X3 @ Z3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: num] :
% 12.45/6.92 ! [X3: num] :
% 12.45/6.92 ( ( ord_less_num @ X3 @ Z3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ X5 @ Z2 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3264_minf_I2_J,axiom,
% 12.45/6.92 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 12.45/6.92 ( ? [Z3: nat] :
% 12.45/6.92 ! [X3: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X3 @ Z3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: nat] :
% 12.45/6.92 ! [X3: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X3 @ Z3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3265_minf_I2_J,axiom,
% 12.45/6.92 ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
% 12.45/6.92 ( ? [Z3: int] :
% 12.45/6.92 ! [X3: int] :
% 12.45/6.92 ( ( ord_less_int @ X3 @ Z3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: int] :
% 12.45/6.92 ! [X3: int] :
% 12.45/6.92 ( ( ord_less_int @ X3 @ Z3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3266_minf_I1_J,axiom,
% 12.45/6.92 ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
% 12.45/6.92 ( ? [Z3: real] :
% 12.45/6.92 ! [X3: real] :
% 12.45/6.92 ( ( ord_less_real @ X3 @ Z3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: real] :
% 12.45/6.92 ! [X3: real] :
% 12.45/6.92 ( ( ord_less_real @ X3 @ Z3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3267_minf_I1_J,axiom,
% 12.45/6.92 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 12.45/6.92 ( ? [Z3: rat] :
% 12.45/6.92 ! [X3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X3 @ Z3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: rat] :
% 12.45/6.92 ! [X3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X3 @ Z3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3268_minf_I1_J,axiom,
% 12.45/6.92 ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
% 12.45/6.92 ( ? [Z3: num] :
% 12.45/6.92 ! [X3: num] :
% 12.45/6.92 ( ( ord_less_num @ X3 @ Z3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: num] :
% 12.45/6.92 ! [X3: num] :
% 12.45/6.92 ( ( ord_less_num @ X3 @ Z3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ X5 @ Z2 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3269_minf_I1_J,axiom,
% 12.45/6.92 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 12.45/6.92 ( ? [Z3: nat] :
% 12.45/6.92 ! [X3: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X3 @ Z3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: nat] :
% 12.45/6.92 ! [X3: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X3 @ Z3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3270_minf_I1_J,axiom,
% 12.45/6.92 ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
% 12.45/6.92 ( ? [Z3: int] :
% 12.45/6.92 ! [X3: int] :
% 12.45/6.92 ( ( ord_less_int @ X3 @ Z3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: int] :
% 12.45/6.92 ! [X3: int] :
% 12.45/6.92 ( ( ord_less_int @ X3 @ Z3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3271_pinf_I7_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.45/6.92 => ( ord_less_real @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3272_pinf_I7_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.45/6.92 => ( ord_less_rat @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3273_pinf_I7_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ Z2 @ X5 )
% 12.45/6.92 => ( ord_less_num @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3274_pinf_I7_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.45/6.92 => ( ord_less_nat @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3275_pinf_I7_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.45/6.92 => ( ord_less_int @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3276_pinf_I5_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.45/6.92 => ~ ( ord_less_real @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3277_pinf_I5_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.45/6.92 => ~ ( ord_less_rat @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3278_pinf_I5_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ Z2 @ X5 )
% 12.45/6.92 => ~ ( ord_less_num @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3279_pinf_I5_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.45/6.92 => ~ ( ord_less_nat @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3280_pinf_I5_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.45/6.92 => ~ ( ord_less_int @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3281_pinf_I4_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3282_pinf_I4_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3283_pinf_I4_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ Z2 @ X5 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3284_pinf_I4_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3285_pinf_I4_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3286_pinf_I3_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3287_pinf_I3_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3288_pinf_I3_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ Z2 @ X5 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3289_pinf_I3_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3290_pinf_I3_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.45/6.92 => ( X5 != T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3291_pinf_I2_J,axiom,
% 12.45/6.92 ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
% 12.45/6.92 ( ? [Z3: real] :
% 12.45/6.92 ! [X3: real] :
% 12.45/6.92 ( ( ord_less_real @ Z3 @ X3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: real] :
% 12.45/6.92 ! [X3: real] :
% 12.45/6.92 ( ( ord_less_real @ Z3 @ X3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3292_pinf_I2_J,axiom,
% 12.45/6.92 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 12.45/6.92 ( ? [Z3: rat] :
% 12.45/6.92 ! [X3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z3 @ X3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: rat] :
% 12.45/6.92 ! [X3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z3 @ X3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3293_pinf_I2_J,axiom,
% 12.45/6.92 ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
% 12.45/6.92 ( ? [Z3: num] :
% 12.45/6.92 ! [X3: num] :
% 12.45/6.92 ( ( ord_less_num @ Z3 @ X3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: num] :
% 12.45/6.92 ! [X3: num] :
% 12.45/6.92 ( ( ord_less_num @ Z3 @ X3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ Z2 @ X5 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3294_pinf_I2_J,axiom,
% 12.45/6.92 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 12.45/6.92 ( ? [Z3: nat] :
% 12.45/6.92 ! [X3: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z3 @ X3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: nat] :
% 12.45/6.92 ! [X3: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z3 @ X3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3295_pinf_I2_J,axiom,
% 12.45/6.92 ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
% 12.45/6.92 ( ? [Z3: int] :
% 12.45/6.92 ! [X3: int] :
% 12.45/6.92 ( ( ord_less_int @ Z3 @ X3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: int] :
% 12.45/6.92 ! [X3: int] :
% 12.45/6.92 ( ( ord_less_int @ Z3 @ X3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3296_pinf_I1_J,axiom,
% 12.45/6.92 ! [P: real > $o,P4: real > $o,Q: real > $o,Q6: real > $o] :
% 12.45/6.92 ( ? [Z3: real] :
% 12.45/6.92 ! [X3: real] :
% 12.45/6.92 ( ( ord_less_real @ Z3 @ X3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: real] :
% 12.45/6.92 ! [X3: real] :
% 12.45/6.92 ( ( ord_less_real @ Z3 @ X3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3297_pinf_I1_J,axiom,
% 12.45/6.92 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 12.45/6.92 ( ? [Z3: rat] :
% 12.45/6.92 ! [X3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z3 @ X3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: rat] :
% 12.45/6.92 ! [X3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z3 @ X3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3298_pinf_I1_J,axiom,
% 12.45/6.92 ! [P: num > $o,P4: num > $o,Q: num > $o,Q6: num > $o] :
% 12.45/6.92 ( ? [Z3: num] :
% 12.45/6.92 ! [X3: num] :
% 12.45/6.92 ( ( ord_less_num @ Z3 @ X3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: num] :
% 12.45/6.92 ! [X3: num] :
% 12.45/6.92 ( ( ord_less_num @ Z3 @ X3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ Z2 @ X5 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3299_pinf_I1_J,axiom,
% 12.45/6.92 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 12.45/6.92 ( ? [Z3: nat] :
% 12.45/6.92 ! [X3: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z3 @ X3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: nat] :
% 12.45/6.92 ! [X3: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z3 @ X3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3300_pinf_I1_J,axiom,
% 12.45/6.92 ! [P: int > $o,P4: int > $o,Q: int > $o,Q6: int > $o] :
% 12.45/6.92 ( ? [Z3: int] :
% 12.45/6.92 ! [X3: int] :
% 12.45/6.92 ( ( ord_less_int @ Z3 @ X3 )
% 12.45/6.92 => ( ( P @ X3 )
% 12.45/6.92 = ( P4 @ X3 ) ) )
% 12.45/6.92 => ( ? [Z3: int] :
% 12.45/6.92 ! [X3: int] :
% 12.45/6.92 ( ( ord_less_int @ Z3 @ X3 )
% 12.45/6.92 => ( ( Q @ X3 )
% 12.45/6.92 = ( Q6 @ X3 ) ) )
% 12.45/6.92 => ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.45/6.92 => ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P4 @ X5 )
% 12.45/6.92 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3301_vebt__buildup_Ocases,axiom,
% 12.45/6.92 ! [X4: nat] :
% 12.45/6.92 ( ( X4 != zero_zero_nat )
% 12.45/6.92 => ( ( X4
% 12.45/6.92 != ( suc @ zero_zero_nat ) )
% 12.45/6.92 => ~ ! [Va: nat] :
% 12.45/6.92 ( X4
% 12.45/6.92 != ( suc @ ( suc @ Va ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3302_Euclid__induct,axiom,
% 12.45/6.92 ! [P: nat > nat > $o,A: nat,B2: nat] :
% 12.45/6.92 ( ! [A5: nat,B5: nat] :
% 12.45/6.92 ( ( P @ A5 @ B5 )
% 12.45/6.92 = ( P @ B5 @ A5 ) )
% 12.45/6.92 => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
% 12.45/6.92 => ( ! [A5: nat,B5: nat] :
% 12.45/6.92 ( ( P @ A5 @ B5 )
% 12.45/6.92 => ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
% 12.45/6.92 => ( P @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3303_minf_I8_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.45/6.92 => ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3304_minf_I8_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.45/6.92 => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3305_minf_I8_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ X5 @ Z2 )
% 12.45/6.92 => ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3306_minf_I8_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.45/6.92 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3307_minf_I8_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.45/6.92 => ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3308_minf_I6_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.45/6.92 => ( ord_less_eq_real @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3309_minf_I6_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.45/6.92 => ( ord_less_eq_rat @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3310_minf_I6_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ X5 @ Z2 )
% 12.45/6.92 => ( ord_less_eq_num @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3311_minf_I6_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.45/6.92 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3312_minf_I6_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.45/6.92 => ( ord_less_eq_int @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3313_pinf_I8_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.45/6.92 => ( ord_less_eq_real @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3314_pinf_I8_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.45/6.92 => ( ord_less_eq_rat @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3315_pinf_I8_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ Z2 @ X5 )
% 12.45/6.92 => ( ord_less_eq_num @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3316_pinf_I8_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.45/6.92 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3317_pinf_I8_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.45/6.92 => ( ord_less_eq_int @ T @ X5 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3318_pinf_I6_J,axiom,
% 12.45/6.92 ! [T: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.45/6.92 => ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3319_pinf_I6_J,axiom,
% 12.45/6.92 ! [T: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.45/6.92 => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3320_pinf_I6_J,axiom,
% 12.45/6.92 ! [T: num] :
% 12.45/6.92 ? [Z2: num] :
% 12.45/6.92 ! [X5: num] :
% 12.45/6.92 ( ( ord_less_num @ Z2 @ X5 )
% 12.45/6.92 => ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3321_pinf_I6_J,axiom,
% 12.45/6.92 ! [T: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.45/6.92 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3322_pinf_I6_J,axiom,
% 12.45/6.92 ! [T: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.45/6.92 => ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3323_inf__period_I2_J,axiom,
% 12.45/6.92 ! [P: real > $o,D4: real,Q: real > $o] :
% 12.45/6.92 ( ! [X3: real,K2: real] :
% 12.45/6.92 ( ( P @ X3 )
% 12.45/6.92 = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ( ! [X3: real,K2: real] :
% 12.45/6.92 ( ( Q @ X3 )
% 12.45/6.92 = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ! [X5: real,K4: real] :
% 12.45/6.92 ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
% 12.45/6.92 | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3324_inf__period_I2_J,axiom,
% 12.45/6.92 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 12.45/6.92 ( ! [X3: rat,K2: rat] :
% 12.45/6.92 ( ( P @ X3 )
% 12.45/6.92 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ( ! [X3: rat,K2: rat] :
% 12.45/6.92 ( ( Q @ X3 )
% 12.45/6.92 = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ! [X5: rat,K4: rat] :
% 12.45/6.92 ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
% 12.45/6.92 | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3325_inf__period_I2_J,axiom,
% 12.45/6.92 ! [P: int > $o,D4: int,Q: int > $o] :
% 12.45/6.92 ( ! [X3: int,K2: int] :
% 12.45/6.92 ( ( P @ X3 )
% 12.45/6.92 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ( ! [X3: int,K2: int] :
% 12.45/6.92 ( ( Q @ X3 )
% 12.45/6.92 = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ! [X5: int,K4: int] :
% 12.45/6.92 ( ( ( P @ X5 )
% 12.45/6.92 | ( Q @ X5 ) )
% 12.45/6.92 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
% 12.45/6.92 | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3326_inf__period_I1_J,axiom,
% 12.45/6.92 ! [P: real > $o,D4: real,Q: real > $o] :
% 12.45/6.92 ( ! [X3: real,K2: real] :
% 12.45/6.92 ( ( P @ X3 )
% 12.45/6.92 = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ( ! [X3: real,K2: real] :
% 12.45/6.92 ( ( Q @ X3 )
% 12.45/6.92 = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ! [X5: real,K4: real] :
% 12.45/6.92 ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
% 12.45/6.92 & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3327_inf__period_I1_J,axiom,
% 12.45/6.92 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 12.45/6.92 ( ! [X3: rat,K2: rat] :
% 12.45/6.92 ( ( P @ X3 )
% 12.45/6.92 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ( ! [X3: rat,K2: rat] :
% 12.45/6.92 ( ( Q @ X3 )
% 12.45/6.92 = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ! [X5: rat,K4: rat] :
% 12.45/6.92 ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
% 12.45/6.92 & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3328_inf__period_I1_J,axiom,
% 12.45/6.92 ! [P: int > $o,D4: int,Q: int > $o] :
% 12.45/6.92 ( ! [X3: int,K2: int] :
% 12.45/6.92 ( ( P @ X3 )
% 12.45/6.92 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ( ! [X3: int,K2: int] :
% 12.45/6.92 ( ( Q @ X3 )
% 12.45/6.92 = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 12.45/6.92 => ! [X5: int,K4: int] :
% 12.45/6.92 ( ( ( P @ X5 )
% 12.45/6.92 & ( Q @ X5 ) )
% 12.45/6.92 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
% 12.45/6.92 & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3329_dvd__productE,axiom,
% 12.45/6.92 ! [P2: nat,A: nat,B2: nat] :
% 12.45/6.92 ( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A @ B2 ) )
% 12.45/6.92 => ~ ! [X3: nat,Y4: nat] :
% 12.45/6.92 ( ( P2
% 12.45/6.92 = ( times_times_nat @ X3 @ Y4 ) )
% 12.45/6.92 => ( ( dvd_dvd_nat @ X3 @ A )
% 12.45/6.92 => ~ ( dvd_dvd_nat @ Y4 @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3330_dvd__productE,axiom,
% 12.45/6.92 ! [P2: int,A: int,B2: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ P2 @ ( times_times_int @ A @ B2 ) )
% 12.45/6.92 => ~ ! [X3: int,Y4: int] :
% 12.45/6.92 ( ( P2
% 12.45/6.92 = ( times_times_int @ X3 @ Y4 ) )
% 12.45/6.92 => ( ( dvd_dvd_int @ X3 @ A )
% 12.45/6.92 => ~ ( dvd_dvd_int @ Y4 @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3331_division__decomp,axiom,
% 12.45/6.92 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.92 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 12.45/6.92 => ? [B6: nat,C4: nat] :
% 12.45/6.92 ( ( A
% 12.45/6.92 = ( times_times_nat @ B6 @ C4 ) )
% 12.45/6.92 & ( dvd_dvd_nat @ B6 @ B2 )
% 12.45/6.92 & ( dvd_dvd_nat @ C4 @ C ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3332_division__decomp,axiom,
% 12.45/6.92 ! [A: int,B2: int,C: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
% 12.45/6.92 => ? [B6: int,C4: int] :
% 12.45/6.92 ( ( A
% 12.45/6.92 = ( times_times_int @ B6 @ C4 ) )
% 12.45/6.92 & ( dvd_dvd_int @ B6 @ B2 )
% 12.45/6.92 & ( dvd_dvd_int @ C4 @ C ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3333_dvd__pos__nat,axiom,
% 12.45/6.92 ! [N: nat,M: nat] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.92 => ( ( dvd_dvd_nat @ M @ N )
% 12.45/6.92 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3334_gcd__nat__induct,axiom,
% 12.45/6.92 ! [P: nat > nat > $o,M: nat,N: nat] :
% 12.45/6.92 ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 12.45/6.92 => ( ! [M4: nat,N4: nat] :
% 12.45/6.92 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 12.45/6.92 => ( ( P @ N4 @ ( modulo_modulo_nat @ M4 @ N4 ) )
% 12.45/6.92 => ( P @ M4 @ N4 ) ) )
% 12.45/6.92 => ( P @ M @ N ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3335_pinf_I9_J,axiom,
% 12.45/6.92 ! [D: code_integer,S: code_integer] :
% 12.45/6.92 ? [Z2: code_integer] :
% 12.45/6.92 ! [X5: code_integer] :
% 12.45/6.92 ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
% 12.45/6.92 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 12.45/6.92 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3336_pinf_I9_J,axiom,
% 12.45/6.92 ! [D: real,S: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.45/6.92 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 12.45/6.92 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3337_pinf_I9_J,axiom,
% 12.45/6.92 ! [D: rat,S: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.45/6.92 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 12.45/6.92 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3338_pinf_I9_J,axiom,
% 12.45/6.92 ! [D: nat,S: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.45/6.92 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 12.45/6.92 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3339_pinf_I9_J,axiom,
% 12.45/6.92 ! [D: int,S: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.45/6.92 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 12.45/6.92 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3340_pinf_I10_J,axiom,
% 12.45/6.92 ! [D: code_integer,S: code_integer] :
% 12.45/6.92 ? [Z2: code_integer] :
% 12.45/6.92 ! [X5: code_integer] :
% 12.45/6.92 ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
% 12.45/6.92 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3341_pinf_I10_J,axiom,
% 12.45/6.92 ! [D: real,S: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.45/6.92 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3342_pinf_I10_J,axiom,
% 12.45/6.92 ! [D: rat,S: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.45/6.92 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3343_pinf_I10_J,axiom,
% 12.45/6.92 ! [D: nat,S: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.45/6.92 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3344_pinf_I10_J,axiom,
% 12.45/6.92 ! [D: int,S: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.45/6.92 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3345_minf_I9_J,axiom,
% 12.45/6.92 ! [D: code_integer,S: code_integer] :
% 12.45/6.92 ? [Z2: code_integer] :
% 12.45/6.92 ! [X5: code_integer] :
% 12.45/6.92 ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
% 12.45/6.92 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 12.45/6.92 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3346_minf_I9_J,axiom,
% 12.45/6.92 ! [D: real,S: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.45/6.92 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 12.45/6.92 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3347_minf_I9_J,axiom,
% 12.45/6.92 ! [D: rat,S: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.45/6.92 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 12.45/6.92 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3348_minf_I9_J,axiom,
% 12.45/6.92 ! [D: nat,S: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.45/6.92 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 12.45/6.92 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3349_minf_I9_J,axiom,
% 12.45/6.92 ! [D: int,S: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.45/6.92 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 12.45/6.92 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3350_minf_I10_J,axiom,
% 12.45/6.92 ! [D: code_integer,S: code_integer] :
% 12.45/6.92 ? [Z2: code_integer] :
% 12.45/6.92 ! [X5: code_integer] :
% 12.45/6.92 ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
% 12.45/6.92 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3351_minf_I10_J,axiom,
% 12.45/6.92 ! [D: real,S: real] :
% 12.45/6.92 ? [Z2: real] :
% 12.45/6.92 ! [X5: real] :
% 12.45/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.45/6.92 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3352_minf_I10_J,axiom,
% 12.45/6.92 ! [D: rat,S: rat] :
% 12.45/6.92 ? [Z2: rat] :
% 12.45/6.92 ! [X5: rat] :
% 12.45/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.45/6.92 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3353_minf_I10_J,axiom,
% 12.45/6.92 ! [D: nat,S: nat] :
% 12.45/6.92 ? [Z2: nat] :
% 12.45/6.92 ! [X5: nat] :
% 12.45/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.45/6.92 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3354_minf_I10_J,axiom,
% 12.45/6.92 ! [D: int,S: int] :
% 12.45/6.92 ? [Z2: int] :
% 12.45/6.92 ! [X5: int] :
% 12.45/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.45/6.92 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 12.45/6.92 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3355_bezout__lemma__nat,axiom,
% 12.45/6.92 ! [D: nat,A: nat,B2: nat,X4: nat,Y3: nat] :
% 12.45/6.92 ( ( dvd_dvd_nat @ D @ A )
% 12.45/6.92 => ( ( dvd_dvd_nat @ D @ B2 )
% 12.45/6.92 => ( ( ( ( times_times_nat @ A @ X4 )
% 12.45/6.92 = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D ) )
% 12.45/6.92 | ( ( times_times_nat @ B2 @ X4 )
% 12.45/6.92 = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
% 12.45/6.92 => ? [X3: nat,Y4: nat] :
% 12.45/6.92 ( ( dvd_dvd_nat @ D @ A )
% 12.45/6.92 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B2 ) )
% 12.45/6.92 & ( ( ( times_times_nat @ A @ X3 )
% 12.45/6.92 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ Y4 ) @ D ) )
% 12.45/6.92 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ X3 )
% 12.45/6.92 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3356_bezout__add__nat,axiom,
% 12.45/6.92 ! [A: nat,B2: nat] :
% 12.45/6.92 ? [D3: nat,X3: nat,Y4: nat] :
% 12.45/6.92 ( ( dvd_dvd_nat @ D3 @ A )
% 12.45/6.92 & ( dvd_dvd_nat @ D3 @ B2 )
% 12.45/6.92 & ( ( ( times_times_nat @ A @ X3 )
% 12.45/6.92 = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) )
% 12.45/6.92 | ( ( times_times_nat @ B2 @ X3 )
% 12.45/6.92 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3357_bezout__add__strong__nat,axiom,
% 12.45/6.92 ! [A: nat,B2: nat] :
% 12.45/6.92 ( ( A != zero_zero_nat )
% 12.45/6.92 => ? [D3: nat,X3: nat,Y4: nat] :
% 12.45/6.92 ( ( dvd_dvd_nat @ D3 @ A )
% 12.45/6.92 & ( dvd_dvd_nat @ D3 @ B2 )
% 12.45/6.92 & ( ( times_times_nat @ A @ X3 )
% 12.45/6.92 = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3358_bezout1__nat,axiom,
% 12.45/6.92 ! [A: nat,B2: nat] :
% 12.45/6.92 ? [D3: nat,X3: nat,Y4: nat] :
% 12.45/6.92 ( ( dvd_dvd_nat @ D3 @ A )
% 12.45/6.92 & ( dvd_dvd_nat @ D3 @ B2 )
% 12.45/6.92 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B2 @ Y4 ) )
% 12.45/6.92 = D3 )
% 12.45/6.92 | ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
% 12.45/6.92 = D3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3359_Bolzano,axiom,
% 12.45/6.92 ! [A: real,B2: real,P: real > real > $o] :
% 12.45/6.92 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.92 => ( ! [A5: real,B5: real,C3: real] :
% 12.45/6.92 ( ( P @ A5 @ B5 )
% 12.45/6.92 => ( ( P @ B5 @ C3 )
% 12.45/6.92 => ( ( ord_less_eq_real @ A5 @ B5 )
% 12.45/6.92 => ( ( ord_less_eq_real @ B5 @ C3 )
% 12.45/6.92 => ( P @ A5 @ C3 ) ) ) ) )
% 12.45/6.92 => ( ! [X3: real] :
% 12.45/6.92 ( ( ord_less_eq_real @ A @ X3 )
% 12.45/6.92 => ( ( ord_less_eq_real @ X3 @ B2 )
% 12.45/6.92 => ? [D5: real] :
% 12.45/6.92 ( ( ord_less_real @ zero_zero_real @ D5 )
% 12.45/6.92 & ! [A5: real,B5: real] :
% 12.45/6.92 ( ( ( ord_less_eq_real @ A5 @ X3 )
% 12.45/6.92 & ( ord_less_eq_real @ X3 @ B5 )
% 12.45/6.92 & ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D5 ) )
% 12.45/6.92 => ( P @ A5 @ B5 ) ) ) ) )
% 12.45/6.92 => ( P @ A @ B2 ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3360_mult__le__cancel__iff1,axiom,
% 12.45/6.92 ! [Z: real,X4: real,Y3: real] :
% 12.45/6.92 ( ( ord_less_real @ zero_zero_real @ Z )
% 12.45/6.92 => ( ( ord_less_eq_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 12.45/6.92 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3361_mult__le__cancel__iff1,axiom,
% 12.45/6.92 ! [Z: rat,X4: rat,Y3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 12.45/6.92 => ( ( ord_less_eq_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 12.45/6.92 = ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3362_mult__le__cancel__iff1,axiom,
% 12.45/6.92 ! [Z: int,X4: int,Y3: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.45/6.92 => ( ( ord_less_eq_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 12.45/6.92 = ( ord_less_eq_int @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3363_mult__le__cancel__iff2,axiom,
% 12.45/6.92 ! [Z: real,X4: real,Y3: real] :
% 12.45/6.92 ( ( ord_less_real @ zero_zero_real @ Z )
% 12.45/6.92 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X4 ) @ ( times_times_real @ Z @ Y3 ) )
% 12.45/6.92 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3364_mult__le__cancel__iff2,axiom,
% 12.45/6.92 ! [Z: rat,X4: rat,Y3: rat] :
% 12.45/6.92 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 12.45/6.92 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X4 ) @ ( times_times_rat @ Z @ Y3 ) )
% 12.45/6.92 = ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3365_mult__le__cancel__iff2,axiom,
% 12.45/6.92 ! [Z: int,X4: int,Y3: int] :
% 12.45/6.92 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.45/6.92 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X4 ) @ ( times_times_int @ Z @ Y3 ) )
% 12.45/6.92 = ( ord_less_eq_int @ X4 @ Y3 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3366_triangle__def,axiom,
% 12.45/6.92 ( nat_triangle
% 12.45/6.92 = ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3367_signed__take__bit__rec,axiom,
% 12.45/6.92 ( bit_ri6519982836138164636nteger
% 12.45/6.92 = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3368_signed__take__bit__rec,axiom,
% 12.45/6.92 ( bit_ri631733984087533419it_int
% 12.45/6.92 = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3369_verit__minus__simplify_I4_J,axiom,
% 12.45/6.92 ! [B2: real] :
% 12.45/6.92 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3370_verit__minus__simplify_I4_J,axiom,
% 12.45/6.92 ! [B2: int] :
% 12.45/6.92 ( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3371_verit__minus__simplify_I4_J,axiom,
% 12.45/6.92 ! [B2: complex] :
% 12.45/6.92 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3372_verit__minus__simplify_I4_J,axiom,
% 12.45/6.92 ! [B2: code_integer] :
% 12.45/6.92 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3373_verit__minus__simplify_I4_J,axiom,
% 12.45/6.92 ! [B2: rat] :
% 12.45/6.92 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3374_add_Oinverse__inverse,axiom,
% 12.45/6.92 ! [A: real] :
% 12.45/6.92 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 12.45/6.92 = A ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3375_add_Oinverse__inverse,axiom,
% 12.45/6.92 ! [A: int] :
% 12.45/6.92 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 12.45/6.92 = A ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3376_add_Oinverse__inverse,axiom,
% 12.45/6.92 ! [A: complex] :
% 12.45/6.92 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 12.45/6.92 = A ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3377_add_Oinverse__inverse,axiom,
% 12.45/6.92 ! [A: code_integer] :
% 12.45/6.92 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.92 = A ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3378_add_Oinverse__inverse,axiom,
% 12.45/6.92 ! [A: rat] :
% 12.45/6.92 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 12.45/6.92 = A ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3379_neg__equal__iff__equal,axiom,
% 12.45/6.92 ! [A: real,B2: real] :
% 12.45/6.92 ( ( ( uminus_uminus_real @ A )
% 12.45/6.92 = ( uminus_uminus_real @ B2 ) )
% 12.45/6.92 = ( A = B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3380_neg__equal__iff__equal,axiom,
% 12.45/6.92 ! [A: int,B2: int] :
% 12.45/6.92 ( ( ( uminus_uminus_int @ A )
% 12.45/6.92 = ( uminus_uminus_int @ B2 ) )
% 12.45/6.92 = ( A = B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3381_neg__equal__iff__equal,axiom,
% 12.45/6.92 ! [A: complex,B2: complex] :
% 12.45/6.92 ( ( ( uminus1482373934393186551omplex @ A )
% 12.45/6.92 = ( uminus1482373934393186551omplex @ B2 ) )
% 12.45/6.92 = ( A = B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3382_neg__equal__iff__equal,axiom,
% 12.45/6.92 ! [A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( ( uminus1351360451143612070nteger @ A )
% 12.45/6.92 = ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.92 = ( A = B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3383_neg__equal__iff__equal,axiom,
% 12.45/6.92 ! [A: rat,B2: rat] :
% 12.45/6.92 ( ( ( uminus_uminus_rat @ A )
% 12.45/6.92 = ( uminus_uminus_rat @ B2 ) )
% 12.45/6.92 = ( A = B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3384_neg__le__iff__le,axiom,
% 12.45/6.92 ! [B2: real,A: real] :
% 12.45/6.92 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
% 12.45/6.92 = ( ord_less_eq_real @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3385_neg__le__iff__le,axiom,
% 12.45/6.92 ! [B2: code_integer,A: code_integer] :
% 12.45/6.92 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.92 = ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3386_neg__le__iff__le,axiom,
% 12.45/6.92 ! [B2: rat,A: rat] :
% 12.45/6.92 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
% 12.45/6.92 = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3387_neg__le__iff__le,axiom,
% 12.45/6.92 ! [B2: int,A: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
% 12.45/6.92 = ( ord_less_eq_int @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3388_add_Oinverse__neutral,axiom,
% 12.45/6.92 ( ( uminus_uminus_real @ zero_zero_real )
% 12.45/6.92 = zero_zero_real ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3389_add_Oinverse__neutral,axiom,
% 12.45/6.92 ( ( uminus_uminus_int @ zero_zero_int )
% 12.45/6.92 = zero_zero_int ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3390_add_Oinverse__neutral,axiom,
% 12.45/6.92 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 12.45/6.92 = zero_zero_complex ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3391_add_Oinverse__neutral,axiom,
% 12.45/6.92 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 12.45/6.92 = zero_z3403309356797280102nteger ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3392_add_Oinverse__neutral,axiom,
% 12.45/6.92 ( ( uminus_uminus_rat @ zero_zero_rat )
% 12.45/6.92 = zero_zero_rat ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3393_neg__0__equal__iff__equal,axiom,
% 12.45/6.92 ! [A: real] :
% 12.45/6.92 ( ( zero_zero_real
% 12.45/6.92 = ( uminus_uminus_real @ A ) )
% 12.45/6.92 = ( zero_zero_real = A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3394_neg__0__equal__iff__equal,axiom,
% 12.45/6.92 ! [A: int] :
% 12.45/6.92 ( ( zero_zero_int
% 12.45/6.92 = ( uminus_uminus_int @ A ) )
% 12.45/6.92 = ( zero_zero_int = A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3395_neg__0__equal__iff__equal,axiom,
% 12.45/6.92 ! [A: complex] :
% 12.45/6.92 ( ( zero_zero_complex
% 12.45/6.92 = ( uminus1482373934393186551omplex @ A ) )
% 12.45/6.92 = ( zero_zero_complex = A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3396_neg__0__equal__iff__equal,axiom,
% 12.45/6.92 ! [A: code_integer] :
% 12.45/6.92 ( ( zero_z3403309356797280102nteger
% 12.45/6.92 = ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.92 = ( zero_z3403309356797280102nteger = A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3397_neg__0__equal__iff__equal,axiom,
% 12.45/6.92 ! [A: rat] :
% 12.45/6.92 ( ( zero_zero_rat
% 12.45/6.92 = ( uminus_uminus_rat @ A ) )
% 12.45/6.92 = ( zero_zero_rat = A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3398_neg__equal__0__iff__equal,axiom,
% 12.45/6.92 ! [A: real] :
% 12.45/6.92 ( ( ( uminus_uminus_real @ A )
% 12.45/6.92 = zero_zero_real )
% 12.45/6.92 = ( A = zero_zero_real ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3399_neg__equal__0__iff__equal,axiom,
% 12.45/6.92 ! [A: int] :
% 12.45/6.92 ( ( ( uminus_uminus_int @ A )
% 12.45/6.92 = zero_zero_int )
% 12.45/6.92 = ( A = zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3400_neg__equal__0__iff__equal,axiom,
% 12.45/6.92 ! [A: complex] :
% 12.45/6.92 ( ( ( uminus1482373934393186551omplex @ A )
% 12.45/6.92 = zero_zero_complex )
% 12.45/6.92 = ( A = zero_zero_complex ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3401_neg__equal__0__iff__equal,axiom,
% 12.45/6.92 ! [A: code_integer] :
% 12.45/6.92 ( ( ( uminus1351360451143612070nteger @ A )
% 12.45/6.92 = zero_z3403309356797280102nteger )
% 12.45/6.92 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3402_neg__equal__0__iff__equal,axiom,
% 12.45/6.92 ! [A: rat] :
% 12.45/6.92 ( ( ( uminus_uminus_rat @ A )
% 12.45/6.92 = zero_zero_rat )
% 12.45/6.92 = ( A = zero_zero_rat ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3403_equal__neg__zero,axiom,
% 12.45/6.92 ! [A: real] :
% 12.45/6.92 ( ( A
% 12.45/6.92 = ( uminus_uminus_real @ A ) )
% 12.45/6.92 = ( A = zero_zero_real ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3404_equal__neg__zero,axiom,
% 12.45/6.92 ! [A: int] :
% 12.45/6.92 ( ( A
% 12.45/6.92 = ( uminus_uminus_int @ A ) )
% 12.45/6.92 = ( A = zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3405_equal__neg__zero,axiom,
% 12.45/6.92 ! [A: code_integer] :
% 12.45/6.92 ( ( A
% 12.45/6.92 = ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.92 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3406_equal__neg__zero,axiom,
% 12.45/6.92 ! [A: rat] :
% 12.45/6.92 ( ( A
% 12.45/6.92 = ( uminus_uminus_rat @ A ) )
% 12.45/6.92 = ( A = zero_zero_rat ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3407_neg__equal__zero,axiom,
% 12.45/6.92 ! [A: real] :
% 12.45/6.92 ( ( ( uminus_uminus_real @ A )
% 12.45/6.92 = A )
% 12.45/6.92 = ( A = zero_zero_real ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3408_neg__equal__zero,axiom,
% 12.45/6.92 ! [A: int] :
% 12.45/6.92 ( ( ( uminus_uminus_int @ A )
% 12.45/6.92 = A )
% 12.45/6.92 = ( A = zero_zero_int ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3409_neg__equal__zero,axiom,
% 12.45/6.92 ! [A: code_integer] :
% 12.45/6.92 ( ( ( uminus1351360451143612070nteger @ A )
% 12.45/6.92 = A )
% 12.45/6.92 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3410_neg__equal__zero,axiom,
% 12.45/6.92 ! [A: rat] :
% 12.45/6.92 ( ( ( uminus_uminus_rat @ A )
% 12.45/6.92 = A )
% 12.45/6.92 = ( A = zero_zero_rat ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3411_neg__less__iff__less,axiom,
% 12.45/6.92 ! [B2: real,A: real] :
% 12.45/6.92 ( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
% 12.45/6.92 = ( ord_less_real @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3412_neg__less__iff__less,axiom,
% 12.45/6.92 ! [B2: int,A: int] :
% 12.45/6.92 ( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
% 12.45/6.92 = ( ord_less_int @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3413_neg__less__iff__less,axiom,
% 12.45/6.92 ! [B2: code_integer,A: code_integer] :
% 12.45/6.92 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.92 = ( ord_le6747313008572928689nteger @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3414_neg__less__iff__less,axiom,
% 12.45/6.92 ! [B2: rat,A: rat] :
% 12.45/6.92 ( ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
% 12.45/6.92 = ( ord_less_rat @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3415_neg__numeral__eq__iff,axiom,
% 12.45/6.92 ! [M: num,N: num] :
% 12.45/6.92 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 12.45/6.92 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.92 = ( M = N ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3416_neg__numeral__eq__iff,axiom,
% 12.45/6.92 ! [M: num,N: num] :
% 12.45/6.92 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 12.45/6.92 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.92 = ( M = N ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3417_neg__numeral__eq__iff,axiom,
% 12.45/6.92 ! [M: num,N: num] :
% 12.45/6.92 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 12.45/6.92 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.45/6.92 = ( M = N ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3418_neg__numeral__eq__iff,axiom,
% 12.45/6.92 ! [M: num,N: num] :
% 12.45/6.92 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 12.45/6.92 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.92 = ( M = N ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3419_neg__numeral__eq__iff,axiom,
% 12.45/6.92 ! [M: num,N: num] :
% 12.45/6.92 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 12.45/6.92 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.92 = ( M = N ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3420_mult__minus__right,axiom,
% 12.45/6.92 ! [A: real,B2: real] :
% 12.45/6.92 ( ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) )
% 12.45/6.92 = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3421_mult__minus__right,axiom,
% 12.45/6.92 ! [A: int,B2: int] :
% 12.45/6.92 ( ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.92 = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3422_mult__minus__right,axiom,
% 12.45/6.92 ! [A: complex,B2: complex] :
% 12.45/6.92 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 12.45/6.92 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3423_mult__minus__right,axiom,
% 12.45/6.92 ! [A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.92 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3424_mult__minus__right,axiom,
% 12.45/6.92 ! [A: rat,B2: rat] :
% 12.45/6.92 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 12.45/6.92 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3425_minus__mult__minus,axiom,
% 12.45/6.92 ! [A: real,B2: real] :
% 12.45/6.92 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 12.45/6.92 = ( times_times_real @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3426_minus__mult__minus,axiom,
% 12.45/6.92 ! [A: int,B2: int] :
% 12.45/6.92 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.92 = ( times_times_int @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3427_minus__mult__minus,axiom,
% 12.45/6.92 ! [A: complex,B2: complex] :
% 12.45/6.92 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 12.45/6.92 = ( times_times_complex @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3428_minus__mult__minus,axiom,
% 12.45/6.92 ! [A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.92 = ( times_3573771949741848930nteger @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3429_minus__mult__minus,axiom,
% 12.45/6.92 ! [A: rat,B2: rat] :
% 12.45/6.92 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 12.45/6.92 = ( times_times_rat @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3430_mult__minus__left,axiom,
% 12.45/6.92 ! [A: real,B2: real] :
% 12.45/6.92 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
% 12.45/6.92 = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3431_mult__minus__left,axiom,
% 12.45/6.92 ! [A: int,B2: int] :
% 12.45/6.92 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.92 = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3432_mult__minus__left,axiom,
% 12.45/6.92 ! [A: complex,B2: complex] :
% 12.45/6.92 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 12.45/6.92 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3433_mult__minus__left,axiom,
% 12.45/6.92 ! [A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 12.45/6.92 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3434_mult__minus__left,axiom,
% 12.45/6.92 ! [A: rat,B2: rat] :
% 12.45/6.92 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 12.45/6.92 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3435_minus__add__distrib,axiom,
% 12.45/6.92 ! [A: real,B2: real] :
% 12.45/6.92 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 12.45/6.92 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3436_minus__add__distrib,axiom,
% 12.45/6.92 ! [A: int,B2: int] :
% 12.45/6.92 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 12.45/6.92 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3437_minus__add__distrib,axiom,
% 12.45/6.92 ! [A: complex,B2: complex] :
% 12.45/6.92 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 12.45/6.92 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3438_minus__add__distrib,axiom,
% 12.45/6.92 ! [A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 12.45/6.92 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3439_minus__add__distrib,axiom,
% 12.45/6.92 ! [A: rat,B2: rat] :
% 12.45/6.92 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 12.45/6.92 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3440_minus__add__cancel,axiom,
% 12.45/6.92 ! [A: real,B2: real] :
% 12.45/6.92 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3441_minus__add__cancel,axiom,
% 12.45/6.92 ! [A: int,B2: int] :
% 12.45/6.92 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3442_minus__add__cancel,axiom,
% 12.45/6.92 ! [A: complex,B2: complex] :
% 12.45/6.92 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3443_minus__add__cancel,axiom,
% 12.45/6.92 ! [A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3444_minus__add__cancel,axiom,
% 12.45/6.92 ! [A: rat,B2: rat] :
% 12.45/6.92 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3445_add__minus__cancel,axiom,
% 12.45/6.92 ! [A: real,B2: real] :
% 12.45/6.92 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3446_add__minus__cancel,axiom,
% 12.45/6.92 ! [A: int,B2: int] :
% 12.45/6.92 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3447_add__minus__cancel,axiom,
% 12.45/6.92 ! [A: complex,B2: complex] :
% 12.45/6.92 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3448_add__minus__cancel,axiom,
% 12.45/6.92 ! [A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3449_add__minus__cancel,axiom,
% 12.45/6.92 ! [A: rat,B2: rat] :
% 12.45/6.92 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) )
% 12.45/6.92 = B2 ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3450_minus__diff__eq,axiom,
% 12.45/6.92 ! [A: real,B2: real] :
% 12.45/6.92 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) )
% 12.45/6.92 = ( minus_minus_real @ B2 @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3451_minus__diff__eq,axiom,
% 12.45/6.92 ! [A: int,B2: int] :
% 12.45/6.92 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) )
% 12.45/6.92 = ( minus_minus_int @ B2 @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3452_minus__diff__eq,axiom,
% 12.45/6.92 ! [A: complex,B2: complex] :
% 12.45/6.92 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) )
% 12.45/6.92 = ( minus_minus_complex @ B2 @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3453_minus__diff__eq,axiom,
% 12.45/6.92 ! [A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 12.45/6.92 = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3454_minus__diff__eq,axiom,
% 12.45/6.92 ! [A: rat,B2: rat] :
% 12.45/6.92 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) )
% 12.45/6.92 = ( minus_minus_rat @ B2 @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3455_div__minus__minus,axiom,
% 12.45/6.92 ! [A: int,B2: int] :
% 12.45/6.92 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.92 = ( divide_divide_int @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3456_div__minus__minus,axiom,
% 12.45/6.92 ! [A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.92 = ( divide6298287555418463151nteger @ A @ B2 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3457_dvd__minus__iff,axiom,
% 12.45/6.92 ! [X4: real,Y3: real] :
% 12.45/6.92 ( ( dvd_dvd_real @ X4 @ ( uminus_uminus_real @ Y3 ) )
% 12.45/6.92 = ( dvd_dvd_real @ X4 @ Y3 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3458_dvd__minus__iff,axiom,
% 12.45/6.92 ! [X4: int,Y3: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ X4 @ ( uminus_uminus_int @ Y3 ) )
% 12.45/6.92 = ( dvd_dvd_int @ X4 @ Y3 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3459_dvd__minus__iff,axiom,
% 12.45/6.92 ! [X4: complex,Y3: complex] :
% 12.45/6.92 ( ( dvd_dvd_complex @ X4 @ ( uminus1482373934393186551omplex @ Y3 ) )
% 12.45/6.92 = ( dvd_dvd_complex @ X4 @ Y3 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3460_dvd__minus__iff,axiom,
% 12.45/6.92 ! [X4: code_integer,Y3: code_integer] :
% 12.45/6.92 ( ( dvd_dvd_Code_integer @ X4 @ ( uminus1351360451143612070nteger @ Y3 ) )
% 12.45/6.92 = ( dvd_dvd_Code_integer @ X4 @ Y3 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3461_dvd__minus__iff,axiom,
% 12.45/6.92 ! [X4: rat,Y3: rat] :
% 12.45/6.92 ( ( dvd_dvd_rat @ X4 @ ( uminus_uminus_rat @ Y3 ) )
% 12.45/6.92 = ( dvd_dvd_rat @ X4 @ Y3 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3462_minus__dvd__iff,axiom,
% 12.45/6.92 ! [X4: real,Y3: real] :
% 12.45/6.92 ( ( dvd_dvd_real @ ( uminus_uminus_real @ X4 ) @ Y3 )
% 12.45/6.92 = ( dvd_dvd_real @ X4 @ Y3 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3463_minus__dvd__iff,axiom,
% 12.45/6.92 ! [X4: int,Y3: int] :
% 12.45/6.92 ( ( dvd_dvd_int @ ( uminus_uminus_int @ X4 ) @ Y3 )
% 12.45/6.92 = ( dvd_dvd_int @ X4 @ Y3 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3464_minus__dvd__iff,axiom,
% 12.45/6.92 ! [X4: complex,Y3: complex] :
% 12.45/6.92 ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X4 ) @ Y3 )
% 12.45/6.92 = ( dvd_dvd_complex @ X4 @ Y3 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3465_minus__dvd__iff,axiom,
% 12.45/6.92 ! [X4: code_integer,Y3: code_integer] :
% 12.45/6.92 ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X4 ) @ Y3 )
% 12.45/6.92 = ( dvd_dvd_Code_integer @ X4 @ Y3 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3466_minus__dvd__iff,axiom,
% 12.45/6.92 ! [X4: rat,Y3: rat] :
% 12.45/6.92 ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X4 ) @ Y3 )
% 12.45/6.92 = ( dvd_dvd_rat @ X4 @ Y3 ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3467_mod__minus__minus,axiom,
% 12.45/6.92 ! [A: int,B2: int] :
% 12.45/6.92 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.92 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3468_mod__minus__minus,axiom,
% 12.45/6.92 ! [A: code_integer,B2: code_integer] :
% 12.45/6.92 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.92 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3469_real__add__minus__iff,axiom,
% 12.45/6.92 ! [X4: real,A: real] :
% 12.45/6.92 ( ( ( plus_plus_real @ X4 @ ( uminus_uminus_real @ A ) )
% 12.45/6.92 = zero_zero_real )
% 12.45/6.92 = ( X4 = A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3470_neg__less__eq__nonneg,axiom,
% 12.45/6.92 ! [A: real] :
% 12.45/6.92 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 12.45/6.92 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3471_neg__less__eq__nonneg,axiom,
% 12.45/6.92 ! [A: code_integer] :
% 12.45/6.92 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 12.45/6.92 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3472_neg__less__eq__nonneg,axiom,
% 12.45/6.92 ! [A: rat] :
% 12.45/6.92 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 12.45/6.92 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3473_neg__less__eq__nonneg,axiom,
% 12.45/6.92 ! [A: int] :
% 12.45/6.92 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 12.45/6.92 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3474_less__eq__neg__nonpos,axiom,
% 12.45/6.92 ! [A: real] :
% 12.45/6.92 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 12.45/6.92 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3475_less__eq__neg__nonpos,axiom,
% 12.45/6.92 ! [A: code_integer] :
% 12.45/6.92 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.92 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.45/6.92
% 12.45/6.92 thf(fact_3476_less__eq__neg__nonpos,axiom,
% 12.45/6.92 ! [A: rat] :
% 12.45/6.93 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 12.45/6.93 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3477_less__eq__neg__nonpos,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 12.45/6.93 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3478_neg__le__0__iff__le,axiom,
% 12.45/6.93 ! [A: real] :
% 12.45/6.93 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 12.45/6.93 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3479_neg__le__0__iff__le,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 12.45/6.93 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3480_neg__le__0__iff__le,axiom,
% 12.45/6.93 ! [A: rat] :
% 12.45/6.93 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 12.45/6.93 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3481_neg__le__0__iff__le,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 12.45/6.93 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3482_neg__0__le__iff__le,axiom,
% 12.45/6.93 ! [A: real] :
% 12.45/6.93 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 12.45/6.93 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3483_neg__0__le__iff__le,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.93 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3484_neg__0__le__iff__le,axiom,
% 12.45/6.93 ! [A: rat] :
% 12.45/6.93 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 12.45/6.93 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3485_neg__0__le__iff__le,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 12.45/6.93 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3486_neg__less__0__iff__less,axiom,
% 12.45/6.93 ! [A: real] :
% 12.45/6.93 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 12.45/6.93 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3487_neg__less__0__iff__less,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 12.45/6.93 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3488_neg__less__0__iff__less,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 12.45/6.93 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3489_neg__less__0__iff__less,axiom,
% 12.45/6.93 ! [A: rat] :
% 12.45/6.93 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 12.45/6.93 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3490_neg__0__less__iff__less,axiom,
% 12.45/6.93 ! [A: real] :
% 12.45/6.93 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 12.45/6.93 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3491_neg__0__less__iff__less,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 12.45/6.93 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3492_neg__0__less__iff__less,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.93 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3493_neg__0__less__iff__less,axiom,
% 12.45/6.93 ! [A: rat] :
% 12.45/6.93 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 12.45/6.93 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3494_neg__less__pos,axiom,
% 12.45/6.93 ! [A: real] :
% 12.45/6.93 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 12.45/6.93 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3495_neg__less__pos,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 12.45/6.93 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3496_neg__less__pos,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 12.45/6.93 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3497_neg__less__pos,axiom,
% 12.45/6.93 ! [A: rat] :
% 12.45/6.93 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 12.45/6.93 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3498_less__neg__neg,axiom,
% 12.45/6.93 ! [A: real] :
% 12.45/6.93 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 12.45/6.93 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3499_less__neg__neg,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 12.45/6.93 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3500_less__neg__neg,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.93 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3501_less__neg__neg,axiom,
% 12.45/6.93 ! [A: rat] :
% 12.45/6.93 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 12.45/6.93 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3502_ab__left__minus,axiom,
% 12.45/6.93 ! [A: real] :
% 12.45/6.93 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 12.45/6.93 = zero_zero_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3503_ab__left__minus,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 12.45/6.93 = zero_zero_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3504_ab__left__minus,axiom,
% 12.45/6.93 ! [A: complex] :
% 12.45/6.93 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 12.45/6.93 = zero_zero_complex ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3505_ab__left__minus,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 12.45/6.93 = zero_z3403309356797280102nteger ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3506_ab__left__minus,axiom,
% 12.45/6.93 ! [A: rat] :
% 12.45/6.93 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 12.45/6.93 = zero_zero_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3507_add_Oright__inverse,axiom,
% 12.45/6.93 ! [A: real] :
% 12.45/6.93 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 12.45/6.93 = zero_zero_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3508_add_Oright__inverse,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 12.45/6.93 = zero_zero_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3509_add_Oright__inverse,axiom,
% 12.45/6.93 ! [A: complex] :
% 12.45/6.93 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 12.45/6.93 = zero_zero_complex ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3510_add_Oright__inverse,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.93 = zero_z3403309356797280102nteger ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3511_add_Oright__inverse,axiom,
% 12.45/6.93 ! [A: rat] :
% 12.45/6.93 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 12.45/6.93 = zero_zero_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3512_diff__0,axiom,
% 12.45/6.93 ! [A: real] :
% 12.45/6.93 ( ( minus_minus_real @ zero_zero_real @ A )
% 12.45/6.93 = ( uminus_uminus_real @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3513_diff__0,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( minus_minus_int @ zero_zero_int @ A )
% 12.45/6.93 = ( uminus_uminus_int @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3514_diff__0,axiom,
% 12.45/6.93 ! [A: complex] :
% 12.45/6.93 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3515_diff__0,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3516_diff__0,axiom,
% 12.45/6.93 ! [A: rat] :
% 12.45/6.93 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 12.45/6.93 = ( uminus_uminus_rat @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3517_verit__minus__simplify_I3_J,axiom,
% 12.45/6.93 ! [B2: real] :
% 12.45/6.93 ( ( minus_minus_real @ zero_zero_real @ B2 )
% 12.45/6.93 = ( uminus_uminus_real @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3518_verit__minus__simplify_I3_J,axiom,
% 12.45/6.93 ! [B2: int] :
% 12.45/6.93 ( ( minus_minus_int @ zero_zero_int @ B2 )
% 12.45/6.93 = ( uminus_uminus_int @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3519_verit__minus__simplify_I3_J,axiom,
% 12.45/6.93 ! [B2: complex] :
% 12.45/6.93 ( ( minus_minus_complex @ zero_zero_complex @ B2 )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3520_verit__minus__simplify_I3_J,axiom,
% 12.45/6.93 ! [B2: code_integer] :
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B2 )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3521_verit__minus__simplify_I3_J,axiom,
% 12.45/6.93 ! [B2: rat] :
% 12.45/6.93 ( ( minus_minus_rat @ zero_zero_rat @ B2 )
% 12.45/6.93 = ( uminus_uminus_rat @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3522_add__neg__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.93 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3523_add__neg__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.93 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3524_add__neg__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3525_add__neg__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3526_add__neg__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.93 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3527_mult__minus1,axiom,
% 12.45/6.93 ! [Z: real] :
% 12.45/6.93 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 12.45/6.93 = ( uminus_uminus_real @ Z ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3528_mult__minus1,axiom,
% 12.45/6.93 ! [Z: int] :
% 12.45/6.93 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 12.45/6.93 = ( uminus_uminus_int @ Z ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3529_mult__minus1,axiom,
% 12.45/6.93 ! [Z: complex] :
% 12.45/6.93 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3530_mult__minus1,axiom,
% 12.45/6.93 ! [Z: code_integer] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3531_mult__minus1,axiom,
% 12.45/6.93 ! [Z: rat] :
% 12.45/6.93 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 12.45/6.93 = ( uminus_uminus_rat @ Z ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3532_mult__minus1__right,axiom,
% 12.45/6.93 ! [Z: real] :
% 12.45/6.93 ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.93 = ( uminus_uminus_real @ Z ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3533_mult__minus1__right,axiom,
% 12.45/6.93 ! [Z: int] :
% 12.45/6.93 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 = ( uminus_uminus_int @ Z ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3534_mult__minus1__right,axiom,
% 12.45/6.93 ! [Z: complex] :
% 12.45/6.93 ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3535_mult__minus1__right,axiom,
% 12.45/6.93 ! [Z: code_integer] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3536_mult__minus1__right,axiom,
% 12.45/6.93 ! [Z: rat] :
% 12.45/6.93 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.93 = ( uminus_uminus_rat @ Z ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3537_uminus__add__conv__diff,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 )
% 12.45/6.93 = ( minus_minus_real @ B2 @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3538_uminus__add__conv__diff,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.93 = ( minus_minus_int @ B2 @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3539_uminus__add__conv__diff,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 12.45/6.93 = ( minus_minus_complex @ B2 @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3540_uminus__add__conv__diff,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 12.45/6.93 = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3541_uminus__add__conv__diff,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 12.45/6.93 = ( minus_minus_rat @ B2 @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3542_diff__minus__eq__add,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B2 ) )
% 12.45/6.93 = ( plus_plus_real @ A @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3543_diff__minus__eq__add,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.93 = ( plus_plus_int @ A @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3544_diff__minus__eq__add,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 12.45/6.93 = ( plus_plus_complex @ A @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3545_diff__minus__eq__add,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.93 = ( plus_p5714425477246183910nteger @ A @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3546_diff__minus__eq__add,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 12.45/6.93 = ( plus_plus_rat @ A @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3547_div__minus1__right,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 = ( uminus_uminus_int @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3548_div__minus1__right,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3549_divide__minus1,axiom,
% 12.45/6.93 ! [X4: real] :
% 12.45/6.93 ( ( divide_divide_real @ X4 @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.93 = ( uminus_uminus_real @ X4 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3550_divide__minus1,axiom,
% 12.45/6.93 ! [X4: complex] :
% 12.45/6.93 ( ( divide1717551699836669952omplex @ X4 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ X4 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3551_divide__minus1,axiom,
% 12.45/6.93 ! [X4: rat] :
% 12.45/6.93 ( ( divide_divide_rat @ X4 @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.93 = ( uminus_uminus_rat @ X4 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3552_minus__mod__self1,axiom,
% 12.45/6.93 ! [B2: int,A: int] :
% 12.45/6.93 ( ( modulo_modulo_int @ ( minus_minus_int @ B2 @ A ) @ B2 )
% 12.45/6.93 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3553_minus__mod__self1,axiom,
% 12.45/6.93 ! [B2: code_integer,A: code_integer] :
% 12.45/6.93 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B2 @ A ) @ B2 )
% 12.45/6.93 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3554_signed__take__bit__of__minus__1,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3555_signed__take__bit__of__minus__1,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3556_dbl__simps_I1_J,axiom,
% 12.45/6.93 ! [K: num] :
% 12.45/6.93 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 12.45/6.93 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3557_dbl__simps_I1_J,axiom,
% 12.45/6.93 ! [K: num] :
% 12.45/6.93 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/6.93 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3558_dbl__simps_I1_J,axiom,
% 12.45/6.93 ! [K: num] :
% 12.45/6.93 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3559_dbl__simps_I1_J,axiom,
% 12.45/6.93 ! [K: num] :
% 12.45/6.93 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3560_dbl__simps_I1_J,axiom,
% 12.45/6.93 ! [K: num] :
% 12.45/6.93 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 12.45/6.93 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3561_triangle__Suc,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( nat_triangle @ ( suc @ N ) )
% 12.45/6.93 = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3562_add__neg__numeral__special_I8_J,axiom,
% 12.45/6.93 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 12.45/6.93 = zero_zero_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3563_add__neg__numeral__special_I8_J,axiom,
% 12.45/6.93 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 12.45/6.93 = zero_zero_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3564_add__neg__numeral__special_I8_J,axiom,
% 12.45/6.93 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 12.45/6.93 = zero_zero_complex ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3565_add__neg__numeral__special_I8_J,axiom,
% 12.45/6.93 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 12.45/6.93 = zero_z3403309356797280102nteger ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3566_add__neg__numeral__special_I8_J,axiom,
% 12.45/6.93 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 12.45/6.93 = zero_zero_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3567_add__neg__numeral__special_I7_J,axiom,
% 12.45/6.93 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.93 = zero_zero_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3568_add__neg__numeral__special_I7_J,axiom,
% 12.45/6.93 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 = zero_zero_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3569_add__neg__numeral__special_I7_J,axiom,
% 12.45/6.93 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.93 = zero_zero_complex ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3570_add__neg__numeral__special_I7_J,axiom,
% 12.45/6.93 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.93 = zero_z3403309356797280102nteger ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3571_add__neg__numeral__special_I7_J,axiom,
% 12.45/6.93 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.93 = zero_zero_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3572_diff__numeral__special_I12_J,axiom,
% 12.45/6.93 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.93 = zero_zero_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3573_diff__numeral__special_I12_J,axiom,
% 12.45/6.93 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 = zero_zero_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3574_diff__numeral__special_I12_J,axiom,
% 12.45/6.93 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.93 = zero_zero_complex ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3575_diff__numeral__special_I12_J,axiom,
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.93 = zero_z3403309356797280102nteger ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3576_diff__numeral__special_I12_J,axiom,
% 12.45/6.93 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.93 = zero_zero_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3577_neg__one__eq__numeral__iff,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( ( uminus_uminus_real @ one_one_real )
% 12.45/6.93 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.93 = ( N = one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3578_neg__one__eq__numeral__iff,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( ( uminus_uminus_int @ one_one_int )
% 12.45/6.93 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.93 = ( N = one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3579_neg__one__eq__numeral__iff,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.45/6.93 = ( N = one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3580_neg__one__eq__numeral__iff,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.93 = ( N = one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3581_neg__one__eq__numeral__iff,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( ( uminus_uminus_rat @ one_one_rat )
% 12.45/6.93 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.93 = ( N = one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3582_numeral__eq__neg__one__iff,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 12.45/6.93 = ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.93 = ( N = one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3583_numeral__eq__neg__one__iff,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 12.45/6.93 = ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 = ( N = one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3584_numeral__eq__neg__one__iff,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.93 = ( N = one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3585_numeral__eq__neg__one__iff,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.93 = ( N = one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3586_numeral__eq__neg__one__iff,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 12.45/6.93 = ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.93 = ( N = one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3587_minus__one__mult__self,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 12.45/6.93 = one_one_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3588_minus__one__mult__self,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 12.45/6.93 = one_one_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3589_minus__one__mult__self,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 12.45/6.93 = one_one_complex ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3590_minus__one__mult__self,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 12.45/6.93 = one_one_Code_integer ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3591_minus__one__mult__self,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 12.45/6.93 = one_one_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3592_left__minus__one__mult__self,axiom,
% 12.45/6.93 ! [N: nat,A: real] :
% 12.45/6.93 ( ( 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 ) )
% 12.45/6.93 = A ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3593_left__minus__one__mult__self,axiom,
% 12.45/6.93 ! [N: nat,A: int] :
% 12.45/6.93 ( ( 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 ) )
% 12.45/6.93 = A ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3594_left__minus__one__mult__self,axiom,
% 12.45/6.93 ! [N: nat,A: complex] :
% 12.45/6.93 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 12.45/6.93 = A ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3595_left__minus__one__mult__self,axiom,
% 12.45/6.93 ! [N: nat,A: code_integer] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 12.45/6.93 = A ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3596_left__minus__one__mult__self,axiom,
% 12.45/6.93 ! [N: nat,A: rat] :
% 12.45/6.93 ( ( 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 ) )
% 12.45/6.93 = A ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3597_mod__minus1__right,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 = zero_zero_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3598_mod__minus1__right,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.93 = zero_z3403309356797280102nteger ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3599_semiring__norm_I168_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: real] :
% 12.45/6.93 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3600_semiring__norm_I168_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: int] :
% 12.45/6.93 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3601_semiring__norm_I168_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: complex] :
% 12.45/6.93 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3602_semiring__norm_I168_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: code_integer] :
% 12.45/6.93 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3603_semiring__norm_I168_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: rat] :
% 12.45/6.93 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3604_diff__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 12.45/6.93 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3605_diff__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.45/6.93 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3606_diff__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3607_diff__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3608_diff__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 12.45/6.93 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3609_diff__numeral__simps_I2_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.93 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3610_diff__numeral__simps_I2_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.93 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3611_diff__numeral__simps_I2_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.45/6.93 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3612_diff__numeral__simps_I2_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.93 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3613_diff__numeral__simps_I2_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.93 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3614_semiring__norm_I172_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: real] :
% 12.45/6.93 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3615_semiring__norm_I172_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: int] :
% 12.45/6.93 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3616_semiring__norm_I172_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: complex] :
% 12.45/6.93 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3617_semiring__norm_I172_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: code_integer] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3618_semiring__norm_I172_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: rat] :
% 12.45/6.93 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3619_semiring__norm_I171_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: real] :
% 12.45/6.93 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3620_semiring__norm_I171_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: int] :
% 12.45/6.93 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3621_semiring__norm_I171_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: complex] :
% 12.45/6.93 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3622_semiring__norm_I171_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: code_integer] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3623_semiring__norm_I171_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: rat] :
% 12.45/6.93 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 12.45/6.93 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3624_semiring__norm_I170_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: real] :
% 12.45/6.93 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
% 12.45/6.93 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3625_semiring__norm_I170_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: int] :
% 12.45/6.93 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
% 12.45/6.93 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3626_semiring__norm_I170_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: complex] :
% 12.45/6.93 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y3 ) )
% 12.45/6.93 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3627_semiring__norm_I170_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: code_integer] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y3 ) )
% 12.45/6.93 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3628_semiring__norm_I170_J,axiom,
% 12.45/6.93 ! [V: num,W: num,Y3: rat] :
% 12.45/6.93 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y3 ) )
% 12.45/6.93 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3629_mult__neg__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.93 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3630_mult__neg__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.93 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3631_mult__neg__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3632_mult__neg__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3633_mult__neg__numeral__simps_I3_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.93 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3634_mult__neg__numeral__simps_I2_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 12.45/6.93 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3635_mult__neg__numeral__simps_I2_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.45/6.93 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3636_mult__neg__numeral__simps_I2_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3637_mult__neg__numeral__simps_I2_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3638_mult__neg__numeral__simps_I2_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 12.45/6.93 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3639_mult__neg__numeral__simps_I1_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.93 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3640_mult__neg__numeral__simps_I1_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.93 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3641_mult__neg__numeral__simps_I1_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.45/6.93 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3642_mult__neg__numeral__simps_I1_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.93 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3643_mult__neg__numeral__simps_I1_J,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.93 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3644_neg__numeral__le__iff,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.93 = ( ord_less_eq_num @ N @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3645_neg__numeral__le__iff,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.93 = ( ord_less_eq_num @ N @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3646_neg__numeral__le__iff,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.93 = ( ord_less_eq_num @ N @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3647_neg__numeral__le__iff,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.93 = ( ord_less_eq_num @ N @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3648_neg__numeral__less__iff,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.93 = ( ord_less_num @ N @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3649_neg__numeral__less__iff,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.93 = ( ord_less_num @ N @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3650_neg__numeral__less__iff,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.93 = ( ord_less_num @ N @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3651_neg__numeral__less__iff,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.93 = ( ord_less_num @ N @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3652_not__neg__one__le__neg__numeral__iff,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 12.45/6.93 = ( M != one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3653_not__neg__one__le__neg__numeral__iff,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 12.45/6.93 = ( M != one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3654_not__neg__one__le__neg__numeral__iff,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 12.45/6.93 = ( M != one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3655_not__neg__one__le__neg__numeral__iff,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 12.45/6.93 = ( M != one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3656_divide__le__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [B2: real,W: num,A: real] :
% 12.45/6.93 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 12.45/6.93 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3657_divide__le__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [B2: rat,W: num,A: rat] :
% 12.45/6.93 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 12.45/6.93 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3658_le__divide__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [A: real,B2: real,W: num] :
% 12.45/6.93 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 12.45/6.93 = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3659_le__divide__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [A: rat,B2: rat,W: num] :
% 12.45/6.93 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 12.45/6.93 = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3660_divide__eq__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [B2: real,W: num,A: real] :
% 12.45/6.93 ( ( ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.93 = A )
% 12.45/6.93 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.45/6.93 != zero_zero_real )
% 12.45/6.93 => ( B2
% 12.45/6.93 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 12.45/6.93 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.45/6.93 = zero_zero_real )
% 12.45/6.93 => ( A = zero_zero_real ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3661_divide__eq__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [B2: complex,W: num,A: complex] :
% 12.45/6.93 ( ( ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.45/6.93 = A )
% 12.45/6.93 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.45/6.93 != zero_zero_complex )
% 12.45/6.93 => ( B2
% 12.45/6.93 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 12.45/6.93 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.45/6.93 = zero_zero_complex )
% 12.45/6.93 => ( A = zero_zero_complex ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3662_divide__eq__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [B2: rat,W: num,A: rat] :
% 12.45/6.93 ( ( ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.45/6.93 = A )
% 12.45/6.93 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.45/6.93 != zero_zero_rat )
% 12.45/6.93 => ( B2
% 12.45/6.93 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 12.45/6.93 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.45/6.93 = zero_zero_rat )
% 12.45/6.93 => ( A = zero_zero_rat ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3663_eq__divide__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [A: real,B2: real,W: num] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 12.45/6.93 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.45/6.93 != zero_zero_real )
% 12.45/6.93 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.93 = B2 ) )
% 12.45/6.93 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.45/6.93 = zero_zero_real )
% 12.45/6.93 => ( A = zero_zero_real ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3664_eq__divide__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [A: complex,B2: complex,W: num] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 12.45/6.93 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.45/6.93 != zero_zero_complex )
% 12.45/6.93 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.45/6.93 = B2 ) )
% 12.45/6.93 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.45/6.93 = zero_zero_complex )
% 12.45/6.93 => ( A = zero_zero_complex ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3665_eq__divide__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [A: rat,B2: rat,W: num] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 12.45/6.93 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.45/6.93 != zero_zero_rat )
% 12.45/6.93 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.45/6.93 = B2 ) )
% 12.45/6.93 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.45/6.93 = zero_zero_rat )
% 12.45/6.93 => ( A = zero_zero_rat ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3666_neg__numeral__less__neg__one__iff,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.93 = ( M != one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3667_neg__numeral__less__neg__one__iff,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 = ( M != one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3668_neg__numeral__less__neg__one__iff,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.93 = ( M != one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3669_neg__numeral__less__neg__one__iff,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.93 = ( M != one ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3670_divide__less__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [B2: real,W: num,A: real] :
% 12.45/6.93 ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 12.45/6.93 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3671_divide__less__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [B2: rat,W: num,A: rat] :
% 12.45/6.93 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 12.45/6.93 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3672_less__divide__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [A: real,B2: real,W: num] :
% 12.45/6.93 ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 12.45/6.93 = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3673_less__divide__eq__numeral1_I2_J,axiom,
% 12.45/6.93 ! [A: rat,B2: rat,W: num] :
% 12.45/6.93 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 12.45/6.93 = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3674_power2__minus,axiom,
% 12.45/6.93 ! [A: real] :
% 12.45/6.93 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.93 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3675_power2__minus,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.93 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3676_power2__minus,axiom,
% 12.45/6.93 ! [A: complex] :
% 12.45/6.93 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.93 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3677_power2__minus,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.93 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3678_power2__minus,axiom,
% 12.45/6.93 ! [A: rat] :
% 12.45/6.93 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.93 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3679_add__neg__numeral__special_I9_J,axiom,
% 12.45/6.93 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.93 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3680_add__neg__numeral__special_I9_J,axiom,
% 12.45/6.93 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3681_add__neg__numeral__special_I9_J,axiom,
% 12.45/6.93 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3682_add__neg__numeral__special_I9_J,axiom,
% 12.45/6.93 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3683_add__neg__numeral__special_I9_J,axiom,
% 12.45/6.93 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.93 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3684_diff__numeral__special_I11_J,axiom,
% 12.45/6.93 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.93 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3685_diff__numeral__special_I11_J,axiom,
% 12.45/6.93 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3686_diff__numeral__special_I11_J,axiom,
% 12.45/6.93 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.93 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3687_diff__numeral__special_I11_J,axiom,
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.93 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3688_diff__numeral__special_I11_J,axiom,
% 12.45/6.93 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.93 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3689_diff__numeral__special_I10_J,axiom,
% 12.45/6.93 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 12.45/6.93 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3690_diff__numeral__special_I10_J,axiom,
% 12.45/6.93 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 12.45/6.93 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3691_diff__numeral__special_I10_J,axiom,
% 12.45/6.93 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3692_diff__numeral__special_I10_J,axiom,
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3693_diff__numeral__special_I10_J,axiom,
% 12.45/6.93 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 12.45/6.93 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3694_minus__1__div__2__eq,axiom,
% 12.45/6.93 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.93 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3695_minus__1__div__2__eq,axiom,
% 12.45/6.93 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3696_minus__1__mod__2__eq,axiom,
% 12.45/6.93 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.93 = one_one_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3697_minus__1__mod__2__eq,axiom,
% 12.45/6.93 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.93 = one_one_Code_integer ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3698_bits__minus__1__mod__2__eq,axiom,
% 12.45/6.93 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.93 = one_one_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3699_bits__minus__1__mod__2__eq,axiom,
% 12.45/6.93 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.93 = one_one_Code_integer ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3700_Power_Oring__1__class_Opower__minus__even,axiom,
% 12.45/6.93 ! [A: real,N: nat] :
% 12.45/6.93 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.93 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3701_Power_Oring__1__class_Opower__minus__even,axiom,
% 12.45/6.93 ! [A: int,N: nat] :
% 12.45/6.93 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.93 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3702_Power_Oring__1__class_Opower__minus__even,axiom,
% 12.45/6.93 ! [A: complex,N: nat] :
% 12.45/6.93 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.93 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3703_Power_Oring__1__class_Opower__minus__even,axiom,
% 12.45/6.93 ! [A: code_integer,N: nat] :
% 12.45/6.93 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.93 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3704_Power_Oring__1__class_Opower__minus__even,axiom,
% 12.45/6.93 ! [A: rat,N: nat] :
% 12.45/6.93 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.93 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3705_power__minus__odd,axiom,
% 12.45/6.93 ! [N: nat,A: real] :
% 12.45/6.93 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 12.45/6.93 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3706_power__minus__odd,axiom,
% 12.45/6.93 ! [N: nat,A: int] :
% 12.45/6.93 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 12.45/6.93 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3707_power__minus__odd,axiom,
% 12.45/6.93 ! [N: nat,A: complex] :
% 12.45/6.93 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3708_power__minus__odd,axiom,
% 12.45/6.93 ! [N: nat,A: code_integer] :
% 12.45/6.93 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3709_power__minus__odd,axiom,
% 12.45/6.93 ! [N: nat,A: rat] :
% 12.45/6.93 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 12.45/6.93 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3710_Parity_Oring__1__class_Opower__minus__even,axiom,
% 12.45/6.93 ! [N: nat,A: real] :
% 12.45/6.93 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 12.45/6.93 = ( power_power_real @ A @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3711_Parity_Oring__1__class_Opower__minus__even,axiom,
% 12.45/6.93 ! [N: nat,A: int] :
% 12.45/6.93 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 12.45/6.93 = ( power_power_int @ A @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3712_Parity_Oring__1__class_Opower__minus__even,axiom,
% 12.45/6.93 ! [N: nat,A: complex] :
% 12.45/6.93 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 12.45/6.93 = ( power_power_complex @ A @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3713_Parity_Oring__1__class_Opower__minus__even,axiom,
% 12.45/6.93 ! [N: nat,A: code_integer] :
% 12.45/6.93 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 12.45/6.93 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3714_Parity_Oring__1__class_Opower__minus__even,axiom,
% 12.45/6.93 ! [N: nat,A: rat] :
% 12.45/6.93 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 12.45/6.93 = ( power_power_rat @ A @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3715_diff__numeral__special_I3_J,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.93 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3716_diff__numeral__special_I3_J,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.93 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3717_diff__numeral__special_I3_J,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.45/6.93 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3718_diff__numeral__special_I3_J,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.93 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3719_diff__numeral__special_I3_J,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.93 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3720_diff__numeral__special_I4_J,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 12.45/6.93 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3721_diff__numeral__special_I4_J,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 12.45/6.93 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3722_diff__numeral__special_I4_J,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3723_diff__numeral__special_I4_J,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3724_diff__numeral__special_I4_J,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 12.45/6.93 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3725_signed__take__bit__Suc__minus__bit0,axiom,
% 12.45/6.93 ! [N: nat,K: num] :
% 12.45/6.93 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.45/6.93 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3726_dbl__simps_I4_J,axiom,
% 12.45/6.93 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.93 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3727_dbl__simps_I4_J,axiom,
% 12.45/6.93 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3728_dbl__simps_I4_J,axiom,
% 12.45/6.93 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3729_dbl__simps_I4_J,axiom,
% 12.45/6.93 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3730_dbl__simps_I4_J,axiom,
% 12.45/6.93 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.93 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3731_power__minus1__even,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.93 = one_one_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3732_power__minus1__even,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.93 = one_one_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3733_power__minus1__even,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.93 = one_one_complex ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3734_power__minus1__even,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.93 = one_one_Code_integer ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3735_power__minus1__even,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.93 = one_one_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3736_neg__one__odd__power,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 12.45/6.93 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3737_neg__one__odd__power,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 12.45/6.93 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3738_neg__one__odd__power,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3739_neg__one__odd__power,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3740_neg__one__odd__power,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 12.45/6.93 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3741_neg__one__even__power,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 12.45/6.93 = one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3742_neg__one__even__power,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 12.45/6.93 = one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3743_neg__one__even__power,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 12.45/6.93 = one_one_complex ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3744_neg__one__even__power,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 12.45/6.93 = one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3745_neg__one__even__power,axiom,
% 12.45/6.93 ! [N: nat] :
% 12.45/6.93 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.93 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 12.45/6.93 = one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3746_signed__take__bit__0,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3747_signed__take__bit__0,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 12.45/6.93 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3748_verit__negate__coefficient_I3_J,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( A = B2 )
% 12.45/6.93 => ( ( uminus_uminus_real @ A )
% 12.45/6.93 = ( uminus_uminus_real @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3749_verit__negate__coefficient_I3_J,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( A = B2 )
% 12.45/6.93 => ( ( uminus_uminus_int @ A )
% 12.45/6.93 = ( uminus_uminus_int @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3750_verit__negate__coefficient_I3_J,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( A = B2 )
% 12.45/6.93 => ( ( uminus1351360451143612070nteger @ A )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3751_verit__negate__coefficient_I3_J,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( A = B2 )
% 12.45/6.93 => ( ( uminus_uminus_rat @ A )
% 12.45/6.93 = ( uminus_uminus_rat @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3752_equation__minus__iff,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus_uminus_real @ B2 ) )
% 12.45/6.93 = ( B2
% 12.45/6.93 = ( uminus_uminus_real @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3753_equation__minus__iff,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus_uminus_int @ B2 ) )
% 12.45/6.93 = ( B2
% 12.45/6.93 = ( uminus_uminus_int @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3754_equation__minus__iff,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus1482373934393186551omplex @ B2 ) )
% 12.45/6.93 = ( B2
% 12.45/6.93 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3755_equation__minus__iff,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.93 = ( B2
% 12.45/6.93 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3756_equation__minus__iff,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus_uminus_rat @ B2 ) )
% 12.45/6.93 = ( B2
% 12.45/6.93 = ( uminus_uminus_rat @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3757_minus__equation__iff,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ( uminus_uminus_real @ A )
% 12.45/6.93 = B2 )
% 12.45/6.93 = ( ( uminus_uminus_real @ B2 )
% 12.45/6.93 = A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3758_minus__equation__iff,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( ( uminus_uminus_int @ A )
% 12.45/6.93 = B2 )
% 12.45/6.93 = ( ( uminus_uminus_int @ B2 )
% 12.45/6.93 = A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3759_minus__equation__iff,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( ( uminus1482373934393186551omplex @ A )
% 12.45/6.93 = B2 )
% 12.45/6.93 = ( ( uminus1482373934393186551omplex @ B2 )
% 12.45/6.93 = A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3760_minus__equation__iff,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( ( uminus1351360451143612070nteger @ A )
% 12.45/6.93 = B2 )
% 12.45/6.93 = ( ( uminus1351360451143612070nteger @ B2 )
% 12.45/6.93 = A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3761_minus__equation__iff,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ( uminus_uminus_rat @ A )
% 12.45/6.93 = B2 )
% 12.45/6.93 = ( ( uminus_uminus_rat @ B2 )
% 12.45/6.93 = A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3762_le__imp__neg__le,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.93 => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3763_le__imp__neg__le,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 12.45/6.93 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3764_le__imp__neg__le,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.93 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3765_le__imp__neg__le,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.93 => ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3766_minus__le__iff,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
% 12.45/6.93 = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3767_minus__le__iff,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 12.45/6.93 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3768_minus__le__iff,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 12.45/6.93 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3769_minus__le__iff,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.93 = ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3770_le__minus__iff,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B2 ) )
% 12.45/6.93 = ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3771_le__minus__iff,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.93 = ( ord_le3102999989581377725nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3772_le__minus__iff,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 12.45/6.93 = ( ord_less_eq_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3773_le__minus__iff,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.93 = ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3774_minus__less__iff,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 )
% 12.45/6.93 = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3775_minus__less__iff,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.93 = ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3776_minus__less__iff,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 12.45/6.93 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3777_minus__less__iff,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 12.45/6.93 = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3778_less__minus__iff,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B2 ) )
% 12.45/6.93 = ( ord_less_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3779_less__minus__iff,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.93 = ( ord_less_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3780_less__minus__iff,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.93 = ( ord_le6747313008572928689nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3781_less__minus__iff,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 12.45/6.93 = ( ord_less_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3782_verit__negate__coefficient_I2_J,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ord_less_real @ A @ B2 )
% 12.45/6.93 => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3783_verit__negate__coefficient_I2_J,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( ord_less_int @ A @ B2 )
% 12.45/6.93 => ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3784_verit__negate__coefficient_I2_J,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 12.45/6.93 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3785_verit__negate__coefficient_I2_J,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.93 => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3786_neg__numeral__neq__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 12.45/6.93 != ( numeral_numeral_real @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3787_neg__numeral__neq__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 12.45/6.93 != ( numeral_numeral_int @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3788_neg__numeral__neq__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 12.45/6.93 != ( numera6690914467698888265omplex @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3789_neg__numeral__neq__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 12.45/6.93 != ( numera6620942414471956472nteger @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3790_neg__numeral__neq__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 12.45/6.93 != ( numeral_numeral_rat @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3791_numeral__neq__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( numeral_numeral_real @ M )
% 12.45/6.93 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3792_numeral__neq__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( numeral_numeral_int @ M )
% 12.45/6.93 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3793_numeral__neq__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( numera6690914467698888265omplex @ M )
% 12.45/6.93 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3794_numeral__neq__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( numera6620942414471956472nteger @ M )
% 12.45/6.93 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3795_numeral__neq__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ( ( numeral_numeral_rat @ M )
% 12.45/6.93 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3796_minus__mult__commute,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
% 12.45/6.93 = ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3797_minus__mult__commute,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.93 = ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3798_minus__mult__commute,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 12.45/6.93 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3799_minus__mult__commute,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 12.45/6.93 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3800_minus__mult__commute,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 12.45/6.93 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3801_square__eq__iff,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ( times_times_real @ A @ A )
% 12.45/6.93 = ( times_times_real @ B2 @ B2 ) )
% 12.45/6.93 = ( ( A = B2 )
% 12.45/6.93 | ( A
% 12.45/6.93 = ( uminus_uminus_real @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3802_square__eq__iff,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( ( times_times_int @ A @ A )
% 12.45/6.93 = ( times_times_int @ B2 @ B2 ) )
% 12.45/6.93 = ( ( A = B2 )
% 12.45/6.93 | ( A
% 12.45/6.93 = ( uminus_uminus_int @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3803_square__eq__iff,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( ( times_times_complex @ A @ A )
% 12.45/6.93 = ( times_times_complex @ B2 @ B2 ) )
% 12.45/6.93 = ( ( A = B2 )
% 12.45/6.93 | ( A
% 12.45/6.93 = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3804_square__eq__iff,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( ( times_3573771949741848930nteger @ A @ A )
% 12.45/6.93 = ( times_3573771949741848930nteger @ B2 @ B2 ) )
% 12.45/6.93 = ( ( A = B2 )
% 12.45/6.93 | ( A
% 12.45/6.93 = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3805_square__eq__iff,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ( times_times_rat @ A @ A )
% 12.45/6.93 = ( times_times_rat @ B2 @ B2 ) )
% 12.45/6.93 = ( ( A = B2 )
% 12.45/6.93 | ( A
% 12.45/6.93 = ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3806_one__neq__neg__one,axiom,
% 12.45/6.93 ( one_one_real
% 12.45/6.93 != ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3807_one__neq__neg__one,axiom,
% 12.45/6.93 ( one_one_int
% 12.45/6.93 != ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3808_one__neq__neg__one,axiom,
% 12.45/6.93 ( one_one_complex
% 12.45/6.93 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3809_one__neq__neg__one,axiom,
% 12.45/6.93 ( one_one_Code_integer
% 12.45/6.93 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3810_one__neq__neg__one,axiom,
% 12.45/6.93 ( one_one_rat
% 12.45/6.93 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3811_add_Oinverse__distrib__swap,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 12.45/6.93 = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3812_add_Oinverse__distrib__swap,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 12.45/6.93 = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3813_add_Oinverse__distrib__swap,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 12.45/6.93 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3814_add_Oinverse__distrib__swap,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 12.45/6.93 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3815_add_Oinverse__distrib__swap,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 12.45/6.93 = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3816_group__cancel_Oneg1,axiom,
% 12.45/6.93 ! [A3: real,K: real,A: real] :
% 12.45/6.93 ( ( A3
% 12.45/6.93 = ( plus_plus_real @ K @ A ) )
% 12.45/6.93 => ( ( uminus_uminus_real @ A3 )
% 12.45/6.93 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3817_group__cancel_Oneg1,axiom,
% 12.45/6.93 ! [A3: int,K: int,A: int] :
% 12.45/6.93 ( ( A3
% 12.45/6.93 = ( plus_plus_int @ K @ A ) )
% 12.45/6.93 => ( ( uminus_uminus_int @ A3 )
% 12.45/6.93 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3818_group__cancel_Oneg1,axiom,
% 12.45/6.93 ! [A3: complex,K: complex,A: complex] :
% 12.45/6.93 ( ( A3
% 12.45/6.93 = ( plus_plus_complex @ K @ A ) )
% 12.45/6.93 => ( ( uminus1482373934393186551omplex @ A3 )
% 12.45/6.93 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3819_group__cancel_Oneg1,axiom,
% 12.45/6.93 ! [A3: code_integer,K: code_integer,A: code_integer] :
% 12.45/6.93 ( ( A3
% 12.45/6.93 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 12.45/6.93 => ( ( uminus1351360451143612070nteger @ A3 )
% 12.45/6.93 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3820_group__cancel_Oneg1,axiom,
% 12.45/6.93 ! [A3: rat,K: rat,A: rat] :
% 12.45/6.93 ( ( A3
% 12.45/6.93 = ( plus_plus_rat @ K @ A ) )
% 12.45/6.93 => ( ( uminus_uminus_rat @ A3 )
% 12.45/6.93 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3821_is__num__normalize_I8_J,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 12.45/6.93 = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3822_is__num__normalize_I8_J,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 12.45/6.93 = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3823_is__num__normalize_I8_J,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 12.45/6.93 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3824_is__num__normalize_I8_J,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 12.45/6.93 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3825_is__num__normalize_I8_J,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 12.45/6.93 = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3826_minus__diff__commute,axiom,
% 12.45/6.93 ! [B2: real,A: real] :
% 12.45/6.93 ( ( minus_minus_real @ ( uminus_uminus_real @ B2 ) @ A )
% 12.45/6.93 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3827_minus__diff__commute,axiom,
% 12.45/6.93 ! [B2: int,A: int] :
% 12.45/6.93 ( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A )
% 12.45/6.93 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3828_minus__diff__commute,axiom,
% 12.45/6.93 ! [B2: complex,A: complex] :
% 12.45/6.93 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ A )
% 12.45/6.93 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3829_minus__diff__commute,axiom,
% 12.45/6.93 ! [B2: code_integer,A: code_integer] :
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A )
% 12.45/6.93 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3830_minus__diff__commute,axiom,
% 12.45/6.93 ! [B2: rat,A: rat] :
% 12.45/6.93 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B2 ) @ A )
% 12.45/6.93 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3831_minus__diff__minus,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 12.45/6.93 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3832_minus__diff__minus,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.93 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3833_minus__diff__minus,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3834_minus__diff__minus,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3835_minus__diff__minus,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 12.45/6.93 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3836_div__minus__right,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.93 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3837_div__minus__right,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.93 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3838_minus__divide__right,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.93 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3839_minus__divide__right,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.45/6.93 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3840_minus__divide__right,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 12.45/6.93 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3841_minus__divide__divide,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 12.45/6.93 = ( divide_divide_real @ A @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3842_minus__divide__divide,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 12.45/6.93 = ( divide1717551699836669952omplex @ A @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3843_minus__divide__divide,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 12.45/6.93 = ( divide_divide_rat @ A @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3844_minus__divide__left,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.93 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3845_minus__divide__left,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.45/6.93 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3846_minus__divide__left,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 12.45/6.93 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3847_mod__minus__eq,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) @ B2 )
% 12.45/6.93 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3848_mod__minus__eq,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ B2 )
% 12.45/6.93 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3849_mod__minus__cong,axiom,
% 12.45/6.93 ! [A: int,B2: int,A2: int] :
% 12.45/6.93 ( ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.93 = ( modulo_modulo_int @ A2 @ B2 ) )
% 12.45/6.93 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.93 = ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3850_mod__minus__cong,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer,A2: code_integer] :
% 12.45/6.93 ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 12.45/6.93 = ( modulo364778990260209775nteger @ A2 @ B2 ) )
% 12.45/6.93 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 12.45/6.93 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3851_mod__minus__right,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.93 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3852_mod__minus__right,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3853_signed__take__bit__minus,axiom,
% 12.45/6.93 ! [N: nat,K: int] :
% 12.45/6.93 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 12.45/6.93 = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3854_neg__numeral__le__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3855_neg__numeral__le__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3856_neg__numeral__le__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3857_neg__numeral__le__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3858_not__numeral__le__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3859_not__numeral__le__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3860_not__numeral__le__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3861_not__numeral__le__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3862_zero__neq__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( zero_zero_real
% 12.45/6.93 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3863_zero__neq__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( zero_zero_int
% 12.45/6.93 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3864_zero__neq__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( zero_zero_complex
% 12.45/6.93 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3865_zero__neq__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( zero_z3403309356797280102nteger
% 12.45/6.93 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3866_zero__neq__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( zero_zero_rat
% 12.45/6.93 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3867_not__numeral__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3868_not__numeral__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3869_not__numeral__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3870_not__numeral__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] :
% 12.45/6.93 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3871_neg__numeral__less__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3872_neg__numeral__less__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3873_neg__numeral__less__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3874_neg__numeral__less__numeral,axiom,
% 12.45/6.93 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3875_le__minus__one__simps_I2_J,axiom,
% 12.45/6.93 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3876_le__minus__one__simps_I2_J,axiom,
% 12.45/6.93 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3877_le__minus__one__simps_I2_J,axiom,
% 12.45/6.93 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3878_le__minus__one__simps_I2_J,axiom,
% 12.45/6.93 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3879_le__minus__one__simps_I4_J,axiom,
% 12.45/6.93 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3880_le__minus__one__simps_I4_J,axiom,
% 12.45/6.93 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3881_le__minus__one__simps_I4_J,axiom,
% 12.45/6.93 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3882_le__minus__one__simps_I4_J,axiom,
% 12.45/6.93 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3883_zero__neq__neg__one,axiom,
% 12.45/6.93 ( zero_zero_real
% 12.45/6.93 != ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3884_zero__neq__neg__one,axiom,
% 12.45/6.93 ( zero_zero_int
% 12.45/6.93 != ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3885_zero__neq__neg__one,axiom,
% 12.45/6.93 ( zero_zero_complex
% 12.45/6.93 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3886_zero__neq__neg__one,axiom,
% 12.45/6.93 ( zero_z3403309356797280102nteger
% 12.45/6.93 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3887_zero__neq__neg__one,axiom,
% 12.45/6.93 ( zero_zero_rat
% 12.45/6.93 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3888_neg__eq__iff__add__eq__0,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ( uminus_uminus_real @ A )
% 12.45/6.93 = B2 )
% 12.45/6.93 = ( ( plus_plus_real @ A @ B2 )
% 12.45/6.93 = zero_zero_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3889_neg__eq__iff__add__eq__0,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( ( uminus_uminus_int @ A )
% 12.45/6.93 = B2 )
% 12.45/6.93 = ( ( plus_plus_int @ A @ B2 )
% 12.45/6.93 = zero_zero_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3890_neg__eq__iff__add__eq__0,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( ( uminus1482373934393186551omplex @ A )
% 12.45/6.93 = B2 )
% 12.45/6.93 = ( ( plus_plus_complex @ A @ B2 )
% 12.45/6.93 = zero_zero_complex ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3891_neg__eq__iff__add__eq__0,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( ( uminus1351360451143612070nteger @ A )
% 12.45/6.93 = B2 )
% 12.45/6.93 = ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 12.45/6.93 = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3892_neg__eq__iff__add__eq__0,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ( uminus_uminus_rat @ A )
% 12.45/6.93 = B2 )
% 12.45/6.93 = ( ( plus_plus_rat @ A @ B2 )
% 12.45/6.93 = zero_zero_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3893_eq__neg__iff__add__eq__0,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus_uminus_real @ B2 ) )
% 12.45/6.93 = ( ( plus_plus_real @ A @ B2 )
% 12.45/6.93 = zero_zero_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3894_eq__neg__iff__add__eq__0,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus_uminus_int @ B2 ) )
% 12.45/6.93 = ( ( plus_plus_int @ A @ B2 )
% 12.45/6.93 = zero_zero_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3895_eq__neg__iff__add__eq__0,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus1482373934393186551omplex @ B2 ) )
% 12.45/6.93 = ( ( plus_plus_complex @ A @ B2 )
% 12.45/6.93 = zero_zero_complex ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3896_eq__neg__iff__add__eq__0,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.93 = ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 12.45/6.93 = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3897_eq__neg__iff__add__eq__0,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus_uminus_rat @ B2 ) )
% 12.45/6.93 = ( ( plus_plus_rat @ A @ B2 )
% 12.45/6.93 = zero_zero_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3898_add_Oinverse__unique,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ( plus_plus_real @ A @ B2 )
% 12.45/6.93 = zero_zero_real )
% 12.45/6.93 => ( ( uminus_uminus_real @ A )
% 12.45/6.93 = B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3899_add_Oinverse__unique,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( ( plus_plus_int @ A @ B2 )
% 12.45/6.93 = zero_zero_int )
% 12.45/6.93 => ( ( uminus_uminus_int @ A )
% 12.45/6.93 = B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3900_add_Oinverse__unique,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( ( plus_plus_complex @ A @ B2 )
% 12.45/6.93 = zero_zero_complex )
% 12.45/6.93 => ( ( uminus1482373934393186551omplex @ A )
% 12.45/6.93 = B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3901_add_Oinverse__unique,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 12.45/6.93 = zero_z3403309356797280102nteger )
% 12.45/6.93 => ( ( uminus1351360451143612070nteger @ A )
% 12.45/6.93 = B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3902_add_Oinverse__unique,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ( plus_plus_rat @ A @ B2 )
% 12.45/6.93 = zero_zero_rat )
% 12.45/6.93 => ( ( uminus_uminus_rat @ A )
% 12.45/6.93 = B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3903_ab__group__add__class_Oab__left__minus,axiom,
% 12.45/6.93 ! [A: real] :
% 12.45/6.93 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 12.45/6.93 = zero_zero_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3904_ab__group__add__class_Oab__left__minus,axiom,
% 12.45/6.93 ! [A: int] :
% 12.45/6.93 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 12.45/6.93 = zero_zero_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3905_ab__group__add__class_Oab__left__minus,axiom,
% 12.45/6.93 ! [A: complex] :
% 12.45/6.93 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 12.45/6.93 = zero_zero_complex ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3906_ab__group__add__class_Oab__left__minus,axiom,
% 12.45/6.93 ! [A: code_integer] :
% 12.45/6.93 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 12.45/6.93 = zero_z3403309356797280102nteger ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3907_ab__group__add__class_Oab__left__minus,axiom,
% 12.45/6.93 ! [A: rat] :
% 12.45/6.93 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 12.45/6.93 = zero_zero_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3908_add__eq__0__iff,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ( plus_plus_real @ A @ B2 )
% 12.45/6.93 = zero_zero_real )
% 12.45/6.93 = ( B2
% 12.45/6.93 = ( uminus_uminus_real @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3909_add__eq__0__iff,axiom,
% 12.45/6.93 ! [A: int,B2: int] :
% 12.45/6.93 ( ( ( plus_plus_int @ A @ B2 )
% 12.45/6.93 = zero_zero_int )
% 12.45/6.93 = ( B2
% 12.45/6.93 = ( uminus_uminus_int @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3910_add__eq__0__iff,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( ( plus_plus_complex @ A @ B2 )
% 12.45/6.93 = zero_zero_complex )
% 12.45/6.93 = ( B2
% 12.45/6.93 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3911_add__eq__0__iff,axiom,
% 12.45/6.93 ! [A: code_integer,B2: code_integer] :
% 12.45/6.93 ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 12.45/6.93 = zero_z3403309356797280102nteger )
% 12.45/6.93 = ( B2
% 12.45/6.93 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3912_add__eq__0__iff,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ( plus_plus_rat @ A @ B2 )
% 12.45/6.93 = zero_zero_rat )
% 12.45/6.93 = ( B2
% 12.45/6.93 = ( uminus_uminus_rat @ A ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3913_less__minus__one__simps_I2_J,axiom,
% 12.45/6.93 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3914_less__minus__one__simps_I2_J,axiom,
% 12.45/6.93 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3915_less__minus__one__simps_I2_J,axiom,
% 12.45/6.93 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3916_less__minus__one__simps_I2_J,axiom,
% 12.45/6.93 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3917_less__minus__one__simps_I4_J,axiom,
% 12.45/6.93 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3918_less__minus__one__simps_I4_J,axiom,
% 12.45/6.93 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3919_less__minus__one__simps_I4_J,axiom,
% 12.45/6.93 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3920_less__minus__one__simps_I4_J,axiom,
% 12.45/6.93 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3921_numeral__times__minus__swap,axiom,
% 12.45/6.93 ! [W: num,X4: real] :
% 12.45/6.93 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X4 ) )
% 12.45/6.93 = ( times_times_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3922_numeral__times__minus__swap,axiom,
% 12.45/6.93 ! [W: num,X4: int] :
% 12.45/6.93 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X4 ) )
% 12.45/6.93 = ( times_times_int @ X4 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3923_numeral__times__minus__swap,axiom,
% 12.45/6.93 ! [W: num,X4: complex] :
% 12.45/6.93 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X4 ) )
% 12.45/6.93 = ( times_times_complex @ X4 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3924_numeral__times__minus__swap,axiom,
% 12.45/6.93 ! [W: num,X4: code_integer] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X4 ) )
% 12.45/6.93 = ( times_3573771949741848930nteger @ X4 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3925_numeral__times__minus__swap,axiom,
% 12.45/6.93 ! [W: num,X4: rat] :
% 12.45/6.93 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X4 ) )
% 12.45/6.93 = ( times_times_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3926_nonzero__minus__divide__right,axiom,
% 12.45/6.93 ! [B2: real,A: real] :
% 12.45/6.93 ( ( B2 != zero_zero_real )
% 12.45/6.93 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.93 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3927_nonzero__minus__divide__right,axiom,
% 12.45/6.93 ! [B2: complex,A: complex] :
% 12.45/6.93 ( ( B2 != zero_zero_complex )
% 12.45/6.93 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.45/6.93 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3928_nonzero__minus__divide__right,axiom,
% 12.45/6.93 ! [B2: rat,A: rat] :
% 12.45/6.93 ( ( B2 != zero_zero_rat )
% 12.45/6.93 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 12.45/6.93 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3929_nonzero__minus__divide__divide,axiom,
% 12.45/6.93 ! [B2: real,A: real] :
% 12.45/6.93 ( ( B2 != zero_zero_real )
% 12.45/6.93 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 12.45/6.93 = ( divide_divide_real @ A @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3930_nonzero__minus__divide__divide,axiom,
% 12.45/6.93 ! [B2: complex,A: complex] :
% 12.45/6.93 ( ( B2 != zero_zero_complex )
% 12.45/6.93 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 12.45/6.93 = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3931_nonzero__minus__divide__divide,axiom,
% 12.45/6.93 ! [B2: rat,A: rat] :
% 12.45/6.93 ( ( B2 != zero_zero_rat )
% 12.45/6.93 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 12.45/6.93 = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3932_one__neq__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( one_one_real
% 12.45/6.93 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3933_one__neq__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( one_one_int
% 12.45/6.93 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3934_one__neq__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( one_one_complex
% 12.45/6.93 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3935_one__neq__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( one_one_Code_integer
% 12.45/6.93 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3936_one__neq__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( one_one_rat
% 12.45/6.93 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3937_numeral__neq__neg__one,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( numeral_numeral_real @ N )
% 12.45/6.93 != ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3938_numeral__neq__neg__one,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( numeral_numeral_int @ N )
% 12.45/6.93 != ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3939_numeral__neq__neg__one,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( numera6690914467698888265omplex @ N )
% 12.45/6.93 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3940_numeral__neq__neg__one,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( numera6620942414471956472nteger @ N )
% 12.45/6.93 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3941_numeral__neq__neg__one,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ( ( numeral_numeral_rat @ N )
% 12.45/6.93 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3942_square__eq__1__iff,axiom,
% 12.45/6.93 ! [X4: real] :
% 12.45/6.93 ( ( ( times_times_real @ X4 @ X4 )
% 12.45/6.93 = one_one_real )
% 12.45/6.93 = ( ( X4 = one_one_real )
% 12.45/6.93 | ( X4
% 12.45/6.93 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3943_square__eq__1__iff,axiom,
% 12.45/6.93 ! [X4: int] :
% 12.45/6.93 ( ( ( times_times_int @ X4 @ X4 )
% 12.45/6.93 = one_one_int )
% 12.45/6.93 = ( ( X4 = one_one_int )
% 12.45/6.93 | ( X4
% 12.45/6.93 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3944_square__eq__1__iff,axiom,
% 12.45/6.93 ! [X4: complex] :
% 12.45/6.93 ( ( ( times_times_complex @ X4 @ X4 )
% 12.45/6.93 = one_one_complex )
% 12.45/6.93 = ( ( X4 = one_one_complex )
% 12.45/6.93 | ( X4
% 12.45/6.93 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3945_square__eq__1__iff,axiom,
% 12.45/6.93 ! [X4: code_integer] :
% 12.45/6.93 ( ( ( times_3573771949741848930nteger @ X4 @ X4 )
% 12.45/6.93 = one_one_Code_integer )
% 12.45/6.93 = ( ( X4 = one_one_Code_integer )
% 12.45/6.93 | ( X4
% 12.45/6.93 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3946_square__eq__1__iff,axiom,
% 12.45/6.93 ! [X4: rat] :
% 12.45/6.93 ( ( ( times_times_rat @ X4 @ X4 )
% 12.45/6.93 = one_one_rat )
% 12.45/6.93 = ( ( X4 = one_one_rat )
% 12.45/6.93 | ( X4
% 12.45/6.93 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3947_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 12.45/6.93 ( minus_minus_real
% 12.45/6.93 = ( ^ [A4: real,B3: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3948_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 12.45/6.93 ( minus_minus_int
% 12.45/6.93 = ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3949_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 12.45/6.93 ( minus_minus_complex
% 12.45/6.93 = ( ^ [A4: complex,B3: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3950_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 12.45/6.93 ( minus_8373710615458151222nteger
% 12.45/6.93 = ( ^ [A4: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3951_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 12.45/6.93 ( minus_minus_rat
% 12.45/6.93 = ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3952_diff__conv__add__uminus,axiom,
% 12.45/6.93 ( minus_minus_real
% 12.45/6.93 = ( ^ [A4: real,B3: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3953_diff__conv__add__uminus,axiom,
% 12.45/6.93 ( minus_minus_int
% 12.45/6.93 = ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3954_diff__conv__add__uminus,axiom,
% 12.45/6.93 ( minus_minus_complex
% 12.45/6.93 = ( ^ [A4: complex,B3: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3955_diff__conv__add__uminus,axiom,
% 12.45/6.93 ( minus_8373710615458151222nteger
% 12.45/6.93 = ( ^ [A4: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3956_diff__conv__add__uminus,axiom,
% 12.45/6.93 ( minus_minus_rat
% 12.45/6.93 = ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3957_group__cancel_Osub2,axiom,
% 12.45/6.93 ! [B4: real,K: real,B2: real,A: real] :
% 12.45/6.93 ( ( B4
% 12.45/6.93 = ( plus_plus_real @ K @ B2 ) )
% 12.45/6.93 => ( ( minus_minus_real @ A @ B4 )
% 12.45/6.93 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3958_group__cancel_Osub2,axiom,
% 12.45/6.93 ! [B4: int,K: int,B2: int,A: int] :
% 12.45/6.93 ( ( B4
% 12.45/6.93 = ( plus_plus_int @ K @ B2 ) )
% 12.45/6.93 => ( ( minus_minus_int @ A @ B4 )
% 12.45/6.93 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3959_group__cancel_Osub2,axiom,
% 12.45/6.93 ! [B4: complex,K: complex,B2: complex,A: complex] :
% 12.45/6.93 ( ( B4
% 12.45/6.93 = ( plus_plus_complex @ K @ B2 ) )
% 12.45/6.93 => ( ( minus_minus_complex @ A @ B4 )
% 12.45/6.93 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3960_group__cancel_Osub2,axiom,
% 12.45/6.93 ! [B4: code_integer,K: code_integer,B2: code_integer,A: code_integer] :
% 12.45/6.93 ( ( B4
% 12.45/6.93 = ( plus_p5714425477246183910nteger @ K @ B2 ) )
% 12.45/6.93 => ( ( minus_8373710615458151222nteger @ A @ B4 )
% 12.45/6.93 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3961_group__cancel_Osub2,axiom,
% 12.45/6.93 ! [B4: rat,K: rat,B2: rat,A: rat] :
% 12.45/6.93 ( ( B4
% 12.45/6.93 = ( plus_plus_rat @ K @ B2 ) )
% 12.45/6.93 => ( ( minus_minus_rat @ A @ B4 )
% 12.45/6.93 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3962_dvd__div__neg,axiom,
% 12.45/6.93 ! [B2: real,A: real] :
% 12.45/6.93 ( ( dvd_dvd_real @ B2 @ A )
% 12.45/6.93 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) )
% 12.45/6.93 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3963_dvd__div__neg,axiom,
% 12.45/6.93 ! [B2: int,A: int] :
% 12.45/6.93 ( ( dvd_dvd_int @ B2 @ A )
% 12.45/6.93 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.93 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3964_dvd__div__neg,axiom,
% 12.45/6.93 ! [B2: complex,A: complex] :
% 12.45/6.93 ( ( dvd_dvd_complex @ B2 @ A )
% 12.45/6.93 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3965_dvd__div__neg,axiom,
% 12.45/6.93 ! [B2: code_integer,A: code_integer] :
% 12.45/6.93 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 12.45/6.93 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3966_dvd__div__neg,axiom,
% 12.45/6.93 ! [B2: rat,A: rat] :
% 12.45/6.93 ( ( dvd_dvd_rat @ B2 @ A )
% 12.45/6.93 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 12.45/6.93 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3967_dvd__neg__div,axiom,
% 12.45/6.93 ! [B2: real,A: real] :
% 12.45/6.93 ( ( dvd_dvd_real @ B2 @ A )
% 12.45/6.93 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 )
% 12.45/6.93 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3968_dvd__neg__div,axiom,
% 12.45/6.93 ! [B2: int,A: int] :
% 12.45/6.93 ( ( dvd_dvd_int @ B2 @ A )
% 12.45/6.93 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.93 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3969_dvd__neg__div,axiom,
% 12.45/6.93 ! [B2: complex,A: complex] :
% 12.45/6.93 ( ( dvd_dvd_complex @ B2 @ A )
% 12.45/6.93 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3970_dvd__neg__div,axiom,
% 12.45/6.93 ! [B2: code_integer,A: code_integer] :
% 12.45/6.93 ( ( dvd_dvd_Code_integer @ B2 @ A )
% 12.45/6.93 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3971_dvd__neg__div,axiom,
% 12.45/6.93 ! [B2: rat,A: rat] :
% 12.45/6.93 ( ( dvd_dvd_rat @ B2 @ A )
% 12.45/6.93 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 12.45/6.93 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3972_real__minus__mult__self__le,axiom,
% 12.45/6.93 ! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X4 @ X4 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3973_pos__zmult__eq__1__iff__lemma,axiom,
% 12.45/6.93 ! [M: int,N: int] :
% 12.45/6.93 ( ( ( times_times_int @ M @ N )
% 12.45/6.93 = one_one_int )
% 12.45/6.93 => ( ( M = one_one_int )
% 12.45/6.93 | ( M
% 12.45/6.93 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3974_zmult__eq__1__iff,axiom,
% 12.45/6.93 ! [M: int,N: int] :
% 12.45/6.93 ( ( ( times_times_int @ M @ N )
% 12.45/6.93 = one_one_int )
% 12.45/6.93 = ( ( ( M = one_one_int )
% 12.45/6.93 & ( N = one_one_int ) )
% 12.45/6.93 | ( ( M
% 12.45/6.93 = ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.93 & ( N
% 12.45/6.93 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3975_minus__int__code_I2_J,axiom,
% 12.45/6.93 ! [L2: int] :
% 12.45/6.93 ( ( minus_minus_int @ zero_zero_int @ L2 )
% 12.45/6.93 = ( uminus_uminus_int @ L2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3976_zmod__zminus2__not__zero,axiom,
% 12.45/6.93 ! [K: int,L2: int] :
% 12.45/6.93 ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L2 ) )
% 12.45/6.93 != zero_zero_int )
% 12.45/6.93 => ( ( modulo_modulo_int @ K @ L2 )
% 12.45/6.93 != zero_zero_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3977_zmod__zminus1__not__zero,axiom,
% 12.45/6.93 ! [K: int,L2: int] :
% 12.45/6.93 ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
% 12.45/6.93 != zero_zero_int )
% 12.45/6.93 => ( ( modulo_modulo_int @ K @ L2 )
% 12.45/6.93 != zero_zero_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3978_minus__real__def,axiom,
% 12.45/6.93 ( minus_minus_real
% 12.45/6.93 = ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3979_not__zero__le__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3980_not__zero__le__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3981_not__zero__le__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3982_not__zero__le__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3983_neg__numeral__le__zero,axiom,
% 12.45/6.93 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3984_neg__numeral__le__zero,axiom,
% 12.45/6.93 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3985_neg__numeral__le__zero,axiom,
% 12.45/6.93 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3986_neg__numeral__le__zero,axiom,
% 12.45/6.93 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3987_not__zero__less__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3988_not__zero__less__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3989_not__zero__less__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3990_not__zero__less__neg__numeral,axiom,
% 12.45/6.93 ! [N: num] :
% 12.45/6.93 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3991_neg__numeral__less__zero,axiom,
% 12.45/6.93 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3992_neg__numeral__less__zero,axiom,
% 12.45/6.93 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3993_neg__numeral__less__zero,axiom,
% 12.45/6.93 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3994_neg__numeral__less__zero,axiom,
% 12.45/6.93 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3995_le__minus__one__simps_I3_J,axiom,
% 12.45/6.93 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3996_le__minus__one__simps_I3_J,axiom,
% 12.45/6.93 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3997_le__minus__one__simps_I3_J,axiom,
% 12.45/6.93 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3998_le__minus__one__simps_I3_J,axiom,
% 12.45/6.93 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_3999_le__minus__one__simps_I1_J,axiom,
% 12.45/6.93 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4000_le__minus__one__simps_I1_J,axiom,
% 12.45/6.93 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4001_le__minus__one__simps_I1_J,axiom,
% 12.45/6.93 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4002_le__minus__one__simps_I1_J,axiom,
% 12.45/6.93 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4003_less__minus__one__simps_I3_J,axiom,
% 12.45/6.93 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4004_less__minus__one__simps_I3_J,axiom,
% 12.45/6.93 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4005_less__minus__one__simps_I3_J,axiom,
% 12.45/6.93 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4006_less__minus__one__simps_I3_J,axiom,
% 12.45/6.93 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4007_less__minus__one__simps_I1_J,axiom,
% 12.45/6.93 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4008_less__minus__one__simps_I1_J,axiom,
% 12.45/6.93 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4009_less__minus__one__simps_I1_J,axiom,
% 12.45/6.93 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4010_less__minus__one__simps_I1_J,axiom,
% 12.45/6.93 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4011_not__one__le__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4012_not__one__le__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4013_not__one__le__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4014_not__one__le__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4015_not__numeral__le__neg__one,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4016_not__numeral__le__neg__one,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4017_not__numeral__le__neg__one,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4018_not__numeral__le__neg__one,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4019_neg__numeral__le__neg__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4020_neg__numeral__le__neg__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4021_neg__numeral__le__neg__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4022_neg__numeral__le__neg__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4023_neg__one__le__numeral,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4024_neg__one__le__numeral,axiom,
% 12.45/6.93 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4025_neg__one__le__numeral,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4026_neg__one__le__numeral,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4027_neg__numeral__le__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4028_neg__numeral__le__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4029_neg__numeral__le__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4030_neg__numeral__le__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4031_not__neg__one__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4032_not__neg__one__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4033_not__neg__one__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4034_not__neg__one__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4035_not__one__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4036_not__one__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4037_not__one__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4038_not__one__less__neg__numeral,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4039_not__numeral__less__neg__one,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4040_not__numeral__less__neg__one,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4041_not__numeral__less__neg__one,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4042_not__numeral__less__neg__one,axiom,
% 12.45/6.93 ! [M: num] :
% 12.45/6.93 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4043_neg__one__less__numeral,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4044_neg__one__less__numeral,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4045_neg__one__less__numeral,axiom,
% 12.45/6.93 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4046_neg__one__less__numeral,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4047_neg__numeral__less__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4048_neg__numeral__less__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4049_neg__numeral__less__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4050_neg__numeral__less__one,axiom,
% 12.45/6.93 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4051_eq__minus__divide__eq,axiom,
% 12.45/6.93 ! [A: real,B2: real,C: real] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 12.45/6.93 = ( ( ( C != zero_zero_real )
% 12.45/6.93 => ( ( times_times_real @ A @ C )
% 12.45/6.93 = ( uminus_uminus_real @ B2 ) ) )
% 12.45/6.93 & ( ( C = zero_zero_real )
% 12.45/6.93 => ( A = zero_zero_real ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4052_eq__minus__divide__eq,axiom,
% 12.45/6.93 ! [A: complex,B2: complex,C: complex] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) ) )
% 12.45/6.93 = ( ( ( C != zero_zero_complex )
% 12.45/6.93 => ( ( times_times_complex @ A @ C )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ B2 ) ) )
% 12.45/6.93 & ( ( C = zero_zero_complex )
% 12.45/6.93 => ( A = zero_zero_complex ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4053_eq__minus__divide__eq,axiom,
% 12.45/6.93 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.93 ( ( A
% 12.45/6.93 = ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 12.45/6.93 = ( ( ( C != zero_zero_rat )
% 12.45/6.93 => ( ( times_times_rat @ A @ C )
% 12.45/6.93 = ( uminus_uminus_rat @ B2 ) ) )
% 12.45/6.93 & ( ( C = zero_zero_rat )
% 12.45/6.93 => ( A = zero_zero_rat ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4054_minus__divide__eq__eq,axiom,
% 12.45/6.93 ! [B2: real,C: real,A: real] :
% 12.45/6.93 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.93 = A )
% 12.45/6.93 = ( ( ( C != zero_zero_real )
% 12.45/6.93 => ( ( uminus_uminus_real @ B2 )
% 12.45/6.93 = ( times_times_real @ A @ C ) ) )
% 12.45/6.93 & ( ( C = zero_zero_real )
% 12.45/6.93 => ( A = zero_zero_real ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4055_minus__divide__eq__eq,axiom,
% 12.45/6.93 ! [B2: complex,C: complex,A: complex] :
% 12.45/6.93 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 12.45/6.93 = A )
% 12.45/6.93 = ( ( ( C != zero_zero_complex )
% 12.45/6.93 => ( ( uminus1482373934393186551omplex @ B2 )
% 12.45/6.93 = ( times_times_complex @ A @ C ) ) )
% 12.45/6.93 & ( ( C = zero_zero_complex )
% 12.45/6.93 => ( A = zero_zero_complex ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4056_minus__divide__eq__eq,axiom,
% 12.45/6.93 ! [B2: rat,C: rat,A: rat] :
% 12.45/6.93 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.93 = A )
% 12.45/6.93 = ( ( ( C != zero_zero_rat )
% 12.45/6.93 => ( ( uminus_uminus_rat @ B2 )
% 12.45/6.93 = ( times_times_rat @ A @ C ) ) )
% 12.45/6.93 & ( ( C = zero_zero_rat )
% 12.45/6.93 => ( A = zero_zero_rat ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4057_nonzero__neg__divide__eq__eq,axiom,
% 12.45/6.93 ! [B2: real,A: real,C: real] :
% 12.45/6.93 ( ( B2 != zero_zero_real )
% 12.45/6.93 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.93 = C )
% 12.45/6.93 = ( ( uminus_uminus_real @ A )
% 12.45/6.93 = ( times_times_real @ C @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4058_nonzero__neg__divide__eq__eq,axiom,
% 12.45/6.93 ! [B2: complex,A: complex,C: complex] :
% 12.45/6.93 ( ( B2 != zero_zero_complex )
% 12.45/6.93 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.45/6.93 = C )
% 12.45/6.93 = ( ( uminus1482373934393186551omplex @ A )
% 12.45/6.93 = ( times_times_complex @ C @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4059_nonzero__neg__divide__eq__eq,axiom,
% 12.45/6.93 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.93 ( ( B2 != zero_zero_rat )
% 12.45/6.93 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 12.45/6.93 = C )
% 12.45/6.93 = ( ( uminus_uminus_rat @ A )
% 12.45/6.93 = ( times_times_rat @ C @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4060_nonzero__neg__divide__eq__eq2,axiom,
% 12.45/6.93 ! [B2: real,C: real,A: real] :
% 12.45/6.93 ( ( B2 != zero_zero_real )
% 12.45/6.93 => ( ( C
% 12.45/6.93 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) )
% 12.45/6.93 = ( ( times_times_real @ C @ B2 )
% 12.45/6.93 = ( uminus_uminus_real @ A ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4061_nonzero__neg__divide__eq__eq2,axiom,
% 12.45/6.93 ! [B2: complex,C: complex,A: complex] :
% 12.45/6.93 ( ( B2 != zero_zero_complex )
% 12.45/6.93 => ( ( C
% 12.45/6.93 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 12.45/6.93 = ( ( times_times_complex @ C @ B2 )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4062_nonzero__neg__divide__eq__eq2,axiom,
% 12.45/6.93 ! [B2: rat,C: rat,A: rat] :
% 12.45/6.93 ( ( B2 != zero_zero_rat )
% 12.45/6.93 => ( ( C
% 12.45/6.93 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) )
% 12.45/6.93 = ( ( times_times_rat @ C @ B2 )
% 12.45/6.93 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4063_mult__1s__ring__1_I2_J,axiom,
% 12.45/6.93 ! [B2: real] :
% 12.45/6.93 ( ( times_times_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 12.45/6.93 = ( uminus_uminus_real @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4064_mult__1s__ring__1_I2_J,axiom,
% 12.45/6.93 ! [B2: int] :
% 12.45/6.93 ( ( times_times_int @ B2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 12.45/6.93 = ( uminus_uminus_int @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4065_mult__1s__ring__1_I2_J,axiom,
% 12.45/6.93 ! [B2: complex] :
% 12.45/6.93 ( ( times_times_complex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4066_mult__1s__ring__1_I2_J,axiom,
% 12.45/6.93 ! [B2: code_integer] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ B2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4067_mult__1s__ring__1_I2_J,axiom,
% 12.45/6.93 ! [B2: rat] :
% 12.45/6.93 ( ( times_times_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 12.45/6.93 = ( uminus_uminus_rat @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4068_mult__1s__ring__1_I1_J,axiom,
% 12.45/6.93 ! [B2: real] :
% 12.45/6.93 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B2 )
% 12.45/6.93 = ( uminus_uminus_real @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4069_mult__1s__ring__1_I1_J,axiom,
% 12.45/6.93 ! [B2: int] :
% 12.45/6.93 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B2 )
% 12.45/6.93 = ( uminus_uminus_int @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4070_mult__1s__ring__1_I1_J,axiom,
% 12.45/6.93 ! [B2: complex] :
% 12.45/6.93 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B2 )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4071_mult__1s__ring__1_I1_J,axiom,
% 12.45/6.93 ! [B2: code_integer] :
% 12.45/6.93 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B2 )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4072_mult__1s__ring__1_I1_J,axiom,
% 12.45/6.93 ! [B2: rat] :
% 12.45/6.93 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B2 )
% 12.45/6.93 = ( uminus_uminus_rat @ B2 ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4073_divide__eq__minus__1__iff,axiom,
% 12.45/6.93 ! [A: real,B2: real] :
% 12.45/6.93 ( ( ( divide_divide_real @ A @ B2 )
% 12.45/6.93 = ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.93 = ( ( B2 != zero_zero_real )
% 12.45/6.93 & ( A
% 12.45/6.93 = ( uminus_uminus_real @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4074_divide__eq__minus__1__iff,axiom,
% 12.45/6.93 ! [A: complex,B2: complex] :
% 12.45/6.93 ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.93 = ( ( B2 != zero_zero_complex )
% 12.45/6.93 & ( A
% 12.45/6.93 = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4075_divide__eq__minus__1__iff,axiom,
% 12.45/6.93 ! [A: rat,B2: rat] :
% 12.45/6.93 ( ( ( divide_divide_rat @ A @ B2 )
% 12.45/6.93 = ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.93 = ( ( B2 != zero_zero_rat )
% 12.45/6.93 & ( A
% 12.45/6.93 = ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4076_uminus__numeral__One,axiom,
% 12.45/6.93 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 12.45/6.93 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4077_uminus__numeral__One,axiom,
% 12.45/6.93 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 12.45/6.93 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4078_uminus__numeral__One,axiom,
% 12.45/6.93 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 12.45/6.93 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4079_uminus__numeral__One,axiom,
% 12.45/6.93 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 12.45/6.93 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4080_uminus__numeral__One,axiom,
% 12.45/6.93 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 12.45/6.93 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4081_power__minus,axiom,
% 12.45/6.93 ! [A: real,N: nat] :
% 12.45/6.93 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 12.45/6.93 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 12.45/6.93
% 12.45/6.93 thf(fact_4082_power__minus,axiom,
% 12.45/6.93 ! [A: int,N: nat] :
% 12.45/6.93 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 12.45/6.94 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4083_power__minus,axiom,
% 12.45/6.94 ! [A: complex,N: nat] :
% 12.45/6.94 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 12.45/6.94 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4084_power__minus,axiom,
% 12.45/6.94 ! [A: code_integer,N: nat] :
% 12.45/6.94 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 12.45/6.94 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4085_power__minus,axiom,
% 12.45/6.94 ! [A: rat,N: nat] :
% 12.45/6.94 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 12.45/6.94 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4086_power__minus__Bit0,axiom,
% 12.45/6.94 ! [X4: real,K: num] :
% 12.45/6.94 ( ( power_power_real @ ( uminus_uminus_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.45/6.94 = ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4087_power__minus__Bit0,axiom,
% 12.45/6.94 ! [X4: int,K: num] :
% 12.45/6.94 ( ( power_power_int @ ( uminus_uminus_int @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.45/6.94 = ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4088_power__minus__Bit0,axiom,
% 12.45/6.94 ! [X4: complex,K: num] :
% 12.45/6.94 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.45/6.94 = ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4089_power__minus__Bit0,axiom,
% 12.45/6.94 ! [X4: code_integer,K: num] :
% 12.45/6.94 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.45/6.94 = ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4090_power__minus__Bit0,axiom,
% 12.45/6.94 ! [X4: rat,K: num] :
% 12.45/6.94 ( ( power_power_rat @ ( uminus_uminus_rat @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.45/6.94 = ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4091_real__add__less__0__iff,axiom,
% 12.45/6.94 ! [X4: real,Y3: real] :
% 12.45/6.94 ( ( ord_less_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
% 12.45/6.94 = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X4 ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4092_real__0__less__add__iff,axiom,
% 12.45/6.94 ! [X4: real,Y3: real] :
% 12.45/6.94 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.94 = ( ord_less_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4093_real__0__le__add__iff,axiom,
% 12.45/6.94 ! [X4: real,Y3: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.94 = ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4094_real__add__le__0__iff,axiom,
% 12.45/6.94 ! [X4: real,Y3: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
% 12.45/6.94 = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X4 ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4095_zmod__zminus2__eq__if,axiom,
% 12.45/6.94 ! [A: int,B2: int] :
% 12.45/6.94 ( ( ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.94 = zero_zero_int )
% 12.45/6.94 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.94 = zero_zero_int ) )
% 12.45/6.94 & ( ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.94 != zero_zero_int )
% 12.45/6.94 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.94 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4096_zmod__zminus1__eq__if,axiom,
% 12.45/6.94 ! [A: int,B2: int] :
% 12.45/6.94 ( ( ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.94 = zero_zero_int )
% 12.45/6.94 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.94 = zero_zero_int ) )
% 12.45/6.94 & ( ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.94 != zero_zero_int )
% 12.45/6.94 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.94 = ( minus_minus_int @ B2 @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4097_pos__minus__divide__less__eq,axiom,
% 12.45/6.94 ! [C: real,B2: real,A: real] :
% 12.45/6.94 ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4098_pos__minus__divide__less__eq,axiom,
% 12.45/6.94 ! [C: rat,B2: rat,A: rat] :
% 12.45/6.94 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4099_pos__less__minus__divide__eq,axiom,
% 12.45/6.94 ! [C: real,A: real,B2: real] :
% 12.45/6.94 ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 12.45/6.94 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4100_pos__less__minus__divide__eq,axiom,
% 12.45/6.94 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.94 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 12.45/6.94 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4101_neg__minus__divide__less__eq,axiom,
% 12.45/6.94 ! [C: real,B2: real,A: real] :
% 12.45/6.94 ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4102_neg__minus__divide__less__eq,axiom,
% 12.45/6.94 ! [C: rat,B2: rat,A: rat] :
% 12.45/6.94 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4103_neg__less__minus__divide__eq,axiom,
% 12.45/6.94 ! [C: real,A: real,B2: real] :
% 12.45/6.94 ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 12.45/6.94 = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4104_neg__less__minus__divide__eq,axiom,
% 12.45/6.94 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.94 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 12.45/6.94 = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4105_minus__divide__less__eq,axiom,
% 12.45/6.94 ! [B2: real,C: real,A: real] :
% 12.45/6.94 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4106_minus__divide__less__eq,axiom,
% 12.45/6.94 ! [B2: rat,C: rat,A: rat] :
% 12.45/6.94 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4107_less__minus__divide__eq,axiom,
% 12.45/6.94 ! [A: real,B2: real,C: real] :
% 12.45/6.94 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 12.45/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4108_less__minus__divide__eq,axiom,
% 12.45/6.94 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.94 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 12.45/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4109_divide__eq__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [B2: real,C: real,W: num] :
% 12.45/6.94 ( ( ( divide_divide_real @ B2 @ C )
% 12.45/6.94 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.94 = ( ( ( C != zero_zero_real )
% 12.45/6.94 => ( B2
% 12.45/6.94 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 12.45/6.94 & ( ( C = zero_zero_real )
% 12.45/6.94 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.45/6.94 = zero_zero_real ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4110_divide__eq__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [B2: complex,C: complex,W: num] :
% 12.45/6.94 ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.45/6.94 = ( ( ( C != zero_zero_complex )
% 12.45/6.94 => ( B2
% 12.45/6.94 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 12.45/6.94 & ( ( C = zero_zero_complex )
% 12.45/6.94 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.45/6.94 = zero_zero_complex ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4111_divide__eq__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [B2: rat,C: rat,W: num] :
% 12.45/6.94 ( ( ( divide_divide_rat @ B2 @ C )
% 12.45/6.94 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.45/6.94 = ( ( ( C != zero_zero_rat )
% 12.45/6.94 => ( B2
% 12.45/6.94 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 12.45/6.94 & ( ( C = zero_zero_rat )
% 12.45/6.94 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.45/6.94 = zero_zero_rat ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4112_eq__divide__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [W: num,B2: real,C: real] :
% 12.45/6.94 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.45/6.94 = ( divide_divide_real @ B2 @ C ) )
% 12.45/6.94 = ( ( ( C != zero_zero_real )
% 12.45/6.94 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 12.45/6.94 = B2 ) )
% 12.45/6.94 & ( ( C = zero_zero_real )
% 12.45/6.94 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.45/6.94 = zero_zero_real ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4113_eq__divide__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [W: num,B2: complex,C: complex] :
% 12.45/6.94 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.45/6.94 = ( divide1717551699836669952omplex @ B2 @ C ) )
% 12.45/6.94 = ( ( ( C != zero_zero_complex )
% 12.45/6.94 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 12.45/6.94 = B2 ) )
% 12.45/6.94 & ( ( C = zero_zero_complex )
% 12.45/6.94 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.45/6.94 = zero_zero_complex ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4114_eq__divide__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [W: num,B2: rat,C: rat] :
% 12.45/6.94 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.45/6.94 = ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.94 = ( ( ( C != zero_zero_rat )
% 12.45/6.94 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 12.45/6.94 = B2 ) )
% 12.45/6.94 & ( ( C = zero_zero_rat )
% 12.45/6.94 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.45/6.94 = zero_zero_rat ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4115_add__divide__eq__if__simps_I3_J,axiom,
% 12.45/6.94 ! [Z: real,A: real,B2: real] :
% 12.45/6.94 ( ( ( Z = zero_zero_real )
% 12.45/6.94 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 12.45/6.94 = B2 ) )
% 12.45/6.94 & ( ( Z != zero_zero_real )
% 12.45/6.94 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 12.45/6.94 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4116_add__divide__eq__if__simps_I3_J,axiom,
% 12.45/6.94 ! [Z: complex,A: complex,B2: complex] :
% 12.45/6.94 ( ( ( Z = zero_zero_complex )
% 12.45/6.94 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 12.45/6.94 = B2 ) )
% 12.45/6.94 & ( ( Z != zero_zero_complex )
% 12.45/6.94 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 12.45/6.94 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4117_add__divide__eq__if__simps_I3_J,axiom,
% 12.45/6.94 ! [Z: rat,A: rat,B2: rat] :
% 12.45/6.94 ( ( ( Z = zero_zero_rat )
% 12.45/6.94 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 12.45/6.94 = B2 ) )
% 12.45/6.94 & ( ( Z != zero_zero_rat )
% 12.45/6.94 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 12.45/6.94 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4118_minus__divide__add__eq__iff,axiom,
% 12.45/6.94 ! [Z: real,X4: real,Y3: real] :
% 12.45/6.94 ( ( Z != zero_zero_real )
% 12.45/6.94 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X4 @ Z ) ) @ Y3 )
% 12.45/6.94 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4119_minus__divide__add__eq__iff,axiom,
% 12.45/6.94 ! [Z: complex,X4: complex,Y3: complex] :
% 12.45/6.94 ( ( Z != zero_zero_complex )
% 12.45/6.94 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X4 @ Z ) ) @ Y3 )
% 12.45/6.94 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4120_minus__divide__add__eq__iff,axiom,
% 12.45/6.94 ! [Z: rat,X4: rat,Y3: rat] :
% 12.45/6.94 ( ( Z != zero_zero_rat )
% 12.45/6.94 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X4 @ Z ) ) @ Y3 )
% 12.45/6.94 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X4 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4121_add__divide__eq__if__simps_I6_J,axiom,
% 12.45/6.94 ! [Z: real,A: real,B2: real] :
% 12.45/6.94 ( ( ( Z = zero_zero_real )
% 12.45/6.94 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 12.45/6.94 = ( uminus_uminus_real @ B2 ) ) )
% 12.45/6.94 & ( ( Z != zero_zero_real )
% 12.45/6.94 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 12.45/6.94 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4122_add__divide__eq__if__simps_I6_J,axiom,
% 12.45/6.94 ! [Z: complex,A: complex,B2: complex] :
% 12.45/6.94 ( ( ( Z = zero_zero_complex )
% 12.45/6.94 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ B2 ) ) )
% 12.45/6.94 & ( ( Z != zero_zero_complex )
% 12.45/6.94 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 12.45/6.94 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4123_add__divide__eq__if__simps_I6_J,axiom,
% 12.45/6.94 ! [Z: rat,A: rat,B2: rat] :
% 12.45/6.94 ( ( ( Z = zero_zero_rat )
% 12.45/6.94 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 12.45/6.94 = ( uminus_uminus_rat @ B2 ) ) )
% 12.45/6.94 & ( ( Z != zero_zero_rat )
% 12.45/6.94 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 12.45/6.94 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4124_add__divide__eq__if__simps_I5_J,axiom,
% 12.45/6.94 ! [Z: real,A: real,B2: real] :
% 12.45/6.94 ( ( ( Z = zero_zero_real )
% 12.45/6.94 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 12.45/6.94 = ( uminus_uminus_real @ B2 ) ) )
% 12.45/6.94 & ( ( Z != zero_zero_real )
% 12.45/6.94 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 12.45/6.94 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4125_add__divide__eq__if__simps_I5_J,axiom,
% 12.45/6.94 ! [Z: complex,A: complex,B2: complex] :
% 12.45/6.94 ( ( ( Z = zero_zero_complex )
% 12.45/6.94 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ B2 ) ) )
% 12.45/6.94 & ( ( Z != zero_zero_complex )
% 12.45/6.94 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 12.45/6.94 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4126_add__divide__eq__if__simps_I5_J,axiom,
% 12.45/6.94 ! [Z: rat,A: rat,B2: rat] :
% 12.45/6.94 ( ( ( Z = zero_zero_rat )
% 12.45/6.94 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 12.45/6.94 = ( uminus_uminus_rat @ B2 ) ) )
% 12.45/6.94 & ( ( Z != zero_zero_rat )
% 12.45/6.94 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 12.45/6.94 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4127_minus__divide__diff__eq__iff,axiom,
% 12.45/6.94 ! [Z: real,X4: real,Y3: real] :
% 12.45/6.94 ( ( Z != zero_zero_real )
% 12.45/6.94 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X4 @ Z ) ) @ Y3 )
% 12.45/6.94 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4128_minus__divide__diff__eq__iff,axiom,
% 12.45/6.94 ! [Z: complex,X4: complex,Y3: complex] :
% 12.45/6.94 ( ( Z != zero_zero_complex )
% 12.45/6.94 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X4 @ Z ) ) @ Y3 )
% 12.45/6.94 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4129_minus__divide__diff__eq__iff,axiom,
% 12.45/6.94 ! [Z: rat,X4: rat,Y3: rat] :
% 12.45/6.94 ( ( Z != zero_zero_rat )
% 12.45/6.94 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X4 @ Z ) ) @ Y3 )
% 12.45/6.94 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X4 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4130_even__minus,axiom,
% 12.45/6.94 ! [A: int] :
% 12.45/6.94 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 12.45/6.94 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4131_even__minus,axiom,
% 12.45/6.94 ! [A: code_integer] :
% 12.45/6.94 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.94 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4132_power2__eq__iff,axiom,
% 12.45/6.94 ! [X4: real,Y3: real] :
% 12.45/6.94 ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.94 = ( ( X4 = Y3 )
% 12.45/6.94 | ( X4
% 12.45/6.94 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4133_power2__eq__iff,axiom,
% 12.45/6.94 ! [X4: int,Y3: int] :
% 12.45/6.94 ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.94 = ( ( X4 = Y3 )
% 12.45/6.94 | ( X4
% 12.45/6.94 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4134_power2__eq__iff,axiom,
% 12.45/6.94 ! [X4: complex,Y3: complex] :
% 12.45/6.94 ( ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.94 = ( ( X4 = Y3 )
% 12.45/6.94 | ( X4
% 12.45/6.94 = ( uminus1482373934393186551omplex @ Y3 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4135_power2__eq__iff,axiom,
% 12.45/6.94 ! [X4: code_integer,Y3: code_integer] :
% 12.45/6.94 ( ( ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.94 = ( ( X4 = Y3 )
% 12.45/6.94 | ( X4
% 12.45/6.94 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4136_power2__eq__iff,axiom,
% 12.45/6.94 ! [X4: rat,Y3: rat] :
% 12.45/6.94 ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.94 = ( ( X4 = Y3 )
% 12.45/6.94 | ( X4
% 12.45/6.94 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4137_verit__less__mono__div__int2,axiom,
% 12.45/6.94 ! [A3: int,B4: int,N: int] :
% 12.45/6.94 ( ( ord_less_eq_int @ A3 @ B4 )
% 12.45/6.94 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 12.45/6.94 => ( ord_less_eq_int @ ( divide_divide_int @ B4 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4138_div__eq__minus1,axiom,
% 12.45/6.94 ! [B2: int] :
% 12.45/6.94 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.94 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
% 12.45/6.94 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4139_pos__minus__divide__le__eq,axiom,
% 12.45/6.94 ! [C: real,B2: real,A: real] :
% 12.45/6.94 ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4140_pos__minus__divide__le__eq,axiom,
% 12.45/6.94 ! [C: rat,B2: rat,A: rat] :
% 12.45/6.94 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4141_pos__le__minus__divide__eq,axiom,
% 12.45/6.94 ! [C: real,A: real,B2: real] :
% 12.45/6.94 ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 12.45/6.94 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4142_pos__le__minus__divide__eq,axiom,
% 12.45/6.94 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.94 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 12.45/6.94 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4143_neg__minus__divide__le__eq,axiom,
% 12.45/6.94 ! [C: real,B2: real,A: real] :
% 12.45/6.94 ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4144_neg__minus__divide__le__eq,axiom,
% 12.45/6.94 ! [C: rat,B2: rat,A: rat] :
% 12.45/6.94 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4145_neg__le__minus__divide__eq,axiom,
% 12.45/6.94 ! [C: real,A: real,B2: real] :
% 12.45/6.94 ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 12.45/6.94 = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4146_neg__le__minus__divide__eq,axiom,
% 12.45/6.94 ! [C: rat,A: rat,B2: rat] :
% 12.45/6.94 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 12.45/6.94 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4147_minus__divide__le__eq,axiom,
% 12.45/6.94 ! [B2: real,C: real,A: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4148_minus__divide__le__eq,axiom,
% 12.45/6.94 ! [B2: rat,C: rat,A: rat] :
% 12.45/6.94 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 12.45/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4149_le__minus__divide__eq,axiom,
% 12.45/6.94 ! [A: real,B2: real,C: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 12.45/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4150_le__minus__divide__eq,axiom,
% 12.45/6.94 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.94 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 12.45/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4151_less__divide__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [W: num,B2: real,C: real] :
% 12.45/6.94 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4152_less__divide__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [W: num,B2: rat,C: rat] :
% 12.45/6.94 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4153_divide__less__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [B2: real,C: real,W: num] :
% 12.45/6.94 ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4154_divide__less__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [B2: rat,C: rat,W: num] :
% 12.45/6.94 ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.45/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4155_power2__eq__1__iff,axiom,
% 12.45/6.94 ! [A: real] :
% 12.45/6.94 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = one_one_real )
% 12.45/6.94 = ( ( A = one_one_real )
% 12.45/6.94 | ( A
% 12.45/6.94 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4156_power2__eq__1__iff,axiom,
% 12.45/6.94 ! [A: int] :
% 12.45/6.94 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = one_one_int )
% 12.45/6.94 = ( ( A = one_one_int )
% 12.45/6.94 | ( A
% 12.45/6.94 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4157_power2__eq__1__iff,axiom,
% 12.45/6.94 ! [A: complex] :
% 12.45/6.94 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = one_one_complex )
% 12.45/6.94 = ( ( A = one_one_complex )
% 12.45/6.94 | ( A
% 12.45/6.94 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4158_power2__eq__1__iff,axiom,
% 12.45/6.94 ! [A: code_integer] :
% 12.45/6.94 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = one_one_Code_integer )
% 12.45/6.94 = ( ( A = one_one_Code_integer )
% 12.45/6.94 | ( A
% 12.45/6.94 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4159_power2__eq__1__iff,axiom,
% 12.45/6.94 ! [A: rat] :
% 12.45/6.94 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = one_one_rat )
% 12.45/6.94 = ( ( A = one_one_rat )
% 12.45/6.94 | ( A
% 12.45/6.94 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4160_uminus__power__if,axiom,
% 12.45/6.94 ! [N: nat,A: real] :
% 12.45/6.94 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 12.45/6.94 = ( power_power_real @ A @ N ) ) )
% 12.45/6.94 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 12.45/6.94 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4161_uminus__power__if,axiom,
% 12.45/6.94 ! [N: nat,A: int] :
% 12.45/6.94 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 12.45/6.94 = ( power_power_int @ A @ N ) ) )
% 12.45/6.94 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 12.45/6.94 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4162_uminus__power__if,axiom,
% 12.45/6.94 ! [N: nat,A: complex] :
% 12.45/6.94 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 12.45/6.94 = ( power_power_complex @ A @ N ) ) )
% 12.45/6.94 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4163_uminus__power__if,axiom,
% 12.45/6.94 ! [N: nat,A: code_integer] :
% 12.45/6.94 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 12.45/6.94 = ( power_8256067586552552935nteger @ A @ N ) ) )
% 12.45/6.94 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 12.45/6.94 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4164_uminus__power__if,axiom,
% 12.45/6.94 ! [N: nat,A: rat] :
% 12.45/6.94 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 12.45/6.94 = ( power_power_rat @ A @ N ) ) )
% 12.45/6.94 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 12.45/6.94 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4165_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 12.45/6.94 ! [K: nat,N: nat] :
% 12.45/6.94 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.94 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 12.45/6.94 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4166_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 12.45/6.94 ! [K: nat,N: nat] :
% 12.45/6.94 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.94 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 12.45/6.94 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4167_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 12.45/6.94 ! [K: nat,N: nat] :
% 12.45/6.94 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.94 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 12.45/6.94 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4168_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 12.45/6.94 ! [K: nat,N: nat] :
% 12.45/6.94 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.94 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 12.45/6.94 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4169_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 12.45/6.94 ! [K: nat,N: nat] :
% 12.45/6.94 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.94 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 12.45/6.94 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4170_realpow__square__minus__le,axiom,
% 12.45/6.94 ! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4171_minus__mod__int__eq,axiom,
% 12.45/6.94 ! [L2: int,K: int] :
% 12.45/6.94 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 12.45/6.94 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
% 12.45/6.94 = ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4172_zmod__minus1,axiom,
% 12.45/6.94 ! [B2: int] :
% 12.45/6.94 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/6.94 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
% 12.45/6.94 = ( minus_minus_int @ B2 @ one_one_int ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4173_zdiv__zminus1__eq__if,axiom,
% 12.45/6.94 ! [B2: int,A: int] :
% 12.45/6.94 ( ( B2 != zero_zero_int )
% 12.45/6.94 => ( ( ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.94 = zero_zero_int )
% 12.45/6.94 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.94 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) )
% 12.45/6.94 & ( ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.94 != zero_zero_int )
% 12.45/6.94 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.94 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) @ one_one_int ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4174_zdiv__zminus2__eq__if,axiom,
% 12.45/6.94 ! [B2: int,A: int] :
% 12.45/6.94 ( ( B2 != zero_zero_int )
% 12.45/6.94 => ( ( ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.94 = zero_zero_int )
% 12.45/6.94 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.94 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) )
% 12.45/6.94 & ( ( ( modulo_modulo_int @ A @ B2 )
% 12.45/6.94 != zero_zero_int )
% 12.45/6.94 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 12.45/6.94 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) @ one_one_int ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4175_le__divide__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [W: num,B2: real,C: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
% 12.45/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4176_le__divide__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [W: num,B2: rat,C: rat] :
% 12.45/6.94 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
% 12.45/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4177_divide__le__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [B2: real,C: real,W: num] :
% 12.45/6.94 ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.45/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 12.45/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.45/6.94 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4178_divide__le__eq__numeral_I2_J,axiom,
% 12.45/6.94 ! [B2: rat,C: rat,W: num] :
% 12.45/6.94 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.45/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.45/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 12.45/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.45/6.94 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4179_square__le__1,axiom,
% 12.45/6.94 ! [X4: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.94 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.94 => ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4180_square__le__1,axiom,
% 12.45/6.94 ! [X4: code_integer] :
% 12.45/6.94 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X4 )
% 12.45/6.94 => ( ( ord_le3102999989581377725nteger @ X4 @ one_one_Code_integer )
% 12.45/6.94 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4181_square__le__1,axiom,
% 12.45/6.94 ! [X4: rat] :
% 12.45/6.94 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X4 )
% 12.45/6.94 => ( ( ord_less_eq_rat @ X4 @ one_one_rat )
% 12.45/6.94 => ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4182_square__le__1,axiom,
% 12.45/6.94 ! [X4: int] :
% 12.45/6.94 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X4 )
% 12.45/6.94 => ( ( ord_less_eq_int @ X4 @ one_one_int )
% 12.45/6.94 => ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4183_minus__power__mult__self,axiom,
% 12.45/6.94 ! [A: real,N: nat] :
% 12.45/6.94 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 12.45/6.94 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4184_minus__power__mult__self,axiom,
% 12.45/6.94 ! [A: int,N: nat] :
% 12.45/6.94 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 12.45/6.94 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4185_minus__power__mult__self,axiom,
% 12.45/6.94 ! [A: complex,N: nat] :
% 12.45/6.94 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 12.45/6.94 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4186_minus__power__mult__self,axiom,
% 12.45/6.94 ! [A: code_integer,N: nat] :
% 12.45/6.94 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 12.45/6.94 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4187_minus__power__mult__self,axiom,
% 12.45/6.94 ! [A: rat,N: nat] :
% 12.45/6.94 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 12.45/6.94 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4188_minus__one__power__iff,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 12.45/6.94 = one_one_real ) )
% 12.45/6.94 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 12.45/6.94 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4189_minus__one__power__iff,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 12.45/6.94 = one_one_int ) )
% 12.45/6.94 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 12.45/6.94 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4190_minus__one__power__iff,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 12.45/6.94 = one_one_complex ) )
% 12.45/6.94 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4191_minus__one__power__iff,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 12.45/6.94 = one_one_Code_integer ) )
% 12.45/6.94 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 12.45/6.94 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4192_minus__one__power__iff,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 12.45/6.94 = one_one_rat ) )
% 12.45/6.94 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.94 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 12.45/6.94 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4193_minus__1__div__exp__eq__int,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4194_div__pos__neg__trivial,axiom,
% 12.45/6.94 ! [K: int,L2: int] :
% 12.45/6.94 ( ( ord_less_int @ zero_zero_int @ K )
% 12.45/6.94 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 12.45/6.94 => ( ( divide_divide_int @ K @ L2 )
% 12.45/6.94 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4195_signed__take__bit__int__less__eq__self__iff,axiom,
% 12.45/6.94 ! [N: nat,K: int] :
% 12.45/6.94 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 12.45/6.94 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4196_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 12.45/6.94 ! [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 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4197_signed__take__bit__int__greater__self__iff,axiom,
% 12.45/6.94 ! [K: int,N: nat] :
% 12.45/6.94 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 12.45/6.94 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4198_power__minus1__odd,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/6.94 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4199_power__minus1__odd,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/6.94 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4200_power__minus1__odd,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4201_power__minus1__odd,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/6.94 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4202_power__minus1__odd,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/6.94 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4203_int__bit__induct,axiom,
% 12.45/6.94 ! [P: int > $o,K: int] :
% 12.45/6.94 ( ( P @ zero_zero_int )
% 12.45/6.94 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.94 => ( ! [K2: int] :
% 12.45/6.94 ( ( P @ K2 )
% 12.45/6.94 => ( ( K2 != zero_zero_int )
% 12.45/6.94 => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.94 => ( ! [K2: int] :
% 12.45/6.94 ( ( P @ K2 )
% 12.45/6.94 => ( ( K2
% 12.45/6.94 != ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.94 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 12.45/6.94 => ( P @ K ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4204_signed__take__bit__int__eq__self,axiom,
% 12.45/6.94 ! [N: nat,K: int] :
% 12.45/6.94 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 12.45/6.94 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 12.45/6.94 = K ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4205_signed__take__bit__int__eq__self__iff,axiom,
% 12.45/6.94 ! [N: nat,K: int] :
% 12.45/6.94 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 12.45/6.94 = K )
% 12.45/6.94 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 12.45/6.94 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4206_signed__take__bit__int__greater__eq,axiom,
% 12.45/6.94 ! [K: int,N: nat] :
% 12.45/6.94 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/6.94 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4207_mult__less__iff1,axiom,
% 12.45/6.94 ! [Z: real,X4: real,Y3: real] :
% 12.45/6.94 ( ( ord_less_real @ zero_zero_real @ Z )
% 12.45/6.94 => ( ( ord_less_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 12.45/6.94 = ( ord_less_real @ X4 @ Y3 ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4208_mult__less__iff1,axiom,
% 12.45/6.94 ! [Z: rat,X4: rat,Y3: rat] :
% 12.45/6.94 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 12.45/6.94 => ( ( ord_less_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 12.45/6.94 = ( ord_less_rat @ X4 @ Y3 ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4209_mult__less__iff1,axiom,
% 12.45/6.94 ! [Z: int,X4: int,Y3: int] :
% 12.45/6.94 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.45/6.94 => ( ( ord_less_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 12.45/6.94 = ( ord_less_int @ X4 @ Y3 ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4210_compl__le__compl__iff,axiom,
% 12.45/6.94 ! [X4: set_nat,Y3: set_nat] :
% 12.45/6.94 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ ( uminus5710092332889474511et_nat @ Y3 ) )
% 12.45/6.94 = ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4211_signed__take__bit__Suc__minus__bit1,axiom,
% 12.45/6.94 ! [N: nat,K: num] :
% 12.45/6.94 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.45/6.94 = ( 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 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4212_concat__bit__Suc,axiom,
% 12.45/6.94 ! [N: nat,K: int,L2: int] :
% 12.45/6.94 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L2 )
% 12.45/6.94 = ( 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 ) ) ) @ L2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4213_flip__bit__0,axiom,
% 12.45/6.94 ! [A: code_integer] :
% 12.45/6.94 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 12.45/6.94 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4214_flip__bit__0,axiom,
% 12.45/6.94 ! [A: int] :
% 12.45/6.94 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 12.45/6.94 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4215_flip__bit__0,axiom,
% 12.45/6.94 ! [A: nat] :
% 12.45/6.94 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 12.45/6.94 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4216_set__decode__0,axiom,
% 12.45/6.94 ! [X4: nat] :
% 12.45/6.94 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X4 ) )
% 12.45/6.94 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4217_semiring__norm_I90_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( ( bit1 @ M )
% 12.45/6.94 = ( bit1 @ N ) )
% 12.45/6.94 = ( M = N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4218_verit__eq__simplify_I9_J,axiom,
% 12.45/6.94 ! [X32: num,Y32: num] :
% 12.45/6.94 ( ( ( bit1 @ X32 )
% 12.45/6.94 = ( bit1 @ Y32 ) )
% 12.45/6.94 = ( X32 = Y32 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4219_semiring__norm_I88_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( bit0 @ M )
% 12.45/6.94 != ( bit1 @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4220_semiring__norm_I89_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( bit1 @ M )
% 12.45/6.94 != ( bit0 @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4221_semiring__norm_I84_J,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( one
% 12.45/6.94 != ( bit1 @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4222_semiring__norm_I86_J,axiom,
% 12.45/6.94 ! [M: num] :
% 12.45/6.94 ( ( bit1 @ M )
% 12.45/6.94 != one ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4223_of__bool__less__eq__iff,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 12.45/6.94 = ( P
% 12.45/6.94 => Q ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4224_of__bool__less__eq__iff,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 12.45/6.94 = ( P
% 12.45/6.94 => Q ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4225_of__bool__less__eq__iff,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 12.45/6.94 = ( P
% 12.45/6.94 => Q ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4226_of__bool__less__eq__iff,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 12.45/6.94 = ( P
% 12.45/6.94 => Q ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4227_of__bool__eq__0__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ( zero_n3304061248610475627l_real @ P )
% 12.45/6.94 = zero_zero_real )
% 12.45/6.94 = ~ P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4228_of__bool__eq__0__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 12.45/6.94 = zero_zero_rat )
% 12.45/6.94 = ~ P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4229_of__bool__eq__0__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 12.45/6.94 = zero_zero_nat )
% 12.45/6.94 = ~ P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4230_of__bool__eq__0__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ( zero_n2684676970156552555ol_int @ P )
% 12.45/6.94 = zero_zero_int )
% 12.45/6.94 = ~ P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4231_of__bool__eq__0__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ( zero_n356916108424825756nteger @ P )
% 12.45/6.94 = zero_z3403309356797280102nteger )
% 12.45/6.94 = ~ P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4232_of__bool__eq_I1_J,axiom,
% 12.45/6.94 ( ( zero_n3304061248610475627l_real @ $false )
% 12.45/6.94 = zero_zero_real ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4233_of__bool__eq_I1_J,axiom,
% 12.45/6.94 ( ( zero_n2052037380579107095ol_rat @ $false )
% 12.45/6.94 = zero_zero_rat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4234_of__bool__eq_I1_J,axiom,
% 12.45/6.94 ( ( zero_n2687167440665602831ol_nat @ $false )
% 12.45/6.94 = zero_zero_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4235_of__bool__eq_I1_J,axiom,
% 12.45/6.94 ( ( zero_n2684676970156552555ol_int @ $false )
% 12.45/6.94 = zero_zero_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4236_of__bool__eq_I1_J,axiom,
% 12.45/6.94 ( ( zero_n356916108424825756nteger @ $false )
% 12.45/6.94 = zero_z3403309356797280102nteger ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4237_of__bool__less__iff,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 12.45/6.94 = ( ~ P
% 12.45/6.94 & Q ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4238_of__bool__less__iff,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 12.45/6.94 = ( ~ P
% 12.45/6.94 & Q ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4239_of__bool__less__iff,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 12.45/6.94 = ( ~ P
% 12.45/6.94 & Q ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4240_of__bool__less__iff,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 12.45/6.94 = ( ~ P
% 12.45/6.94 & Q ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4241_of__bool__less__iff,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 12.45/6.94 = ( ~ P
% 12.45/6.94 & Q ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4242_of__bool__eq_I2_J,axiom,
% 12.45/6.94 ( ( zero_n1201886186963655149omplex @ $true )
% 12.45/6.94 = one_one_complex ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4243_of__bool__eq_I2_J,axiom,
% 12.45/6.94 ( ( zero_n3304061248610475627l_real @ $true )
% 12.45/6.94 = one_one_real ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4244_of__bool__eq_I2_J,axiom,
% 12.45/6.94 ( ( zero_n2052037380579107095ol_rat @ $true )
% 12.45/6.94 = one_one_rat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4245_of__bool__eq_I2_J,axiom,
% 12.45/6.94 ( ( zero_n2687167440665602831ol_nat @ $true )
% 12.45/6.94 = one_one_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4246_of__bool__eq_I2_J,axiom,
% 12.45/6.94 ( ( zero_n2684676970156552555ol_int @ $true )
% 12.45/6.94 = one_one_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4247_of__bool__eq_I2_J,axiom,
% 12.45/6.94 ( ( zero_n356916108424825756nteger @ $true )
% 12.45/6.94 = one_one_Code_integer ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4248_of__bool__eq__1__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ( zero_n1201886186963655149omplex @ P )
% 12.45/6.94 = one_one_complex )
% 12.45/6.94 = P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4249_of__bool__eq__1__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ( zero_n3304061248610475627l_real @ P )
% 12.45/6.94 = one_one_real )
% 12.45/6.94 = P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4250_of__bool__eq__1__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 12.45/6.94 = one_one_rat )
% 12.45/6.94 = P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4251_of__bool__eq__1__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 12.45/6.94 = one_one_nat )
% 12.45/6.94 = P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4252_of__bool__eq__1__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ( zero_n2684676970156552555ol_int @ P )
% 12.45/6.94 = one_one_int )
% 12.45/6.94 = P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4253_of__bool__eq__1__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ( zero_n356916108424825756nteger @ P )
% 12.45/6.94 = one_one_Code_integer )
% 12.45/6.94 = P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4254_concat__bit__0,axiom,
% 12.45/6.94 ! [K: int,L2: int] :
% 12.45/6.94 ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
% 12.45/6.94 = L2 ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4255_semiring__norm_I73_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/6.94 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4256_semiring__norm_I80_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/6.94 = ( ord_less_num @ M @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4257_zero__less__of__bool__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 12.45/6.94 = P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4258_zero__less__of__bool__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 12.45/6.94 = P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4259_zero__less__of__bool__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.45/6.94 = P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4260_zero__less__of__bool__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 12.45/6.94 = P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4261_zero__less__of__bool__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 12.45/6.94 = P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4262_of__bool__less__one__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 12.45/6.94 = ~ P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4263_of__bool__less__one__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 12.45/6.94 = ~ P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4264_of__bool__less__one__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 12.45/6.94 = ~ P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4265_of__bool__less__one__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 12.45/6.94 = ~ P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4266_of__bool__less__one__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 12.45/6.94 = ~ P ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4267_of__bool__not__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( zero_n1201886186963655149omplex @ ~ P )
% 12.45/6.94 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4268_of__bool__not__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( zero_n3304061248610475627l_real @ ~ P )
% 12.45/6.94 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4269_of__bool__not__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 12.45/6.94 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4270_of__bool__not__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 12.45/6.94 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4271_of__bool__not__iff,axiom,
% 12.45/6.94 ! [P: $o] :
% 12.45/6.94 ( ( zero_n356916108424825756nteger @ ~ P )
% 12.45/6.94 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4272_Suc__0__mod__eq,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.45/6.94 = ( zero_n2687167440665602831ol_nat
% 12.45/6.94 @ ( N
% 12.45/6.94 != ( suc @ zero_zero_nat ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4273_semiring__norm_I9_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.45/6.94 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4274_semiring__norm_I7_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/6.94 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4275_semiring__norm_I15_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.45/6.94 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4276_semiring__norm_I14_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/6.94 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4277_semiring__norm_I72_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/6.94 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4278_semiring__norm_I81_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.45/6.94 = ( ord_less_num @ M @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4279_semiring__norm_I70_J,axiom,
% 12.45/6.94 ! [M: num] :
% 12.45/6.94 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4280_semiring__norm_I77_J,axiom,
% 12.45/6.94 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4281_concat__bit__nonnegative__iff,axiom,
% 12.45/6.94 ! [N: nat,K: int,L2: int] :
% 12.45/6.94 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L2 ) )
% 12.45/6.94 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4282_concat__bit__negative__iff,axiom,
% 12.45/6.94 ! [N: nat,K: int,L2: int] :
% 12.45/6.94 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L2 ) @ zero_zero_int )
% 12.45/6.94 = ( ord_less_int @ L2 @ zero_zero_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4283_zdiv__numeral__Bit1,axiom,
% 12.45/6.94 ! [V: num,W: num] :
% 12.45/6.94 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 12.45/6.94 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4284_semiring__norm_I3_J,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 12.45/6.94 = ( bit1 @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4285_semiring__norm_I4_J,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 12.45/6.94 = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4286_semiring__norm_I5_J,axiom,
% 12.45/6.94 ! [M: num] :
% 12.45/6.94 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 12.45/6.94 = ( bit1 @ M ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4287_semiring__norm_I8_J,axiom,
% 12.45/6.94 ! [M: num] :
% 12.45/6.94 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 12.45/6.94 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4288_semiring__norm_I10_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/6.94 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4289_semiring__norm_I16_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/6.94 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4290_semiring__norm_I74_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.45/6.94 = ( ord_less_num @ M @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4291_semiring__norm_I79_J,axiom,
% 12.45/6.94 ! [M: num,N: num] :
% 12.45/6.94 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/6.94 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4292_odd__of__bool__self,axiom,
% 12.45/6.94 ! [P2: $o] :
% 12.45/6.94 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P2 ) ) )
% 12.45/6.94 = P2 ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4293_odd__of__bool__self,axiom,
% 12.45/6.94 ! [P2: $o] :
% 12.45/6.94 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P2 ) ) )
% 12.45/6.94 = P2 ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4294_odd__of__bool__self,axiom,
% 12.45/6.94 ! [P2: $o] :
% 12.45/6.94 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P2 ) ) )
% 12.45/6.94 = P2 ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4295_of__bool__half__eq__0,axiom,
% 12.45/6.94 ! [B2: $o] :
% 12.45/6.94 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = zero_zero_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4296_of__bool__half__eq__0,axiom,
% 12.45/6.94 ! [B2: $o] :
% 12.45/6.94 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.94 = zero_zero_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4297_of__bool__half__eq__0,axiom,
% 12.45/6.94 ! [B2: $o] :
% 12.45/6.94 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.94 = zero_z3403309356797280102nteger ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4298_Suc__div__eq__add3__div__numeral,axiom,
% 12.45/6.94 ! [M: nat,V: num] :
% 12.45/6.94 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 12.45/6.94 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4299_div__Suc__eq__div__add3,axiom,
% 12.45/6.94 ! [M: nat,N: nat] :
% 12.45/6.94 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 12.45/6.94 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4300_mod__Suc__eq__mod__add3,axiom,
% 12.45/6.94 ! [M: nat,N: nat] :
% 12.45/6.94 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 12.45/6.94 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4301_Suc__mod__eq__add3__mod__numeral,axiom,
% 12.45/6.94 ! [M: nat,V: num] :
% 12.45/6.94 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 12.45/6.94 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4302_set__decode__Suc,axiom,
% 12.45/6.94 ! [N: nat,X4: nat] :
% 12.45/6.94 ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X4 ) )
% 12.45/6.94 = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4303_bits__1__div__exp,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4304_bits__1__div__exp,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4305_bits__1__div__exp,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4306_one__div__2__pow__eq,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4307_one__div__2__pow__eq,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4308_one__div__2__pow__eq,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4309_zmod__numeral__Bit1,axiom,
% 12.45/6.94 ! [V: num,W: num] :
% 12.45/6.94 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 12.45/6.94 = ( 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 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4310_one__mod__2__pow__eq,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4311_one__mod__2__pow__eq,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4312_one__mod__2__pow__eq,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4313_signed__take__bit__Suc__bit1,axiom,
% 12.45/6.94 ! [N: nat,K: num] :
% 12.45/6.94 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 12.45/6.94 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4314_of__bool__eq__iff,axiom,
% 12.45/6.94 ! [P2: $o,Q2: $o] :
% 12.45/6.94 ( ( ( zero_n2687167440665602831ol_nat @ P2 )
% 12.45/6.94 = ( zero_n2687167440665602831ol_nat @ Q2 ) )
% 12.45/6.94 = ( P2 = Q2 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4315_of__bool__eq__iff,axiom,
% 12.45/6.94 ! [P2: $o,Q2: $o] :
% 12.45/6.94 ( ( ( zero_n2684676970156552555ol_int @ P2 )
% 12.45/6.94 = ( zero_n2684676970156552555ol_int @ Q2 ) )
% 12.45/6.94 = ( P2 = Q2 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4316_of__bool__eq__iff,axiom,
% 12.45/6.94 ! [P2: $o,Q2: $o] :
% 12.45/6.94 ( ( ( zero_n356916108424825756nteger @ P2 )
% 12.45/6.94 = ( zero_n356916108424825756nteger @ Q2 ) )
% 12.45/6.94 = ( P2 = Q2 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4317_of__bool__conj,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( zero_n3304061248610475627l_real
% 12.45/6.94 @ ( P
% 12.45/6.94 & Q ) )
% 12.45/6.94 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4318_of__bool__conj,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( zero_n2052037380579107095ol_rat
% 12.45/6.94 @ ( P
% 12.45/6.94 & Q ) )
% 12.45/6.94 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4319_of__bool__conj,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( zero_n2687167440665602831ol_nat
% 12.45/6.94 @ ( P
% 12.45/6.94 & Q ) )
% 12.45/6.94 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4320_of__bool__conj,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( zero_n2684676970156552555ol_int
% 12.45/6.94 @ ( P
% 12.45/6.94 & Q ) )
% 12.45/6.94 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4321_of__bool__conj,axiom,
% 12.45/6.94 ! [P: $o,Q: $o] :
% 12.45/6.94 ( ( zero_n356916108424825756nteger
% 12.45/6.94 @ ( P
% 12.45/6.94 & Q ) )
% 12.45/6.94 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4322_verit__eq__simplify_I14_J,axiom,
% 12.45/6.94 ! [X2: num,X32: num] :
% 12.45/6.94 ( ( bit0 @ X2 )
% 12.45/6.94 != ( bit1 @ X32 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4323_verit__eq__simplify_I12_J,axiom,
% 12.45/6.94 ! [X32: num] :
% 12.45/6.94 ( one
% 12.45/6.94 != ( bit1 @ X32 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4324_concat__bit__assoc,axiom,
% 12.45/6.94 ! [N: nat,K: int,M: nat,L2: int,R2: int] :
% 12.45/6.94 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L2 @ R2 ) )
% 12.45/6.94 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L2 ) @ R2 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4325_subset__decode__imp__le,axiom,
% 12.45/6.94 ! [M: nat,N: nat] :
% 12.45/6.94 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 12.45/6.94 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4326_zero__less__eq__of__bool,axiom,
% 12.45/6.94 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4327_zero__less__eq__of__bool,axiom,
% 12.45/6.94 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4328_zero__less__eq__of__bool,axiom,
% 12.45/6.94 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4329_zero__less__eq__of__bool,axiom,
% 12.45/6.94 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4330_zero__less__eq__of__bool,axiom,
% 12.45/6.94 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4331_of__bool__less__eq__one,axiom,
% 12.45/6.94 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4332_of__bool__less__eq__one,axiom,
% 12.45/6.94 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4333_of__bool__less__eq__one,axiom,
% 12.45/6.94 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4334_of__bool__less__eq__one,axiom,
% 12.45/6.94 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4335_of__bool__less__eq__one,axiom,
% 12.45/6.94 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4336_split__of__bool__asm,axiom,
% 12.45/6.94 ! [P: complex > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
% 12.45/6.94 = ( ~ ( ( P2
% 12.45/6.94 & ~ ( P @ one_one_complex ) )
% 12.45/6.94 | ( ~ P2
% 12.45/6.94 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4337_split__of__bool__asm,axiom,
% 12.45/6.94 ! [P: real > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
% 12.45/6.94 = ( ~ ( ( P2
% 12.45/6.94 & ~ ( P @ one_one_real ) )
% 12.45/6.94 | ( ~ P2
% 12.45/6.94 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4338_split__of__bool__asm,axiom,
% 12.45/6.94 ! [P: rat > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
% 12.45/6.94 = ( ~ ( ( P2
% 12.45/6.94 & ~ ( P @ one_one_rat ) )
% 12.45/6.94 | ( ~ P2
% 12.45/6.94 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4339_split__of__bool__asm,axiom,
% 12.45/6.94 ! [P: nat > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
% 12.45/6.94 = ( ~ ( ( P2
% 12.45/6.94 & ~ ( P @ one_one_nat ) )
% 12.45/6.94 | ( ~ P2
% 12.45/6.94 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4340_split__of__bool__asm,axiom,
% 12.45/6.94 ! [P: int > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
% 12.45/6.94 = ( ~ ( ( P2
% 12.45/6.94 & ~ ( P @ one_one_int ) )
% 12.45/6.94 | ( ~ P2
% 12.45/6.94 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4341_split__of__bool__asm,axiom,
% 12.45/6.94 ! [P: code_integer > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
% 12.45/6.94 = ( ~ ( ( P2
% 12.45/6.94 & ~ ( P @ one_one_Code_integer ) )
% 12.45/6.94 | ( ~ P2
% 12.45/6.94 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4342_split__of__bool,axiom,
% 12.45/6.94 ! [P: complex > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
% 12.45/6.94 = ( ( P2
% 12.45/6.94 => ( P @ one_one_complex ) )
% 12.45/6.94 & ( ~ P2
% 12.45/6.94 => ( P @ zero_zero_complex ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4343_split__of__bool,axiom,
% 12.45/6.94 ! [P: real > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
% 12.45/6.94 = ( ( P2
% 12.45/6.94 => ( P @ one_one_real ) )
% 12.45/6.94 & ( ~ P2
% 12.45/6.94 => ( P @ zero_zero_real ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4344_split__of__bool,axiom,
% 12.45/6.94 ! [P: rat > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
% 12.45/6.94 = ( ( P2
% 12.45/6.94 => ( P @ one_one_rat ) )
% 12.45/6.94 & ( ~ P2
% 12.45/6.94 => ( P @ zero_zero_rat ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4345_split__of__bool,axiom,
% 12.45/6.94 ! [P: nat > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
% 12.45/6.94 = ( ( P2
% 12.45/6.94 => ( P @ one_one_nat ) )
% 12.45/6.94 & ( ~ P2
% 12.45/6.94 => ( P @ zero_zero_nat ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4346_split__of__bool,axiom,
% 12.45/6.94 ! [P: int > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
% 12.45/6.94 = ( ( P2
% 12.45/6.94 => ( P @ one_one_int ) )
% 12.45/6.94 & ( ~ P2
% 12.45/6.94 => ( P @ zero_zero_int ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4347_split__of__bool,axiom,
% 12.45/6.94 ! [P: code_integer > $o,P2: $o] :
% 12.45/6.94 ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
% 12.45/6.94 = ( ( P2
% 12.45/6.94 => ( P @ one_one_Code_integer ) )
% 12.45/6.94 & ( ~ P2
% 12.45/6.94 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4348_of__bool__def,axiom,
% 12.45/6.94 ( zero_n1201886186963655149omplex
% 12.45/6.94 = ( ^ [P3: $o] : ( if_complex @ P3 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4349_of__bool__def,axiom,
% 12.45/6.94 ( zero_n3304061248610475627l_real
% 12.45/6.94 = ( ^ [P3: $o] : ( if_real @ P3 @ one_one_real @ zero_zero_real ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4350_of__bool__def,axiom,
% 12.45/6.94 ( zero_n2052037380579107095ol_rat
% 12.45/6.94 = ( ^ [P3: $o] : ( if_rat @ P3 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4351_of__bool__def,axiom,
% 12.45/6.94 ( zero_n2687167440665602831ol_nat
% 12.45/6.94 = ( ^ [P3: $o] : ( if_nat @ P3 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4352_of__bool__def,axiom,
% 12.45/6.94 ( zero_n2684676970156552555ol_int
% 12.45/6.94 = ( ^ [P3: $o] : ( if_int @ P3 @ one_one_int @ zero_zero_int ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4353_of__bool__def,axiom,
% 12.45/6.94 ( zero_n356916108424825756nteger
% 12.45/6.94 = ( ^ [P3: $o] : ( if_Code_integer @ P3 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4354_num_Oexhaust,axiom,
% 12.45/6.94 ! [Y3: num] :
% 12.45/6.94 ( ( Y3 != one )
% 12.45/6.94 => ( ! [X22: num] :
% 12.45/6.94 ( Y3
% 12.45/6.94 != ( bit0 @ X22 ) )
% 12.45/6.94 => ~ ! [X33: num] :
% 12.45/6.94 ( Y3
% 12.45/6.94 != ( bit1 @ X33 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4355_numeral__Bit1,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 12.45/6.94 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4356_numeral__Bit1,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 12.45/6.94 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4357_numeral__Bit1,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 12.45/6.94 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4358_numeral__Bit1,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 12.45/6.94 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4359_numeral__Bit1,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 12.45/6.94 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4360_eval__nat__numeral_I3_J,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 12.45/6.94 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4361_cong__exp__iff__simps_I10_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num,N: num] :
% 12.45/6.94 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4362_cong__exp__iff__simps_I10_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num,N: num] :
% 12.45/6.94 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4363_cong__exp__iff__simps_I10_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num,N: num] :
% 12.45/6.94 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4364_cong__exp__iff__simps_I12_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num,N: num] :
% 12.45/6.94 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4365_cong__exp__iff__simps_I12_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num,N: num] :
% 12.45/6.94 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4366_cong__exp__iff__simps_I12_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num,N: num] :
% 12.45/6.94 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4367_cong__exp__iff__simps_I13_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num,N: num] :
% 12.45/6.94 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 12.45/6.94 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 12.45/6.94 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4368_cong__exp__iff__simps_I13_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num,N: num] :
% 12.45/6.94 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 12.45/6.94 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 12.45/6.94 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4369_cong__exp__iff__simps_I13_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num,N: num] :
% 12.45/6.94 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 12.45/6.94 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 12.45/6.94 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4370_power__minus__Bit1,axiom,
% 12.45/6.94 ! [X4: real,K: num] :
% 12.45/6.94 ( ( power_power_real @ ( uminus_uminus_real @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.45/6.94 = ( uminus_uminus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4371_power__minus__Bit1,axiom,
% 12.45/6.94 ! [X4: int,K: num] :
% 12.45/6.94 ( ( power_power_int @ ( uminus_uminus_int @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.45/6.94 = ( uminus_uminus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4372_power__minus__Bit1,axiom,
% 12.45/6.94 ! [X4: complex,K: num] :
% 12.45/6.94 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4373_power__minus__Bit1,axiom,
% 12.45/6.94 ! [X4: code_integer,K: num] :
% 12.45/6.94 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.45/6.94 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4374_power__minus__Bit1,axiom,
% 12.45/6.94 ! [X4: rat,K: num] :
% 12.45/6.94 ( ( power_power_rat @ ( uminus_uminus_rat @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.45/6.94 = ( uminus_uminus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4375_numeral__Bit1__div__2,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = ( numeral_numeral_nat @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4376_numeral__Bit1__div__2,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.94 = ( numeral_numeral_int @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4377_odd__numeral,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4378_odd__numeral,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4379_odd__numeral,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4380_cong__exp__iff__simps_I3_J,axiom,
% 12.45/6.94 ! [N: num,Q2: num] :
% 12.45/6.94 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 != zero_zero_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4381_cong__exp__iff__simps_I3_J,axiom,
% 12.45/6.94 ! [N: num,Q2: num] :
% 12.45/6.94 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 != zero_zero_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4382_cong__exp__iff__simps_I3_J,axiom,
% 12.45/6.94 ! [N: num,Q2: num] :
% 12.45/6.94 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 != zero_z3403309356797280102nteger ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4383_power3__eq__cube,axiom,
% 12.45/6.94 ! [A: complex] :
% 12.45/6.94 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.45/6.94 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4384_power3__eq__cube,axiom,
% 12.45/6.94 ! [A: real] :
% 12.45/6.94 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.45/6.94 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4385_power3__eq__cube,axiom,
% 12.45/6.94 ! [A: rat] :
% 12.45/6.94 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.45/6.94 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4386_power3__eq__cube,axiom,
% 12.45/6.94 ! [A: nat] :
% 12.45/6.94 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.45/6.94 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4387_power3__eq__cube,axiom,
% 12.45/6.94 ! [A: int] :
% 12.45/6.94 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.45/6.94 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4388_numeral__3__eq__3,axiom,
% 12.45/6.94 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 12.45/6.94 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4389_Suc3__eq__add__3,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 12.45/6.94 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4390_of__bool__odd__eq__mod__2,axiom,
% 12.45/6.94 ! [A: nat] :
% 12.45/6.94 ( ( zero_n2687167440665602831ol_nat
% 12.45/6.94 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 12.45/6.94 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4391_of__bool__odd__eq__mod__2,axiom,
% 12.45/6.94 ! [A: int] :
% 12.45/6.94 ( ( zero_n2684676970156552555ol_int
% 12.45/6.94 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.45/6.94 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4392_of__bool__odd__eq__mod__2,axiom,
% 12.45/6.94 ! [A: code_integer] :
% 12.45/6.94 ( ( zero_n356916108424825756nteger
% 12.45/6.94 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 12.45/6.94 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4393_cong__exp__iff__simps_I7_J,axiom,
% 12.45/6.94 ! [Q2: num,N: num] :
% 12.45/6.94 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 12.45/6.94 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 12.45/6.94 = zero_zero_nat ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4394_cong__exp__iff__simps_I7_J,axiom,
% 12.45/6.94 ! [Q2: num,N: num] :
% 12.45/6.94 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 12.45/6.94 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 12.45/6.94 = zero_zero_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4395_cong__exp__iff__simps_I7_J,axiom,
% 12.45/6.94 ! [Q2: num,N: num] :
% 12.45/6.94 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 12.45/6.94 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 12.45/6.94 = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4396_cong__exp__iff__simps_I11_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num] :
% 12.45/6.94 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 12.45/6.94 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 12.45/6.94 = zero_zero_nat ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4397_cong__exp__iff__simps_I11_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num] :
% 12.45/6.94 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 12.45/6.94 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 12.45/6.94 = zero_zero_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4398_cong__exp__iff__simps_I11_J,axiom,
% 12.45/6.94 ! [M: num,Q2: num] :
% 12.45/6.94 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.45/6.94 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 12.45/6.94 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 12.45/6.94 = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4399_Suc__div__eq__add3__div,axiom,
% 12.45/6.94 ! [M: nat,N: nat] :
% 12.45/6.94 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 12.45/6.94 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4400_Suc__mod__eq__add3__mod,axiom,
% 12.45/6.94 ! [M: nat,N: nat] :
% 12.45/6.94 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 12.45/6.94 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4401_bits__induct,axiom,
% 12.45/6.94 ! [P: nat > $o,A: nat] :
% 12.45/6.94 ( ! [A5: nat] :
% 12.45/6.94 ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = A5 )
% 12.45/6.94 => ( P @ A5 ) )
% 12.45/6.94 => ( ! [A5: nat,B5: $o] :
% 12.45/6.94 ( ( P @ A5 )
% 12.45/6.94 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 = A5 )
% 12.45/6.94 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 12.45/6.94 => ( P @ A ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4402_bits__induct,axiom,
% 12.45/6.94 ! [P: int > $o,A: int] :
% 12.45/6.94 ( ! [A5: int] :
% 12.45/6.94 ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.94 = A5 )
% 12.45/6.94 => ( P @ A5 ) )
% 12.45/6.94 => ( ! [A5: int,B5: $o] :
% 12.45/6.94 ( ( P @ A5 )
% 12.45/6.94 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.94 = A5 )
% 12.45/6.94 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 12.45/6.94 => ( P @ A ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4403_bits__induct,axiom,
% 12.45/6.94 ! [P: code_integer > $o,A: code_integer] :
% 12.45/6.94 ( ! [A5: code_integer] :
% 12.45/6.94 ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.94 = A5 )
% 12.45/6.94 => ( P @ A5 ) )
% 12.45/6.94 => ( ! [A5: code_integer,B5: $o] :
% 12.45/6.94 ( ( P @ A5 )
% 12.45/6.94 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.94 = A5 )
% 12.45/6.94 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 12.45/6.94 => ( P @ A ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4404_compl__le__swap2,axiom,
% 12.45/6.94 ! [Y3: set_nat,X4: set_nat] :
% 12.45/6.94 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ X4 )
% 12.45/6.94 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ Y3 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4405_compl__le__swap1,axiom,
% 12.45/6.94 ! [Y3: set_nat,X4: set_nat] :
% 12.45/6.94 ( ( ord_less_eq_set_nat @ Y3 @ ( uminus5710092332889474511et_nat @ X4 ) )
% 12.45/6.94 => ( ord_less_eq_set_nat @ X4 @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4406_compl__mono,axiom,
% 12.45/6.94 ! [X4: set_nat,Y3: set_nat] :
% 12.45/6.94 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 12.45/6.94 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ ( uminus5710092332889474511et_nat @ X4 ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4407_exp__mod__exp,axiom,
% 12.45/6.94 ! [M: nat,N: nat] :
% 12.45/6.94 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4408_exp__mod__exp,axiom,
% 12.45/6.94 ! [M: nat,N: nat] :
% 12.45/6.94 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4409_exp__mod__exp,axiom,
% 12.45/6.94 ! [M: nat,N: nat] :
% 12.45/6.94 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4410_exp__div__exp__eq,axiom,
% 12.45/6.94 ! [M: nat,N: nat] :
% 12.45/6.94 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( times_times_nat
% 12.45/6.94 @ ( zero_n2687167440665602831ol_nat
% 12.45/6.94 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 12.45/6.94 != zero_zero_nat )
% 12.45/6.94 & ( ord_less_eq_nat @ N @ M ) ) )
% 12.45/6.94 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4411_exp__div__exp__eq,axiom,
% 12.45/6.94 ! [M: nat,N: nat] :
% 12.45/6.94 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( times_times_int
% 12.45/6.94 @ ( zero_n2684676970156552555ol_int
% 12.45/6.94 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 12.45/6.94 != zero_zero_int )
% 12.45/6.94 & ( ord_less_eq_nat @ N @ M ) ) )
% 12.45/6.94 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4412_exp__div__exp__eq,axiom,
% 12.45/6.94 ! [M: nat,N: nat] :
% 12.45/6.94 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.94 = ( times_3573771949741848930nteger
% 12.45/6.94 @ ( zero_n356916108424825756nteger
% 12.45/6.94 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 12.45/6.94 != zero_z3403309356797280102nteger )
% 12.45/6.94 & ( ord_less_eq_nat @ N @ M ) ) )
% 12.45/6.94 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4413_odd__mod__4__div__2,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.45/6.94 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.45/6.94 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4414_mod__exhaust__less__4,axiom,
% 12.45/6.94 ! [M: nat] :
% 12.45/6.94 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.45/6.94 = zero_zero_nat )
% 12.45/6.94 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.45/6.94 = one_one_nat )
% 12.45/6.94 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.45/6.94 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.94 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.45/6.94 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4415_signed__take__bit__numeral__minus__bit1,axiom,
% 12.45/6.94 ! [L2: num,K: num] :
% 12.45/6.94 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.45/6.94 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4416_dbl__dec__simps_I4_J,axiom,
% 12.45/6.94 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.94 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4417_dbl__dec__simps_I4_J,axiom,
% 12.45/6.94 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.94 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4418_dbl__dec__simps_I4_J,axiom,
% 12.45/6.94 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4419_dbl__dec__simps_I4_J,axiom,
% 12.45/6.94 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.94 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4420_dbl__dec__simps_I4_J,axiom,
% 12.45/6.94 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.94 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4421_signed__take__bit__numeral__bit1,axiom,
% 12.45/6.94 ! [L2: num,K: num] :
% 12.45/6.94 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 12.45/6.94 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4422_dbl__inc__simps_I3_J,axiom,
% 12.45/6.94 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 12.45/6.94 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4423_dbl__inc__simps_I3_J,axiom,
% 12.45/6.94 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 12.45/6.94 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4424_dbl__inc__simps_I3_J,axiom,
% 12.45/6.94 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 12.45/6.94 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4425_dbl__inc__simps_I3_J,axiom,
% 12.45/6.94 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 12.45/6.94 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4426_add__scale__eq__noteq,axiom,
% 12.45/6.94 ! [R2: real,A: real,B2: real,C: real,D: real] :
% 12.45/6.94 ( ( R2 != zero_zero_real )
% 12.45/6.94 => ( ( ( A = B2 )
% 12.45/6.94 & ( C != D ) )
% 12.45/6.94 => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 12.45/6.94 != ( plus_plus_real @ B2 @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4427_add__scale__eq__noteq,axiom,
% 12.45/6.94 ! [R2: rat,A: rat,B2: rat,C: rat,D: rat] :
% 12.45/6.94 ( ( R2 != zero_zero_rat )
% 12.45/6.94 => ( ( ( A = B2 )
% 12.45/6.94 & ( C != D ) )
% 12.45/6.94 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 12.45/6.94 != ( plus_plus_rat @ B2 @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4428_add__scale__eq__noteq,axiom,
% 12.45/6.94 ! [R2: nat,A: nat,B2: nat,C: nat,D: nat] :
% 12.45/6.94 ( ( R2 != zero_zero_nat )
% 12.45/6.94 => ( ( ( A = B2 )
% 12.45/6.94 & ( C != D ) )
% 12.45/6.94 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 12.45/6.94 != ( plus_plus_nat @ B2 @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4429_add__scale__eq__noteq,axiom,
% 12.45/6.94 ! [R2: int,A: int,B2: int,C: int,D: int] :
% 12.45/6.94 ( ( R2 != zero_zero_int )
% 12.45/6.94 => ( ( ( A = B2 )
% 12.45/6.94 & ( C != D ) )
% 12.45/6.94 => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 12.45/6.94 != ( plus_plus_int @ B2 @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4430_num_Osize__gen_I3_J,axiom,
% 12.45/6.94 ! [X32: num] :
% 12.45/6.94 ( ( size_num @ ( bit1 @ X32 ) )
% 12.45/6.94 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4431_dbl__dec__simps_I3_J,axiom,
% 12.45/6.94 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 12.45/6.94 = one_one_complex ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4432_dbl__dec__simps_I3_J,axiom,
% 12.45/6.94 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 12.45/6.94 = one_one_real ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4433_dbl__dec__simps_I3_J,axiom,
% 12.45/6.94 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 12.45/6.94 = one_one_rat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4434_dbl__dec__simps_I3_J,axiom,
% 12.45/6.94 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 12.45/6.94 = one_one_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4435_pred__numeral__simps_I1_J,axiom,
% 12.45/6.94 ( ( pred_numeral @ one )
% 12.45/6.94 = zero_zero_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4436_eq__numeral__Suc,axiom,
% 12.45/6.94 ! [K: num,N: nat] :
% 12.45/6.94 ( ( ( numeral_numeral_nat @ K )
% 12.45/6.94 = ( suc @ N ) )
% 12.45/6.94 = ( ( pred_numeral @ K )
% 12.45/6.94 = N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4437_Suc__eq__numeral,axiom,
% 12.45/6.94 ! [N: nat,K: num] :
% 12.45/6.94 ( ( ( suc @ N )
% 12.45/6.94 = ( numeral_numeral_nat @ K ) )
% 12.45/6.94 = ( N
% 12.45/6.94 = ( pred_numeral @ K ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4438_dbl__inc__simps_I2_J,axiom,
% 12.45/6.94 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 12.45/6.94 = one_one_complex ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4439_dbl__inc__simps_I2_J,axiom,
% 12.45/6.94 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 12.45/6.94 = one_one_real ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4440_dbl__inc__simps_I2_J,axiom,
% 12.45/6.94 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 12.45/6.94 = one_one_rat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4441_dbl__inc__simps_I2_J,axiom,
% 12.45/6.94 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 12.45/6.94 = one_one_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4442_dbl__inc__simps_I4_J,axiom,
% 12.45/6.94 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.94 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4443_dbl__inc__simps_I4_J,axiom,
% 12.45/6.94 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.94 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4444_dbl__inc__simps_I4_J,axiom,
% 12.45/6.94 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4445_dbl__inc__simps_I4_J,axiom,
% 12.45/6.94 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.94 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4446_dbl__inc__simps_I4_J,axiom,
% 12.45/6.94 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.94 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4447_dbl__inc__simps_I5_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 12.45/6.94 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4448_dbl__inc__simps_I5_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 12.45/6.94 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4449_dbl__inc__simps_I5_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 12.45/6.94 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4450_dbl__inc__simps_I5_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 12.45/6.94 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4451_less__numeral__Suc,axiom,
% 12.45/6.94 ! [K: num,N: nat] :
% 12.45/6.94 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 12.45/6.94 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4452_less__Suc__numeral,axiom,
% 12.45/6.94 ! [N: nat,K: num] :
% 12.45/6.94 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.45/6.94 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4453_pred__numeral__simps_I3_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( pred_numeral @ ( bit1 @ K ) )
% 12.45/6.94 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4454_le__numeral__Suc,axiom,
% 12.45/6.94 ! [K: num,N: nat] :
% 12.45/6.94 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 12.45/6.94 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4455_le__Suc__numeral,axiom,
% 12.45/6.94 ! [N: nat,K: num] :
% 12.45/6.94 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.45/6.94 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4456_diff__Suc__numeral,axiom,
% 12.45/6.94 ! [N: nat,K: num] :
% 12.45/6.94 ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.45/6.94 = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4457_diff__numeral__Suc,axiom,
% 12.45/6.94 ! [K: num,N: nat] :
% 12.45/6.94 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 12.45/6.94 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4458_dbl__dec__simps_I2_J,axiom,
% 12.45/6.94 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 12.45/6.94 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4459_dbl__dec__simps_I2_J,axiom,
% 12.45/6.94 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 12.45/6.94 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4460_dbl__dec__simps_I2_J,axiom,
% 12.45/6.94 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4461_dbl__dec__simps_I2_J,axiom,
% 12.45/6.94 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 12.45/6.94 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4462_dbl__dec__simps_I2_J,axiom,
% 12.45/6.94 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 12.45/6.94 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4463_dbl__dec__simps_I1_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 12.45/6.94 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4464_dbl__dec__simps_I1_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/6.94 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4465_dbl__dec__simps_I1_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4466_dbl__dec__simps_I1_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 12.45/6.94 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4467_dbl__dec__simps_I1_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 12.45/6.94 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4468_dbl__inc__simps_I1_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 12.45/6.94 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4469_dbl__inc__simps_I1_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/6.94 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4470_dbl__inc__simps_I1_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 12.45/6.94 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4471_dbl__inc__simps_I1_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 12.45/6.94 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4472_dbl__inc__simps_I1_J,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 12.45/6.94 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4473_signed__take__bit__numeral__bit0,axiom,
% 12.45/6.94 ! [L2: num,K: num] :
% 12.45/6.94 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 12.45/6.94 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4474_signed__take__bit__numeral__minus__bit0,axiom,
% 12.45/6.94 ! [L2: num,K: num] :
% 12.45/6.94 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.45/6.94 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4475_numeral__eq__Suc,axiom,
% 12.45/6.94 ( numeral_numeral_nat
% 12.45/6.94 = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4476_pred__numeral__def,axiom,
% 12.45/6.94 ( pred_numeral
% 12.45/6.94 = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4477_dbl__inc__def,axiom,
% 12.45/6.94 ( neg_nu8557863876264182079omplex
% 12.45/6.94 = ( ^ [X: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4478_dbl__inc__def,axiom,
% 12.45/6.94 ( neg_nu8295874005876285629c_real
% 12.45/6.94 = ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4479_dbl__inc__def,axiom,
% 12.45/6.94 ( neg_nu5219082963157363817nc_rat
% 12.45/6.94 = ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4480_dbl__inc__def,axiom,
% 12.45/6.94 ( neg_nu5851722552734809277nc_int
% 12.45/6.94 = ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4481_num_Osize__gen_I1_J,axiom,
% 12.45/6.94 ( ( size_num @ one )
% 12.45/6.94 = zero_zero_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4482_dbl__dec__def,axiom,
% 12.45/6.94 ( neg_nu6511756317524482435omplex
% 12.45/6.94 = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4483_dbl__dec__def,axiom,
% 12.45/6.94 ( neg_nu6075765906172075777c_real
% 12.45/6.94 = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4484_dbl__dec__def,axiom,
% 12.45/6.94 ( neg_nu3179335615603231917ec_rat
% 12.45/6.94 = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4485_dbl__dec__def,axiom,
% 12.45/6.94 ( neg_nu3811975205180677377ec_int
% 12.45/6.94 = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4486_add__0__iff,axiom,
% 12.45/6.94 ! [B2: real,A: real] :
% 12.45/6.94 ( ( B2
% 12.45/6.94 = ( plus_plus_real @ B2 @ A ) )
% 12.45/6.94 = ( A = zero_zero_real ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4487_add__0__iff,axiom,
% 12.45/6.94 ! [B2: rat,A: rat] :
% 12.45/6.94 ( ( B2
% 12.45/6.94 = ( plus_plus_rat @ B2 @ A ) )
% 12.45/6.94 = ( A = zero_zero_rat ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4488_add__0__iff,axiom,
% 12.45/6.94 ! [B2: nat,A: nat] :
% 12.45/6.94 ( ( B2
% 12.45/6.94 = ( plus_plus_nat @ B2 @ A ) )
% 12.45/6.94 = ( A = zero_zero_nat ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4489_add__0__iff,axiom,
% 12.45/6.94 ! [B2: int,A: int] :
% 12.45/6.94 ( ( B2
% 12.45/6.94 = ( plus_plus_int @ B2 @ A ) )
% 12.45/6.94 = ( A = zero_zero_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4490_crossproduct__noteq,axiom,
% 12.45/6.94 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.94 ( ( ( A != B2 )
% 12.45/6.94 & ( C != D ) )
% 12.45/6.94 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) )
% 12.45/6.94 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4491_crossproduct__noteq,axiom,
% 12.45/6.94 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.45/6.94 ( ( ( A != B2 )
% 12.45/6.94 & ( C != D ) )
% 12.45/6.94 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) )
% 12.45/6.94 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4492_crossproduct__noteq,axiom,
% 12.45/6.94 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.45/6.94 ( ( ( A != B2 )
% 12.45/6.94 & ( C != D ) )
% 12.45/6.94 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) )
% 12.45/6.94 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4493_crossproduct__noteq,axiom,
% 12.45/6.94 ! [A: int,B2: int,C: int,D: int] :
% 12.45/6.94 ( ( ( A != B2 )
% 12.45/6.94 & ( C != D ) )
% 12.45/6.94 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) )
% 12.45/6.94 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4494_crossproduct__eq,axiom,
% 12.45/6.94 ! [W: real,Y3: real,X4: real,Z: real] :
% 12.45/6.94 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X4 @ Z ) )
% 12.45/6.94 = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X4 @ Y3 ) ) )
% 12.45/6.94 = ( ( W = X4 )
% 12.45/6.94 | ( Y3 = Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4495_crossproduct__eq,axiom,
% 12.45/6.94 ! [W: rat,Y3: rat,X4: rat,Z: rat] :
% 12.45/6.94 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y3 ) @ ( times_times_rat @ X4 @ Z ) )
% 12.45/6.94 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X4 @ Y3 ) ) )
% 12.45/6.94 = ( ( W = X4 )
% 12.45/6.94 | ( Y3 = Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4496_crossproduct__eq,axiom,
% 12.45/6.94 ! [W: nat,Y3: nat,X4: nat,Z: nat] :
% 12.45/6.94 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X4 @ Z ) )
% 12.45/6.94 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X4 @ Y3 ) ) )
% 12.45/6.94 = ( ( W = X4 )
% 12.45/6.94 | ( Y3 = Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4497_crossproduct__eq,axiom,
% 12.45/6.94 ! [W: int,Y3: int,X4: int,Z: int] :
% 12.45/6.94 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X4 @ Z ) )
% 12.45/6.94 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X4 @ Y3 ) ) )
% 12.45/6.94 = ( ( W = X4 )
% 12.45/6.94 | ( Y3 = Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4498_eq__diff__eq_H,axiom,
% 12.45/6.94 ! [X4: real,Y3: real,Z: real] :
% 12.45/6.94 ( ( X4
% 12.45/6.94 = ( minus_minus_real @ Y3 @ Z ) )
% 12.45/6.94 = ( Y3
% 12.45/6.94 = ( plus_plus_real @ X4 @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4499_num_Osize__gen_I2_J,axiom,
% 12.45/6.94 ! [X2: num] :
% 12.45/6.94 ( ( size_num @ ( bit0 @ X2 ) )
% 12.45/6.94 = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4500_mask__numeral,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
% 12.45/6.94 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4501_mask__numeral,axiom,
% 12.45/6.94 ! [N: num] :
% 12.45/6.94 ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
% 12.45/6.94 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4502_take__bit__rec,axiom,
% 12.45/6.94 ( bit_se1745604003318907178nteger
% 12.45/6.94 = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4503_take__bit__rec,axiom,
% 12.45/6.94 ( bit_se2923211474154528505it_int
% 12.45/6.94 = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4504_take__bit__rec,axiom,
% 12.45/6.94 ( bit_se2925701944663578781it_nat
% 12.45/6.94 = ( ^ [N2: nat,A4: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4505_and__int__unfold,axiom,
% 12.45/6.94 ( bit_se725231765392027082nd_int
% 12.45/6.94 = ( ^ [K3: int,L: int] :
% 12.45/6.94 ( if_int
% 12.45/6.94 @ ( ( K3 = zero_zero_int )
% 12.45/6.94 | ( L = zero_zero_int ) )
% 12.45/6.94 @ zero_zero_int
% 12.45/6.94 @ ( if_int
% 12.45/6.94 @ ( K3
% 12.45/6.94 = ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.94 @ L
% 12.45/6.94 @ ( if_int
% 12.45/6.94 @ ( L
% 12.45/6.94 = ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.94 @ K3
% 12.45/6.94 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4506_exp__lower__Taylor__quadratic,axiom,
% 12.45/6.94 ! [X4: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.94 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( divide_divide_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X4 ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4507_sqrt__sum__squares__half__less,axiom,
% 12.45/6.94 ! [X4: real,U: real,Y3: real] :
% 12.45/6.94 ( ( ord_less_real @ X4 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.94 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.94 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4508_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.94 ! [A: int,X4: num,N: nat] :
% 12.45/6.94 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) )
% 12.45/6.94 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4509_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.94 ! [A: int,X4: num,N: nat] :
% 12.45/6.94 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) )
% 12.45/6.94 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4510_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.94 ! [A: int,X4: num,N: nat] :
% 12.45/6.94 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) )
% 12.45/6.94 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4511_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.94 ! [A: int,X4: num,N: nat] :
% 12.45/6.94 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) )
% 12.45/6.94 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4512_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 12.45/6.94 ! [X4: num,N: nat,A: int] :
% 12.45/6.94 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 12.45/6.94 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4513_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 12.45/6.94 ! [X4: num,N: nat,A: int] :
% 12.45/6.94 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 12.45/6.94 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4514_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 12.45/6.94 ! [X4: num,N: nat,A: int] :
% 12.45/6.94 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 12.45/6.94 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4515_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 12.45/6.94 ! [X4: num,N: nat,A: int] :
% 12.45/6.94 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 12.45/6.94 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4516_and_Oright__idem,axiom,
% 12.45/6.94 ! [A: int,B2: int] :
% 12.45/6.94 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B2 ) @ B2 )
% 12.45/6.94 = ( bit_se725231765392027082nd_int @ A @ B2 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4517_and_Oright__idem,axiom,
% 12.45/6.94 ! [A: nat,B2: nat] :
% 12.45/6.94 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B2 ) @ B2 )
% 12.45/6.94 = ( bit_se727722235901077358nd_nat @ A @ B2 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4518_and_Oleft__idem,axiom,
% 12.45/6.94 ! [A: int,B2: int] :
% 12.45/6.94 ( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
% 12.45/6.94 = ( bit_se725231765392027082nd_int @ A @ B2 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4519_and_Oleft__idem,axiom,
% 12.45/6.94 ! [A: nat,B2: nat] :
% 12.45/6.94 ( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
% 12.45/6.94 = ( bit_se727722235901077358nd_nat @ A @ B2 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4520_and_Oidem,axiom,
% 12.45/6.94 ! [A: int] :
% 12.45/6.94 ( ( bit_se725231765392027082nd_int @ A @ A )
% 12.45/6.94 = A ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4521_and_Oidem,axiom,
% 12.45/6.94 ! [A: nat] :
% 12.45/6.94 ( ( bit_se727722235901077358nd_nat @ A @ A )
% 12.45/6.94 = A ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4522_real__sqrt__eq__iff,axiom,
% 12.45/6.94 ! [X4: real,Y3: real] :
% 12.45/6.94 ( ( ( sqrt @ X4 )
% 12.45/6.94 = ( sqrt @ Y3 ) )
% 12.45/6.94 = ( X4 = Y3 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4523_mask__nat__positive__iff,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 12.45/6.94 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4524_take__bit__of__0,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 12.45/6.94 = zero_zero_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4525_take__bit__of__0,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 12.45/6.94 = zero_zero_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4526_bit_Oconj__zero__right,axiom,
% 12.45/6.94 ! [X4: int] :
% 12.45/6.94 ( ( bit_se725231765392027082nd_int @ X4 @ zero_zero_int )
% 12.45/6.94 = zero_zero_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4527_bit_Oconj__zero__left,axiom,
% 12.45/6.94 ! [X4: int] :
% 12.45/6.94 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X4 )
% 12.45/6.94 = zero_zero_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4528_zero__and__eq,axiom,
% 12.45/6.94 ! [A: int] :
% 12.45/6.94 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 12.45/6.94 = zero_zero_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4529_zero__and__eq,axiom,
% 12.45/6.94 ! [A: nat] :
% 12.45/6.94 ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 12.45/6.94 = zero_zero_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4530_and__zero__eq,axiom,
% 12.45/6.94 ! [A: int] :
% 12.45/6.94 ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 12.45/6.94 = zero_zero_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4531_and__zero__eq,axiom,
% 12.45/6.94 ! [A: nat] :
% 12.45/6.94 ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 12.45/6.94 = zero_zero_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4532_real__sqrt__zero,axiom,
% 12.45/6.94 ( ( sqrt @ zero_zero_real )
% 12.45/6.94 = zero_zero_real ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4533_real__sqrt__eq__zero__cancel__iff,axiom,
% 12.45/6.94 ! [X4: real] :
% 12.45/6.94 ( ( ( sqrt @ X4 )
% 12.45/6.94 = zero_zero_real )
% 12.45/6.94 = ( X4 = zero_zero_real ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4534_real__sqrt__less__iff,axiom,
% 12.45/6.94 ! [X4: real,Y3: real] :
% 12.45/6.94 ( ( ord_less_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) )
% 12.45/6.94 = ( ord_less_real @ X4 @ Y3 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4535_real__sqrt__le__iff,axiom,
% 12.45/6.94 ! [X4: real,Y3: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) )
% 12.45/6.94 = ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4536_real__sqrt__one,axiom,
% 12.45/6.94 ( ( sqrt @ one_one_real )
% 12.45/6.94 = one_one_real ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4537_real__sqrt__eq__1__iff,axiom,
% 12.45/6.94 ! [X4: real] :
% 12.45/6.94 ( ( ( sqrt @ X4 )
% 12.45/6.94 = one_one_real )
% 12.45/6.94 = ( X4 = one_one_real ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4538_take__bit__and,axiom,
% 12.45/6.94 ! [N: nat,A: int,B2: int] :
% 12.45/6.94 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
% 12.45/6.94 = ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B2 ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4539_take__bit__and,axiom,
% 12.45/6.94 ! [N: nat,A: nat,B2: nat] :
% 12.45/6.94 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
% 12.45/6.94 = ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B2 ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4540_concat__bit__of__zero__2,axiom,
% 12.45/6.94 ! [N: nat,K: int] :
% 12.45/6.94 ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 12.45/6.94 = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4541_take__bit__0,axiom,
% 12.45/6.94 ! [A: int] :
% 12.45/6.94 ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 12.45/6.94 = zero_zero_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4542_take__bit__0,axiom,
% 12.45/6.94 ! [A: nat] :
% 12.45/6.94 ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 12.45/6.94 = zero_zero_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4543_take__bit__Suc__1,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
% 12.45/6.94 = one_one_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4544_take__bit__Suc__1,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
% 12.45/6.94 = one_one_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4545_and_Oleft__neutral,axiom,
% 12.45/6.94 ! [A: code_integer] :
% 12.45/6.94 ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
% 12.45/6.94 = A ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4546_and_Oleft__neutral,axiom,
% 12.45/6.94 ! [A: int] :
% 12.45/6.94 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
% 12.45/6.94 = A ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4547_and_Oright__neutral,axiom,
% 12.45/6.94 ! [A: code_integer] :
% 12.45/6.94 ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.94 = A ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4548_and_Oright__neutral,axiom,
% 12.45/6.94 ! [A: int] :
% 12.45/6.94 ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.94 = A ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4549_bit_Oconj__one__right,axiom,
% 12.45/6.94 ! [X4: code_integer] :
% 12.45/6.94 ( ( bit_se3949692690581998587nteger @ X4 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.94 = X4 ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4550_bit_Oconj__one__right,axiom,
% 12.45/6.94 ! [X4: int] :
% 12.45/6.94 ( ( bit_se725231765392027082nd_int @ X4 @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.94 = X4 ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4551_of__int__eq__numeral__iff,axiom,
% 12.45/6.94 ! [Z: int,N: num] :
% 12.45/6.94 ( ( ( ring_17405671764205052669omplex @ Z )
% 12.45/6.94 = ( numera6690914467698888265omplex @ N ) )
% 12.45/6.94 = ( Z
% 12.45/6.94 = ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4552_of__int__eq__numeral__iff,axiom,
% 12.45/6.94 ! [Z: int,N: num] :
% 12.45/6.94 ( ( ( ring_1_of_int_real @ Z )
% 12.45/6.94 = ( numeral_numeral_real @ N ) )
% 12.45/6.94 = ( Z
% 12.45/6.94 = ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4553_of__int__eq__numeral__iff,axiom,
% 12.45/6.94 ! [Z: int,N: num] :
% 12.45/6.94 ( ( ( ring_1_of_int_rat @ Z )
% 12.45/6.94 = ( numeral_numeral_rat @ N ) )
% 12.45/6.94 = ( Z
% 12.45/6.94 = ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4554_of__int__eq__numeral__iff,axiom,
% 12.45/6.94 ! [Z: int,N: num] :
% 12.45/6.94 ( ( ( ring_1_of_int_int @ Z )
% 12.45/6.94 = ( numeral_numeral_int @ N ) )
% 12.45/6.94 = ( Z
% 12.45/6.94 = ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4555_of__int__numeral,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 12.45/6.94 = ( numera6690914467698888265omplex @ K ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4556_of__int__numeral,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 12.45/6.94 = ( numeral_numeral_real @ K ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4557_of__int__numeral,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 12.45/6.94 = ( numeral_numeral_rat @ K ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4558_of__int__numeral,axiom,
% 12.45/6.94 ! [K: num] :
% 12.45/6.94 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 12.45/6.94 = ( numeral_numeral_int @ K ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4559_of__int__le__iff,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 12.45/6.94 = ( ord_less_eq_int @ W @ Z ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4560_of__int__le__iff,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 12.45/6.94 = ( ord_less_eq_int @ W @ Z ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4561_of__int__le__iff,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 12.45/6.94 = ( ord_less_eq_int @ W @ Z ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4562_take__bit__numeral__1,axiom,
% 12.45/6.94 ! [L2: num] :
% 12.45/6.94 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ one_one_int )
% 12.45/6.94 = one_one_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4563_take__bit__numeral__1,axiom,
% 12.45/6.94 ! [L2: num] :
% 12.45/6.94 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ one_one_nat )
% 12.45/6.94 = one_one_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4564_of__int__less__iff,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 12.45/6.94 = ( ord_less_int @ W @ Z ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4565_of__int__less__iff,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 12.45/6.94 = ( ord_less_int @ W @ Z ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4566_of__int__less__iff,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 12.45/6.94 = ( ord_less_int @ W @ Z ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4567_of__int__1,axiom,
% 12.45/6.94 ( ( ring_17405671764205052669omplex @ one_one_int )
% 12.45/6.94 = one_one_complex ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4568_of__int__1,axiom,
% 12.45/6.94 ( ( ring_1_of_int_int @ one_one_int )
% 12.45/6.94 = one_one_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4569_of__int__1,axiom,
% 12.45/6.94 ( ( ring_1_of_int_real @ one_one_int )
% 12.45/6.94 = one_one_real ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4570_of__int__1,axiom,
% 12.45/6.94 ( ( ring_1_of_int_rat @ one_one_int )
% 12.45/6.94 = one_one_rat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4571_of__int__eq__1__iff,axiom,
% 12.45/6.94 ! [Z: int] :
% 12.45/6.94 ( ( ( ring_17405671764205052669omplex @ Z )
% 12.45/6.94 = one_one_complex )
% 12.45/6.94 = ( Z = one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4572_of__int__eq__1__iff,axiom,
% 12.45/6.94 ! [Z: int] :
% 12.45/6.94 ( ( ( ring_1_of_int_int @ Z )
% 12.45/6.94 = one_one_int )
% 12.45/6.94 = ( Z = one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4573_of__int__eq__1__iff,axiom,
% 12.45/6.94 ! [Z: int] :
% 12.45/6.94 ( ( ( ring_1_of_int_real @ Z )
% 12.45/6.94 = one_one_real )
% 12.45/6.94 = ( Z = one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4574_of__int__eq__1__iff,axiom,
% 12.45/6.94 ! [Z: int] :
% 12.45/6.94 ( ( ( ring_1_of_int_rat @ Z )
% 12.45/6.94 = one_one_rat )
% 12.45/6.94 = ( Z = one_one_int ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4575_of__int__mult,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 12.45/6.94 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4576_of__int__mult,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 12.45/6.94 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4577_of__int__mult,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 12.45/6.94 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4578_of__int__add,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 12.45/6.94 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4579_of__int__add,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 12.45/6.94 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4580_of__int__add,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 12.45/6.94 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4581_real__sqrt__lt__0__iff,axiom,
% 12.45/6.94 ! [X4: real] :
% 12.45/6.94 ( ( ord_less_real @ ( sqrt @ X4 ) @ zero_zero_real )
% 12.45/6.94 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4582_real__sqrt__gt__0__iff,axiom,
% 12.45/6.94 ! [Y3: real] :
% 12.45/6.94 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 12.45/6.94 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4583_real__sqrt__ge__0__iff,axiom,
% 12.45/6.94 ! [Y3: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 12.45/6.94 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4584_real__sqrt__le__0__iff,axiom,
% 12.45/6.94 ! [X4: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ zero_zero_real )
% 12.45/6.94 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4585_mask__eq__0__iff,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( ( bit_se2002935070580805687sk_nat @ N )
% 12.45/6.94 = zero_zero_nat )
% 12.45/6.94 = ( N = zero_zero_nat ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4586_mask__eq__0__iff,axiom,
% 12.45/6.94 ! [N: nat] :
% 12.45/6.94 ( ( ( bit_se2000444600071755411sk_int @ N )
% 12.45/6.94 = zero_zero_int )
% 12.45/6.94 = ( N = zero_zero_nat ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4587_mask__0,axiom,
% 12.45/6.94 ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
% 12.45/6.94 = zero_zero_nat ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4588_mask__0,axiom,
% 12.45/6.94 ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
% 12.45/6.94 = zero_zero_int ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4589_of__int__diff,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 12.45/6.94 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4590_of__int__diff,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 12.45/6.94 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4591_of__int__diff,axiom,
% 12.45/6.94 ! [W: int,Z: int] :
% 12.45/6.94 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 12.45/6.94 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4592_real__sqrt__lt__1__iff,axiom,
% 12.45/6.94 ! [X4: real] :
% 12.45/6.94 ( ( ord_less_real @ ( sqrt @ X4 ) @ one_one_real )
% 12.45/6.94 = ( ord_less_real @ X4 @ one_one_real ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4593_real__sqrt__gt__1__iff,axiom,
% 12.45/6.94 ! [Y3: real] :
% 12.45/6.94 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
% 12.45/6.94 = ( ord_less_real @ one_one_real @ Y3 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4594_real__sqrt__le__1__iff,axiom,
% 12.45/6.94 ! [X4: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ one_one_real )
% 12.45/6.94 = ( ord_less_eq_real @ X4 @ one_one_real ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4595_real__sqrt__ge__1__iff,axiom,
% 12.45/6.94 ! [Y3: real] :
% 12.45/6.94 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
% 12.45/6.94 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4596_of__int__power,axiom,
% 12.45/6.94 ! [Z: int,N: nat] :
% 12.45/6.94 ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
% 12.45/6.94 = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4597_of__int__power,axiom,
% 12.45/6.94 ! [Z: int,N: nat] :
% 12.45/6.94 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 12.45/6.94 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4598_of__int__power,axiom,
% 12.45/6.94 ! [Z: int,N: nat] :
% 12.45/6.94 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 12.45/6.94 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4599_of__int__power,axiom,
% 12.45/6.94 ! [Z: int,N: nat] :
% 12.45/6.94 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
% 12.45/6.94 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4600_of__int__eq__of__int__power__cancel__iff,axiom,
% 12.45/6.94 ! [B2: int,W: nat,X4: int] :
% 12.45/6.94 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W )
% 12.45/6.94 = ( ring_1_of_int_rat @ X4 ) )
% 12.45/6.94 = ( ( power_power_int @ B2 @ W )
% 12.45/6.94 = X4 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4601_of__int__eq__of__int__power__cancel__iff,axiom,
% 12.45/6.94 ! [B2: int,W: nat,X4: int] :
% 12.45/6.94 ( ( ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W )
% 12.45/6.94 = ( ring_1_of_int_real @ X4 ) )
% 12.45/6.94 = ( ( power_power_int @ B2 @ W )
% 12.45/6.94 = X4 ) ) ).
% 12.45/6.94
% 12.45/6.94 thf(fact_4602_of__int__eq__of__int__power__cancel__iff,axiom,
% 12.45/6.94 ! [B2: int,W: nat,X4: int] :
% 12.45/6.94 ( ( ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W )
% 12.45/6.94 = ( ring_1_of_int_int @ X4 ) )
% 12.45/6.95 = ( ( power_power_int @ B2 @ W )
% 12.45/6.95 = X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4603_of__int__eq__of__int__power__cancel__iff,axiom,
% 12.45/6.95 ! [B2: int,W: nat,X4: int] :
% 12.45/6.95 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B2 ) @ W )
% 12.45/6.95 = ( ring_17405671764205052669omplex @ X4 ) )
% 12.45/6.95 = ( ( power_power_int @ B2 @ W )
% 12.45/6.95 = X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4604_of__int__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: int,B2: int,W: nat] :
% 12.45/6.95 ( ( ( ring_1_of_int_rat @ X4 )
% 12.45/6.95 = ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
% 12.45/6.95 = ( X4
% 12.45/6.95 = ( power_power_int @ B2 @ W ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4605_of__int__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: int,B2: int,W: nat] :
% 12.45/6.95 ( ( ( ring_1_of_int_real @ X4 )
% 12.45/6.95 = ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
% 12.45/6.95 = ( X4
% 12.45/6.95 = ( power_power_int @ B2 @ W ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4606_of__int__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: int,B2: int,W: nat] :
% 12.45/6.95 ( ( ( ring_1_of_int_int @ X4 )
% 12.45/6.95 = ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
% 12.45/6.95 = ( X4
% 12.45/6.95 = ( power_power_int @ B2 @ W ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4607_of__int__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: int,B2: int,W: nat] :
% 12.45/6.95 ( ( ( ring_17405671764205052669omplex @ X4 )
% 12.45/6.95 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B2 ) @ W ) )
% 12.45/6.95 = ( X4
% 12.45/6.95 = ( power_power_int @ B2 @ W ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4608_and__nonnegative__int__iff,axiom,
% 12.45/6.95 ! [K: int,L2: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 12.45/6.95 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.95 | ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4609_and__negative__int__iff,axiom,
% 12.45/6.95 ! [K: int,L2: int] :
% 12.45/6.95 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ zero_zero_int )
% 12.45/6.95 = ( ( ord_less_int @ K @ zero_zero_int )
% 12.45/6.95 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4610_take__bit__of__1__eq__0__iff,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 12.45/6.95 = zero_zero_int )
% 12.45/6.95 = ( N = zero_zero_nat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4611_take__bit__of__1__eq__0__iff,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 12.45/6.95 = zero_zero_nat )
% 12.45/6.95 = ( N = zero_zero_nat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4612_and__numerals_I2_J,axiom,
% 12.45/6.95 ! [Y3: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 12.45/6.95 = one_one_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4613_and__numerals_I2_J,axiom,
% 12.45/6.95 ! [Y3: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 12.45/6.95 = one_one_nat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4614_and__numerals_I8_J,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ one_one_int )
% 12.45/6.95 = one_one_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4615_and__numerals_I8_J,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ one_one_nat )
% 12.45/6.95 = one_one_nat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4616_real__sqrt__four,axiom,
% 12.45/6.95 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.45/6.95 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4617_mask__Suc__0,axiom,
% 12.45/6.95 ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
% 12.45/6.95 = one_one_nat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4618_mask__Suc__0,axiom,
% 12.45/6.95 ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
% 12.45/6.95 = one_one_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4619_take__bit__minus__one__eq__mask,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.95 = ( bit_se2119862282449309892nteger @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4620_take__bit__minus__one__eq__mask,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.95 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4621_take__bit__of__Suc__0,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.45/6.95 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4622_and__numerals_I1_J,axiom,
% 12.45/6.95 ! [Y3: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 12.45/6.95 = zero_zero_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4623_and__numerals_I1_J,axiom,
% 12.45/6.95 ! [Y3: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 12.45/6.95 = zero_zero_nat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4624_and__numerals_I5_J,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ one_one_int )
% 12.45/6.95 = zero_zero_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4625_and__numerals_I5_J,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ one_one_nat )
% 12.45/6.95 = zero_zero_nat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4626_and__numerals_I3_J,axiom,
% 12.45/6.95 ! [X4: num,Y3: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 12.45/6.95 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4627_and__numerals_I3_J,axiom,
% 12.45/6.95 ! [X4: num,Y3: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 12.45/6.95 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4628_of__int__le__0__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 12.45/6.95 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4629_of__int__le__0__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 12.45/6.95 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4630_of__int__le__0__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 12.45/6.95 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4631_of__int__0__le__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 12.45/6.95 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4632_of__int__0__le__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 12.45/6.95 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4633_of__int__0__le__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 12.45/6.95 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4634_of__int__0__less__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 12.45/6.95 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4635_of__int__0__less__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 12.45/6.95 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4636_of__int__0__less__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 12.45/6.95 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4637_of__int__less__0__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 12.45/6.95 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4638_of__int__less__0__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 12.45/6.95 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4639_of__int__less__0__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 12.45/6.95 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4640_of__int__le__numeral__iff,axiom,
% 12.45/6.95 ! [Z: int,N: num] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 12.45/6.95 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4641_of__int__le__numeral__iff,axiom,
% 12.45/6.95 ! [Z: int,N: num] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 12.45/6.95 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4642_of__int__le__numeral__iff,axiom,
% 12.45/6.95 ! [Z: int,N: num] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 12.45/6.95 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4643_of__int__numeral__le__iff,axiom,
% 12.45/6.95 ! [N: num,Z: int] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4644_of__int__numeral__le__iff,axiom,
% 12.45/6.95 ! [N: num,Z: int] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4645_of__int__numeral__le__iff,axiom,
% 12.45/6.95 ! [N: num,Z: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4646_of__int__less__numeral__iff,axiom,
% 12.45/6.95 ! [Z: int,N: num] :
% 12.45/6.95 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 12.45/6.95 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4647_of__int__less__numeral__iff,axiom,
% 12.45/6.95 ! [Z: int,N: num] :
% 12.45/6.95 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 12.45/6.95 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4648_of__int__less__numeral__iff,axiom,
% 12.45/6.95 ! [Z: int,N: num] :
% 12.45/6.95 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 12.45/6.95 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4649_of__int__numeral__less__iff,axiom,
% 12.45/6.95 ! [N: num,Z: int] :
% 12.45/6.95 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 12.45/6.95 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4650_of__int__numeral__less__iff,axiom,
% 12.45/6.95 ! [N: num,Z: int] :
% 12.45/6.95 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 12.45/6.95 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4651_of__int__numeral__less__iff,axiom,
% 12.45/6.95 ! [N: num,Z: int] :
% 12.45/6.95 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 12.45/6.95 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4652_of__int__le__1__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 12.45/6.95 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4653_of__int__le__1__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 12.45/6.95 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4654_of__int__le__1__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 12.45/6.95 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4655_of__int__1__le__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 12.45/6.95 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4656_of__int__1__le__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 12.45/6.95 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4657_of__int__1__le__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 12.45/6.95 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4658_of__int__less__1__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 12.45/6.95 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4659_of__int__less__1__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 12.45/6.95 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4660_of__int__less__1__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 12.45/6.95 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4661_of__int__1__less__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 12.45/6.95 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4662_of__int__1__less__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 12.45/6.95 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4663_of__int__1__less__iff,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 12.45/6.95 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4664_take__bit__of__1,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
% 12.45/6.95 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4665_take__bit__of__1,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 12.45/6.95 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4666_take__bit__of__1,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 12.45/6.95 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4667_numeral__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,Y3: int] :
% 12.45/6.95 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N )
% 12.45/6.95 = ( ring_17405671764205052669omplex @ Y3 ) )
% 12.45/6.95 = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 12.45/6.95 = Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4668_numeral__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,Y3: int] :
% 12.45/6.95 ( ( ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N )
% 12.45/6.95 = ( ring_1_of_int_real @ Y3 ) )
% 12.45/6.95 = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 12.45/6.95 = Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4669_numeral__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,Y3: int] :
% 12.45/6.95 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N )
% 12.45/6.95 = ( ring_1_of_int_rat @ Y3 ) )
% 12.45/6.95 = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 12.45/6.95 = Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4670_numeral__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,Y3: int] :
% 12.45/6.95 ( ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 12.45/6.95 = ( ring_1_of_int_int @ Y3 ) )
% 12.45/6.95 = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 12.45/6.95 = Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4671_of__int__eq__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [Y3: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 12.45/6.95 = ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N ) )
% 12.45/6.95 = ( Y3
% 12.45/6.95 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4672_of__int__eq__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [Y3: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ( ring_1_of_int_real @ Y3 )
% 12.45/6.95 = ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 12.45/6.95 = ( Y3
% 12.45/6.95 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4673_of__int__eq__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [Y3: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ( ring_1_of_int_rat @ Y3 )
% 12.45/6.95 = ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 12.45/6.95 = ( Y3
% 12.45/6.95 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4674_of__int__eq__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [Y3: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ( ring_1_of_int_int @ Y3 )
% 12.45/6.95 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
% 12.45/6.95 = ( Y3
% 12.45/6.95 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4675_of__int__le__of__int__power__cancel__iff,axiom,
% 12.45/6.95 ! [B2: int,W: nat,X4: int] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) @ ( ring_1_of_int_real @ X4 ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4676_of__int__le__of__int__power__cancel__iff,axiom,
% 12.45/6.95 ! [B2: int,W: nat,X4: int] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) @ ( ring_1_of_int_rat @ X4 ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4677_of__int__le__of__int__power__cancel__iff,axiom,
% 12.45/6.95 ! [B2: int,W: nat,X4: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) @ ( ring_1_of_int_int @ X4 ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4678_of__int__power__le__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: int,B2: int,W: nat] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
% 12.45/6.95 = ( ord_less_eq_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4679_of__int__power__le__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: int,B2: int,W: nat] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
% 12.45/6.95 = ( ord_less_eq_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4680_of__int__power__le__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: int,B2: int,W: nat] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X4 ) @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
% 12.45/6.95 = ( ord_less_eq_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4681_of__int__less__of__int__power__cancel__iff,axiom,
% 12.45/6.95 ! [B2: int,W: nat,X4: int] :
% 12.45/6.95 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) @ ( ring_1_of_int_real @ X4 ) )
% 12.45/6.95 = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4682_of__int__less__of__int__power__cancel__iff,axiom,
% 12.45/6.95 ! [B2: int,W: nat,X4: int] :
% 12.45/6.95 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) @ ( ring_1_of_int_rat @ X4 ) )
% 12.45/6.95 = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4683_of__int__less__of__int__power__cancel__iff,axiom,
% 12.45/6.95 ! [B2: int,W: nat,X4: int] :
% 12.45/6.95 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) @ ( ring_1_of_int_int @ X4 ) )
% 12.45/6.95 = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4684_of__int__power__less__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: int,B2: int,W: nat] :
% 12.45/6.95 ( ( ord_less_real @ ( ring_1_of_int_real @ X4 ) @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
% 12.45/6.95 = ( ord_less_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4685_of__int__power__less__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: int,B2: int,W: nat] :
% 12.45/6.95 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X4 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
% 12.45/6.95 = ( ord_less_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4686_of__int__power__less__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: int,B2: int,W: nat] :
% 12.45/6.95 ( ( ord_less_int @ ( ring_1_of_int_int @ X4 ) @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
% 12.45/6.95 = ( ord_less_int @ X4 @ ( power_power_int @ B2 @ W ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4687_and__minus__numerals_I6_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 12.45/6.95 = one_one_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4688_and__minus__numerals_I2_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.45/6.95 = one_one_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4689_and__numerals_I4_J,axiom,
% 12.45/6.95 ! [X4: num,Y3: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 12.45/6.95 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4690_and__numerals_I4_J,axiom,
% 12.45/6.95 ! [X4: num,Y3: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 12.45/6.95 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4691_and__numerals_I6_J,axiom,
% 12.45/6.95 ! [X4: num,Y3: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 12.45/6.95 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4692_and__numerals_I6_J,axiom,
% 12.45/6.95 ! [X4: num,Y3: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 12.45/6.95 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4693_even__take__bit__eq,axiom,
% 12.45/6.95 ! [N: nat,A: code_integer] :
% 12.45/6.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 12.45/6.95 = ( ( N = zero_zero_nat )
% 12.45/6.95 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4694_even__take__bit__eq,axiom,
% 12.45/6.95 ! [N: nat,A: int] :
% 12.45/6.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 12.45/6.95 = ( ( N = zero_zero_nat )
% 12.45/6.95 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4695_even__take__bit__eq,axiom,
% 12.45/6.95 ! [N: nat,A: nat] :
% 12.45/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 12.45/6.95 = ( ( N = zero_zero_nat )
% 12.45/6.95 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4696_and__minus__numerals_I5_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 12.45/6.95 = zero_zero_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4697_and__minus__numerals_I1_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.45/6.95 = zero_zero_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4698_take__bit__Suc__0,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 12.45/6.95 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4699_take__bit__Suc__0,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 12.45/6.95 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4700_take__bit__Suc__0,axiom,
% 12.45/6.95 ! [A: nat] :
% 12.45/6.95 ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 12.45/6.95 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4701_real__sqrt__pow2__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ( power_power_real @ ( sqrt @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.95 = X4 )
% 12.45/6.95 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4702_real__sqrt__pow2,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( power_power_real @ ( sqrt @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.95 = X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4703_numeral__power__le__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,A: int] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4704_numeral__power__le__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,A: int] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4705_numeral__power__le__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,A: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4706_of__int__le__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [A: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 12.45/6.95 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4707_of__int__le__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [A: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 12.45/6.95 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4708_of__int__le__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [A: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
% 12.45/6.95 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4709_numeral__power__less__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,A: int] :
% 12.45/6.95 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 12.45/6.95 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4710_numeral__power__less__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,A: int] :
% 12.45/6.95 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 12.45/6.95 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4711_numeral__power__less__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,A: int] :
% 12.45/6.95 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 12.45/6.95 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4712_of__int__less__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [A: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 12.45/6.95 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4713_of__int__less__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [A: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 12.45/6.95 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4714_of__int__less__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [A: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
% 12.45/6.95 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4715_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 12.45/6.95 ! [X4: real,Y3: real,Xa: real,Ya: real] :
% 12.45/6.95 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( 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 ) ) )
% 12.45/6.95 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( 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 ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4716_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [Y3: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ( ring_1_of_int_real @ Y3 )
% 12.45/6.95 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) )
% 12.45/6.95 = ( Y3
% 12.45/6.95 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4717_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [Y3: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ( ring_1_of_int_int @ Y3 )
% 12.45/6.95 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) )
% 12.45/6.95 = ( Y3
% 12.45/6.95 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4718_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [Y3: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 12.45/6.95 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X4 ) ) @ N ) )
% 12.45/6.95 = ( Y3
% 12.45/6.95 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4719_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [Y3: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ( ring_18347121197199848620nteger @ Y3 )
% 12.45/6.95 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) )
% 12.45/6.95 = ( Y3
% 12.45/6.95 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4720_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [Y3: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ( ring_1_of_int_rat @ Y3 )
% 12.45/6.95 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) )
% 12.45/6.95 = ( Y3
% 12.45/6.95 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4721_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,Y3: int] :
% 12.45/6.95 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N )
% 12.45/6.95 = ( ring_1_of_int_real @ Y3 ) )
% 12.45/6.95 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 12.45/6.95 = Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4722_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,Y3: int] :
% 12.45/6.95 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 12.45/6.95 = ( ring_1_of_int_int @ Y3 ) )
% 12.45/6.95 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 12.45/6.95 = Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4723_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,Y3: int] :
% 12.45/6.95 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X4 ) ) @ N )
% 12.45/6.95 = ( ring_17405671764205052669omplex @ Y3 ) )
% 12.45/6.95 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 12.45/6.95 = Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4724_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,Y3: int] :
% 12.45/6.95 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N )
% 12.45/6.95 = ( ring_18347121197199848620nteger @ Y3 ) )
% 12.45/6.95 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 12.45/6.95 = Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4725_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,Y3: int] :
% 12.45/6.95 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N )
% 12.45/6.95 = ( ring_1_of_int_rat @ Y3 ) )
% 12.45/6.95 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N )
% 12.45/6.95 = Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4726_and__numerals_I7_J,axiom,
% 12.45/6.95 ! [X4: num,Y3: num] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 12.45/6.95 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4727_and__numerals_I7_J,axiom,
% 12.45/6.95 ! [X4: num,Y3: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 12.45/6.95 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4728_take__bit__of__exp,axiom,
% 12.45/6.95 ! [M: nat,N: nat] :
% 12.45/6.95 ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.95 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4729_take__bit__of__exp,axiom,
% 12.45/6.95 ! [M: nat,N: nat] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.95 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4730_take__bit__of__exp,axiom,
% 12.45/6.95 ! [M: nat,N: nat] :
% 12.45/6.95 ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.95 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4731_take__bit__of__2,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.95 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4732_take__bit__of__2,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.95 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4733_take__bit__of__2,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.95 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4734_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,A: int] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4735_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,A: int] :
% 12.45/6.95 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4736_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,A: int] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4737_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 12.45/6.95 ! [X4: num,N: nat,A: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 12.45/6.95 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4738_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [A: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N ) )
% 12.45/6.95 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4739_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [A: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N ) )
% 12.45/6.95 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4740_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [A: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N ) )
% 12.45/6.95 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4741_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 12.45/6.95 ! [A: int,X4: num,N: nat] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) )
% 12.45/6.95 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4742_take__bit__eq__mask,axiom,
% 12.45/6.95 ( bit_se2923211474154528505it_int
% 12.45/6.95 = ( ^ [N2: nat,A4: int] : ( bit_se725231765392027082nd_int @ A4 @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4743_take__bit__eq__mask,axiom,
% 12.45/6.95 ( bit_se2925701944663578781it_nat
% 12.45/6.95 = ( ^ [N2: nat,A4: nat] : ( bit_se727722235901077358nd_nat @ A4 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4744_and_Oleft__commute,axiom,
% 12.45/6.95 ! [B2: int,A: int,C: int] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ B2 @ ( bit_se725231765392027082nd_int @ A @ C ) )
% 12.45/6.95 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B2 @ C ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4745_and_Oleft__commute,axiom,
% 12.45/6.95 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ B2 @ ( bit_se727722235901077358nd_nat @ A @ C ) )
% 12.45/6.95 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B2 @ C ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4746_and_Ocommute,axiom,
% 12.45/6.95 ( bit_se725231765392027082nd_int
% 12.45/6.95 = ( ^ [A4: int,B3: int] : ( bit_se725231765392027082nd_int @ B3 @ A4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4747_and_Ocommute,axiom,
% 12.45/6.95 ( bit_se727722235901077358nd_nat
% 12.45/6.95 = ( ^ [A4: nat,B3: nat] : ( bit_se727722235901077358nd_nat @ B3 @ A4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4748_take__bit__of__int,axiom,
% 12.45/6.95 ! [N: nat,K: int] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
% 12.45/6.95 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4749_of__int__mask__eq,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
% 12.45/6.95 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4750_and_Oassoc,axiom,
% 12.45/6.95 ! [A: int,B2: int,C: int] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B2 ) @ C )
% 12.45/6.95 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B2 @ C ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4751_and_Oassoc,axiom,
% 12.45/6.95 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B2 ) @ C )
% 12.45/6.95 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B2 @ C ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4752_of__int__and__eq,axiom,
% 12.45/6.95 ! [K: int,L2: int] :
% 12.45/6.95 ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 12.45/6.95 = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4753_take__bit__add,axiom,
% 12.45/6.95 ! [N: nat,A: int,B2: int] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B2 ) ) )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4754_take__bit__add,axiom,
% 12.45/6.95 ! [N: nat,A: nat,B2: nat] :
% 12.45/6.95 ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B2 ) ) )
% 12.45/6.95 = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4755_take__bit__tightened,axiom,
% 12.45/6.95 ! [N: nat,A: int,B2: int,M: nat] :
% 12.45/6.95 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ N @ B2 ) )
% 12.45/6.95 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ M @ A )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ M @ B2 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4756_take__bit__tightened,axiom,
% 12.45/6.95 ! [N: nat,A: nat,B2: nat,M: nat] :
% 12.45/6.95 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 12.45/6.95 = ( bit_se2925701944663578781it_nat @ N @ B2 ) )
% 12.45/6.95 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.95 => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 12.45/6.95 = ( bit_se2925701944663578781it_nat @ M @ B2 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4757_take__bit__nat__less__eq__self,axiom,
% 12.45/6.95 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4758_take__bit__tightened__less__eq__nat,axiom,
% 12.45/6.95 ! [M: nat,N: nat,Q2: nat] :
% 12.45/6.95 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.95 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4759_real__sqrt__less__mono,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.95 => ( ord_less_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4760_real__sqrt__le__mono,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.95 => ( ord_less_eq_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4761_mult__of__int__commute,axiom,
% 12.45/6.95 ! [X4: int,Y3: real] :
% 12.45/6.95 ( ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ Y3 )
% 12.45/6.95 = ( times_times_real @ Y3 @ ( ring_1_of_int_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4762_mult__of__int__commute,axiom,
% 12.45/6.95 ! [X4: int,Y3: rat] :
% 12.45/6.95 ( ( times_times_rat @ ( ring_1_of_int_rat @ X4 ) @ Y3 )
% 12.45/6.95 = ( times_times_rat @ Y3 @ ( ring_1_of_int_rat @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4763_mult__of__int__commute,axiom,
% 12.45/6.95 ! [X4: int,Y3: int] :
% 12.45/6.95 ( ( times_times_int @ ( ring_1_of_int_int @ X4 ) @ Y3 )
% 12.45/6.95 = ( times_times_int @ Y3 @ ( ring_1_of_int_int @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4764_real__sqrt__divide,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( sqrt @ ( divide_divide_real @ X4 @ Y3 ) )
% 12.45/6.95 = ( divide_divide_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4765_real__sqrt__mult,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( sqrt @ ( times_times_real @ X4 @ Y3 ) )
% 12.45/6.95 = ( times_times_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4766_real__sqrt__power,axiom,
% 12.45/6.95 ! [X4: real,K: nat] :
% 12.45/6.95 ( ( sqrt @ ( power_power_real @ X4 @ K ) )
% 12.45/6.95 = ( power_power_real @ ( sqrt @ X4 ) @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4767_real__sqrt__minus,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( sqrt @ ( uminus_uminus_real @ X4 ) )
% 12.45/6.95 = ( uminus_uminus_real @ ( sqrt @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4768_take__bit__minus,axiom,
% 12.45/6.95 ! [N: nat,K: int] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4769_take__bit__mult,axiom,
% 12.45/6.95 ! [N: nat,K: int,L2: int] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4770_take__bit__diff,axiom,
% 12.45/6.95 ! [N: nat,K: int,L2: int] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4771_concat__bit__take__bit__eq,axiom,
% 12.45/6.95 ! [N: nat,B2: int] :
% 12.45/6.95 ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B2 ) )
% 12.45/6.95 = ( bit_concat_bit @ N @ B2 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4772_concat__bit__eq__iff,axiom,
% 12.45/6.95 ! [N: nat,K: int,L2: int,R2: int,S: int] :
% 12.45/6.95 ( ( ( bit_concat_bit @ N @ K @ L2 )
% 12.45/6.95 = ( bit_concat_bit @ N @ R2 @ S ) )
% 12.45/6.95 = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ N @ R2 ) )
% 12.45/6.95 & ( L2 = S ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4773_less__eq__mask,axiom,
% 12.45/6.95 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4774_take__bit__eq__mask__iff,axiom,
% 12.45/6.95 ! [N: nat,K: int] :
% 12.45/6.95 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.45/6.95 = ( bit_se2000444600071755411sk_int @ N ) )
% 12.45/6.95 = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 12.45/6.95 = zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4775_and__eq__minus__1__iff,axiom,
% 12.45/6.95 ! [A: code_integer,B2: code_integer] :
% 12.45/6.95 ( ( ( bit_se3949692690581998587nteger @ A @ B2 )
% 12.45/6.95 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.95 = ( ( A
% 12.45/6.95 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.95 & ( B2
% 12.45/6.95 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4776_and__eq__minus__1__iff,axiom,
% 12.45/6.95 ! [A: int,B2: int] :
% 12.45/6.95 ( ( ( bit_se725231765392027082nd_int @ A @ B2 )
% 12.45/6.95 = ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.95 = ( ( A
% 12.45/6.95 = ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.95 & ( B2
% 12.45/6.95 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4777_real__sqrt__gt__zero,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4778_real__sqrt__ge__zero,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4779_real__sqrt__eq__zero__cancel,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ( sqrt @ X4 )
% 12.45/6.95 = zero_zero_real )
% 12.45/6.95 => ( X4 = zero_zero_real ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4780_real__sqrt__ge__one,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ one_one_real @ X4 )
% 12.45/6.95 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4781_take__bit__tightened__less__eq__int,axiom,
% 12.45/6.95 ! [M: nat,N: nat,K: int] :
% 12.45/6.95 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.95 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4782_AND__upper2_H,axiom,
% 12.45/6.95 ! [Y3: int,Z: int,X4: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.95 => ( ( ord_less_eq_int @ Y3 @ Z )
% 12.45/6.95 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ Z ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4783_AND__upper1_H,axiom,
% 12.45/6.95 ! [Y3: int,Z: int,Ya: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.95 => ( ( ord_less_eq_int @ Y3 @ Z )
% 12.45/6.95 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4784_AND__upper2,axiom,
% 12.45/6.95 ! [Y3: int,X4: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.95 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4785_AND__upper1,axiom,
% 12.45/6.95 ! [X4: int,Y3: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.95 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4786_AND__lower,axiom,
% 12.45/6.95 ! [X4: int,Y3: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.95 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4787_take__bit__int__less__eq__self__iff,axiom,
% 12.45/6.95 ! [N: nat,K: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 12.45/6.95 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4788_take__bit__nonnegative,axiom,
% 12.45/6.95 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4789_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 12.45/6.95 ! [N: nat,A: int,B2: int] :
% 12.45/6.95 ( ( ( bit_ri631733984087533419it_int @ N @ A )
% 12.45/6.95 = ( bit_ri631733984087533419it_int @ N @ B2 ) )
% 12.45/6.95 = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4790_not__take__bit__negative,axiom,
% 12.45/6.95 ! [N: nat,K: int] :
% 12.45/6.95 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4791_take__bit__int__greater__self__iff,axiom,
% 12.45/6.95 ! [K: int,N: nat] :
% 12.45/6.95 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 12.45/6.95 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4792_signed__take__bit__take__bit,axiom,
% 12.45/6.95 ! [M: nat,N: nat,A: int] :
% 12.45/6.95 ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 12.45/6.95 = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4793_take__bit__unset__bit__eq,axiom,
% 12.45/6.95 ! [N: nat,M: nat,A: int] :
% 12.45/6.95 ( ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 12.45/6.95 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 12.45/6.95 = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4794_take__bit__unset__bit__eq,axiom,
% 12.45/6.95 ! [N: nat,M: nat,A: nat] :
% 12.45/6.95 ( ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 12.45/6.95 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 12.45/6.95 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 12.45/6.95 = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4795_take__bit__set__bit__eq,axiom,
% 12.45/6.95 ! [N: nat,M: nat,A: int] :
% 12.45/6.95 ( ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 12.45/6.95 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 12.45/6.95 = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4796_take__bit__set__bit__eq,axiom,
% 12.45/6.95 ! [N: nat,M: nat,A: nat] :
% 12.45/6.95 ( ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 12.45/6.95 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 12.45/6.95 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 12.45/6.95 = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4797_take__bit__flip__bit__eq,axiom,
% 12.45/6.95 ! [N: nat,M: nat,A: int] :
% 12.45/6.95 ( ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 12.45/6.95 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 12.45/6.95 = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4798_take__bit__flip__bit__eq,axiom,
% 12.45/6.95 ! [N: nat,M: nat,A: nat] :
% 12.45/6.95 ( ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 12.45/6.95 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 12.45/6.95 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.45/6.95 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 12.45/6.95 = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4799_mask__nonnegative__int,axiom,
% 12.45/6.95 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4800_not__mask__negative__int,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4801_real__div__sqrt,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( divide_divide_real @ X4 @ ( sqrt @ X4 ) )
% 12.45/6.95 = ( sqrt @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4802_sqrt__add__le__add__sqrt,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X4 ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4803_take__bit__signed__take__bit,axiom,
% 12.45/6.95 ! [M: nat,N: nat,A: int] :
% 12.45/6.95 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 12.45/6.95 = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4804_le__real__sqrt__sumsq,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] : ( ord_less_eq_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4805_and__less__eq,axiom,
% 12.45/6.95 ! [L2: int,K: int] :
% 12.45/6.95 ( ( ord_less_int @ L2 @ zero_zero_int )
% 12.45/6.95 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4806_AND__upper1_H_H,axiom,
% 12.45/6.95 ! [Y3: int,Z: int,Ya: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.95 => ( ( ord_less_int @ Y3 @ Z )
% 12.45/6.95 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4807_AND__upper2_H_H,axiom,
% 12.45/6.95 ! [Y3: int,Z: int,X4: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.95 => ( ( ord_less_int @ Y3 @ Z )
% 12.45/6.95 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ Z ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4808_take__bit__decr__eq,axiom,
% 12.45/6.95 ! [N: nat,K: int] :
% 12.45/6.95 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.45/6.95 != zero_zero_int )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 12.45/6.95 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4809_real__of__int__div4,axiom,
% 12.45/6.95 ! [N: int,X4: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4810_real__of__int__div,axiom,
% 12.45/6.95 ! [D: int,N: int] :
% 12.45/6.95 ( ( dvd_dvd_int @ D @ N )
% 12.45/6.95 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 12.45/6.95 = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4811_less__mask,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.45/6.95 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4812_take__bit__eq__mask__iff__exp__dvd,axiom,
% 12.45/6.95 ! [N: nat,K: int] :
% 12.45/6.95 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.45/6.95 = ( bit_se2000444600071755411sk_int @ N ) )
% 12.45/6.95 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4813_even__and__iff,axiom,
% 12.45/6.95 ! [A: code_integer,B2: code_integer] :
% 12.45/6.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B2 ) )
% 12.45/6.95 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.95 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4814_even__and__iff,axiom,
% 12.45/6.95 ! [A: int,B2: int] :
% 12.45/6.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
% 12.45/6.95 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.95 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4815_even__and__iff,axiom,
% 12.45/6.95 ! [A: nat,B2: nat] :
% 12.45/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
% 12.45/6.95 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.95 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4816_sqrt2__less__2,axiom,
% 12.45/6.95 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4817_of__int__nonneg,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.45/6.95 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4818_of__int__nonneg,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.45/6.95 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4819_of__int__nonneg,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.45/6.95 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4820_of__int__pos,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.45/6.95 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4821_of__int__pos,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.45/6.95 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4822_of__int__pos,axiom,
% 12.45/6.95 ! [Z: int] :
% 12.45/6.95 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.45/6.95 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4823_even__and__iff__int,axiom,
% 12.45/6.95 ! [K: int,L2: int] :
% 12.45/6.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 12.45/6.95 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 12.45/6.95 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4824_of__int__neg__numeral,axiom,
% 12.45/6.95 ! [K: num] :
% 12.45/6.95 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/6.95 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4825_of__int__neg__numeral,axiom,
% 12.45/6.95 ! [K: num] :
% 12.45/6.95 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/6.95 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4826_of__int__neg__numeral,axiom,
% 12.45/6.95 ! [K: num] :
% 12.45/6.95 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/6.95 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4827_of__int__neg__numeral,axiom,
% 12.45/6.95 ! [K: num] :
% 12.45/6.95 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/6.95 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4828_of__int__neg__numeral,axiom,
% 12.45/6.95 ! [K: num] :
% 12.45/6.95 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/6.95 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4829_int__le__real__less,axiom,
% 12.45/6.95 ( ord_less_eq_int
% 12.45/6.95 = ( ^ [N2: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4830_int__less__real__le,axiom,
% 12.45/6.95 ( ord_less_int
% 12.45/6.95 = ( ^ [N2: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4831_real__of__int__div__aux,axiom,
% 12.45/6.95 ! [X4: int,D: int] :
% 12.45/6.95 ( ( divide_divide_real @ ( ring_1_of_int_real @ X4 ) @ ( ring_1_of_int_real @ D ) )
% 12.45/6.95 = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X4 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X4 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4832_take__bit__Suc__bit0,axiom,
% 12.45/6.95 ! [N: nat,K: num] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 12.45/6.95 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4833_take__bit__Suc__bit0,axiom,
% 12.45/6.95 ! [N: nat,K: num] :
% 12.45/6.95 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.45/6.95 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4834_take__bit__eq__mod,axiom,
% 12.45/6.95 ( bit_se1745604003318907178nteger
% 12.45/6.95 = ( ^ [N2: nat,A4: code_integer] : ( modulo364778990260209775nteger @ A4 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4835_take__bit__eq__mod,axiom,
% 12.45/6.95 ( bit_se2923211474154528505it_int
% 12.45/6.95 = ( ^ [N2: nat,A4: int] : ( modulo_modulo_int @ A4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4836_take__bit__eq__mod,axiom,
% 12.45/6.95 ( bit_se2925701944663578781it_nat
% 12.45/6.95 = ( ^ [N2: nat,A4: nat] : ( modulo_modulo_nat @ A4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4837_one__and__eq,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
% 12.45/6.95 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4838_one__and__eq,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
% 12.45/6.95 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4839_one__and__eq,axiom,
% 12.45/6.95 ! [A: nat] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
% 12.45/6.95 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4840_and__one__eq,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
% 12.45/6.95 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4841_and__one__eq,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
% 12.45/6.95 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4842_and__one__eq,axiom,
% 12.45/6.95 ! [A: nat] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
% 12.45/6.95 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4843_take__bit__nat__eq__self__iff,axiom,
% 12.45/6.95 ! [N: nat,M: nat] :
% 12.45/6.95 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 12.45/6.95 = M )
% 12.45/6.95 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4844_take__bit__nat__less__exp,axiom,
% 12.45/6.95 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4845_take__bit__nat__eq__self,axiom,
% 12.45/6.95 ! [M: nat,N: nat] :
% 12.45/6.95 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.95 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 12.45/6.95 = M ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4846_real__less__rsqrt,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 12.45/6.95 => ( ord_less_real @ X4 @ ( sqrt @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4847_real__le__rsqrt,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 12.45/6.95 => ( ord_less_eq_real @ X4 @ ( sqrt @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4848_sqrt__le__D,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ Y3 )
% 12.45/6.95 => ( ord_less_eq_real @ X4 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4849_take__bit__nat__def,axiom,
% 12.45/6.95 ( bit_se2925701944663578781it_nat
% 12.45/6.95 = ( ^ [N2: nat,M6: nat] : ( modulo_modulo_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4850_take__bit__int__less__exp,axiom,
% 12.45/6.95 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4851_take__bit__int__def,axiom,
% 12.45/6.95 ( bit_se2923211474154528505it_int
% 12.45/6.95 = ( ^ [N2: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4852_real__of__int__div2,axiom,
% 12.45/6.95 ! [N: int,X4: 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 @ X4 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4853_real__of__int__div3,axiom,
% 12.45/6.95 ! [N: int,X4: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X4 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) ) @ one_one_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4854_take__bit__eq__0__iff,axiom,
% 12.45/6.95 ! [N: nat,A: code_integer] :
% 12.45/6.95 ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 12.45/6.95 = zero_z3403309356797280102nteger )
% 12.45/6.95 = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4855_take__bit__eq__0__iff,axiom,
% 12.45/6.95 ! [N: nat,A: int] :
% 12.45/6.95 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 12.45/6.95 = zero_zero_int )
% 12.45/6.95 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4856_take__bit__eq__0__iff,axiom,
% 12.45/6.95 ! [N: nat,A: nat] :
% 12.45/6.95 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 12.45/6.95 = zero_zero_nat )
% 12.45/6.95 = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4857_take__bit__numeral__bit0,axiom,
% 12.45/6.95 ! [L2: num,K: num] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 12.45/6.95 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4858_take__bit__numeral__bit0,axiom,
% 12.45/6.95 ! [L2: num,K: num] :
% 12.45/6.95 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.45/6.95 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4859_take__bit__nat__less__self__iff,axiom,
% 12.45/6.95 ! [N: nat,M: nat] :
% 12.45/6.95 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 12.45/6.95 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4860_real__sqrt__unique,axiom,
% 12.45/6.95 ! [Y3: real,X4: real] :
% 12.45/6.95 ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.95 = X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ( sqrt @ X4 )
% 12.45/6.95 = Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4861_real__le__lsqrt,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ( ord_less_eq_real @ X4 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.95 => ( ord_less_eq_real @ ( sqrt @ X4 ) @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4862_lemma__real__divide__sqrt__less,axiom,
% 12.45/6.95 ! [U: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ U )
% 12.45/6.95 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4863_Suc__mask__eq__exp,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 12.45/6.95 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4864_real__sqrt__sum__squares__eq__cancel2,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.95 = Y3 )
% 12.45/6.95 => ( X4 = zero_zero_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4865_real__sqrt__sum__squares__eq__cancel,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.95 = X4 )
% 12.45/6.95 => ( Y3 = zero_zero_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4866_mask__nat__less__exp,axiom,
% 12.45/6.95 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4867_real__sqrt__sum__squares__triangle__ineq,axiom,
% 12.45/6.95 ! [A: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B2 @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4868_real__sqrt__sum__squares__ge2,axiom,
% 12.45/6.95 ! [Y3: real,X4: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4869_real__sqrt__sum__squares__ge1,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] : ( ord_less_eq_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4870_take__bit__Suc__minus__bit0,axiom,
% 12.45/6.95 ! [N: nat,K: num] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.45/6.95 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4871_even__of__int__iff,axiom,
% 12.45/6.95 ! [K: int] :
% 12.45/6.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 12.45/6.95 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4872_even__of__int__iff,axiom,
% 12.45/6.95 ! [K: int] :
% 12.45/6.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 12.45/6.95 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4873_take__bit__int__greater__eq__self__iff,axiom,
% 12.45/6.95 ! [K: int,N: nat] :
% 12.45/6.95 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 12.45/6.95 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4874_take__bit__int__less__self__iff,axiom,
% 12.45/6.95 ! [N: nat,K: int] :
% 12.45/6.95 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 12.45/6.95 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4875_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
% 12.45/6.95 = ( N = zero_zero_nat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4876_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
% 12.45/6.95 = ( N = zero_zero_nat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4877_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
% 12.45/6.95 = ( N = zero_zero_nat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4878_real__less__lsqrt,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ( ord_less_real @ X4 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.95 => ( ord_less_real @ ( sqrt @ X4 ) @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4879_sqrt__sum__squares__le__sum,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4880_sqrt__even__pow2,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.95 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.95 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4881_take__bit__int__eq__self__iff,axiom,
% 12.45/6.95 ! [N: nat,K: int] :
% 12.45/6.95 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.45/6.95 = K )
% 12.45/6.95 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.95 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4882_take__bit__int__eq__self,axiom,
% 12.45/6.95 ! [K: int,N: nat] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.95 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.45/6.95 = K ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4883_and__int__rec,axiom,
% 12.45/6.95 ( bit_se725231765392027082nd_int
% 12.45/6.95 = ( ^ [K3: int,L: int] :
% 12.45/6.95 ( plus_plus_int
% 12.45/6.95 @ ( zero_n2684676970156552555ol_int
% 12.45/6.95 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 12.45/6.95 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 12.45/6.95 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4884_mask__nat__def,axiom,
% 12.45/6.95 ( bit_se2002935070580805687sk_nat
% 12.45/6.95 = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4885_take__bit__numeral__minus__bit0,axiom,
% 12.45/6.95 ! [L2: num,K: num] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.45/6.95 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4886_mask__half__int,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.95 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4887_take__bit__incr__eq,axiom,
% 12.45/6.95 ! [N: nat,K: int] :
% 12.45/6.95 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.45/6.95 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 12.45/6.95 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4888_mask__int__def,axiom,
% 12.45/6.95 ( bit_se2000444600071755411sk_int
% 12.45/6.95 = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4889_take__bit__Suc__minus__1__eq,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.95 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4890_take__bit__Suc__minus__1__eq,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.95 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4891_take__bit__Suc__bit1,axiom,
% 12.45/6.95 ! [N: nat,K: num] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 12.45/6.95 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4892_take__bit__Suc__bit1,axiom,
% 12.45/6.95 ! [N: nat,K: num] :
% 12.45/6.95 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.45/6.95 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4893_take__bit__numeral__minus__1__eq,axiom,
% 12.45/6.95 ! [K: num] :
% 12.45/6.95 ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.95 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4894_take__bit__numeral__minus__1__eq,axiom,
% 12.45/6.95 ! [K: num] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.95 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4895_take__bit__Suc,axiom,
% 12.45/6.95 ! [N: nat,A: code_integer] :
% 12.45/6.95 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
% 12.45/6.95 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4896_take__bit__Suc,axiom,
% 12.45/6.95 ! [N: nat,A: int] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 12.45/6.95 = ( 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 ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4897_take__bit__Suc,axiom,
% 12.45/6.95 ! [N: nat,A: nat] :
% 12.45/6.95 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
% 12.45/6.95 = ( 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 ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4898_mask__eq__exp__minus__1,axiom,
% 12.45/6.95 ( bit_se2002935070580805687sk_nat
% 12.45/6.95 = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4899_mask__eq__exp__minus__1,axiom,
% 12.45/6.95 ( bit_se2000444600071755411sk_int
% 12.45/6.95 = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4900_real__sqrt__power__even,axiom,
% 12.45/6.95 ! [N: nat,X4: real] :
% 12.45/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( power_power_real @ ( sqrt @ X4 ) @ N )
% 12.45/6.95 = ( power_power_real @ X4 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4901_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 12.45/6.95 ! [X4: real,Y3: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( 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 ) ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4902_arith__geo__mean__sqrt,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X4 @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4903_take__bit__int__less__eq,axiom,
% 12.45/6.95 ! [N: nat,K: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 12.45/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.95 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4904_take__bit__int__greater__eq,axiom,
% 12.45/6.95 ! [K: int,N: nat] :
% 12.45/6.95 ( ( ord_less_int @ K @ zero_zero_int )
% 12.45/6.95 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4905_signed__take__bit__eq__take__bit__shift,axiom,
% 12.45/6.95 ( bit_ri631733984087533419it_int
% 12.45/6.95 = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4906_stable__imp__take__bit__eq,axiom,
% 12.45/6.95 ! [A: code_integer,N: nat] :
% 12.45/6.95 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.95 = A )
% 12.45/6.95 => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.95 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 12.45/6.95 = zero_z3403309356797280102nteger ) )
% 12.45/6.95 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.95 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 12.45/6.95 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4907_stable__imp__take__bit__eq,axiom,
% 12.45/6.95 ! [A: int,N: nat] :
% 12.45/6.95 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.95 = A )
% 12.45/6.95 => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 12.45/6.95 = zero_zero_int ) )
% 12.45/6.95 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 12.45/6.95 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4908_stable__imp__take__bit__eq,axiom,
% 12.45/6.95 ! [A: nat,N: nat] :
% 12.45/6.95 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.95 = A )
% 12.45/6.95 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.95 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 12.45/6.95 = zero_zero_nat ) )
% 12.45/6.95 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.95 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 12.45/6.95 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4909_take__bit__numeral__bit1,axiom,
% 12.45/6.95 ! [L2: num,K: num] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 12.45/6.95 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4910_take__bit__numeral__bit1,axiom,
% 12.45/6.95 ! [L2: num,K: num] :
% 12.45/6.95 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.45/6.95 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4911_take__bit__minus__small__eq,axiom,
% 12.45/6.95 ! [K: int,N: nat] :
% 12.45/6.95 ( ( ord_less_int @ zero_zero_int @ K )
% 12.45/6.95 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.95 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 12.45/6.95 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4912_one__le__exp__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X4 ) )
% 12.45/6.95 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4913_exp__le__one__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( exp_real @ X4 ) @ one_one_real )
% 12.45/6.95 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4914_exp__less__one__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ ( exp_real @ X4 ) @ one_one_real )
% 12.45/6.95 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4915_one__less__exp__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ one_one_real @ ( exp_real @ X4 ) )
% 12.45/6.95 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4916_real__exp__bound__lemma,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.95 => ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4917_exp__bound,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.95 => ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4918_arsinh__real__aux,axiom,
% 12.45/6.95 ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4919_exp__less__cancel__iff,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
% 12.45/6.95 = ( ord_less_real @ X4 @ Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4920_exp__less__mono,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.95 => ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4921_exp__le__cancel__iff,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
% 12.45/6.95 = ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4922_exp__zero,axiom,
% 12.45/6.95 ( ( exp_complex @ zero_zero_complex )
% 12.45/6.95 = one_one_complex ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4923_exp__zero,axiom,
% 12.45/6.95 ( ( exp_real @ zero_zero_real )
% 12.45/6.95 = one_one_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4924_exp__eq__one__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ( exp_real @ X4 )
% 12.45/6.95 = one_one_real )
% 12.45/6.95 = ( X4 = zero_zero_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4925_and__nat__numerals_I3_J,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 12.45/6.95 = zero_zero_nat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4926_and__nat__numerals_I1_J,axiom,
% 12.45/6.95 ! [Y3: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 12.45/6.95 = zero_zero_nat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4927_and__nat__numerals_I2_J,axiom,
% 12.45/6.95 ! [Y3: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 12.45/6.95 = one_one_nat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4928_and__nat__numerals_I4_J,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 12.45/6.95 = one_one_nat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4929_Suc__0__and__eq,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.45/6.95 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4930_and__Suc__0__eq,axiom,
% 12.45/6.95 ! [N: nat] :
% 12.45/6.95 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.45/6.95 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4931_exp__times__arg__commute,axiom,
% 12.45/6.95 ! [A3: complex] :
% 12.45/6.95 ( ( times_times_complex @ ( exp_complex @ A3 ) @ A3 )
% 12.45/6.95 = ( times_times_complex @ A3 @ ( exp_complex @ A3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4932_exp__times__arg__commute,axiom,
% 12.45/6.95 ! [A3: real] :
% 12.45/6.95 ( ( times_times_real @ ( exp_real @ A3 ) @ A3 )
% 12.45/6.95 = ( times_times_real @ A3 @ ( exp_real @ A3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4933_exp__less__cancel,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
% 12.45/6.95 => ( ord_less_real @ X4 @ Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4934_and__nat__unfold,axiom,
% 12.45/6.95 ( bit_se727722235901077358nd_nat
% 12.45/6.95 = ( ^ [M6: nat,N2: nat] :
% 12.45/6.95 ( if_nat
% 12.45/6.95 @ ( ( M6 = zero_zero_nat )
% 12.45/6.95 | ( N2 = zero_zero_nat ) )
% 12.45/6.95 @ zero_zero_nat
% 12.45/6.95 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4935_and__nat__rec,axiom,
% 12.45/6.95 ( bit_se727722235901077358nd_nat
% 12.45/6.95 = ( ^ [M6: nat,N2: nat] :
% 12.45/6.95 ( plus_plus_nat
% 12.45/6.95 @ ( zero_n2687167440665602831ol_nat
% 12.45/6.95 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 12.45/6.95 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 12.45/6.95 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4936_mult__exp__exp,axiom,
% 12.45/6.95 ! [X4: complex,Y3: complex] :
% 12.45/6.95 ( ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y3 ) )
% 12.45/6.95 = ( exp_complex @ ( plus_plus_complex @ X4 @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4937_mult__exp__exp,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) )
% 12.45/6.95 = ( exp_real @ ( plus_plus_real @ X4 @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4938_exp__add__commuting,axiom,
% 12.45/6.95 ! [X4: complex,Y3: complex] :
% 12.45/6.95 ( ( ( times_times_complex @ X4 @ Y3 )
% 12.45/6.95 = ( times_times_complex @ Y3 @ X4 ) )
% 12.45/6.95 => ( ( exp_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
% 12.45/6.95 = ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4939_exp__add__commuting,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ( times_times_real @ X4 @ Y3 )
% 12.45/6.95 = ( times_times_real @ Y3 @ X4 ) )
% 12.45/6.95 => ( ( exp_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.95 = ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4940_exp__diff,axiom,
% 12.45/6.95 ! [X4: complex,Y3: complex] :
% 12.45/6.95 ( ( exp_complex @ ( minus_minus_complex @ X4 @ Y3 ) )
% 12.45/6.95 = ( divide1717551699836669952omplex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4941_exp__diff,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( exp_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 12.45/6.95 = ( divide_divide_real @ ( exp_real @ X4 ) @ ( exp_real @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4942_not__exp__less__zero,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ~ ( ord_less_real @ ( exp_real @ X4 ) @ zero_zero_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4943_exp__gt__zero,axiom,
% 12.45/6.95 ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4944_exp__total,axiom,
% 12.45/6.95 ! [Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ? [X3: real] :
% 12.45/6.95 ( ( exp_real @ X3 )
% 12.45/6.95 = Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4945_exp__ge__zero,axiom,
% 12.45/6.95 ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4946_not__exp__le__zero,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ~ ( ord_less_eq_real @ ( exp_real @ X4 ) @ zero_zero_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4947_exp__minus__inverse,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) )
% 12.45/6.95 = one_one_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4948_exp__minus__inverse,axiom,
% 12.45/6.95 ! [X4: complex] :
% 12.45/6.95 ( ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) )
% 12.45/6.95 = one_one_complex ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4949_exp__gt__one,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ord_less_real @ one_one_real @ ( exp_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4950_exp__ge__add__one__self,axiom,
% 12.45/6.95 ! [X4: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( exp_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4951_exp__ge__add__one__self__aux,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( exp_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4952_lemma__exp__total,axiom,
% 12.45/6.95 ! [Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ one_one_real @ Y3 )
% 12.45/6.95 => ? [X3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 12.45/6.95 & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y3 @ one_one_real ) )
% 12.45/6.95 & ( ( exp_real @ X3 )
% 12.45/6.95 = Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4953_exp__le,axiom,
% 12.45/6.95 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4954_exp__double,axiom,
% 12.45/6.95 ! [Z: complex] :
% 12.45/6.95 ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 12.45/6.95 = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4955_exp__double,axiom,
% 12.45/6.95 ! [Z: real] :
% 12.45/6.95 ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 12.45/6.95 = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4956_exp__half__le2,axiom,
% 12.45/6.95 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4957_arcosh__1,axiom,
% 12.45/6.95 ( ( arcosh_real @ one_one_real )
% 12.45/6.95 = zero_zero_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4958_floor__exists1,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ? [X3: int] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X4 )
% 12.45/6.95 & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 12.45/6.95 & ! [Y5: int] :
% 12.45/6.95 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X4 )
% 12.45/6.95 & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 12.45/6.95 => ( Y5 = X3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4959_floor__exists1,axiom,
% 12.45/6.95 ! [X4: rat] :
% 12.45/6.95 ? [X3: int] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X4 )
% 12.45/6.95 & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 12.45/6.95 & ! [Y5: int] :
% 12.45/6.95 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X4 )
% 12.45/6.95 & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 12.45/6.95 => ( Y5 = X3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4960_floor__exists,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ? [Z2: int] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X4 )
% 12.45/6.95 & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4961_floor__exists,axiom,
% 12.45/6.95 ! [X4: rat] :
% 12.45/6.95 ? [Z2: int] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X4 )
% 12.45/6.95 & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4962_take__bit__numeral__minus__bit1,axiom,
% 12.45/6.95 ! [L2: num,K: num] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.45/6.95 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4963_take__bit__Suc__minus__bit1,axiom,
% 12.45/6.95 ! [N: nat,K: num] :
% 12.45/6.95 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.45/6.95 = ( 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 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4964_tanh__real__altdef,axiom,
% 12.45/6.95 ( tanh_real
% 12.45/6.95 = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4965_ln__one__minus__pos__lower__bound,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.95 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X4 ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4966_tanh__real__less__iff,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) )
% 12.45/6.95 = ( ord_less_real @ X4 @ Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4967_tanh__real__le__iff,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) )
% 12.45/6.95 = ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4968_ln__one,axiom,
% 12.45/6.95 ( ( ln_ln_real @ one_one_real )
% 12.45/6.95 = zero_zero_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4969_ln__inj__iff,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ( ( ln_ln_real @ X4 )
% 12.45/6.95 = ( ln_ln_real @ Y3 ) )
% 12.45/6.95 = ( X4 = Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4970_ln__less__cancel__iff,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) )
% 12.45/6.95 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4971_tanh__real__neg__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ ( tanh_real @ X4 ) @ zero_zero_real )
% 12.45/6.95 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4972_tanh__real__pos__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X4 ) )
% 12.45/6.95 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4973_tanh__real__nonpos__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( tanh_real @ X4 ) @ zero_zero_real )
% 12.45/6.95 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4974_tanh__real__nonneg__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X4 ) )
% 12.45/6.95 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4975_pred__numeral__inc,axiom,
% 12.45/6.95 ! [K: num] :
% 12.45/6.95 ( ( pred_numeral @ ( inc @ K ) )
% 12.45/6.95 = ( numeral_numeral_nat @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4976_ln__le__cancel__iff,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) )
% 12.45/6.95 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4977_ln__eq__zero__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ( ln_ln_real @ X4 )
% 12.45/6.95 = zero_zero_real )
% 12.45/6.95 = ( X4 = one_one_real ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4978_ln__gt__zero__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
% 12.45/6.95 = ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4979_ln__less__zero__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
% 12.45/6.95 = ( ord_less_real @ X4 @ one_one_real ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4980_exp__ln,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( exp_real @ ( ln_ln_real @ X4 ) )
% 12.45/6.95 = X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4981_exp__ln__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ( exp_real @ ( ln_ln_real @ X4 ) )
% 12.45/6.95 = X4 )
% 12.45/6.95 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4982_ln__le__zero__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
% 12.45/6.95 = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4983_ln__ge__zero__iff,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
% 12.45/6.95 = ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4984_add__neg__numeral__special_I5_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.95 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4985_add__neg__numeral__special_I5_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.95 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4986_add__neg__numeral__special_I5_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.45/6.95 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4987_add__neg__numeral__special_I5_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.95 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4988_add__neg__numeral__special_I5_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.95 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4989_add__neg__numeral__special_I6_J,axiom,
% 12.45/6.95 ! [M: num] :
% 12.45/6.95 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.95 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4990_add__neg__numeral__special_I6_J,axiom,
% 12.45/6.95 ! [M: num] :
% 12.45/6.95 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.95 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4991_add__neg__numeral__special_I6_J,axiom,
% 12.45/6.95 ! [M: num] :
% 12.45/6.95 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.95 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4992_add__neg__numeral__special_I6_J,axiom,
% 12.45/6.95 ! [M: num] :
% 12.45/6.95 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.95 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4993_add__neg__numeral__special_I6_J,axiom,
% 12.45/6.95 ! [M: num] :
% 12.45/6.95 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.95 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4994_diff__numeral__special_I5_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 12.45/6.95 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4995_diff__numeral__special_I5_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 12.45/6.95 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4996_diff__numeral__special_I5_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 12.45/6.95 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4997_diff__numeral__special_I5_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 12.45/6.95 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4998_diff__numeral__special_I5_J,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 12.45/6.95 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_4999_diff__numeral__special_I6_J,axiom,
% 12.45/6.95 ! [M: num] :
% 12.45/6.95 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.95 = ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5000_diff__numeral__special_I6_J,axiom,
% 12.45/6.95 ! [M: num] :
% 12.45/6.95 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.95 = ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5001_diff__numeral__special_I6_J,axiom,
% 12.45/6.95 ! [M: num] :
% 12.45/6.95 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.95 = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5002_diff__numeral__special_I6_J,axiom,
% 12.45/6.95 ! [M: num] :
% 12.45/6.95 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.95 = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5003_diff__numeral__special_I6_J,axiom,
% 12.45/6.95 ! [M: num] :
% 12.45/6.95 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.95 = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5004_num__induct,axiom,
% 12.45/6.95 ! [P: num > $o,X4: num] :
% 12.45/6.95 ( ( P @ one )
% 12.45/6.95 => ( ! [X3: num] :
% 12.45/6.95 ( ( P @ X3 )
% 12.45/6.95 => ( P @ ( inc @ X3 ) ) )
% 12.45/6.95 => ( P @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5005_add__inc,axiom,
% 12.45/6.95 ! [X4: num,Y3: num] :
% 12.45/6.95 ( ( plus_plus_num @ X4 @ ( inc @ Y3 ) )
% 12.45/6.95 = ( inc @ ( plus_plus_num @ X4 @ Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5006_ln__less__self,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ord_less_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5007_tanh__real__lt__1,axiom,
% 12.45/6.95 ! [X4: real] : ( ord_less_real @ ( tanh_real @ X4 ) @ one_one_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5008_inc_Osimps_I1_J,axiom,
% 12.45/6.95 ( ( inc @ one )
% 12.45/6.95 = ( bit0 @ one ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5009_inc_Osimps_I2_J,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( inc @ ( bit0 @ X4 ) )
% 12.45/6.95 = ( bit1 @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5010_inc_Osimps_I3_J,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( inc @ ( bit1 @ X4 ) )
% 12.45/6.95 = ( bit0 @ ( inc @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5011_ln__bound,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5012_ln__gt__zero,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/6.95 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5013_ln__less__zero,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/6.95 => ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5014_ln__gt__zero__imp__gt__one,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
% 12.45/6.95 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5015_ln__ge__zero,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ one_one_real @ X4 )
% 12.45/6.95 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5016_add__One,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( plus_plus_num @ X4 @ one )
% 12.45/6.95 = ( inc @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5017_mult__inc,axiom,
% 12.45/6.95 ! [X4: num,Y3: num] :
% 12.45/6.95 ( ( times_times_num @ X4 @ ( inc @ Y3 ) )
% 12.45/6.95 = ( plus_plus_num @ ( times_times_num @ X4 @ Y3 ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5018_tanh__real__gt__neg1,axiom,
% 12.45/6.95 ! [X4: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5019_ln__ge__zero__imp__ge__one,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
% 12.45/6.95 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5020_ln__add__one__self__le__self,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5021_ln__mult,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ( ln_ln_real @ ( times_times_real @ X4 @ Y3 ) )
% 12.45/6.95 = ( plus_plus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5022_ln__eq__minus__one,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ( ln_ln_real @ X4 )
% 12.45/6.95 = ( minus_minus_real @ X4 @ one_one_real ) )
% 12.45/6.95 => ( X4 = one_one_real ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5023_ln__div,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ( ln_ln_real @ ( divide_divide_real @ X4 @ Y3 ) )
% 12.45/6.95 = ( minus_minus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5024_ln__ge__iff,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X4 ) )
% 12.45/6.95 = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5025_ln__x__over__x__mono,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.95 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5026_numeral__inc,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( numera6690914467698888265omplex @ ( inc @ X4 ) )
% 12.45/6.95 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X4 ) @ one_one_complex ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5027_numeral__inc,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( numeral_numeral_real @ ( inc @ X4 ) )
% 12.45/6.95 = ( plus_plus_real @ ( numeral_numeral_real @ X4 ) @ one_one_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5028_numeral__inc,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( numeral_numeral_rat @ ( inc @ X4 ) )
% 12.45/6.95 = ( plus_plus_rat @ ( numeral_numeral_rat @ X4 ) @ one_one_rat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5029_numeral__inc,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( numeral_numeral_nat @ ( inc @ X4 ) )
% 12.45/6.95 = ( plus_plus_nat @ ( numeral_numeral_nat @ X4 ) @ one_one_nat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5030_numeral__inc,axiom,
% 12.45/6.95 ! [X4: num] :
% 12.45/6.95 ( ( numeral_numeral_int @ ( inc @ X4 ) )
% 12.45/6.95 = ( plus_plus_int @ ( numeral_numeral_int @ X4 ) @ one_one_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5031_ln__2__less__1,axiom,
% 12.45/6.95 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5032_ln__le__minus__one,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5033_ln__diff__le,axiom,
% 12.45/6.95 ! [X4: real,Y3: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.95 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X4 @ Y3 ) @ Y3 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5034_ln__add__one__self__le__self2,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.95 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5035_tanh__ln__real,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( tanh_real @ ( ln_ln_real @ X4 ) )
% 12.45/6.95 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5036_arcosh__real__def,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ one_one_real @ X4 )
% 12.45/6.95 => ( ( arcosh_real @ X4 )
% 12.45/6.95 = ( ln_ln_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5037_ln__one__minus__pos__upper__bound,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/6.95 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X4 ) ) @ ( uminus_uminus_real @ X4 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5038_ln__sqrt,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ln_ln_real @ ( sqrt @ X4 ) )
% 12.45/6.95 = ( divide_divide_real @ ( ln_ln_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5039_exists__least__lemma,axiom,
% 12.45/6.95 ! [P: nat > $o] :
% 12.45/6.95 ( ~ ( P @ zero_zero_nat )
% 12.45/6.95 => ( ? [X_12: nat] : ( P @ X_12 )
% 12.45/6.95 => ? [N4: nat] :
% 12.45/6.95 ( ~ ( P @ N4 )
% 12.45/6.95 & ( P @ ( suc @ N4 ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5040_ex__le__of__int,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ? [Z2: int] : ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z2 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5041_ex__le__of__int,axiom,
% 12.45/6.95 ! [X4: rat] :
% 12.45/6.95 ? [Z2: int] : ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z2 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5042_ex__of__int__less,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X4 ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5043_ex__of__int__less,axiom,
% 12.45/6.95 ! [X4: rat] :
% 12.45/6.95 ? [Z2: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X4 ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5044_ex__less__of__int,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ? [Z2: int] : ( ord_less_real @ X4 @ ( ring_1_of_int_real @ Z2 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5045_ex__less__of__int,axiom,
% 12.45/6.95 ! [X4: rat] :
% 12.45/6.95 ? [Z2: int] : ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ Z2 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5046_tanh__altdef,axiom,
% 12.45/6.95 ( tanh_real
% 12.45/6.95 = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5047_tanh__altdef,axiom,
% 12.45/6.95 ( tanh_complex
% 12.45/6.95 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5048_ln__one__plus__pos__lower__bound,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.95 => ( ord_less_eq_real @ ( minus_minus_real @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5049_artanh__def,axiom,
% 12.45/6.95 ( artanh_real
% 12.45/6.95 = ( ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X ) @ ( minus_minus_real @ one_one_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5050_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.95 => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.95 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5051_round__unique,axiom,
% 12.45/6.95 ! [X4: real,Y3: int] :
% 12.45/6.95 ( ( ord_less_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
% 12.45/6.95 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.45/6.95 => ( ( archim8280529875227126926d_real @ X4 )
% 12.45/6.95 = Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5052_round__unique,axiom,
% 12.45/6.95 ! [X4: rat,Y3: int] :
% 12.45/6.95 ( ( ord_less_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
% 12.45/6.95 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.95 => ( ( archim7778729529865785530nd_rat @ X4 )
% 12.45/6.95 = Y3 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5053_arsinh__real__def,axiom,
% 12.45/6.95 ( arsinh_real
% 12.45/6.95 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5054_abs__ln__one__plus__x__minus__x__bound,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.95 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5055_of__int__round__gt,axiom,
% 12.45/6.95 ! [X4: real] : ( ord_less_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5056_of__int__round__gt,axiom,
% 12.45/6.95 ! [X4: rat] : ( ord_less_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5057_of__int__round__ge,axiom,
% 12.45/6.95 ! [X4: real] : ( ord_less_eq_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5058_of__int__round__ge,axiom,
% 12.45/6.95 ! [X4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5059_abs__abs,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 12.45/6.95 = ( abs_abs_real @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5060_abs__abs,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 12.45/6.95 = ( abs_abs_int @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5061_abs__abs,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 12.45/6.95 = ( abs_abs_Code_integer @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5062_abs__abs,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 12.45/6.95 = ( abs_abs_rat @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5063_abs__idempotent,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 12.45/6.95 = ( abs_abs_real @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5064_abs__idempotent,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 12.45/6.95 = ( abs_abs_int @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5065_abs__idempotent,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 12.45/6.95 = ( abs_abs_Code_integer @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5066_abs__idempotent,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 12.45/6.95 = ( abs_abs_rat @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5067_abs__0,axiom,
% 12.45/6.95 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 12.45/6.95 = zero_z3403309356797280102nteger ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5068_abs__0,axiom,
% 12.45/6.95 ( ( abs_abs_real @ zero_zero_real )
% 12.45/6.95 = zero_zero_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5069_abs__0,axiom,
% 12.45/6.95 ( ( abs_abs_rat @ zero_zero_rat )
% 12.45/6.95 = zero_zero_rat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5070_abs__0,axiom,
% 12.45/6.95 ( ( abs_abs_int @ zero_zero_int )
% 12.45/6.95 = zero_zero_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5071_abs__0__eq,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( zero_z3403309356797280102nteger
% 12.45/6.95 = ( abs_abs_Code_integer @ A ) )
% 12.45/6.95 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5072_abs__0__eq,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( zero_zero_real
% 12.45/6.95 = ( abs_abs_real @ A ) )
% 12.45/6.95 = ( A = zero_zero_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5073_abs__0__eq,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( zero_zero_rat
% 12.45/6.95 = ( abs_abs_rat @ A ) )
% 12.45/6.95 = ( A = zero_zero_rat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5074_abs__0__eq,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( zero_zero_int
% 12.45/6.95 = ( abs_abs_int @ A ) )
% 12.45/6.95 = ( A = zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5075_abs__eq__0,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( ( abs_abs_Code_integer @ A )
% 12.45/6.95 = zero_z3403309356797280102nteger )
% 12.45/6.95 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5076_abs__eq__0,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( ( abs_abs_real @ A )
% 12.45/6.95 = zero_zero_real )
% 12.45/6.95 = ( A = zero_zero_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5077_abs__eq__0,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( ( abs_abs_rat @ A )
% 12.45/6.95 = zero_zero_rat )
% 12.45/6.95 = ( A = zero_zero_rat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5078_abs__eq__0,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( ( abs_abs_int @ A )
% 12.45/6.95 = zero_zero_int )
% 12.45/6.95 = ( A = zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5079_abs__zero,axiom,
% 12.45/6.95 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 12.45/6.95 = zero_z3403309356797280102nteger ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5080_abs__zero,axiom,
% 12.45/6.95 ( ( abs_abs_real @ zero_zero_real )
% 12.45/6.95 = zero_zero_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5081_abs__zero,axiom,
% 12.45/6.95 ( ( abs_abs_rat @ zero_zero_rat )
% 12.45/6.95 = zero_zero_rat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5082_abs__zero,axiom,
% 12.45/6.95 ( ( abs_abs_int @ zero_zero_int )
% 12.45/6.95 = zero_zero_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5083_abs__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 12.45/6.95 = ( numera6620942414471956472nteger @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5084_abs__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 12.45/6.95 = ( numeral_numeral_real @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5085_abs__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 12.45/6.95 = ( numeral_numeral_rat @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5086_abs__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 12.45/6.95 = ( numeral_numeral_int @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5087_abs__mult__self__eq,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 12.45/6.95 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5088_abs__mult__self__eq,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 12.45/6.95 = ( times_times_real @ A @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5089_abs__mult__self__eq,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 12.45/6.95 = ( times_times_rat @ A @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5090_abs__mult__self__eq,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 12.45/6.95 = ( times_times_int @ A @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5091_abs__1,axiom,
% 12.45/6.95 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 12.45/6.95 = one_one_Code_integer ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5092_abs__1,axiom,
% 12.45/6.95 ( ( abs_abs_complex @ one_one_complex )
% 12.45/6.95 = one_one_complex ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5093_abs__1,axiom,
% 12.45/6.95 ( ( abs_abs_real @ one_one_real )
% 12.45/6.95 = one_one_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5094_abs__1,axiom,
% 12.45/6.95 ( ( abs_abs_rat @ one_one_rat )
% 12.45/6.95 = one_one_rat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5095_abs__1,axiom,
% 12.45/6.95 ( ( abs_abs_int @ one_one_int )
% 12.45/6.95 = one_one_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5096_abs__add__abs,axiom,
% 12.45/6.95 ! [A: code_integer,B2: code_integer] :
% 12.45/6.95 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) )
% 12.45/6.95 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5097_abs__add__abs,axiom,
% 12.45/6.95 ! [A: real,B2: real] :
% 12.45/6.95 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) )
% 12.45/6.95 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5098_abs__add__abs,axiom,
% 12.45/6.95 ! [A: rat,B2: rat] :
% 12.45/6.95 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) )
% 12.45/6.95 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5099_abs__add__abs,axiom,
% 12.45/6.95 ! [A: int,B2: int] :
% 12.45/6.95 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) )
% 12.45/6.95 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5100_abs__divide,axiom,
% 12.45/6.95 ! [A: complex,B2: complex] :
% 12.45/6.95 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.45/6.95 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5101_abs__divide,axiom,
% 12.45/6.95 ! [A: real,B2: real] :
% 12.45/6.95 ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.95 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5102_abs__divide,axiom,
% 12.45/6.95 ! [A: rat,B2: rat] :
% 12.45/6.95 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
% 12.45/6.95 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5103_abs__minus,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 12.45/6.95 = ( abs_abs_real @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5104_abs__minus,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 12.45/6.95 = ( abs_abs_int @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5105_abs__minus,axiom,
% 12.45/6.95 ! [A: complex] :
% 12.45/6.95 ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 12.45/6.95 = ( abs_abs_complex @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5106_abs__minus,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.95 = ( abs_abs_Code_integer @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5107_abs__minus,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 12.45/6.95 = ( abs_abs_rat @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5108_abs__minus__cancel,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 12.45/6.95 = ( abs_abs_real @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5109_abs__minus__cancel,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 12.45/6.95 = ( abs_abs_int @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5110_abs__minus__cancel,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.95 = ( abs_abs_Code_integer @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5111_abs__minus__cancel,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 12.45/6.95 = ( abs_abs_rat @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5112_abs__dvd__iff,axiom,
% 12.45/6.95 ! [M: real,K: real] :
% 12.45/6.95 ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 12.45/6.95 = ( dvd_dvd_real @ M @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5113_abs__dvd__iff,axiom,
% 12.45/6.95 ! [M: int,K: int] :
% 12.45/6.95 ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 12.45/6.95 = ( dvd_dvd_int @ M @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5114_abs__dvd__iff,axiom,
% 12.45/6.95 ! [M: code_integer,K: code_integer] :
% 12.45/6.95 ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 12.45/6.95 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5115_abs__dvd__iff,axiom,
% 12.45/6.95 ! [M: rat,K: rat] :
% 12.45/6.95 ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 12.45/6.95 = ( dvd_dvd_rat @ M @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5116_dvd__abs__iff,axiom,
% 12.45/6.95 ! [M: real,K: real] :
% 12.45/6.95 ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 12.45/6.95 = ( dvd_dvd_real @ M @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5117_dvd__abs__iff,axiom,
% 12.45/6.95 ! [M: int,K: int] :
% 12.45/6.95 ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 12.45/6.95 = ( dvd_dvd_int @ M @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5118_dvd__abs__iff,axiom,
% 12.45/6.95 ! [M: code_integer,K: code_integer] :
% 12.45/6.95 ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 12.45/6.95 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5119_dvd__abs__iff,axiom,
% 12.45/6.95 ! [M: rat,K: rat] :
% 12.45/6.95 ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 12.45/6.95 = ( dvd_dvd_rat @ M @ K ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5120_abs__bool__eq,axiom,
% 12.45/6.95 ! [P: $o] :
% 12.45/6.95 ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 12.45/6.95 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5121_abs__bool__eq,axiom,
% 12.45/6.95 ! [P: $o] :
% 12.45/6.95 ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 12.45/6.95 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5122_abs__bool__eq,axiom,
% 12.45/6.95 ! [P: $o] :
% 12.45/6.95 ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 12.45/6.95 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5123_abs__bool__eq,axiom,
% 12.45/6.95 ! [P: $o] :
% 12.45/6.95 ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 12.45/6.95 = ( zero_n356916108424825756nteger @ P ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5124_abs__le__zero__iff,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 12.45/6.95 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5125_abs__le__zero__iff,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 12.45/6.95 = ( A = zero_zero_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5126_abs__le__zero__iff,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 12.45/6.95 = ( A = zero_zero_rat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5127_abs__le__zero__iff,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 12.45/6.95 = ( A = zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5128_abs__le__self__iff,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 12.45/6.95 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5129_abs__le__self__iff,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 12.45/6.95 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5130_abs__le__self__iff,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 12.45/6.95 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5131_abs__le__self__iff,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 12.45/6.95 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5132_abs__of__nonneg,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.45/6.95 => ( ( abs_abs_Code_integer @ A )
% 12.45/6.95 = A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5133_abs__of__nonneg,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.95 => ( ( abs_abs_real @ A )
% 12.45/6.95 = A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5134_abs__of__nonneg,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.95 => ( ( abs_abs_rat @ A )
% 12.45/6.95 = A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5135_abs__of__nonneg,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.95 => ( ( abs_abs_int @ A )
% 12.45/6.95 = A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5136_zero__less__abs__iff,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 12.45/6.95 = ( A != zero_z3403309356797280102nteger ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5137_zero__less__abs__iff,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 12.45/6.95 = ( A != zero_zero_real ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5138_zero__less__abs__iff,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 12.45/6.95 = ( A != zero_zero_rat ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5139_zero__less__abs__iff,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 12.45/6.95 = ( A != zero_zero_int ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5140_abs__neg__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.95 = ( numeral_numeral_real @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5141_abs__neg__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.95 = ( numeral_numeral_int @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5142_abs__neg__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.95 = ( numera6620942414471956472nteger @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5143_abs__neg__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.95 = ( numeral_numeral_rat @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5144_abs__neg__one,axiom,
% 12.45/6.95 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.95 = one_one_real ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5145_abs__neg__one,axiom,
% 12.45/6.95 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.95 = one_one_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5146_abs__neg__one,axiom,
% 12.45/6.95 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.95 = one_one_Code_integer ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5147_abs__neg__one,axiom,
% 12.45/6.95 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.95 = one_one_rat ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5148_abs__power__minus,axiom,
% 12.45/6.95 ! [A: real,N: nat] :
% 12.45/6.95 ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 12.45/6.95 = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5149_abs__power__minus,axiom,
% 12.45/6.95 ! [A: int,N: nat] :
% 12.45/6.95 ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 12.45/6.95 = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5150_abs__power__minus,axiom,
% 12.45/6.95 ! [A: code_integer,N: nat] :
% 12.45/6.95 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 12.45/6.95 = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5151_abs__power__minus,axiom,
% 12.45/6.95 ! [A: rat,N: nat] :
% 12.45/6.95 ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 12.45/6.95 = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5152_real__sqrt__abs2,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( sqrt @ ( times_times_real @ X4 @ X4 ) )
% 12.45/6.95 = ( abs_abs_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5153_real__sqrt__mult__self,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 12.45/6.95 = ( abs_abs_real @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5154_round__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 12.45/6.95 = ( numeral_numeral_int @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5155_round__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 12.45/6.95 = ( numeral_numeral_int @ N ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5156_round__1,axiom,
% 12.45/6.95 ( ( archim8280529875227126926d_real @ one_one_real )
% 12.45/6.95 = one_one_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5157_round__1,axiom,
% 12.45/6.95 ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 12.45/6.95 = one_one_int ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5158_zero__le__divide__abs__iff,axiom,
% 12.45/6.95 ! [A: real,B2: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) )
% 12.45/6.95 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.95 | ( B2 = zero_zero_real ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5159_zero__le__divide__abs__iff,axiom,
% 12.45/6.95 ! [A: rat,B2: rat] :
% 12.45/6.95 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) )
% 12.45/6.95 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.95 | ( B2 = zero_zero_rat ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5160_divide__le__0__abs__iff,axiom,
% 12.45/6.95 ! [A: real,B2: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) @ zero_zero_real )
% 12.45/6.95 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.95 | ( B2 = zero_zero_real ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5161_divide__le__0__abs__iff,axiom,
% 12.45/6.95 ! [A: rat,B2: rat] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) @ zero_zero_rat )
% 12.45/6.95 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.95 | ( B2 = zero_zero_rat ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5162_abs__of__nonpos,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.95 => ( ( abs_abs_real @ A )
% 12.45/6.95 = ( uminus_uminus_real @ A ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5163_abs__of__nonpos,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 12.45/6.95 => ( ( abs_abs_Code_integer @ A )
% 12.45/6.95 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5164_abs__of__nonpos,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.45/6.95 => ( ( abs_abs_rat @ A )
% 12.45/6.95 = ( uminus_uminus_rat @ A ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5165_abs__of__nonpos,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.45/6.95 => ( ( abs_abs_int @ A )
% 12.45/6.95 = ( uminus_uminus_int @ A ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5166_artanh__minus__real,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.95 => ( ( artanh_real @ ( uminus_uminus_real @ X4 ) )
% 12.45/6.95 = ( uminus_uminus_real @ ( artanh_real @ X4 ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5167_zero__less__power__abs__iff,axiom,
% 12.45/6.95 ! [A: code_integer,N: nat] :
% 12.45/6.95 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 12.45/6.95 = ( ( A != zero_z3403309356797280102nteger )
% 12.45/6.95 | ( N = zero_zero_nat ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5168_zero__less__power__abs__iff,axiom,
% 12.45/6.95 ! [A: real,N: nat] :
% 12.45/6.95 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 12.45/6.95 = ( ( A != zero_zero_real )
% 12.45/6.95 | ( N = zero_zero_nat ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5169_zero__less__power__abs__iff,axiom,
% 12.45/6.95 ! [A: rat,N: nat] :
% 12.45/6.95 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 12.45/6.95 = ( ( A != zero_zero_rat )
% 12.45/6.95 | ( N = zero_zero_nat ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5170_zero__less__power__abs__iff,axiom,
% 12.45/6.95 ! [A: int,N: nat] :
% 12.45/6.95 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 12.45/6.95 = ( ( A != zero_zero_int )
% 12.45/6.95 | ( N = zero_zero_nat ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5171_abs__power2,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.95 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5172_abs__power2,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.95 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5173_abs__power2,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.95 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5174_abs__power2,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.95 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5175_power2__abs,axiom,
% 12.45/6.95 ! [A: code_integer] :
% 12.45/6.95 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.95 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5176_power2__abs,axiom,
% 12.45/6.95 ! [A: rat] :
% 12.45/6.95 ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.95 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5177_power2__abs,axiom,
% 12.45/6.95 ! [A: real] :
% 12.45/6.95 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.95 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5178_power2__abs,axiom,
% 12.45/6.95 ! [A: int] :
% 12.45/6.95 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.95 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5179_round__neg__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.95 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5180_round__neg__numeral,axiom,
% 12.45/6.95 ! [N: num] :
% 12.45/6.95 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.45/6.95 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5181_power__even__abs__numeral,axiom,
% 12.45/6.95 ! [W: num,A: code_integer] :
% 12.45/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.45/6.95 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 12.45/6.95 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5182_power__even__abs__numeral,axiom,
% 12.45/6.95 ! [W: num,A: rat] :
% 12.45/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.45/6.95 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 12.45/6.95 = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5183_power__even__abs__numeral,axiom,
% 12.45/6.95 ! [W: num,A: real] :
% 12.45/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.45/6.95 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 12.45/6.95 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5184_power__even__abs__numeral,axiom,
% 12.45/6.95 ! [W: num,A: int] :
% 12.45/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.45/6.95 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 12.45/6.95 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5185_real__sqrt__abs,axiom,
% 12.45/6.95 ! [X4: real] :
% 12.45/6.95 ( ( sqrt @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.95 = ( abs_abs_real @ X4 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5186_abs__ge__self,axiom,
% 12.45/6.95 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5187_abs__ge__self,axiom,
% 12.45/6.95 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5188_abs__ge__self,axiom,
% 12.45/6.95 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5189_abs__ge__self,axiom,
% 12.45/6.95 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5190_abs__le__D1,axiom,
% 12.45/6.95 ! [A: real,B2: real] :
% 12.45/6.95 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 12.45/6.95 => ( ord_less_eq_real @ A @ B2 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5191_abs__le__D1,axiom,
% 12.45/6.95 ! [A: code_integer,B2: code_integer] :
% 12.45/6.95 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 12.45/6.95 => ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5192_abs__le__D1,axiom,
% 12.45/6.95 ! [A: rat,B2: rat] :
% 12.45/6.95 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 12.45/6.95 => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 12.45/6.95
% 12.45/6.95 thf(fact_5193_abs__le__D1,axiom,
% 12.45/6.95 ! [A: int,B2: int] :
% 12.45/6.95 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 12.45/6.96 => ( ord_less_eq_int @ A @ B2 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5194_abs__eq__0__iff,axiom,
% 12.45/6.96 ! [A: code_integer] :
% 12.45/6.96 ( ( ( abs_abs_Code_integer @ A )
% 12.45/6.96 = zero_z3403309356797280102nteger )
% 12.45/6.96 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5195_abs__eq__0__iff,axiom,
% 12.45/6.96 ! [A: real] :
% 12.45/6.96 ( ( ( abs_abs_real @ A )
% 12.45/6.96 = zero_zero_real )
% 12.45/6.96 = ( A = zero_zero_real ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5196_abs__eq__0__iff,axiom,
% 12.45/6.96 ! [A: rat] :
% 12.45/6.96 ( ( ( abs_abs_rat @ A )
% 12.45/6.96 = zero_zero_rat )
% 12.45/6.96 = ( A = zero_zero_rat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5197_abs__eq__0__iff,axiom,
% 12.45/6.96 ! [A: int] :
% 12.45/6.96 ( ( ( abs_abs_int @ A )
% 12.45/6.96 = zero_zero_int )
% 12.45/6.96 = ( A = zero_zero_int ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5198_abs__mult,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] :
% 12.45/6.96 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 12.45/6.96 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5199_abs__mult,axiom,
% 12.45/6.96 ! [A: real,B2: real] :
% 12.45/6.96 ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.96 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5200_abs__mult,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] :
% 12.45/6.96 ( ( abs_abs_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.96 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5201_abs__mult,axiom,
% 12.45/6.96 ! [A: int,B2: int] :
% 12.45/6.96 ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
% 12.45/6.96 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5202_abs__one,axiom,
% 12.45/6.96 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 12.45/6.96 = one_one_Code_integer ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5203_abs__one,axiom,
% 12.45/6.96 ( ( abs_abs_real @ one_one_real )
% 12.45/6.96 = one_one_real ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5204_abs__one,axiom,
% 12.45/6.96 ( ( abs_abs_rat @ one_one_rat )
% 12.45/6.96 = one_one_rat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5205_abs__one,axiom,
% 12.45/6.96 ( ( abs_abs_int @ one_one_int )
% 12.45/6.96 = one_one_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5206_abs__minus__commute,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] :
% 12.45/6.96 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 12.45/6.96 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5207_abs__minus__commute,axiom,
% 12.45/6.96 ! [A: real,B2: real] :
% 12.45/6.96 ( ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) )
% 12.45/6.96 = ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5208_abs__minus__commute,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] :
% 12.45/6.96 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) )
% 12.45/6.96 = ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5209_abs__minus__commute,axiom,
% 12.45/6.96 ! [A: int,B2: int] :
% 12.45/6.96 ( ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) )
% 12.45/6.96 = ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5210_abs__eq__iff,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] :
% 12.45/6.96 ( ( ( abs_abs_real @ X4 )
% 12.45/6.96 = ( abs_abs_real @ Y3 ) )
% 12.45/6.96 = ( ( X4 = Y3 )
% 12.45/6.96 | ( X4
% 12.45/6.96 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5211_abs__eq__iff,axiom,
% 12.45/6.96 ! [X4: int,Y3: int] :
% 12.45/6.96 ( ( ( abs_abs_int @ X4 )
% 12.45/6.96 = ( abs_abs_int @ Y3 ) )
% 12.45/6.96 = ( ( X4 = Y3 )
% 12.45/6.96 | ( X4
% 12.45/6.96 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5212_abs__eq__iff,axiom,
% 12.45/6.96 ! [X4: code_integer,Y3: code_integer] :
% 12.45/6.96 ( ( ( abs_abs_Code_integer @ X4 )
% 12.45/6.96 = ( abs_abs_Code_integer @ Y3 ) )
% 12.45/6.96 = ( ( X4 = Y3 )
% 12.45/6.96 | ( X4
% 12.45/6.96 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5213_abs__eq__iff,axiom,
% 12.45/6.96 ! [X4: rat,Y3: rat] :
% 12.45/6.96 ( ( ( abs_abs_rat @ X4 )
% 12.45/6.96 = ( abs_abs_rat @ Y3 ) )
% 12.45/6.96 = ( ( X4 = Y3 )
% 12.45/6.96 | ( X4
% 12.45/6.96 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5214_power__abs,axiom,
% 12.45/6.96 ! [A: code_integer,N: nat] :
% 12.45/6.96 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 12.45/6.96 = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5215_power__abs,axiom,
% 12.45/6.96 ! [A: rat,N: nat] :
% 12.45/6.96 ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
% 12.45/6.96 = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5216_power__abs,axiom,
% 12.45/6.96 ! [A: real,N: nat] :
% 12.45/6.96 ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 12.45/6.96 = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5217_power__abs,axiom,
% 12.45/6.96 ! [A: int,N: nat] :
% 12.45/6.96 ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 12.45/6.96 = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5218_dvd__if__abs__eq,axiom,
% 12.45/6.96 ! [L2: real,K: real] :
% 12.45/6.96 ( ( ( abs_abs_real @ L2 )
% 12.45/6.96 = ( abs_abs_real @ K ) )
% 12.45/6.96 => ( dvd_dvd_real @ L2 @ K ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5219_dvd__if__abs__eq,axiom,
% 12.45/6.96 ! [L2: int,K: int] :
% 12.45/6.96 ( ( ( abs_abs_int @ L2 )
% 12.45/6.96 = ( abs_abs_int @ K ) )
% 12.45/6.96 => ( dvd_dvd_int @ L2 @ K ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5220_dvd__if__abs__eq,axiom,
% 12.45/6.96 ! [L2: code_integer,K: code_integer] :
% 12.45/6.96 ( ( ( abs_abs_Code_integer @ L2 )
% 12.45/6.96 = ( abs_abs_Code_integer @ K ) )
% 12.45/6.96 => ( dvd_dvd_Code_integer @ L2 @ K ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5221_dvd__if__abs__eq,axiom,
% 12.45/6.96 ! [L2: rat,K: rat] :
% 12.45/6.96 ( ( ( abs_abs_rat @ L2 )
% 12.45/6.96 = ( abs_abs_rat @ K ) )
% 12.45/6.96 => ( dvd_dvd_rat @ L2 @ K ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5222_round__diff__minimal,axiom,
% 12.45/6.96 ! [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 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5223_round__diff__minimal,axiom,
% 12.45/6.96 ! [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 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5224_abs__ge__zero,axiom,
% 12.45/6.96 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5225_abs__ge__zero,axiom,
% 12.45/6.96 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5226_abs__ge__zero,axiom,
% 12.45/6.96 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5227_abs__ge__zero,axiom,
% 12.45/6.96 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5228_abs__not__less__zero,axiom,
% 12.45/6.96 ! [A: code_integer] :
% 12.45/6.96 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5229_abs__not__less__zero,axiom,
% 12.45/6.96 ! [A: real] :
% 12.45/6.96 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5230_abs__not__less__zero,axiom,
% 12.45/6.96 ! [A: rat] :
% 12.45/6.96 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5231_abs__not__less__zero,axiom,
% 12.45/6.96 ! [A: int] :
% 12.45/6.96 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5232_abs__of__pos,axiom,
% 12.45/6.96 ! [A: code_integer] :
% 12.45/6.96 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 12.45/6.96 => ( ( abs_abs_Code_integer @ A )
% 12.45/6.96 = A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5233_abs__of__pos,axiom,
% 12.45/6.96 ! [A: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.96 => ( ( abs_abs_real @ A )
% 12.45/6.96 = A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5234_abs__of__pos,axiom,
% 12.45/6.96 ! [A: rat] :
% 12.45/6.96 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.96 => ( ( abs_abs_rat @ A )
% 12.45/6.96 = A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5235_abs__of__pos,axiom,
% 12.45/6.96 ! [A: int] :
% 12.45/6.96 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.96 => ( ( abs_abs_int @ A )
% 12.45/6.96 = A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5236_abs__triangle__ineq,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5237_abs__triangle__ineq,axiom,
% 12.45/6.96 ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5238_abs__triangle__ineq,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5239_abs__triangle__ineq,axiom,
% 12.45/6.96 ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5240_abs__mult__less,axiom,
% 12.45/6.96 ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
% 12.45/6.96 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 12.45/6.96 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B2 ) @ D )
% 12.45/6.96 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5241_abs__mult__less,axiom,
% 12.45/6.96 ! [A: real,C: real,B2: real,D: real] :
% 12.45/6.96 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 12.45/6.96 => ( ( ord_less_real @ ( abs_abs_real @ B2 ) @ D )
% 12.45/6.96 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5242_abs__mult__less,axiom,
% 12.45/6.96 ! [A: rat,C: rat,B2: rat,D: rat] :
% 12.45/6.96 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 12.45/6.96 => ( ( ord_less_rat @ ( abs_abs_rat @ B2 ) @ D )
% 12.45/6.96 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5243_abs__mult__less,axiom,
% 12.45/6.96 ! [A: int,C: int,B2: int,D: int] :
% 12.45/6.96 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 12.45/6.96 => ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D )
% 12.45/6.96 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5244_abs__triangle__ineq2,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5245_abs__triangle__ineq2,axiom,
% 12.45/6.96 ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5246_abs__triangle__ineq2,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5247_abs__triangle__ineq2,axiom,
% 12.45/6.96 ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5248_abs__triangle__ineq3,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5249_abs__triangle__ineq3,axiom,
% 12.45/6.96 ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5250_abs__triangle__ineq3,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5251_abs__triangle__ineq3,axiom,
% 12.45/6.96 ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5252_abs__triangle__ineq2__sym,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5253_abs__triangle__ineq2__sym,axiom,
% 12.45/6.96 ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5254_abs__triangle__ineq2__sym,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5255_abs__triangle__ineq2__sym,axiom,
% 12.45/6.96 ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5256_nonzero__abs__divide,axiom,
% 12.45/6.96 ! [B2: real,A: real] :
% 12.45/6.96 ( ( B2 != zero_zero_real )
% 12.45/6.96 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.96 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5257_nonzero__abs__divide,axiom,
% 12.45/6.96 ! [B2: rat,A: rat] :
% 12.45/6.96 ( ( B2 != zero_zero_rat )
% 12.45/6.96 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
% 12.45/6.96 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5258_abs__ge__minus__self,axiom,
% 12.45/6.96 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5259_abs__ge__minus__self,axiom,
% 12.45/6.96 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5260_abs__ge__minus__self,axiom,
% 12.45/6.96 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5261_abs__ge__minus__self,axiom,
% 12.45/6.96 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5262_abs__le__iff,axiom,
% 12.45/6.96 ! [A: real,B2: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 12.45/6.96 = ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.96 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5263_abs__le__iff,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] :
% 12.45/6.96 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 12.45/6.96 = ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 12.45/6.96 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5264_abs__le__iff,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 12.45/6.96 = ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.96 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5265_abs__le__iff,axiom,
% 12.45/6.96 ! [A: int,B2: int] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 12.45/6.96 = ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.96 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5266_abs__le__D2,axiom,
% 12.45/6.96 ! [A: real,B2: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 12.45/6.96 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5267_abs__le__D2,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] :
% 12.45/6.96 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 12.45/6.96 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5268_abs__le__D2,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 12.45/6.96 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5269_abs__le__D2,axiom,
% 12.45/6.96 ! [A: int,B2: int] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 12.45/6.96 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5270_abs__leI,axiom,
% 12.45/6.96 ! [A: real,B2: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.96 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
% 12.45/6.96 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5271_abs__leI,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] :
% 12.45/6.96 ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 12.45/6.96 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 12.45/6.96 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5272_abs__leI,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.96 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 12.45/6.96 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5273_abs__leI,axiom,
% 12.45/6.96 ! [A: int,B2: int] :
% 12.45/6.96 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.96 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
% 12.45/6.96 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5274_abs__less__iff,axiom,
% 12.45/6.96 ! [A: real,B2: real] :
% 12.45/6.96 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B2 )
% 12.45/6.96 = ( ( ord_less_real @ A @ B2 )
% 12.45/6.96 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5275_abs__less__iff,axiom,
% 12.45/6.96 ! [A: int,B2: int] :
% 12.45/6.96 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B2 )
% 12.45/6.96 = ( ( ord_less_int @ A @ B2 )
% 12.45/6.96 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5276_abs__less__iff,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] :
% 12.45/6.96 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 12.45/6.96 = ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 12.45/6.96 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5277_abs__less__iff,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] :
% 12.45/6.96 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B2 )
% 12.45/6.96 = ( ( ord_less_rat @ A @ B2 )
% 12.45/6.96 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5278_dense__eq0__I,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ! [E2: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.45/6.96 => ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ E2 ) )
% 12.45/6.96 => ( X4 = zero_zero_real ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5279_dense__eq0__I,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ! [E2: rat] :
% 12.45/6.96 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 12.45/6.96 => ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ E2 ) )
% 12.45/6.96 => ( X4 = zero_zero_rat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5280_abs__eq__mult,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] :
% 12.45/6.96 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.45/6.96 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 12.45/6.96 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
% 12.45/6.96 | ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) )
% 12.45/6.96 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 12.45/6.96 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5281_abs__eq__mult,axiom,
% 12.45/6.96 ! [A: real,B2: real] :
% 12.45/6.96 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.96 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 12.45/6.96 & ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.96 | ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
% 12.45/6.96 => ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.96 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5282_abs__eq__mult,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] :
% 12.45/6.96 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.96 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 12.45/6.96 & ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.96 | ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
% 12.45/6.96 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.96 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5283_abs__eq__mult,axiom,
% 12.45/6.96 ! [A: int,B2: int] :
% 12.45/6.96 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.96 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 12.45/6.96 & ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.96 | ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
% 12.45/6.96 => ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
% 12.45/6.96 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5284_abs__mult__pos,axiom,
% 12.45/6.96 ! [X4: code_integer,Y3: code_integer] :
% 12.45/6.96 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 12.45/6.96 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X4 )
% 12.45/6.96 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5285_abs__mult__pos,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X4 )
% 12.45/6.96 = ( abs_abs_real @ ( times_times_real @ Y3 @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5286_abs__mult__pos,axiom,
% 12.45/6.96 ! [X4: rat,Y3: rat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.45/6.96 => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X4 )
% 12.45/6.96 = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5287_abs__mult__pos,axiom,
% 12.45/6.96 ! [X4: int,Y3: int] :
% 12.45/6.96 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.96 => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X4 )
% 12.45/6.96 = ( abs_abs_int @ ( times_times_int @ Y3 @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5288_abs__minus__le__zero,axiom,
% 12.45/6.96 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5289_abs__minus__le__zero,axiom,
% 12.45/6.96 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5290_abs__minus__le__zero,axiom,
% 12.45/6.96 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5291_abs__minus__le__zero,axiom,
% 12.45/6.96 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5292_eq__abs__iff_H,axiom,
% 12.45/6.96 ! [A: real,B2: real] :
% 12.45/6.96 ( ( A
% 12.45/6.96 = ( abs_abs_real @ B2 ) )
% 12.45/6.96 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.96 & ( ( B2 = A )
% 12.45/6.96 | ( B2
% 12.45/6.96 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5293_eq__abs__iff_H,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] :
% 12.45/6.96 ( ( A
% 12.45/6.96 = ( abs_abs_Code_integer @ B2 ) )
% 12.45/6.96 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.45/6.96 & ( ( B2 = A )
% 12.45/6.96 | ( B2
% 12.45/6.96 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5294_eq__abs__iff_H,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] :
% 12.45/6.96 ( ( A
% 12.45/6.96 = ( abs_abs_rat @ B2 ) )
% 12.45/6.96 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.96 & ( ( B2 = A )
% 12.45/6.96 | ( B2
% 12.45/6.96 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5295_eq__abs__iff_H,axiom,
% 12.45/6.96 ! [A: int,B2: int] :
% 12.45/6.96 ( ( A
% 12.45/6.96 = ( abs_abs_int @ B2 ) )
% 12.45/6.96 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.45/6.96 & ( ( B2 = A )
% 12.45/6.96 | ( B2
% 12.45/6.96 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5296_abs__eq__iff_H,axiom,
% 12.45/6.96 ! [A: real,B2: real] :
% 12.45/6.96 ( ( ( abs_abs_real @ A )
% 12.45/6.96 = B2 )
% 12.45/6.96 = ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.96 & ( ( A = B2 )
% 12.45/6.96 | ( A
% 12.45/6.96 = ( uminus_uminus_real @ B2 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5297_abs__eq__iff_H,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] :
% 12.45/6.96 ( ( ( abs_abs_Code_integer @ A )
% 12.45/6.96 = B2 )
% 12.45/6.96 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
% 12.45/6.96 & ( ( A = B2 )
% 12.45/6.96 | ( A
% 12.45/6.96 = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5298_abs__eq__iff_H,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] :
% 12.45/6.96 ( ( ( abs_abs_rat @ A )
% 12.45/6.96 = B2 )
% 12.45/6.96 = ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.96 & ( ( A = B2 )
% 12.45/6.96 | ( A
% 12.45/6.96 = ( uminus_uminus_rat @ B2 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5299_abs__eq__iff_H,axiom,
% 12.45/6.96 ! [A: int,B2: int] :
% 12.45/6.96 ( ( ( abs_abs_int @ A )
% 12.45/6.96 = B2 )
% 12.45/6.96 = ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.96 & ( ( A = B2 )
% 12.45/6.96 | ( A
% 12.45/6.96 = ( uminus_uminus_int @ B2 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5300_abs__div__pos,axiom,
% 12.45/6.96 ! [Y3: real,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.96 => ( ( divide_divide_real @ ( abs_abs_real @ X4 ) @ Y3 )
% 12.45/6.96 = ( abs_abs_real @ ( divide_divide_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5301_abs__div__pos,axiom,
% 12.45/6.96 ! [Y3: rat,X4: rat] :
% 12.45/6.96 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 12.45/6.96 => ( ( divide_divide_rat @ ( abs_abs_rat @ X4 ) @ Y3 )
% 12.45/6.96 = ( abs_abs_rat @ ( divide_divide_rat @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5302_zero__le__power__abs,axiom,
% 12.45/6.96 ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5303_zero__le__power__abs,axiom,
% 12.45/6.96 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5304_zero__le__power__abs,axiom,
% 12.45/6.96 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5305_zero__le__power__abs,axiom,
% 12.45/6.96 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5306_abs__if,axiom,
% 12.45/6.96 ( abs_abs_real
% 12.45/6.96 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5307_abs__if,axiom,
% 12.45/6.96 ( abs_abs_int
% 12.45/6.96 = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5308_abs__if,axiom,
% 12.45/6.96 ( abs_abs_Code_integer
% 12.45/6.96 = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5309_abs__if,axiom,
% 12.45/6.96 ( abs_abs_rat
% 12.45/6.96 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5310_abs__if__raw,axiom,
% 12.45/6.96 ( abs_abs_real
% 12.45/6.96 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5311_abs__if__raw,axiom,
% 12.45/6.96 ( abs_abs_int
% 12.45/6.96 = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5312_abs__if__raw,axiom,
% 12.45/6.96 ( abs_abs_Code_integer
% 12.45/6.96 = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5313_abs__if__raw,axiom,
% 12.45/6.96 ( abs_abs_rat
% 12.45/6.96 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5314_abs__of__neg,axiom,
% 12.45/6.96 ! [A: real] :
% 12.45/6.96 ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.96 => ( ( abs_abs_real @ A )
% 12.45/6.96 = ( uminus_uminus_real @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5315_abs__of__neg,axiom,
% 12.45/6.96 ! [A: int] :
% 12.45/6.96 ( ( ord_less_int @ A @ zero_zero_int )
% 12.45/6.96 => ( ( abs_abs_int @ A )
% 12.45/6.96 = ( uminus_uminus_int @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5316_abs__of__neg,axiom,
% 12.45/6.96 ! [A: code_integer] :
% 12.45/6.96 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 12.45/6.96 => ( ( abs_abs_Code_integer @ A )
% 12.45/6.96 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5317_abs__of__neg,axiom,
% 12.45/6.96 ! [A: rat] :
% 12.45/6.96 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.96 => ( ( abs_abs_rat @ A )
% 12.45/6.96 = ( uminus_uminus_rat @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5318_abs__triangle__ineq4,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5319_abs__triangle__ineq4,axiom,
% 12.45/6.96 ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5320_abs__triangle__ineq4,axiom,
% 12.45/6.96 ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5321_abs__triangle__ineq4,axiom,
% 12.45/6.96 ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5322_abs__diff__triangle__ineq,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ D ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5323_abs__diff__triangle__ineq,axiom,
% 12.45/6.96 ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5324_abs__diff__triangle__ineq,axiom,
% 12.45/6.96 ! [A: rat,B2: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5325_abs__diff__triangle__ineq,axiom,
% 12.45/6.96 ! [A: int,B2: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5326_abs__diff__le__iff,axiom,
% 12.45/6.96 ! [X4: code_integer,A: code_integer,R2: code_integer] :
% 12.45/6.96 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ A ) ) @ R2 )
% 12.45/6.96 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X4 )
% 12.45/6.96 & ( ord_le3102999989581377725nteger @ X4 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5327_abs__diff__le__iff,axiom,
% 12.45/6.96 ! [X4: real,A: real,R2: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ A ) ) @ R2 )
% 12.45/6.96 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X4 )
% 12.45/6.96 & ( ord_less_eq_real @ X4 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5328_abs__diff__le__iff,axiom,
% 12.45/6.96 ! [X4: rat,A: rat,R2: rat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ A ) ) @ R2 )
% 12.45/6.96 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X4 )
% 12.45/6.96 & ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5329_abs__diff__le__iff,axiom,
% 12.45/6.96 ! [X4: int,A: int,R2: int] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ A ) ) @ R2 )
% 12.45/6.96 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X4 )
% 12.45/6.96 & ( ord_less_eq_int @ X4 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5330_abs__diff__less__iff,axiom,
% 12.45/6.96 ! [X4: code_integer,A: code_integer,R2: code_integer] :
% 12.45/6.96 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ A ) ) @ R2 )
% 12.45/6.96 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X4 )
% 12.45/6.96 & ( ord_le6747313008572928689nteger @ X4 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5331_abs__diff__less__iff,axiom,
% 12.45/6.96 ! [X4: real,A: real,R2: real] :
% 12.45/6.96 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ A ) ) @ R2 )
% 12.45/6.96 = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X4 )
% 12.45/6.96 & ( ord_less_real @ X4 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5332_abs__diff__less__iff,axiom,
% 12.45/6.96 ! [X4: rat,A: rat,R2: rat] :
% 12.45/6.96 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ A ) ) @ R2 )
% 12.45/6.96 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X4 )
% 12.45/6.96 & ( ord_less_rat @ X4 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5333_abs__diff__less__iff,axiom,
% 12.45/6.96 ! [X4: int,A: int,R2: int] :
% 12.45/6.96 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ A ) ) @ R2 )
% 12.45/6.96 = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X4 )
% 12.45/6.96 & ( ord_less_int @ X4 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5334_round__mono,axiom,
% 12.45/6.96 ! [X4: rat,Y3: rat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.96 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X4 ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5335_abs__real__def,axiom,
% 12.45/6.96 ( abs_abs_real
% 12.45/6.96 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5336_lemma__interval__lt,axiom,
% 12.45/6.96 ! [A: real,X4: real,B2: real] :
% 12.45/6.96 ( ( ord_less_real @ A @ X4 )
% 12.45/6.96 => ( ( ord_less_real @ X4 @ B2 )
% 12.45/6.96 => ? [D3: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ D3 )
% 12.45/6.96 & ! [Y5: real] :
% 12.45/6.96 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y5 ) ) @ D3 )
% 12.45/6.96 => ( ( ord_less_real @ A @ Y5 )
% 12.45/6.96 & ( ord_less_real @ Y5 @ B2 ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5337_sin__bound__lemma,axiom,
% 12.45/6.96 ! [X4: real,Y3: real,U: real,V: real] :
% 12.45/6.96 ( ( X4 = Y3 )
% 12.45/6.96 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 12.45/6.96 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X4 @ U ) @ Y3 ) ) @ V ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5338_abs__add__one__gt__zero,axiom,
% 12.45/6.96 ! [X4: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5339_abs__add__one__gt__zero,axiom,
% 12.45/6.96 ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5340_abs__add__one__gt__zero,axiom,
% 12.45/6.96 ! [X4: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5341_abs__add__one__gt__zero,axiom,
% 12.45/6.96 ! [X4: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5342_of__int__leD,axiom,
% 12.45/6.96 ! [N: int,X4: code_integer] :
% 12.45/6.96 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X4 )
% 12.45/6.96 => ( ( N = zero_zero_int )
% 12.45/6.96 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5343_of__int__leD,axiom,
% 12.45/6.96 ! [N: int,X4: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X4 )
% 12.45/6.96 => ( ( N = zero_zero_int )
% 12.45/6.96 | ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5344_of__int__leD,axiom,
% 12.45/6.96 ! [N: int,X4: rat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X4 )
% 12.45/6.96 => ( ( N = zero_zero_int )
% 12.45/6.96 | ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5345_of__int__leD,axiom,
% 12.45/6.96 ! [N: int,X4: int] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X4 )
% 12.45/6.96 => ( ( N = zero_zero_int )
% 12.45/6.96 | ( ord_less_eq_int @ one_one_int @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5346_of__int__lessD,axiom,
% 12.45/6.96 ! [N: int,X4: code_integer] :
% 12.45/6.96 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X4 )
% 12.45/6.96 => ( ( N = zero_zero_int )
% 12.45/6.96 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5347_of__int__lessD,axiom,
% 12.45/6.96 ! [N: int,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X4 )
% 12.45/6.96 => ( ( N = zero_zero_int )
% 12.45/6.96 | ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5348_of__int__lessD,axiom,
% 12.45/6.96 ! [N: int,X4: rat] :
% 12.45/6.96 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X4 )
% 12.45/6.96 => ( ( N = zero_zero_int )
% 12.45/6.96 | ( ord_less_rat @ one_one_rat @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5349_of__int__lessD,axiom,
% 12.45/6.96 ! [N: int,X4: int] :
% 12.45/6.96 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X4 )
% 12.45/6.96 => ( ( N = zero_zero_int )
% 12.45/6.96 | ( ord_less_int @ one_one_int @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5350_lemma__interval,axiom,
% 12.45/6.96 ! [A: real,X4: real,B2: real] :
% 12.45/6.96 ( ( ord_less_real @ A @ X4 )
% 12.45/6.96 => ( ( ord_less_real @ X4 @ B2 )
% 12.45/6.96 => ? [D3: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ D3 )
% 12.45/6.96 & ! [Y5: real] :
% 12.45/6.96 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y5 ) ) @ D3 )
% 12.45/6.96 => ( ( ord_less_eq_real @ A @ Y5 )
% 12.45/6.96 & ( ord_less_eq_real @ Y5 @ B2 ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5351_of__int__round__abs__le,axiom,
% 12.45/6.96 ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) @ X4 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5352_of__int__round__abs__le,axiom,
% 12.45/6.96 ! [X4: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) @ X4 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5353_round__unique_H,axiom,
% 12.45/6.96 ! [X4: real,N: int] :
% 12.45/6.96 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.96 => ( ( archim8280529875227126926d_real @ X4 )
% 12.45/6.96 = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5354_round__unique_H,axiom,
% 12.45/6.96 ! [X4: rat,N: int] :
% 12.45/6.96 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 12.45/6.96 => ( ( archim7778729529865785530nd_rat @ X4 )
% 12.45/6.96 = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5355_abs__le__square__iff,axiom,
% 12.45/6.96 ! [X4: code_integer,Y3: code_integer] :
% 12.45/6.96 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ ( abs_abs_Code_integer @ Y3 ) )
% 12.45/6.96 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5356_abs__le__square__iff,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ Y3 ) )
% 12.45/6.96 = ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5357_abs__le__square__iff,axiom,
% 12.45/6.96 ! [X4: rat,Y3: rat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ ( abs_abs_rat @ Y3 ) )
% 12.45/6.96 = ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5358_abs__le__square__iff,axiom,
% 12.45/6.96 ! [X4: int,Y3: int] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ ( abs_abs_int @ Y3 ) )
% 12.45/6.96 = ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5359_abs__square__eq__1,axiom,
% 12.45/6.96 ! [X4: code_integer] :
% 12.45/6.96 ( ( ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.96 = one_one_Code_integer )
% 12.45/6.96 = ( ( abs_abs_Code_integer @ X4 )
% 12.45/6.96 = one_one_Code_integer ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5360_abs__square__eq__1,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.96 = one_one_rat )
% 12.45/6.96 = ( ( abs_abs_rat @ X4 )
% 12.45/6.96 = one_one_rat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5361_abs__square__eq__1,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.96 = one_one_real )
% 12.45/6.96 = ( ( abs_abs_real @ X4 )
% 12.45/6.96 = one_one_real ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5362_abs__square__eq__1,axiom,
% 12.45/6.96 ! [X4: int] :
% 12.45/6.96 ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.96 = one_one_int )
% 12.45/6.96 = ( ( abs_abs_int @ X4 )
% 12.45/6.96 = one_one_int ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5363_power__even__abs,axiom,
% 12.45/6.96 ! [N: nat,A: code_integer] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 12.45/6.96 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5364_power__even__abs,axiom,
% 12.45/6.96 ! [N: nat,A: rat] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 12.45/6.96 = ( power_power_rat @ A @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5365_power__even__abs,axiom,
% 12.45/6.96 ! [N: nat,A: real] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 12.45/6.96 = ( power_power_real @ A @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5366_power__even__abs,axiom,
% 12.45/6.96 ! [N: nat,A: int] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 12.45/6.96 = ( power_power_int @ A @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5367_power2__le__iff__abs__le,axiom,
% 12.45/6.96 ! [Y3: code_integer,X4: code_integer] :
% 12.45/6.96 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
% 12.45/6.96 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.96 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ Y3 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5368_power2__le__iff__abs__le,axiom,
% 12.45/6.96 ! [Y3: real,X4: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.96 => ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.96 = ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ Y3 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5369_power2__le__iff__abs__le,axiom,
% 12.45/6.96 ! [Y3: rat,X4: rat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 12.45/6.96 => ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.96 = ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ Y3 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5370_power2__le__iff__abs__le,axiom,
% 12.45/6.96 ! [Y3: int,X4: int] :
% 12.45/6.96 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.96 => ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.96 = ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ Y3 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5371_abs__square__le__1,axiom,
% 12.45/6.96 ! [X4: code_integer] :
% 12.45/6.96 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 12.45/6.96 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5372_abs__square__le__1,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 12.45/6.96 = ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5373_abs__square__le__1,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 12.45/6.96 = ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5374_abs__square__le__1,axiom,
% 12.45/6.96 ! [X4: int] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 12.45/6.96 = ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ one_one_int ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5375_abs__square__less__1,axiom,
% 12.45/6.96 ! [X4: code_integer] :
% 12.45/6.96 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 12.45/6.96 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5376_abs__square__less__1,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 12.45/6.96 = ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5377_abs__square__less__1,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ord_less_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 12.45/6.96 = ( ord_less_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5378_abs__square__less__1,axiom,
% 12.45/6.96 ! [X4: int] :
% 12.45/6.96 ( ( ord_less_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 12.45/6.96 = ( ord_less_int @ ( abs_abs_int @ X4 ) @ one_one_int ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5379_power__mono__even,axiom,
% 12.45/6.96 ! [N: nat,A: code_integer,B2: code_integer] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) )
% 12.45/6.96 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5380_power__mono__even,axiom,
% 12.45/6.96 ! [N: nat,A: real,B2: real] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) )
% 12.45/6.96 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5381_power__mono__even,axiom,
% 12.45/6.96 ! [N: nat,A: rat,B2: rat] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) )
% 12.45/6.96 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5382_power__mono__even,axiom,
% 12.45/6.96 ! [N: nat,A: int,B2: int] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) )
% 12.45/6.96 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5383_sqrt__ge__absD,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( sqrt @ Y3 ) )
% 12.45/6.96 => ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5384_real__sqrt__ge__abs1,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5385_real__sqrt__ge__abs2,axiom,
% 12.45/6.96 ! [Y3: real,X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5386_sqrt__sum__squares__le__sum__abs,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ Y3 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5387_cos__x__y__le__one,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5388_real__sqrt__sum__squares__less,axiom,
% 12.45/6.96 ! [X4: real,U: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.45/6.96 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.45/6.96 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5389_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.96 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5390_of__int__round__le,axiom,
% 12.45/6.96 ! [X4: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5391_of__int__round__le,axiom,
% 12.45/6.96 ! [X4: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5392_abs__sqrt__wlog,axiom,
% 12.45/6.96 ! [P: code_integer > code_integer > $o,X4: code_integer] :
% 12.45/6.96 ( ! [X3: code_integer] :
% 12.45/6.96 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 12.45/6.96 => ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.96 => ( P @ ( abs_abs_Code_integer @ X4 ) @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5393_abs__sqrt__wlog,axiom,
% 12.45/6.96 ! [P: real > real > $o,X4: real] :
% 12.45/6.96 ( ! [X3: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 12.45/6.96 => ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.96 => ( P @ ( abs_abs_real @ X4 ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5394_abs__sqrt__wlog,axiom,
% 12.45/6.96 ! [P: rat > rat > $o,X4: rat] :
% 12.45/6.96 ( ! [X3: rat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 12.45/6.96 => ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.96 => ( P @ ( abs_abs_rat @ X4 ) @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5395_abs__sqrt__wlog,axiom,
% 12.45/6.96 ! [P: int > int > $o,X4: int] :
% 12.45/6.96 ( ! [X3: int] :
% 12.45/6.96 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 12.45/6.96 => ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.96 => ( P @ ( abs_abs_int @ X4 ) @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5396_arctan__double,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.96 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X4 ) )
% 12.45/6.96 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5397_arctan__half,axiom,
% 12.45/6.96 ( arctan
% 12.45/6.96 = ( ^ [X: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5398_log__base__10__eq1,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.96 = ( 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 @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5399_signed__take__bit__eq__take__bit__minus,axiom,
% 12.45/6.96 ( bit_ri631733984087533419it_int
% 12.45/6.96 = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5400_log__base__10__eq2,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.96 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5401_exp__ge__one__minus__x__over__n__power__n,axiom,
% 12.45/6.96 ! [X4: real,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.96 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.96 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5402_of__nat__eq__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ( semiri8010041392384452111omplex @ M )
% 12.45/6.96 = ( semiri8010041392384452111omplex @ N ) )
% 12.45/6.96 = ( M = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5403_of__nat__eq__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ( semiri5074537144036343181t_real @ M )
% 12.45/6.96 = ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.96 = ( M = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5404_of__nat__eq__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ( semiri681578069525770553at_rat @ M )
% 12.45/6.96 = ( semiri681578069525770553at_rat @ N ) )
% 12.45/6.96 = ( M = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5405_of__nat__eq__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ( semiri1316708129612266289at_nat @ M )
% 12.45/6.96 = ( semiri1316708129612266289at_nat @ N ) )
% 12.45/6.96 = ( M = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5406_of__nat__eq__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ( semiri1314217659103216013at_int @ M )
% 12.45/6.96 = ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.96 = ( M = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5407_abs__of__nat,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 12.45/6.96 = ( semiri4939895301339042750nteger @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5408_abs__of__nat,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.96 = ( semiri5074537144036343181t_real @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5409_abs__of__nat,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 12.45/6.96 = ( semiri681578069525770553at_rat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5410_abs__of__nat,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.96 = ( semiri1314217659103216013at_int @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5411_of__nat__eq__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( ( semiri8010041392384452111omplex @ M )
% 12.45/6.96 = zero_zero_complex )
% 12.45/6.96 = ( M = zero_zero_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5412_of__nat__eq__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( ( semiri5074537144036343181t_real @ M )
% 12.45/6.96 = zero_zero_real )
% 12.45/6.96 = ( M = zero_zero_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5413_of__nat__eq__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( ( semiri681578069525770553at_rat @ M )
% 12.45/6.96 = zero_zero_rat )
% 12.45/6.96 = ( M = zero_zero_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5414_of__nat__eq__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( ( semiri1316708129612266289at_nat @ M )
% 12.45/6.96 = zero_zero_nat )
% 12.45/6.96 = ( M = zero_zero_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5415_of__nat__eq__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( ( semiri1314217659103216013at_int @ M )
% 12.45/6.96 = zero_zero_int )
% 12.45/6.96 = ( M = zero_zero_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5416_of__nat__0__eq__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( zero_zero_complex
% 12.45/6.96 = ( semiri8010041392384452111omplex @ N ) )
% 12.45/6.96 = ( zero_zero_nat = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5417_of__nat__0__eq__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( zero_zero_real
% 12.45/6.96 = ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.96 = ( zero_zero_nat = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5418_of__nat__0__eq__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( zero_zero_rat
% 12.45/6.96 = ( semiri681578069525770553at_rat @ N ) )
% 12.45/6.96 = ( zero_zero_nat = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5419_of__nat__0__eq__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( zero_zero_nat
% 12.45/6.96 = ( semiri1316708129612266289at_nat @ N ) )
% 12.45/6.96 = ( zero_zero_nat = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5420_of__nat__0__eq__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( zero_zero_int
% 12.45/6.96 = ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.96 = ( zero_zero_nat = N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5421_of__nat__0,axiom,
% 12.45/6.96 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 12.45/6.96 = zero_zero_complex ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5422_of__nat__0,axiom,
% 12.45/6.96 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 12.45/6.96 = zero_zero_real ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5423_of__nat__0,axiom,
% 12.45/6.96 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 12.45/6.96 = zero_zero_rat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5424_of__nat__0,axiom,
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 12.45/6.96 = zero_zero_nat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5425_of__nat__0,axiom,
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 12.45/6.96 = zero_zero_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5426_of__nat__less__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.96 = ( ord_less_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5427_of__nat__less__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 12.45/6.96 = ( ord_less_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5428_of__nat__less__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 12.45/6.96 = ( ord_less_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5429_of__nat__less__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.96 = ( ord_less_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5430_of__nat__numeral,axiom,
% 12.45/6.96 ! [N: num] :
% 12.45/6.96 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 12.45/6.96 = ( numera6690914467698888265omplex @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5431_of__nat__numeral,axiom,
% 12.45/6.96 ! [N: num] :
% 12.45/6.96 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 12.45/6.96 = ( numeral_numeral_real @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5432_of__nat__numeral,axiom,
% 12.45/6.96 ! [N: num] :
% 12.45/6.96 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 12.45/6.96 = ( numeral_numeral_rat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5433_of__nat__numeral,axiom,
% 12.45/6.96 ! [N: num] :
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 12.45/6.96 = ( numeral_numeral_nat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5434_of__nat__numeral,axiom,
% 12.45/6.96 ! [N: num] :
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 12.45/6.96 = ( numeral_numeral_int @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5435_of__nat__le__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.96 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5436_of__nat__le__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 12.45/6.96 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5437_of__nat__le__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 12.45/6.96 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5438_of__nat__le__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.96 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5439_of__nat__add,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
% 12.45/6.96 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5440_of__nat__add,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 12.45/6.96 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5441_of__nat__add,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 12.45/6.96 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5442_of__nat__add,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 12.45/6.96 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5443_of__nat__add,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 12.45/6.96 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5444_of__nat__mult,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 12.45/6.96 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5445_of__nat__mult,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 12.45/6.96 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5446_of__nat__mult,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 12.45/6.96 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5447_of__nat__mult,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 12.45/6.96 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5448_of__nat__mult,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 12.45/6.96 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5449_of__nat__1,axiom,
% 12.45/6.96 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 12.45/6.96 = one_one_complex ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5450_of__nat__1,axiom,
% 12.45/6.96 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 12.45/6.96 = one_one_real ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5451_of__nat__1,axiom,
% 12.45/6.96 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 12.45/6.96 = one_one_rat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5452_of__nat__1,axiom,
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 12.45/6.96 = one_one_nat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5453_of__nat__1,axiom,
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 12.45/6.96 = one_one_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5454_of__nat__1__eq__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( one_one_complex
% 12.45/6.96 = ( semiri8010041392384452111omplex @ N ) )
% 12.45/6.96 = ( N = one_one_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5455_of__nat__1__eq__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( one_one_real
% 12.45/6.96 = ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.96 = ( N = one_one_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5456_of__nat__1__eq__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( one_one_rat
% 12.45/6.96 = ( semiri681578069525770553at_rat @ N ) )
% 12.45/6.96 = ( N = one_one_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5457_of__nat__1__eq__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( one_one_nat
% 12.45/6.96 = ( semiri1316708129612266289at_nat @ N ) )
% 12.45/6.96 = ( N = one_one_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5458_of__nat__1__eq__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( one_one_int
% 12.45/6.96 = ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.96 = ( N = one_one_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5459_of__nat__eq__1__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ( semiri8010041392384452111omplex @ N )
% 12.45/6.96 = one_one_complex )
% 12.45/6.96 = ( N = one_one_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5460_of__nat__eq__1__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ( semiri5074537144036343181t_real @ N )
% 12.45/6.96 = one_one_real )
% 12.45/6.96 = ( N = one_one_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5461_of__nat__eq__1__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ( semiri681578069525770553at_rat @ N )
% 12.45/6.96 = one_one_rat )
% 12.45/6.96 = ( N = one_one_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5462_of__nat__eq__1__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ( semiri1316708129612266289at_nat @ N )
% 12.45/6.96 = one_one_nat )
% 12.45/6.96 = ( N = one_one_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5463_of__nat__eq__1__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ( semiri1314217659103216013at_int @ N )
% 12.45/6.96 = one_one_int )
% 12.45/6.96 = ( N = one_one_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5464_of__nat__power,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 12.45/6.96 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5465_of__nat__power,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 12.45/6.96 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5466_of__nat__power,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 12.45/6.96 = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5467_of__nat__power,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 12.45/6.96 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5468_of__nat__power,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 12.45/6.96 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5469_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W )
% 12.45/6.96 = ( semiri8010041392384452111omplex @ X4 ) )
% 12.45/6.96 = ( ( power_power_nat @ B2 @ W )
% 12.45/6.96 = X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5470_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W )
% 12.45/6.96 = ( semiri5074537144036343181t_real @ X4 ) )
% 12.45/6.96 = ( ( power_power_nat @ B2 @ W )
% 12.45/6.96 = X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5471_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W )
% 12.45/6.96 = ( semiri681578069525770553at_rat @ X4 ) )
% 12.45/6.96 = ( ( power_power_nat @ B2 @ W )
% 12.45/6.96 = X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5472_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W )
% 12.45/6.96 = ( semiri1316708129612266289at_nat @ X4 ) )
% 12.45/6.96 = ( ( power_power_nat @ B2 @ W )
% 12.45/6.96 = X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5473_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W )
% 12.45/6.96 = ( semiri1314217659103216013at_int @ X4 ) )
% 12.45/6.96 = ( ( power_power_nat @ B2 @ W )
% 12.45/6.96 = X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5474_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ( semiri8010041392384452111omplex @ X4 )
% 12.45/6.96 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W ) )
% 12.45/6.96 = ( X4
% 12.45/6.96 = ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5475_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ( semiri5074537144036343181t_real @ X4 )
% 12.45/6.96 = ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 12.45/6.96 = ( X4
% 12.45/6.96 = ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5476_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ( semiri681578069525770553at_rat @ X4 )
% 12.45/6.96 = ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
% 12.45/6.96 = ( X4
% 12.45/6.96 = ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5477_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ( semiri1316708129612266289at_nat @ X4 )
% 12.45/6.96 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 12.45/6.96 = ( X4
% 12.45/6.96 = ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5478_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ( semiri1314217659103216013at_int @ X4 )
% 12.45/6.96 = ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 12.45/6.96 = ( X4
% 12.45/6.96 = ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5479_zabs__less__one__iff,axiom,
% 12.45/6.96 ! [Z: int] :
% 12.45/6.96 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 12.45/6.96 = ( Z = zero_zero_int ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5480_log__one,axiom,
% 12.45/6.96 ! [A: real] :
% 12.45/6.96 ( ( log @ A @ one_one_real )
% 12.45/6.96 = zero_zero_real ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5481_arctan__less__zero__iff,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_real @ ( arctan @ X4 ) @ zero_zero_real )
% 12.45/6.96 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5482_zero__less__arctan__iff,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ ( arctan @ X4 ) )
% 12.45/6.96 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5483_zero__le__arctan__iff,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5484_arctan__le__zero__iff,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( arctan @ X4 ) @ zero_zero_real )
% 12.45/6.96 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5485_of__nat__of__bool,axiom,
% 12.45/6.96 ! [P: $o] :
% 12.45/6.96 ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.45/6.96 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5486_of__nat__of__bool,axiom,
% 12.45/6.96 ! [P: $o] :
% 12.45/6.96 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.45/6.96 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5487_of__nat__of__bool,axiom,
% 12.45/6.96 ! [P: $o] :
% 12.45/6.96 ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.45/6.96 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5488_of__nat__of__bool,axiom,
% 12.45/6.96 ! [P: $o] :
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.45/6.96 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5489_of__nat__of__bool,axiom,
% 12.45/6.96 ! [P: $o] :
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.45/6.96 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5490_of__nat__of__bool,axiom,
% 12.45/6.96 ! [P: $o] :
% 12.45/6.96 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.45/6.96 = ( zero_n356916108424825756nteger @ P ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5491_of__nat__le__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 12.45/6.96 = ( M = zero_zero_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5492_of__nat__le__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 12.45/6.96 = ( M = zero_zero_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5493_of__nat__le__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 12.45/6.96 = ( M = zero_zero_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5494_of__nat__le__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 12.45/6.96 = ( M = zero_zero_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5495_of__nat__Suc,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 12.45/6.96 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5496_of__nat__Suc,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 12.45/6.96 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5497_of__nat__Suc,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 12.45/6.96 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5498_of__nat__Suc,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 12.45/6.96 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5499_of__nat__Suc,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 12.45/6.96 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5500_bit__numeral__Bit0__Suc__iff,axiom,
% 12.45/6.96 ! [M: num,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 12.45/6.96 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5501_bit__numeral__Bit0__Suc__iff,axiom,
% 12.45/6.96 ! [M: num,N: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 12.45/6.96 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5502_bit__numeral__Bit1__Suc__iff,axiom,
% 12.45/6.96 ! [M: num,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 12.45/6.96 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5503_bit__numeral__Bit1__Suc__iff,axiom,
% 12.45/6.96 ! [M: num,N: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 12.45/6.96 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5504_log__eq__one,axiom,
% 12.45/6.96 ! [A: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.96 => ( ( A != one_one_real )
% 12.45/6.96 => ( ( log @ A @ A )
% 12.45/6.96 = one_one_real ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5505_log__less__cancel__iff,axiom,
% 12.45/6.96 ! [A: real,X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.96 => ( ( ord_less_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) )
% 12.45/6.96 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5506_log__less__one__cancel__iff,axiom,
% 12.45/6.96 ! [A: real,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_real @ ( log @ A @ X4 ) @ one_one_real )
% 12.45/6.96 = ( ord_less_real @ X4 @ A ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5507_one__less__log__cancel__iff,axiom,
% 12.45/6.96 ! [A: real,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_real @ one_one_real @ ( log @ A @ X4 ) )
% 12.45/6.96 = ( ord_less_real @ A @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5508_log__less__zero__cancel__iff,axiom,
% 12.45/6.96 ! [A: real,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_real @ ( log @ A @ X4 ) @ zero_zero_real )
% 12.45/6.96 = ( ord_less_real @ X4 @ one_one_real ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5509_zero__less__log__cancel__iff,axiom,
% 12.45/6.96 ! [A: real,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X4 ) )
% 12.45/6.96 = ( ord_less_real @ one_one_real @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5510_signed__take__bit__nonnegative__iff,axiom,
% 12.45/6.96 ! [N: nat,K: int] :
% 12.45/6.96 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 12.45/6.96 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5511_signed__take__bit__negative__iff,axiom,
% 12.45/6.96 ! [N: nat,K: int] :
% 12.45/6.96 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 12.45/6.96 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5512_of__nat__0__less__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.96 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5513_of__nat__0__less__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 12.45/6.96 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5514_of__nat__0__less__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 12.45/6.96 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5515_of__nat__0__less__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.96 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5516_of__nat__less__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X4 ) )
% 12.45/6.96 = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5517_of__nat__less__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X4 ) )
% 12.45/6.96 = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5518_of__nat__less__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X4 ) )
% 12.45/6.96 = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5519_of__nat__less__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X4 ) )
% 12.45/6.96 = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5520_of__nat__power__less__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 12.45/6.96 = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5521_of__nat__power__less__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
% 12.45/6.96 = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5522_of__nat__power__less__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 12.45/6.96 = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5523_of__nat__power__less__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 12.45/6.96 = ( ord_less_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5524_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [Y3: nat,X4: num,N: nat] :
% 12.45/6.96 ( ( ( semiri8010041392384452111omplex @ Y3 )
% 12.45/6.96 = ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N ) )
% 12.45/6.96 = ( Y3
% 12.45/6.96 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5525_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [Y3: nat,X4: num,N: nat] :
% 12.45/6.96 ( ( ( semiri5074537144036343181t_real @ Y3 )
% 12.45/6.96 = ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 12.45/6.96 = ( Y3
% 12.45/6.96 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5526_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [Y3: nat,X4: num,N: nat] :
% 12.45/6.96 ( ( ( semiri681578069525770553at_rat @ Y3 )
% 12.45/6.96 = ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 12.45/6.96 = ( Y3
% 12.45/6.96 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5527_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [Y3: nat,X4: num,N: nat] :
% 12.45/6.96 ( ( ( semiri1316708129612266289at_nat @ Y3 )
% 12.45/6.96 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
% 12.45/6.96 = ( Y3
% 12.45/6.96 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5528_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [Y3: nat,X4: num,N: nat] :
% 12.45/6.96 ( ( ( semiri1314217659103216013at_int @ Y3 )
% 12.45/6.96 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) )
% 12.45/6.96 = ( Y3
% 12.45/6.96 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5529_numeral__power__eq__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: num,N: nat,Y3: nat] :
% 12.45/6.96 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N )
% 12.45/6.96 = ( semiri8010041392384452111omplex @ Y3 ) )
% 12.45/6.96 = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 12.45/6.96 = Y3 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5530_numeral__power__eq__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: num,N: nat,Y3: nat] :
% 12.45/6.96 ( ( ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N )
% 12.45/6.96 = ( semiri5074537144036343181t_real @ Y3 ) )
% 12.45/6.96 = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 12.45/6.96 = Y3 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5531_numeral__power__eq__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: num,N: nat,Y3: nat] :
% 12.45/6.96 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N )
% 12.45/6.96 = ( semiri681578069525770553at_rat @ Y3 ) )
% 12.45/6.96 = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 12.45/6.96 = Y3 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5532_numeral__power__eq__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: num,N: nat,Y3: nat] :
% 12.45/6.96 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 12.45/6.96 = ( semiri1316708129612266289at_nat @ Y3 ) )
% 12.45/6.96 = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 12.45/6.96 = Y3 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5533_numeral__power__eq__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: num,N: nat,Y3: nat] :
% 12.45/6.96 ( ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 12.45/6.96 = ( semiri1314217659103216013at_int @ Y3 ) )
% 12.45/6.96 = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 12.45/6.96 = Y3 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5534_of__nat__le__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5535_of__nat__le__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5536_of__nat__le__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5537_of__nat__le__of__nat__power__cancel__iff,axiom,
% 12.45/6.96 ! [B2: nat,W: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5538_of__nat__power__le__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 12.45/6.96 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5539_of__nat__power__le__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
% 12.45/6.96 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5540_of__nat__power__le__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 12.45/6.96 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5541_of__nat__power__le__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,B2: nat,W: nat] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 12.45/6.96 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5542_numeral__less__real__of__nat__iff,axiom,
% 12.45/6.96 ! [W: num,N: nat] :
% 12.45/6.96 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.96 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5543_real__of__nat__less__numeral__iff,axiom,
% 12.45/6.96 ! [N: nat,W: num] :
% 12.45/6.96 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 12.45/6.96 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5544_numeral__le__real__of__nat__iff,axiom,
% 12.45/6.96 ! [N: num,M: nat] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 12.45/6.96 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5545_zero__le__log__cancel__iff,axiom,
% 12.45/6.96 ! [A: real,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5546_log__le__zero__cancel__iff,axiom,
% 12.45/6.96 ! [A: real,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ zero_zero_real )
% 12.45/6.96 = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5547_one__le__log__cancel__iff,axiom,
% 12.45/6.96 ! [A: real,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_real @ A @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5548_log__le__one__cancel__iff,axiom,
% 12.45/6.96 ! [A: real,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ one_one_real )
% 12.45/6.96 = ( ord_less_eq_real @ X4 @ A ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5549_log__le__cancel__iff,axiom,
% 12.45/6.96 ! [A: real,X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.96 => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) )
% 12.45/6.96 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5550_bit__numeral__simps_I2_J,axiom,
% 12.45/6.96 ! [W: num,N: num] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 12.45/6.96 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5551_bit__numeral__simps_I2_J,axiom,
% 12.45/6.96 ! [W: num,N: num] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 12.45/6.96 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5552_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 12.45/6.96 ! [W: num,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 12.45/6.96 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5553_bit__numeral__simps_I3_J,axiom,
% 12.45/6.96 ! [W: num,N: num] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 12.45/6.96 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5554_bit__numeral__simps_I3_J,axiom,
% 12.45/6.96 ! [W: num,N: num] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 12.45/6.96 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5555_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 12.45/6.96 ! [W: num,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 12.45/6.96 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5556_of__nat__zero__less__power__iff,axiom,
% 12.45/6.96 ! [X4: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X4 ) @ N ) )
% 12.45/6.96 = ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 12.45/6.96 | ( N = zero_zero_nat ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5557_of__nat__zero__less__power__iff,axiom,
% 12.45/6.96 ! [X4: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X4 ) @ N ) )
% 12.45/6.96 = ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 12.45/6.96 | ( N = zero_zero_nat ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5558_of__nat__zero__less__power__iff,axiom,
% 12.45/6.96 ! [X4: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ N ) )
% 12.45/6.96 = ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 12.45/6.96 | ( N = zero_zero_nat ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5559_of__nat__zero__less__power__iff,axiom,
% 12.45/6.96 ! [X4: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X4 ) @ N ) )
% 12.45/6.96 = ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 12.45/6.96 | ( N = zero_zero_nat ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5560_bit__0,axiom,
% 12.45/6.96 ! [A: code_integer] :
% 12.45/6.96 ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 12.45/6.96 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5561_bit__0,axiom,
% 12.45/6.96 ! [A: int] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 12.45/6.96 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5562_bit__0,axiom,
% 12.45/6.96 ! [A: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 12.45/6.96 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5563_log__pow__cancel,axiom,
% 12.45/6.96 ! [A: real,B2: nat] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.96 => ( ( A != one_one_real )
% 12.45/6.96 => ( ( log @ A @ ( power_power_real @ A @ B2 ) )
% 12.45/6.96 = ( semiri5074537144036343181t_real @ B2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5564_bit__minus__numeral__int_I1_J,axiom,
% 12.45/6.96 ! [W: num,N: num] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 12.45/6.96 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5565_bit__minus__numeral__int_I2_J,axiom,
% 12.45/6.96 ! [W: num,N: num] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 12.45/6.96 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5566_even__of__nat,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 12.45/6.96 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5567_even__of__nat,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 12.45/6.96 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5568_even__of__nat,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.96 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5569_numeral__power__less__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [I: num,N: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X4 ) )
% 12.45/6.96 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5570_numeral__power__less__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [I: num,N: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X4 ) )
% 12.45/6.96 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5571_numeral__power__less__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [I: num,N: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X4 ) )
% 12.45/6.96 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5572_numeral__power__less__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [I: num,N: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X4 ) )
% 12.45/6.96 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5573_of__nat__less__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,I: num,N: nat] :
% 12.45/6.96 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 12.45/6.96 = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5574_of__nat__less__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,I: num,N: nat] :
% 12.45/6.96 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 12.45/6.96 = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5575_of__nat__less__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,I: num,N: nat] :
% 12.45/6.96 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 12.45/6.96 = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5576_of__nat__less__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,I: num,N: nat] :
% 12.45/6.96 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 12.45/6.96 = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5577_numeral__power__le__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [I: num,N: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5578_numeral__power__le__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [I: num,N: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5579_numeral__power__le__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [I: num,N: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5580_numeral__power__le__of__nat__cancel__iff,axiom,
% 12.45/6.96 ! [I: num,N: nat,X4: nat] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5581_of__nat__le__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,I: num,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 12.45/6.96 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5582_of__nat__le__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,I: num,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 12.45/6.96 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5583_of__nat__le__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,I: num,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 12.45/6.96 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5584_of__nat__le__numeral__power__cancel__iff,axiom,
% 12.45/6.96 ! [X4: nat,I: num,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 12.45/6.96 = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5585_bit__mod__2__iff,axiom,
% 12.45/6.96 ! [A: code_integer,N: nat] :
% 12.45/6.96 ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
% 12.45/6.96 = ( ( N = zero_zero_nat )
% 12.45/6.96 & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5586_bit__mod__2__iff,axiom,
% 12.45/6.96 ! [A: int,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
% 12.45/6.96 = ( ( N = zero_zero_nat )
% 12.45/6.96 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5587_bit__mod__2__iff,axiom,
% 12.45/6.96 ! [A: nat,N: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 12.45/6.96 = ( ( N = zero_zero_nat )
% 12.45/6.96 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5588_bit__numeral__iff,axiom,
% 12.45/6.96 ! [M: num,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N )
% 12.45/6.96 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5589_bit__numeral__iff,axiom,
% 12.45/6.96 ! [M: num,N: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N )
% 12.45/6.96 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5590_bit__of__nat__iff__bit,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
% 12.45/6.96 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5591_bit__of__nat__iff__bit,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
% 12.45/6.96 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5592_bit__disjunctive__add__iff,axiom,
% 12.45/6.96 ! [A: int,B2: int,N: nat] :
% 12.45/6.96 ( ! [N4: nat] :
% 12.45/6.96 ( ~ ( bit_se1146084159140164899it_int @ A @ N4 )
% 12.45/6.96 | ~ ( bit_se1146084159140164899it_int @ B2 @ N4 ) )
% 12.45/6.96 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B2 ) @ N )
% 12.45/6.96 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.45/6.96 | ( bit_se1146084159140164899it_int @ B2 @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5593_bit__disjunctive__add__iff,axiom,
% 12.45/6.96 ! [A: nat,B2: nat,N: nat] :
% 12.45/6.96 ( ! [N4: nat] :
% 12.45/6.96 ( ~ ( bit_se1148574629649215175it_nat @ A @ N4 )
% 12.45/6.96 | ~ ( bit_se1148574629649215175it_nat @ B2 @ N4 ) )
% 12.45/6.96 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
% 12.45/6.96 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.45/6.96 | ( bit_se1148574629649215175it_nat @ B2 @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5594_real__arch__simple,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ? [N4: nat] : ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5595_real__arch__simple,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ? [N4: nat] : ( ord_less_eq_rat @ X4 @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5596_reals__Archimedean2,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ? [N4: nat] : ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5597_reals__Archimedean2,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ? [N4: nat] : ( ord_less_rat @ X4 @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5598_mult__of__nat__commute,axiom,
% 12.45/6.96 ! [X4: nat,Y3: complex] :
% 12.45/6.96 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X4 ) @ Y3 )
% 12.45/6.96 = ( times_times_complex @ Y3 @ ( semiri8010041392384452111omplex @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5599_mult__of__nat__commute,axiom,
% 12.45/6.96 ! [X4: nat,Y3: real] :
% 12.45/6.96 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ Y3 )
% 12.45/6.96 = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5600_mult__of__nat__commute,axiom,
% 12.45/6.96 ! [X4: nat,Y3: rat] :
% 12.45/6.96 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X4 ) @ Y3 )
% 12.45/6.96 = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5601_mult__of__nat__commute,axiom,
% 12.45/6.96 ! [X4: nat,Y3: nat] :
% 12.45/6.96 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ Y3 )
% 12.45/6.96 = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5602_mult__of__nat__commute,axiom,
% 12.45/6.96 ! [X4: nat,Y3: int] :
% 12.45/6.96 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X4 ) @ Y3 )
% 12.45/6.96 = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5603_bit__and__iff,axiom,
% 12.45/6.96 ! [A: int,B2: int,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ A @ B2 ) @ N )
% 12.45/6.96 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.45/6.96 & ( bit_se1146084159140164899it_int @ B2 @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5604_bit__and__iff,axiom,
% 12.45/6.96 ! [A: nat,B2: nat,N: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ ( bit_se727722235901077358nd_nat @ A @ B2 ) @ N )
% 12.45/6.96 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.45/6.96 & ( bit_se1148574629649215175it_nat @ B2 @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5605_bit__and__int__iff,axiom,
% 12.45/6.96 ! [K: int,L2: int,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N )
% 12.45/6.96 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 12.45/6.96 & ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5606_arctan__less__iff,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) )
% 12.45/6.96 = ( ord_less_real @ X4 @ Y3 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5607_arctan__monotone,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.96 => ( ord_less_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5608_arctan__le__iff,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) )
% 12.45/6.96 = ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5609_arctan__monotone_H,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.96 => ( ord_less_eq_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5610_bit__unset__bit__iff,axiom,
% 12.45/6.96 ! [M: nat,A: int,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( bit_se4203085406695923979it_int @ M @ A ) @ N )
% 12.45/6.96 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.45/6.96 & ( M != N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5611_bit__unset__bit__iff,axiom,
% 12.45/6.96 ! [M: nat,A: nat,N: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ ( bit_se4205575877204974255it_nat @ M @ A ) @ N )
% 12.45/6.96 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.45/6.96 & ( M != N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5612_less__log__of__power,axiom,
% 12.45/6.96 ! [B2: real,N: nat,M: real] :
% 12.45/6.96 ( ( ord_less_real @ ( power_power_real @ B2 @ N ) @ M )
% 12.45/6.96 => ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.96 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5613_log__of__power__eq,axiom,
% 12.45/6.96 ! [M: nat,B2: real,N: nat] :
% 12.45/6.96 ( ( ( semiri5074537144036343181t_real @ M )
% 12.45/6.96 = ( power_power_real @ B2 @ N ) )
% 12.45/6.96 => ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.96 => ( ( semiri5074537144036343181t_real @ N )
% 12.45/6.96 = ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5614_of__nat__less__of__int__iff,axiom,
% 12.45/6.96 ! [N: nat,X4: int] :
% 12.45/6.96 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X4 ) )
% 12.45/6.96 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5615_of__nat__less__of__int__iff,axiom,
% 12.45/6.96 ! [N: nat,X4: int] :
% 12.45/6.96 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X4 ) )
% 12.45/6.96 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5616_of__nat__less__of__int__iff,axiom,
% 12.45/6.96 ! [N: nat,X4: int] :
% 12.45/6.96 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X4 ) )
% 12.45/6.96 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5617_le__log__of__power,axiom,
% 12.45/6.96 ! [B2: real,N: nat,M: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( power_power_real @ B2 @ N ) @ M )
% 12.45/6.96 => ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.96 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5618_log__base__pow,axiom,
% 12.45/6.96 ! [A: real,N: nat,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.96 => ( ( log @ ( power_power_real @ A @ N ) @ X4 )
% 12.45/6.96 = ( divide_divide_real @ ( log @ A @ X4 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5619_log__nat__power,axiom,
% 12.45/6.96 ! [X4: real,B2: real,N: nat] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( log @ B2 @ ( power_power_real @ X4 @ N ) )
% 12.45/6.96 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5620_not__bit__1__Suc,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5621_not__bit__1__Suc,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5622_bit__1__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
% 12.45/6.96 = ( N = zero_zero_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5623_bit__1__iff,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
% 12.45/6.96 = ( N = zero_zero_nat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5624_bit__numeral__simps_I1_J,axiom,
% 12.45/6.96 ! [N: num] :
% 12.45/6.96 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5625_bit__numeral__simps_I1_J,axiom,
% 12.45/6.96 ! [N: num] :
% 12.45/6.96 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5626_of__nat__0__le__iff,axiom,
% 12.45/6.96 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5627_of__nat__0__le__iff,axiom,
% 12.45/6.96 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5628_of__nat__0__le__iff,axiom,
% 12.45/6.96 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5629_of__nat__0__le__iff,axiom,
% 12.45/6.96 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5630_of__nat__less__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5631_of__nat__less__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5632_of__nat__less__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5633_of__nat__less__0__iff,axiom,
% 12.45/6.96 ! [M: nat] :
% 12.45/6.96 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5634_bit__take__bit__iff,axiom,
% 12.45/6.96 ! [M: nat,A: int,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
% 12.45/6.96 = ( ( ord_less_nat @ N @ M )
% 12.45/6.96 & ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5635_bit__take__bit__iff,axiom,
% 12.45/6.96 ! [M: nat,A: nat,N: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
% 12.45/6.96 = ( ( ord_less_nat @ N @ M )
% 12.45/6.96 & ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5636_of__nat__neq__0,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 12.45/6.96 != zero_zero_complex ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5637_of__nat__neq__0,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 12.45/6.96 != zero_zero_real ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5638_of__nat__neq__0,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 12.45/6.96 != zero_zero_rat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5639_of__nat__neq__0,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 12.45/6.96 != zero_zero_nat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5640_of__nat__neq__0,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 12.45/6.96 != zero_zero_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5641_bit__of__bool__iff,axiom,
% 12.45/6.96 ! [B2: $o,N: nat] :
% 12.45/6.96 ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B2 ) @ N )
% 12.45/6.96 = ( B2
% 12.45/6.96 & ( N = zero_zero_nat ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5642_bit__of__bool__iff,axiom,
% 12.45/6.96 ! [B2: $o,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ N )
% 12.45/6.96 = ( B2
% 12.45/6.96 & ( N = zero_zero_nat ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5643_bit__of__bool__iff,axiom,
% 12.45/6.96 ! [B2: $o,N: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ N )
% 12.45/6.96 = ( B2
% 12.45/6.96 & ( N = zero_zero_nat ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5644_div__mult2__eq_H,axiom,
% 12.45/6.96 ! [A: nat,M: nat,N: nat] :
% 12.45/6.96 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 12.45/6.96 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5645_div__mult2__eq_H,axiom,
% 12.45/6.96 ! [A: int,M: nat,N: nat] :
% 12.45/6.96 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.45/6.96 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5646_less__imp__of__nat__less,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_nat @ M @ N )
% 12.45/6.96 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5647_less__imp__of__nat__less,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_nat @ M @ N )
% 12.45/6.96 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5648_less__imp__of__nat__less,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_nat @ M @ N )
% 12.45/6.96 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5649_less__imp__of__nat__less,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_nat @ M @ N )
% 12.45/6.96 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5650_of__nat__less__imp__less,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.96 => ( ord_less_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5651_of__nat__less__imp__less,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 12.45/6.96 => ( ord_less_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5652_of__nat__less__imp__less,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 12.45/6.96 => ( ord_less_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5653_of__nat__less__imp__less,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.96 => ( ord_less_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5654_of__nat__mono,axiom,
% 12.45/6.96 ! [I: nat,J: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.45/6.96 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5655_of__nat__mono,axiom,
% 12.45/6.96 ! [I: nat,J: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.45/6.96 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5656_of__nat__mono,axiom,
% 12.45/6.96 ! [I: nat,J: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.45/6.96 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5657_of__nat__mono,axiom,
% 12.45/6.96 ! [I: nat,J: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.45/6.96 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5658_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 12.45/6.96 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5659_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 12.45/6.96 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5660_log__def,axiom,
% 12.45/6.96 ( log
% 12.45/6.96 = ( ^ [A4: real,X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ A4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5661_of__nat__dvd__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 12.45/6.96 = ( dvd_dvd_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5662_of__nat__dvd__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 12.45/6.96 = ( dvd_dvd_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5663_of__nat__dvd__iff,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.96 = ( dvd_dvd_nat @ M @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5664_signed__take__bit__eq__if__positive,axiom,
% 12.45/6.96 ! [A: int,N: nat] :
% 12.45/6.96 ( ~ ( bit_se1146084159140164899it_int @ A @ N )
% 12.45/6.96 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 12.45/6.96 = ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5665_abs__zmult__eq__1,axiom,
% 12.45/6.96 ! [M: int,N: int] :
% 12.45/6.96 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 12.45/6.96 = one_one_int )
% 12.45/6.96 => ( ( abs_abs_int @ M )
% 12.45/6.96 = one_one_int ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5666_of__nat__mod,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 12.45/6.96 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5667_of__nat__mod,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 12.45/6.96 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5668_of__nat__mod,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 12.45/6.96 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5669_abs__div,axiom,
% 12.45/6.96 ! [Y3: int,X4: int] :
% 12.45/6.96 ( ( dvd_dvd_int @ Y3 @ X4 )
% 12.45/6.96 => ( ( abs_abs_int @ ( divide_divide_int @ X4 @ Y3 ) )
% 12.45/6.96 = ( divide_divide_int @ ( abs_abs_int @ X4 ) @ ( abs_abs_int @ Y3 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5670_log2__of__power__eq,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( M
% 12.45/6.96 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.96 => ( ( semiri5074537144036343181t_real @ N )
% 12.45/6.96 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5671_log__of__power__less,axiom,
% 12.45/6.96 ! [M: nat,B2: real,N: nat] :
% 12.45/6.96 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
% 12.45/6.96 => ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.96 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.96 => ( ord_less_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5672_take__bit__of__nat,axiom,
% 12.45/6.96 ! [N: nat,M: nat] :
% 12.45/6.96 ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 12.45/6.96 = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5673_take__bit__of__nat,axiom,
% 12.45/6.96 ! [N: nat,M: nat] :
% 12.45/6.96 ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 12.45/6.96 = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5674_of__nat__and__eq,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 12.45/6.96 = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5675_of__nat__and__eq,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 12.45/6.96 = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5676_of__nat__mask__eq,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 12.45/6.96 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5677_of__nat__mask__eq,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
% 12.45/6.96 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5678_log__of__power__le,axiom,
% 12.45/6.96 ! [M: nat,B2: real,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
% 12.45/6.96 => ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.96 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.96 => ( ord_less_eq_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5679_ex__less__of__nat__mult,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ? [N4: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5680_ex__less__of__nat__mult,axiom,
% 12.45/6.96 ! [X4: rat,Y3: rat] :
% 12.45/6.96 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 12.45/6.96 => ? [N4: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N4 ) @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5681_of__nat__diff,axiom,
% 12.45/6.96 ! [N: nat,M: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.96 => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.96 = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5682_of__nat__diff,axiom,
% 12.45/6.96 ! [N: nat,M: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.96 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.96 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5683_of__nat__diff,axiom,
% 12.45/6.96 ! [N: nat,M: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.96 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.96 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5684_of__nat__diff,axiom,
% 12.45/6.96 ! [N: nat,M: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.96 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.96 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5685_of__nat__diff,axiom,
% 12.45/6.96 ! [N: nat,M: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.96 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 12.45/6.96 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5686_bit__not__int__iff_H,axiom,
% 12.45/6.96 ! [K: int,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 12.45/6.96 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5687_exp__of__nat__mult,axiom,
% 12.45/6.96 ! [N: nat,X4: complex] :
% 12.45/6.96 ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X4 ) )
% 12.45/6.96 = ( power_power_complex @ ( exp_complex @ X4 ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5688_exp__of__nat__mult,axiom,
% 12.45/6.96 ! [N: nat,X4: real] :
% 12.45/6.96 ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) )
% 12.45/6.96 = ( power_power_real @ ( exp_real @ X4 ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5689_exp__of__nat2__mult,axiom,
% 12.45/6.96 ! [X4: complex,N: nat] :
% 12.45/6.96 ( ( exp_complex @ ( times_times_complex @ X4 @ ( semiri8010041392384452111omplex @ N ) ) )
% 12.45/6.96 = ( power_power_complex @ ( exp_complex @ X4 ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5690_exp__of__nat2__mult,axiom,
% 12.45/6.96 ! [X4: real,N: nat] :
% 12.45/6.96 ( ( exp_real @ ( times_times_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.45/6.96 = ( power_power_real @ ( exp_real @ X4 ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5691_zabs__def,axiom,
% 12.45/6.96 ( abs_abs_int
% 12.45/6.96 = ( ^ [I2: int] : ( if_int @ ( ord_less_int @ I2 @ zero_zero_int ) @ ( uminus_uminus_int @ I2 ) @ I2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5692_reals__Archimedean3,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ! [Y5: real] :
% 12.45/6.96 ? [N4: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5693_log__ln,axiom,
% 12.45/6.96 ( ln_ln_real
% 12.45/6.96 = ( log @ ( exp_real @ one_one_real ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5694_real__of__nat__div4,axiom,
% 12.45/6.96 ! [N: nat,X4: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5695_abs__mod__less,axiom,
% 12.45/6.96 ! [L2: int,K: int] :
% 12.45/6.96 ( ( L2 != zero_zero_int )
% 12.45/6.96 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5696_real__of__nat__div,axiom,
% 12.45/6.96 ! [D: nat,N: nat] :
% 12.45/6.96 ( ( dvd_dvd_nat @ D @ N )
% 12.45/6.96 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 12.45/6.96 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5697_less__log2__of__power,axiom,
% 12.45/6.96 ! [N: nat,M: nat] :
% 12.45/6.96 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 12.45/6.96 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5698_le__log2__of__power,axiom,
% 12.45/6.96 ! [N: nat,M: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 12.45/6.96 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5699_flip__bit__eq__if,axiom,
% 12.45/6.96 ( bit_se2159334234014336723it_int
% 12.45/6.96 = ( ^ [N2: nat,A4: int] : ( if_nat_int_int @ ( bit_se1146084159140164899it_int @ A4 @ N2 ) @ bit_se4203085406695923979it_int @ bit_se7879613467334960850it_int @ N2 @ A4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5700_flip__bit__eq__if,axiom,
% 12.45/6.96 ( bit_se2161824704523386999it_nat
% 12.45/6.96 = ( ^ [N2: nat,A4: nat] : ( if_nat_nat_nat @ ( bit_se1148574629649215175it_nat @ A4 @ N2 ) @ bit_se4205575877204974255it_nat @ bit_se7882103937844011126it_nat @ N2 @ A4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5701_log2__of__power__less,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.96 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.96 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5702_log__base__change,axiom,
% 12.45/6.96 ! [A: real,B2: real,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.96 => ( ( A != one_one_real )
% 12.45/6.96 => ( ( log @ B2 @ X4 )
% 12.45/6.96 = ( divide_divide_real @ ( log @ A @ X4 ) @ ( log @ A @ B2 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5703_mod__mult2__eq_H,axiom,
% 12.45/6.96 ! [A: code_integer,M: nat,N: nat] :
% 12.45/6.96 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 12.45/6.96 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5704_mod__mult2__eq_H,axiom,
% 12.45/6.96 ! [A: nat,M: nat,N: nat] :
% 12.45/6.96 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 12.45/6.96 = ( 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 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5705_mod__mult2__eq_H,axiom,
% 12.45/6.96 ! [A: int,M: nat,N: nat] :
% 12.45/6.96 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.45/6.96 = ( 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 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5706_bit__imp__take__bit__positive,axiom,
% 12.45/6.96 ! [N: nat,M: nat,K: int] :
% 12.45/6.96 ( ( ord_less_nat @ N @ M )
% 12.45/6.96 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 12.45/6.96 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5707_field__char__0__class_Oof__nat__div,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 12.45/6.96 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5708_field__char__0__class_Oof__nat__div,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 12.45/6.96 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5709_field__char__0__class_Oof__nat__div,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 12.45/6.96 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5710_bit__concat__bit__iff,axiom,
% 12.45/6.96 ! [M: nat,K: int,L2: int,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N )
% 12.45/6.96 = ( ( ( ord_less_nat @ N @ M )
% 12.45/6.96 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 12.45/6.96 | ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.96 & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5711_nat__less__real__le,axiom,
% 12.45/6.96 ( ord_less_nat
% 12.45/6.96 = ( ^ [N2: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5712_nat__le__real__less,axiom,
% 12.45/6.96 ( ord_less_eq_nat
% 12.45/6.96 = ( ^ [N2: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5713_zdvd__mult__cancel1,axiom,
% 12.45/6.96 ! [M: int,N: int] :
% 12.45/6.96 ( ( M != zero_zero_int )
% 12.45/6.96 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 12.45/6.96 = ( ( abs_abs_int @ N )
% 12.45/6.96 = one_one_int ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5714_log2__of__power__le,axiom,
% 12.45/6.96 ! [M: nat,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.96 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.96 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5715_real__of__nat__div__aux,axiom,
% 12.45/6.96 ! [X4: nat,D: nat] :
% 12.45/6.96 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( semiri5074537144036343181t_real @ D ) )
% 12.45/6.96 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X4 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X4 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5716_signed__take__bit__eq__concat__bit,axiom,
% 12.45/6.96 ( bit_ri631733984087533419it_int
% 12.45/6.96 = ( ^ [N2: nat,K3: int] : ( bit_concat_bit @ N2 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5717_exp__eq__0__imp__not__bit,axiom,
% 12.45/6.96 ! [N: nat,A: int] :
% 12.45/6.96 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 = zero_zero_int )
% 12.45/6.96 => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5718_exp__eq__0__imp__not__bit,axiom,
% 12.45/6.96 ! [N: nat,A: nat] :
% 12.45/6.96 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 = zero_zero_nat )
% 12.45/6.96 => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5719_bit__Suc,axiom,
% 12.45/6.96 ! [A: int,N: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
% 12.45/6.96 = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5720_bit__Suc,axiom,
% 12.45/6.96 ! [A: nat,N: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
% 12.45/6.96 = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5721_stable__imp__bit__iff__odd,axiom,
% 12.45/6.96 ! [A: code_integer,N: nat] :
% 12.45/6.96 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.96 = A )
% 12.45/6.96 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 12.45/6.96 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5722_stable__imp__bit__iff__odd,axiom,
% 12.45/6.96 ! [A: int,N: nat] :
% 12.45/6.96 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.96 = A )
% 12.45/6.96 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.45/6.96 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5723_stable__imp__bit__iff__odd,axiom,
% 12.45/6.96 ! [A: nat,N: nat] :
% 12.45/6.96 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.96 = A )
% 12.45/6.96 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.45/6.96 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5724_bit__iff__idd__imp__stable,axiom,
% 12.45/6.96 ! [A: code_integer] :
% 12.45/6.96 ( ! [N4: nat] :
% 12.45/6.96 ( ( bit_se9216721137139052372nteger @ A @ N4 )
% 12.45/6.96 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 12.45/6.96 => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/6.96 = A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5725_bit__iff__idd__imp__stable,axiom,
% 12.45/6.96 ! [A: int] :
% 12.45/6.96 ( ! [N4: nat] :
% 12.45/6.96 ( ( bit_se1146084159140164899it_int @ A @ N4 )
% 12.45/6.96 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 12.45/6.96 => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.96 = A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5726_bit__iff__idd__imp__stable,axiom,
% 12.45/6.96 ! [A: nat] :
% 12.45/6.96 ( ! [N4: nat] :
% 12.45/6.96 ( ( bit_se1148574629649215175it_nat @ A @ N4 )
% 12.45/6.96 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 12.45/6.96 => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.96 = A ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5727_log__mult,axiom,
% 12.45/6.96 ! [A: real,X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.96 => ( ( A != one_one_real )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.96 => ( ( log @ A @ ( times_times_real @ X4 @ Y3 ) )
% 12.45/6.96 = ( plus_plus_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5728_nat__approx__posE,axiom,
% 12.45/6.96 ! [E: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ E )
% 12.45/6.96 => ~ ! [N4: nat] :
% 12.45/6.96 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ E ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5729_nat__approx__posE,axiom,
% 12.45/6.96 ! [E: rat] :
% 12.45/6.96 ( ( ord_less_rat @ zero_zero_rat @ E )
% 12.45/6.96 => ~ ! [N4: nat] :
% 12.45/6.96 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ E ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5730_of__nat__less__two__power,axiom,
% 12.45/6.96 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5731_of__nat__less__two__power,axiom,
% 12.45/6.96 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5732_of__nat__less__two__power,axiom,
% 12.45/6.96 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5733_int__bit__bound,axiom,
% 12.45/6.96 ! [K: int] :
% 12.45/6.96 ~ ! [N4: nat] :
% 12.45/6.96 ( ! [M5: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ N4 @ M5 )
% 12.45/6.96 => ( ( bit_se1146084159140164899it_int @ K @ M5 )
% 12.45/6.96 = ( bit_se1146084159140164899it_int @ K @ N4 ) ) )
% 12.45/6.96 => ~ ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 12.45/6.96 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N4 @ one_one_nat ) )
% 12.45/6.96 = ( ~ ( bit_se1146084159140164899it_int @ K @ N4 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5734_log__divide,axiom,
% 12.45/6.96 ! [A: real,X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.96 => ( ( A != one_one_real )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.96 => ( ( log @ A @ ( divide_divide_real @ X4 @ Y3 ) )
% 12.45/6.96 = ( minus_minus_real @ ( log @ A @ X4 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5735_inverse__of__nat__le,axiom,
% 12.45/6.96 ! [N: nat,M: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.96 => ( ( N != zero_zero_nat )
% 12.45/6.96 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5736_inverse__of__nat__le,axiom,
% 12.45/6.96 ! [N: nat,M: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.96 => ( ( N != zero_zero_nat )
% 12.45/6.96 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5737_exp__divide__power__eq,axiom,
% 12.45/6.96 ! [N: nat,X4: complex] :
% 12.45/6.96 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.96 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X4 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 12.45/6.96 = ( exp_complex @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5738_exp__divide__power__eq,axiom,
% 12.45/6.96 ! [N: nat,X4: real] :
% 12.45/6.96 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.96 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 12.45/6.96 = ( exp_real @ X4 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5739_even__abs__add__iff,axiom,
% 12.45/6.96 ! [K: int,L2: int] :
% 12.45/6.96 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
% 12.45/6.96 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5740_even__add__abs__iff,axiom,
% 12.45/6.96 ! [K: int,L2: int] :
% 12.45/6.96 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
% 12.45/6.96 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5741_real__archimedian__rdiv__eq__0,axiom,
% 12.45/6.96 ! [X4: real,C: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.45/6.96 => ( ! [M4: nat] :
% 12.45/6.96 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 12.45/6.96 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X4 ) @ C ) )
% 12.45/6.96 => ( X4 = zero_zero_real ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5742_real__of__nat__div2,axiom,
% 12.45/6.96 ! [N: nat,X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5743_real__of__nat__div3,axiom,
% 12.45/6.96 ! [N: nat,X4: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) ) @ one_one_real ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5744_ln__realpow,axiom,
% 12.45/6.96 ! [X4: real,N: nat] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( ln_ln_real @ ( power_power_real @ X4 @ N ) )
% 12.45/6.96 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5745_bit__iff__odd,axiom,
% 12.45/6.96 ( bit_se9216721137139052372nteger
% 12.45/6.96 = ( ^ [A4: code_integer,N2: nat] :
% 12.45/6.96 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A4 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5746_bit__iff__odd,axiom,
% 12.45/6.96 ( bit_se1146084159140164899it_int
% 12.45/6.96 = ( ^ [A4: int,N2: nat] :
% 12.45/6.96 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5747_bit__iff__odd,axiom,
% 12.45/6.96 ( bit_se1148574629649215175it_nat
% 12.45/6.96 = ( ^ [A4: nat,N2: nat] :
% 12.45/6.96 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5748_and__exp__eq__0__iff__not__bit,axiom,
% 12.45/6.96 ! [A: int,N: nat] :
% 12.45/6.96 ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.96 = zero_zero_int )
% 12.45/6.96 = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5749_and__exp__eq__0__iff__not__bit,axiom,
% 12.45/6.96 ! [A: nat,N: nat] :
% 12.45/6.96 ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.96 = zero_zero_nat )
% 12.45/6.96 = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5750_log__eq__div__ln__mult__log,axiom,
% 12.45/6.96 ! [A: real,B2: real,X4: real] :
% 12.45/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.96 => ( ( A != one_one_real )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.96 => ( ( B2 != one_one_real )
% 12.45/6.96 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ( log @ A @ X4 )
% 12.45/6.96 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( ln_ln_real @ A ) ) @ ( log @ B2 @ X4 ) ) ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5751_bit__int__def,axiom,
% 12.45/6.96 ( bit_se1146084159140164899it_int
% 12.45/6.96 = ( ^ [K3: int,N2: nat] :
% 12.45/6.96 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5752_nat__intermed__int__val,axiom,
% 12.45/6.96 ! [M: nat,N: nat,F: nat > int,K: int] :
% 12.45/6.96 ( ! [I3: nat] :
% 12.45/6.96 ( ( ( ord_less_eq_nat @ M @ I3 )
% 12.45/6.96 & ( ord_less_nat @ I3 @ N ) )
% 12.45/6.96 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 12.45/6.96 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.96 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 12.45/6.96 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 12.45/6.96 => ? [I3: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ M @ I3 )
% 12.45/6.96 & ( ord_less_eq_nat @ I3 @ N )
% 12.45/6.96 & ( ( F @ I3 )
% 12.45/6.96 = K ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5753_decr__lemma,axiom,
% 12.45/6.96 ! [D: int,X4: int,Z: int] :
% 12.45/6.96 ( ( ord_less_int @ zero_zero_int @ D )
% 12.45/6.96 => ( ord_less_int @ ( minus_minus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5754_incr__lemma,axiom,
% 12.45/6.96 ! [D: int,Z: int,X4: int] :
% 12.45/6.96 ( ( ord_less_int @ zero_zero_int @ D )
% 12.45/6.96 => ( ord_less_int @ Z @ ( plus_plus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5755_linear__plus__1__le__power,axiom,
% 12.45/6.96 ! [X4: real,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X4 @ one_one_real ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5756_Bernoulli__inequality,axiom,
% 12.45/6.96 ! [X4: real,N: nat] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X4 ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5757_even__bit__succ__iff,axiom,
% 12.45/6.96 ! [A: code_integer,N: nat] :
% 12.45/6.96 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.96 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
% 12.45/6.96 = ( ( bit_se9216721137139052372nteger @ A @ N )
% 12.45/6.96 | ( N = zero_zero_nat ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5758_even__bit__succ__iff,axiom,
% 12.45/6.96 ! [A: int,N: nat] :
% 12.45/6.96 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.96 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
% 12.45/6.96 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.45/6.96 | ( N = zero_zero_nat ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5759_even__bit__succ__iff,axiom,
% 12.45/6.96 ! [A: nat,N: nat] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.96 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
% 12.45/6.96 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.45/6.96 | ( N = zero_zero_nat ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5760_odd__bit__iff__bit__pred,axiom,
% 12.45/6.96 ! [A: code_integer,N: nat] :
% 12.45/6.96 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.96 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 12.45/6.96 = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
% 12.45/6.96 | ( N = zero_zero_nat ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5761_odd__bit__iff__bit__pred,axiom,
% 12.45/6.96 ! [A: int,N: nat] :
% 12.45/6.96 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.96 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.45/6.96 = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
% 12.45/6.96 | ( N = zero_zero_nat ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5762_odd__bit__iff__bit__pred,axiom,
% 12.45/6.96 ! [A: nat,N: nat] :
% 12.45/6.96 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.96 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.45/6.96 = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
% 12.45/6.96 | ( N = zero_zero_nat ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5763_nat__ivt__aux,axiom,
% 12.45/6.96 ! [N: nat,F: nat > int,K: int] :
% 12.45/6.96 ( ! [I3: nat] :
% 12.45/6.96 ( ( ord_less_nat @ I3 @ N )
% 12.45/6.96 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 12.45/6.96 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 12.45/6.96 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 12.45/6.96 => ? [I3: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ I3 @ N )
% 12.45/6.96 & ( ( F @ I3 )
% 12.45/6.96 = K ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5764_bit__sum__mult__2__cases,axiom,
% 12.45/6.96 ! [A: code_integer,B2: code_integer,N: nat] :
% 12.45/6.96 ( ! [J2: nat] :
% 12.45/6.96 ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
% 12.45/6.96 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ N )
% 12.45/6.96 = ( ( ( N = zero_zero_nat )
% 12.45/6.96 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 12.45/6.96 & ( ( N != zero_zero_nat )
% 12.45/6.96 => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5765_bit__sum__mult__2__cases,axiom,
% 12.45/6.96 ! [A: int,B2: int,N: nat] :
% 12.45/6.96 ( ! [J2: nat] :
% 12.45/6.96 ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
% 12.45/6.96 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ N )
% 12.45/6.96 = ( ( ( N = zero_zero_nat )
% 12.45/6.96 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.45/6.96 & ( ( N != zero_zero_nat )
% 12.45/6.96 => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5766_bit__sum__mult__2__cases,axiom,
% 12.45/6.96 ! [A: nat,B2: nat,N: nat] :
% 12.45/6.96 ( ! [J2: nat] :
% 12.45/6.96 ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
% 12.45/6.96 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ N )
% 12.45/6.96 = ( ( ( N = zero_zero_nat )
% 12.45/6.96 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 12.45/6.96 & ( ( N != zero_zero_nat )
% 12.45/6.96 => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5767_bit__rec,axiom,
% 12.45/6.96 ( bit_se9216721137139052372nteger
% 12.45/6.96 = ( ^ [A4: code_integer,N2: nat] :
% 12.45/6.96 ( ( ( N2 = zero_zero_nat )
% 12.45/6.96 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) )
% 12.45/6.96 & ( ( N2 != zero_zero_nat )
% 12.45/6.96 => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5768_bit__rec,axiom,
% 12.45/6.96 ( bit_se1146084159140164899it_int
% 12.45/6.96 = ( ^ [A4: int,N2: nat] :
% 12.45/6.96 ( ( ( N2 = zero_zero_nat )
% 12.45/6.96 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) )
% 12.45/6.96 & ( ( N2 != zero_zero_nat )
% 12.45/6.96 => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5769_bit__rec,axiom,
% 12.45/6.96 ( bit_se1148574629649215175it_nat
% 12.45/6.96 = ( ^ [A4: nat,N2: nat] :
% 12.45/6.96 ( ( ( N2 = zero_zero_nat )
% 12.45/6.96 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) )
% 12.45/6.96 & ( ( N2 != zero_zero_nat )
% 12.45/6.96 => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5770_nat0__intermed__int__val,axiom,
% 12.45/6.96 ! [N: nat,F: nat > int,K: int] :
% 12.45/6.96 ( ! [I3: nat] :
% 12.45/6.96 ( ( ord_less_nat @ I3 @ N )
% 12.45/6.96 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 12.45/6.96 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 12.45/6.96 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 12.45/6.96 => ? [I3: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ I3 @ N )
% 12.45/6.96 & ( ( F @ I3 )
% 12.45/6.96 = K ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5771_set__bit__eq,axiom,
% 12.45/6.96 ( bit_se7879613467334960850it_int
% 12.45/6.96 = ( ^ [N2: nat,K3: int] :
% 12.45/6.96 ( plus_plus_int @ K3
% 12.45/6.96 @ ( times_times_int
% 12.45/6.96 @ ( zero_n2684676970156552555ol_int
% 12.45/6.96 @ ~ ( bit_se1146084159140164899it_int @ K3 @ N2 ) )
% 12.45/6.96 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5772_unset__bit__eq,axiom,
% 12.45/6.96 ( bit_se4203085406695923979it_int
% 12.45/6.96 = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5773_arctan__add,axiom,
% 12.45/6.96 ! [X4: real,Y3: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.96 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 12.45/6.96 => ( ( plus_plus_real @ ( arctan @ X4 ) @ ( arctan @ Y3 ) )
% 12.45/6.96 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5774_take__bit__Suc__from__most,axiom,
% 12.45/6.96 ! [N: nat,K: int] :
% 12.45/6.96 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 12.45/6.96 = ( 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 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5775_Bernoulli__inequality__even,axiom,
% 12.45/6.96 ! [N: nat,X4: real] :
% 12.45/6.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X4 ) @ N ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5776_exp__ge__one__plus__x__over__n__power__n,axiom,
% 12.45/6.96 ! [N: nat,X4: real] :
% 12.45/6.96 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X4 )
% 12.45/6.96 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.96 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X4 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5777_lemma__termdiff3,axiom,
% 12.45/6.96 ! [H2: real,Z: real,K5: real,N: nat] :
% 12.45/6.96 ( ( H2 != zero_zero_real )
% 12.45/6.96 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 12.45/6.96 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
% 12.45/6.96 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5778_lemma__termdiff3,axiom,
% 12.45/6.96 ! [H2: complex,Z: complex,K5: real,N: nat] :
% 12.45/6.96 ( ( H2 != zero_zero_complex )
% 12.45/6.96 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 12.45/6.96 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
% 12.45/6.96 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5779_ceiling__log2__div2,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.45/6.96 = ( 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 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5780_pochhammer__double,axiom,
% 12.45/6.96 ! [Z: complex,N: nat] :
% 12.45/6.96 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.96 = ( 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 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5781_pochhammer__double,axiom,
% 12.45/6.96 ! [Z: real,N: nat] :
% 12.45/6.96 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.96 = ( 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 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5782_pochhammer__double,axiom,
% 12.45/6.96 ! [Z: rat,N: nat] :
% 12.45/6.96 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.96 = ( 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 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5783_central__binomial__lower__bound,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.96 => ( 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 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5784_floor__log2__div2,axiom,
% 12.45/6.96 ! [N: nat] :
% 12.45/6.96 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.96 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.45/6.96 = ( 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 ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5785_machin,axiom,
% 12.45/6.96 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.45/6.96 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5786_of__int__floor__cancel,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.96 = X4 )
% 12.45/6.96 = ( ? [N2: int] :
% 12.45/6.96 ( X4
% 12.45/6.96 = ( ring_1_of_int_real @ N2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5787_of__int__floor__cancel,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.96 = X4 )
% 12.45/6.96 = ( ? [N2: int] :
% 12.45/6.96 ( X4
% 12.45/6.96 = ( ring_1_of_int_rat @ N2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5788_of__int__ceiling__cancel,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) )
% 12.45/6.96 = X4 )
% 12.45/6.96 = ( ? [N2: int] :
% 12.45/6.96 ( X4
% 12.45/6.96 = ( ring_1_of_int_rat @ N2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5789_of__int__ceiling__cancel,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) )
% 12.45/6.96 = X4 )
% 12.45/6.96 = ( ? [N2: int] :
% 12.45/6.96 ( X4
% 12.45/6.96 = ( ring_1_of_int_real @ N2 ) ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5790_int__eq__iff__numeral,axiom,
% 12.45/6.96 ! [M: nat,V: num] :
% 12.45/6.96 ( ( ( semiri1314217659103216013at_int @ M )
% 12.45/6.96 = ( numeral_numeral_int @ V ) )
% 12.45/6.96 = ( M
% 12.45/6.96 = ( numeral_numeral_nat @ V ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5791_floor__numeral,axiom,
% 12.45/6.96 ! [V: num] :
% 12.45/6.96 ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
% 12.45/6.96 = ( numeral_numeral_int @ V ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5792_floor__numeral,axiom,
% 12.45/6.96 ! [V: num] :
% 12.45/6.96 ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
% 12.45/6.96 = ( numeral_numeral_int @ V ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5793_floor__one,axiom,
% 12.45/6.96 ( ( archim6058952711729229775r_real @ one_one_real )
% 12.45/6.96 = one_one_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5794_floor__one,axiom,
% 12.45/6.96 ( ( archim3151403230148437115or_rat @ one_one_rat )
% 12.45/6.96 = one_one_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5795_ceiling__numeral,axiom,
% 12.45/6.96 ! [V: num] :
% 12.45/6.96 ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
% 12.45/6.96 = ( numeral_numeral_int @ V ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5796_ceiling__numeral,axiom,
% 12.45/6.96 ! [V: num] :
% 12.45/6.96 ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
% 12.45/6.96 = ( numeral_numeral_int @ V ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5797_pochhammer__0,axiom,
% 12.45/6.96 ! [A: complex] :
% 12.45/6.96 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 12.45/6.96 = one_one_complex ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5798_pochhammer__0,axiom,
% 12.45/6.96 ! [A: real] :
% 12.45/6.96 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 12.45/6.96 = one_one_real ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5799_pochhammer__0,axiom,
% 12.45/6.96 ! [A: rat] :
% 12.45/6.96 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 12.45/6.96 = one_one_rat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5800_pochhammer__0,axiom,
% 12.45/6.96 ! [A: nat] :
% 12.45/6.96 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 12.45/6.96 = one_one_nat ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5801_pochhammer__0,axiom,
% 12.45/6.96 ! [A: int] :
% 12.45/6.96 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 12.45/6.96 = one_one_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5802_ceiling__one,axiom,
% 12.45/6.96 ( ( archim2889992004027027881ng_rat @ one_one_rat )
% 12.45/6.96 = one_one_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5803_ceiling__one,axiom,
% 12.45/6.96 ( ( archim7802044766580827645g_real @ one_one_real )
% 12.45/6.96 = one_one_int ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5804_negative__zless,axiom,
% 12.45/6.96 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5805_ceiling__add__of__int,axiom,
% 12.45/6.96 ! [X4: rat,Z: int] :
% 12.45/6.96 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) )
% 12.45/6.96 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5806_ceiling__add__of__int,axiom,
% 12.45/6.96 ! [X4: real,Z: int] :
% 12.45/6.96 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) )
% 12.45/6.96 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ Z ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5807_floor__diff__of__int,axiom,
% 12.45/6.96 ! [X4: real,Z: int] :
% 12.45/6.96 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) )
% 12.45/6.96 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X4 ) @ Z ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5808_floor__diff__of__int,axiom,
% 12.45/6.96 ! [X4: rat,Z: int] :
% 12.45/6.96 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) )
% 12.45/6.96 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5809_ceiling__diff__of__int,axiom,
% 12.45/6.96 ! [X4: rat,Z: int] :
% 12.45/6.96 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) )
% 12.45/6.96 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5810_ceiling__diff__of__int,axiom,
% 12.45/6.96 ! [X4: real,Z: int] :
% 12.45/6.96 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) )
% 12.45/6.96 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ Z ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5811_zero__le__floor,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5812_zero__le__floor,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_rat @ zero_zero_rat @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5813_floor__less__zero,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ zero_zero_int )
% 12.45/6.96 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5814_floor__less__zero,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ zero_zero_int )
% 12.45/6.96 = ( ord_less_rat @ X4 @ zero_zero_rat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5815_numeral__le__floor,axiom,
% 12.45/6.96 ! [V: num,X4: real] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5816_numeral__le__floor,axiom,
% 12.45/6.96 ! [V: num,X4: rat] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5817_zero__less__floor,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_real @ one_one_real @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5818_zero__less__floor,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5819_floor__le__zero,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ zero_zero_int )
% 12.45/6.96 = ( ord_less_real @ X4 @ one_one_real ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5820_floor__le__zero,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ zero_zero_int )
% 12.45/6.96 = ( ord_less_rat @ X4 @ one_one_rat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5821_ceiling__le__zero,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ zero_zero_int )
% 12.45/6.96 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5822_ceiling__le__zero,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ zero_zero_int )
% 12.45/6.96 = ( ord_less_eq_rat @ X4 @ zero_zero_rat ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5823_floor__less__numeral,axiom,
% 12.45/6.96 ! [X4: real,V: num] :
% 12.45/6.96 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) )
% 12.45/6.96 = ( ord_less_real @ X4 @ ( numeral_numeral_real @ V ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5824_floor__less__numeral,axiom,
% 12.45/6.96 ! [X4: rat,V: num] :
% 12.45/6.96 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
% 12.45/6.96 = ( ord_less_rat @ X4 @ ( numeral_numeral_rat @ V ) ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5825_zero__less__ceiling,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X4 ) )
% 12.45/6.96 = ( ord_less_rat @ zero_zero_rat @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5826_zero__less__ceiling,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X4 ) )
% 12.45/6.96 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5827_one__le__floor,axiom,
% 12.45/6.96 ! [X4: real] :
% 12.45/6.96 ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_real @ one_one_real @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5828_one__le__floor,axiom,
% 12.45/6.96 ! [X4: rat] :
% 12.45/6.96 ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.96 = ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ).
% 12.45/6.96
% 12.45/6.96 thf(fact_5829_ceiling__le__numeral,axiom,
% 12.45/6.97 ! [X4: real,V: num] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5830_ceiling__le__numeral,axiom,
% 12.45/6.97 ! [X4: rat,V: num] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
% 12.45/6.97 = ( ord_less_eq_rat @ X4 @ ( numeral_numeral_rat @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5831_floor__less__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
% 12.45/6.97 = ( ord_less_real @ X4 @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5832_floor__less__one,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
% 12.45/6.97 = ( ord_less_rat @ X4 @ one_one_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5833_ceiling__less__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5834_ceiling__less__one,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int )
% 12.45/6.97 = ( ord_less_eq_rat @ X4 @ zero_zero_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5835_one__le__ceiling,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_rat @ zero_zero_rat @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5836_one__le__ceiling,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X4 ) )
% 12.45/6.97 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5837_numeral__less__ceiling,axiom,
% 12.45/6.97 ! [V: num,X4: real] :
% 12.45/6.97 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X4 ) )
% 12.45/6.97 = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5838_numeral__less__ceiling,axiom,
% 12.45/6.97 ! [V: num,X4: rat] :
% 12.45/6.97 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5839_floor__neg__numeral,axiom,
% 12.45/6.97 ! [V: num] :
% 12.45/6.97 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.45/6.97 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5840_floor__neg__numeral,axiom,
% 12.45/6.97 ! [V: num] :
% 12.45/6.97 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.45/6.97 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5841_ceiling__le__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5842_ceiling__le__one,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int )
% 12.45/6.97 = ( ord_less_eq_rat @ X4 @ one_one_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5843_one__less__ceiling,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_rat @ one_one_rat @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5844_one__less__ceiling,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X4 ) )
% 12.45/6.97 = ( ord_less_real @ one_one_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5845_ceiling__add__numeral,axiom,
% 12.45/6.97 ! [X4: real,V: num] :
% 12.45/6.97 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
% 12.45/6.97 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5846_ceiling__add__numeral,axiom,
% 12.45/6.97 ! [X4: rat,V: num] :
% 12.45/6.97 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
% 12.45/6.97 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5847_floor__diff__numeral,axiom,
% 12.45/6.97 ! [X4: real,V: num] :
% 12.45/6.97 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
% 12.45/6.97 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5848_floor__diff__numeral,axiom,
% 12.45/6.97 ! [X4: rat,V: num] :
% 12.45/6.97 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
% 12.45/6.97 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5849_ceiling__neg__numeral,axiom,
% 12.45/6.97 ! [V: num] :
% 12.45/6.97 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.45/6.97 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5850_ceiling__neg__numeral,axiom,
% 12.45/6.97 ! [V: num] :
% 12.45/6.97 ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.45/6.97 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5851_ceiling__add__one,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) )
% 12.45/6.97 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5852_ceiling__add__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ one_one_real ) )
% 12.45/6.97 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5853_floor__diff__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X4 @ one_one_real ) )
% 12.45/6.97 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5854_floor__diff__one,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X4 @ one_one_rat ) )
% 12.45/6.97 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5855_ceiling__diff__numeral,axiom,
% 12.45/6.97 ! [X4: real,V: num] :
% 12.45/6.97 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
% 12.45/6.97 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5856_ceiling__diff__numeral,axiom,
% 12.45/6.97 ! [X4: rat,V: num] :
% 12.45/6.97 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
% 12.45/6.97 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5857_ceiling__diff__one,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X4 @ one_one_rat ) )
% 12.45/6.97 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5858_ceiling__diff__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X4 @ one_one_real ) )
% 12.45/6.97 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5859_floor__numeral__power,axiom,
% 12.45/6.97 ! [X4: num,N: nat] :
% 12.45/6.97 ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 12.45/6.97 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5860_floor__numeral__power,axiom,
% 12.45/6.97 ! [X4: num,N: nat] :
% 12.45/6.97 ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 12.45/6.97 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5861_ceiling__numeral__power,axiom,
% 12.45/6.97 ! [X4: num,N: nat] :
% 12.45/6.97 ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N ) )
% 12.45/6.97 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5862_ceiling__numeral__power,axiom,
% 12.45/6.97 ! [X4: num,N: nat] :
% 12.45/6.97 ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N ) )
% 12.45/6.97 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5863_floor__divide__eq__div__numeral,axiom,
% 12.45/6.97 ! [A: num,B2: num] :
% 12.45/6.97 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
% 12.45/6.97 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5864_ceiling__less__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ zero_zero_int )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5865_ceiling__less__zero,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ zero_zero_int )
% 12.45/6.97 = ( ord_less_eq_rat @ X4 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5866_zero__le__ceiling,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X4 ) )
% 12.45/6.97 = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5867_zero__le__ceiling,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5868_ceiling__divide__eq__div__numeral,axiom,
% 12.45/6.97 ! [A: num,B2: num] :
% 12.45/6.97 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
% 12.45/6.97 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5869_numeral__less__floor,axiom,
% 12.45/6.97 ! [V: num,X4: real] :
% 12.45/6.97 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5870_numeral__less__floor,axiom,
% 12.45/6.97 ! [V: num,X4: rat] :
% 12.45/6.97 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5871_floor__le__numeral,axiom,
% 12.45/6.97 ! [X4: real,V: num] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) )
% 12.45/6.97 = ( ord_less_real @ X4 @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5872_floor__le__numeral,axiom,
% 12.45/6.97 ! [X4: rat,V: num] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
% 12.45/6.97 = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5873_one__less__floor,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5874_one__less__floor,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5875_floor__le__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
% 12.45/6.97 = ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5876_floor__le__one,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
% 12.45/6.97 = ( ord_less_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5877_ceiling__less__numeral,axiom,
% 12.45/6.97 ! [X4: real,V: num] :
% 12.45/6.97 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5878_ceiling__less__numeral,axiom,
% 12.45/6.97 ! [X4: rat,V: num] :
% 12.45/6.97 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
% 12.45/6.97 = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5879_numeral__le__ceiling,axiom,
% 12.45/6.97 ! [V: num,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X4 ) )
% 12.45/6.97 = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5880_numeral__le__ceiling,axiom,
% 12.45/6.97 ! [V: num,X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5881_neg__numeral__le__floor,axiom,
% 12.45/6.97 ! [V: num,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5882_neg__numeral__le__floor,axiom,
% 12.45/6.97 ! [V: num,X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5883_floor__less__neg__numeral,axiom,
% 12.45/6.97 ! [X4: real,V: num] :
% 12.45/6.97 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.45/6.97 = ( ord_less_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5884_floor__less__neg__numeral,axiom,
% 12.45/6.97 ! [X4: rat,V: num] :
% 12.45/6.97 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.45/6.97 = ( ord_less_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5885_ceiling__le__neg__numeral,axiom,
% 12.45/6.97 ! [X4: real,V: num] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5886_ceiling__le__neg__numeral,axiom,
% 12.45/6.97 ! [X4: rat,V: num] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.45/6.97 = ( ord_less_eq_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5887_neg__numeral__less__ceiling,axiom,
% 12.45/6.97 ! [V: num,X4: real] :
% 12.45/6.97 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X4 ) )
% 12.45/6.97 = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5888_neg__numeral__less__ceiling,axiom,
% 12.45/6.97 ! [V: num,X4: rat] :
% 12.45/6.97 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5889_floor__one__divide__eq__div__numeral,axiom,
% 12.45/6.97 ! [B2: num] :
% 12.45/6.97 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) )
% 12.45/6.97 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5890_floor__minus__divide__eq__div__numeral,axiom,
% 12.45/6.97 ! [A: num,B2: num] :
% 12.45/6.97 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
% 12.45/6.97 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5891_ceiling__minus__divide__eq__div__numeral,axiom,
% 12.45/6.97 ! [A: num,B2: num] :
% 12.45/6.97 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
% 12.45/6.97 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5892_neg__numeral__less__floor,axiom,
% 12.45/6.97 ! [V: num,X4: real] :
% 12.45/6.97 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5893_neg__numeral__less__floor,axiom,
% 12.45/6.97 ! [V: num,X4: rat] :
% 12.45/6.97 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5894_floor__le__neg__numeral,axiom,
% 12.45/6.97 ! [X4: real,V: num] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.45/6.97 = ( ord_less_real @ X4 @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5895_floor__le__neg__numeral,axiom,
% 12.45/6.97 ! [X4: rat,V: num] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.45/6.97 = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5896_ceiling__less__neg__numeral,axiom,
% 12.45/6.97 ! [X4: real,V: num] :
% 12.45/6.97 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5897_ceiling__less__neg__numeral,axiom,
% 12.45/6.97 ! [X4: rat,V: num] :
% 12.45/6.97 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.45/6.97 = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5898_neg__numeral__le__ceiling,axiom,
% 12.45/6.97 ! [V: num,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X4 ) )
% 12.45/6.97 = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5899_neg__numeral__le__ceiling,axiom,
% 12.45/6.97 ! [V: num,X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5900_floor__minus__one__divide__eq__div__numeral,axiom,
% 12.45/6.97 ! [B2: num] :
% 12.45/6.97 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) ) )
% 12.45/6.97 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5901_int__if,axiom,
% 12.45/6.97 ! [P: $o,A: nat,B2: nat] :
% 12.45/6.97 ( ( P
% 12.45/6.97 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
% 12.45/6.97 = ( semiri1314217659103216013at_int @ A ) ) )
% 12.45/6.97 & ( ~ P
% 12.45/6.97 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
% 12.45/6.97 = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5902_nat__int__comparison_I1_J,axiom,
% 12.45/6.97 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 12.45/6.97 = ( ^ [A4: nat,B3: nat] :
% 12.45/6.97 ( ( semiri1314217659103216013at_int @ A4 )
% 12.45/6.97 = ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5903_int__diff__cases,axiom,
% 12.45/6.97 ! [Z: int] :
% 12.45/6.97 ~ ! [M4: nat,N4: nat] :
% 12.45/6.97 ( Z
% 12.45/6.97 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5904_ceiling__altdef,axiom,
% 12.45/6.97 ( archim7802044766580827645g_real
% 12.45/6.97 = ( ^ [X: real] :
% 12.45/6.97 ( if_int
% 12.45/6.97 @ ( X
% 12.45/6.97 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
% 12.45/6.97 @ ( archim6058952711729229775r_real @ X )
% 12.45/6.97 @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5905_ceiling__altdef,axiom,
% 12.45/6.97 ( archim2889992004027027881ng_rat
% 12.45/6.97 = ( ^ [X: rat] :
% 12.45/6.97 ( if_int
% 12.45/6.97 @ ( X
% 12.45/6.97 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
% 12.45/6.97 @ ( archim3151403230148437115or_rat @ X )
% 12.45/6.97 @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5906_ceiling__diff__floor__le__1,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim6058952711729229775r_real @ X4 ) ) @ one_one_int ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5907_ceiling__diff__floor__le__1,axiom,
% 12.45/6.97 ! [X4: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim3151403230148437115or_rat @ X4 ) ) @ one_one_int ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5908_floor__mono,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.97 => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5909_floor__mono,axiom,
% 12.45/6.97 ! [X4: rat,Y3: rat] :
% 12.45/6.97 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.97 => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5910_of__int__floor__le,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) @ X4 ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5911_of__int__floor__le,axiom,
% 12.45/6.97 ! [X4: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) @ X4 ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5912_floor__less__cancel,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) )
% 12.45/6.97 => ( ord_less_real @ X4 @ Y3 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5913_floor__less__cancel,axiom,
% 12.45/6.97 ! [X4: rat,Y3: rat] :
% 12.45/6.97 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) )
% 12.45/6.97 => ( ord_less_rat @ X4 @ Y3 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5914_not__bit__Suc__0__Suc,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5915_bit__Suc__0__iff,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.45/6.97 = ( N = zero_zero_nat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5916_ceiling__mono,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ Y3 @ X4 )
% 12.45/6.97 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y3 ) @ ( archim7802044766580827645g_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5917_ceiling__mono,axiom,
% 12.45/6.97 ! [Y3: rat,X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_rat @ Y3 @ X4 )
% 12.45/6.97 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5918_le__of__int__ceiling,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5919_le__of__int__ceiling,axiom,
% 12.45/6.97 ! [X4: rat] : ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5920_ceiling__less__cancel,axiom,
% 12.45/6.97 ! [X4: rat,Y3: rat] :
% 12.45/6.97 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
% 12.45/6.97 => ( ord_less_rat @ X4 @ Y3 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5921_ceiling__less__cancel,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim7802044766580827645g_real @ Y3 ) )
% 12.45/6.97 => ( ord_less_real @ X4 @ Y3 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5922_int__ops_I1_J,axiom,
% 12.45/6.97 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 12.45/6.97 = zero_zero_int ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5923_complex__mod__minus__le__complex__mod,axiom,
% 12.45/6.97 ! [X4: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5924_int__ops_I3_J,axiom,
% 12.45/6.97 ! [N: num] :
% 12.45/6.97 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 12.45/6.97 = ( numeral_numeral_int @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5925_pochhammer__pos,axiom,
% 12.45/6.97 ! [X4: real,N: nat] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X4 @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5926_pochhammer__pos,axiom,
% 12.45/6.97 ! [X4: rat,N: nat] :
% 12.45/6.97 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 12.45/6.97 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X4 @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5927_pochhammer__pos,axiom,
% 12.45/6.97 ! [X4: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 12.45/6.97 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X4 @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5928_pochhammer__pos,axiom,
% 12.45/6.97 ! [X4: int,N: nat] :
% 12.45/6.97 ( ( ord_less_int @ zero_zero_int @ X4 )
% 12.45/6.97 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X4 @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5929_nat__int__comparison_I2_J,axiom,
% 12.45/6.97 ( ord_less_nat
% 12.45/6.97 = ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5930_int__of__nat__induct,axiom,
% 12.45/6.97 ! [P: int > $o,Z: int] :
% 12.45/6.97 ( ! [N4: nat] : ( P @ ( semiri1314217659103216013at_int @ N4 ) )
% 12.45/6.97 => ( ! [N4: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
% 12.45/6.97 => ( P @ Z ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5931_int__cases,axiom,
% 12.45/6.97 ! [Z: int] :
% 12.45/6.97 ( ! [N4: nat] :
% 12.45/6.97 ( Z
% 12.45/6.97 != ( semiri1314217659103216013at_int @ N4 ) )
% 12.45/6.97 => ~ ! [N4: nat] :
% 12.45/6.97 ( Z
% 12.45/6.97 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5932_pi__not__less__zero,axiom,
% 12.45/6.97 ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5933_pi__gt__zero,axiom,
% 12.45/6.97 ord_less_real @ zero_zero_real @ pi ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5934_pi__ge__zero,axiom,
% 12.45/6.97 ord_less_eq_real @ zero_zero_real @ pi ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5935_nat__int__comparison_I3_J,axiom,
% 12.45/6.97 ( ord_less_eq_nat
% 12.45/6.97 = ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5936_zle__int,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.97 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5937_complex__mod__triangle__ineq2,axiom,
% 12.45/6.97 ! [B2: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B2 @ A ) ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5938_pochhammer__eq__0__mono,axiom,
% 12.45/6.97 ! [A: real,N: nat,M: nat] :
% 12.45/6.97 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.97 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 12.45/6.97 = zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5939_pochhammer__eq__0__mono,axiom,
% 12.45/6.97 ! [A: rat,N: nat,M: nat] :
% 12.45/6.97 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 12.45/6.97 = zero_zero_rat )
% 12.45/6.97 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.97 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 12.45/6.97 = zero_zero_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5940_pochhammer__neq__0__mono,axiom,
% 12.45/6.97 ! [A: real,M: nat,N: nat] :
% 12.45/6.97 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.97 => ( ( comm_s7457072308508201937r_real @ A @ N )
% 12.45/6.97 != zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5941_pochhammer__neq__0__mono,axiom,
% 12.45/6.97 ! [A: rat,M: nat,N: nat] :
% 12.45/6.97 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 12.45/6.97 != zero_zero_rat )
% 12.45/6.97 => ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.97 => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 12.45/6.97 != zero_zero_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5942_int__ops_I5_J,axiom,
% 12.45/6.97 ! [A: nat,B2: nat] :
% 12.45/6.97 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B2 ) )
% 12.45/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5943_int__plus,axiom,
% 12.45/6.97 ! [N: nat,M: nat] :
% 12.45/6.97 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 12.45/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5944_zadd__int__left,axiom,
% 12.45/6.97 ! [M: nat,N: nat,Z: int] :
% 12.45/6.97 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 12.45/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5945_int__ops_I2_J,axiom,
% 12.45/6.97 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 12.45/6.97 = one_one_int ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5946_int__ops_I7_J,axiom,
% 12.45/6.97 ! [A: nat,B2: nat] :
% 12.45/6.97 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B2 ) )
% 12.45/6.97 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5947_zle__iff__zadd,axiom,
% 12.45/6.97 ( ord_less_eq_int
% 12.45/6.97 = ( ^ [W2: int,Z5: int] :
% 12.45/6.97 ? [N2: nat] :
% 12.45/6.97 ( Z5
% 12.45/6.97 = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5948_not__int__zless__negative,axiom,
% 12.45/6.97 ! [N: nat,M: nat] :
% 12.45/6.97 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5949_zdiv__int,axiom,
% 12.45/6.97 ! [A: nat,B2: nat] :
% 12.45/6.97 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B2 ) )
% 12.45/6.97 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5950_zmod__int,axiom,
% 12.45/6.97 ! [A: nat,B2: nat] :
% 12.45/6.97 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B2 ) )
% 12.45/6.97 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5951_not__bit__Suc__0__numeral,axiom,
% 12.45/6.97 ! [N: num] :
% 12.45/6.97 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5952_floor__divide__of__nat__eq,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.45/6.97 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5953_floor__divide__of__nat__eq,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
% 12.45/6.97 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5954_le__floor__iff,axiom,
% 12.45/6.97 ! [Z: int,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5955_le__floor__iff,axiom,
% 12.45/6.97 ! [Z: int,X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5956_floor__less__iff,axiom,
% 12.45/6.97 ! [X4: real,Z: int] :
% 12.45/6.97 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
% 12.45/6.97 = ( ord_less_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5957_floor__less__iff,axiom,
% 12.45/6.97 ! [X4: rat,Z: int] :
% 12.45/6.97 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
% 12.45/6.97 = ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5958_le__floor__add,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5959_le__floor__add,axiom,
% 12.45/6.97 ! [X4: rat,Y3: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5960_int__add__floor,axiom,
% 12.45/6.97 ! [Z: int,X4: real] :
% 12.45/6.97 ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.97 = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5961_int__add__floor,axiom,
% 12.45/6.97 ! [Z: int,X4: rat] :
% 12.45/6.97 ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.97 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5962_floor__add__int,axiom,
% 12.45/6.97 ! [X4: real,Z: int] :
% 12.45/6.97 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
% 12.45/6.97 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5963_floor__add__int,axiom,
% 12.45/6.97 ! [X4: rat,Z: int] :
% 12.45/6.97 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
% 12.45/6.97 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5964_floor__divide__of__int__eq,axiom,
% 12.45/6.97 ! [K: int,L2: int] :
% 12.45/6.97 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L2 ) ) )
% 12.45/6.97 = ( divide_divide_int @ K @ L2 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5965_floor__divide__of__int__eq,axiom,
% 12.45/6.97 ! [K: int,L2: int] :
% 12.45/6.97 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L2 ) ) )
% 12.45/6.97 = ( divide_divide_int @ K @ L2 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5966_floor__power,axiom,
% 12.45/6.97 ! [X4: real,N: nat] :
% 12.45/6.97 ( ( X4
% 12.45/6.97 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) )
% 12.45/6.97 => ( ( archim6058952711729229775r_real @ ( power_power_real @ X4 @ N ) )
% 12.45/6.97 = ( power_power_int @ ( archim6058952711729229775r_real @ X4 ) @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5967_floor__power,axiom,
% 12.45/6.97 ! [X4: rat,N: nat] :
% 12.45/6.97 ( ( X4
% 12.45/6.97 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) )
% 12.45/6.97 => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X4 @ N ) )
% 12.45/6.97 = ( power_power_int @ ( archim3151403230148437115or_rat @ X4 ) @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5968_ceiling__le,axiom,
% 12.45/6.97 ! [X4: real,A: int] :
% 12.45/6.97 ( ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ A ) )
% 12.45/6.97 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5969_ceiling__le,axiom,
% 12.45/6.97 ! [X4: rat,A: int] :
% 12.45/6.97 ( ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ A ) )
% 12.45/6.97 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5970_ceiling__le__iff,axiom,
% 12.45/6.97 ! [X4: real,Z: int] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ Z )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5971_ceiling__le__iff,axiom,
% 12.45/6.97 ! [X4: rat,Z: int] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z )
% 12.45/6.97 = ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5972_less__ceiling__iff,axiom,
% 12.45/6.97 ! [Z: int,X4: rat] :
% 12.45/6.97 ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5973_less__ceiling__iff,axiom,
% 12.45/6.97 ! [Z: int,X4: real] :
% 12.45/6.97 ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X4 ) )
% 12.45/6.97 = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5974_ceiling__add__le,axiom,
% 12.45/6.97 ! [X4: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5975_ceiling__add__le,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim7802044766580827645g_real @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5976_pochhammer__nonneg,axiom,
% 12.45/6.97 ! [X4: real,N: nat] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X4 @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5977_pochhammer__nonneg,axiom,
% 12.45/6.97 ! [X4: rat,N: nat] :
% 12.45/6.97 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 12.45/6.97 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X4 @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5978_pochhammer__nonneg,axiom,
% 12.45/6.97 ! [X4: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ X4 )
% 12.45/6.97 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X4 @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5979_pochhammer__nonneg,axiom,
% 12.45/6.97 ! [X4: int,N: nat] :
% 12.45/6.97 ( ( ord_less_int @ zero_zero_int @ X4 )
% 12.45/6.97 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X4 @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5980_int__cases4,axiom,
% 12.45/6.97 ! [M: int] :
% 12.45/6.97 ( ! [N4: nat] :
% 12.45/6.97 ( M
% 12.45/6.97 != ( semiri1314217659103216013at_int @ N4 ) )
% 12.45/6.97 => ~ ! [N4: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 12.45/6.97 => ( M
% 12.45/6.97 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5981_pochhammer__0__left,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( ( N = zero_zero_nat )
% 12.45/6.97 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 12.45/6.97 = one_one_complex ) )
% 12.45/6.97 & ( ( N != zero_zero_nat )
% 12.45/6.97 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 12.45/6.97 = zero_zero_complex ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5982_pochhammer__0__left,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( ( N = zero_zero_nat )
% 12.45/6.97 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 12.45/6.97 = one_one_real ) )
% 12.45/6.97 & ( ( N != zero_zero_nat )
% 12.45/6.97 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 12.45/6.97 = zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5983_pochhammer__0__left,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( ( N = zero_zero_nat )
% 12.45/6.97 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 12.45/6.97 = one_one_rat ) )
% 12.45/6.97 & ( ( N != zero_zero_nat )
% 12.45/6.97 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 12.45/6.97 = zero_zero_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5984_pochhammer__0__left,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( ( N = zero_zero_nat )
% 12.45/6.97 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 12.45/6.97 = one_one_nat ) )
% 12.45/6.97 & ( ( N != zero_zero_nat )
% 12.45/6.97 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 12.45/6.97 = zero_zero_nat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5985_pochhammer__0__left,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( ( N = zero_zero_nat )
% 12.45/6.97 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 12.45/6.97 = one_one_int ) )
% 12.45/6.97 & ( ( N != zero_zero_nat )
% 12.45/6.97 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 12.45/6.97 = zero_zero_int ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5986_int__Suc,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 12.45/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5987_int__ops_I4_J,axiom,
% 12.45/6.97 ! [A: nat] :
% 12.45/6.97 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 12.45/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5988_zless__iff__Suc__zadd,axiom,
% 12.45/6.97 ( ord_less_int
% 12.45/6.97 = ( ^ [W2: int,Z5: int] :
% 12.45/6.97 ? [N2: nat] :
% 12.45/6.97 ( Z5
% 12.45/6.97 = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5989_norm__exp,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X4 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5990_norm__exp,axiom,
% 12.45/6.97 ! [X4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X4 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5991_one__add__floor,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
% 12.45/6.97 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5992_one__add__floor,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
% 12.45/6.97 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5993_of__int__ceiling__le__add__one,axiom,
% 12.45/6.97 ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5994_of__int__ceiling__le__add__one,axiom,
% 12.45/6.97 ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5995_of__int__ceiling__diff__one__le,axiom,
% 12.45/6.97 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5996_of__int__ceiling__diff__one__le,axiom,
% 12.45/6.97 ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5997_floor__eq,axiom,
% 12.45/6.97 ! [N: int,X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 12.45/6.97 => ( ( archim6058952711729229775r_real @ X4 )
% 12.45/6.97 = N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5998_real__of__int__floor__add__one__gt,axiom,
% 12.45/6.97 ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_5999_real__of__int__floor__add__one__ge,axiom,
% 12.45/6.97 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6000_real__of__int__floor__gt__diff__one,axiom,
% 12.45/6.97 ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6001_real__of__int__floor__ge__diff__one,axiom,
% 12.45/6.97 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6002_ceiling__divide__eq__div,axiom,
% 12.45/6.97 ! [A: int,B2: int] :
% 12.45/6.97 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B2 ) ) )
% 12.45/6.97 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6003_ceiling__divide__eq__div,axiom,
% 12.45/6.97 ! [A: int,B2: int] :
% 12.45/6.97 ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B2 ) ) )
% 12.45/6.97 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6004_zero__less__imp__eq__int,axiom,
% 12.45/6.97 ! [K: int] :
% 12.45/6.97 ( ( ord_less_int @ zero_zero_int @ K )
% 12.45/6.97 => ? [N4: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 12.45/6.97 & ( K
% 12.45/6.97 = ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6005_pos__int__cases,axiom,
% 12.45/6.97 ! [K: int] :
% 12.45/6.97 ( ( ord_less_int @ zero_zero_int @ K )
% 12.45/6.97 => ~ ! [N4: nat] :
% 12.45/6.97 ( ( K
% 12.45/6.97 = ( semiri1314217659103216013at_int @ N4 ) )
% 12.45/6.97 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6006_int__cases3,axiom,
% 12.45/6.97 ! [K: int] :
% 12.45/6.97 ( ( K != zero_zero_int )
% 12.45/6.97 => ( ! [N4: nat] :
% 12.45/6.97 ( ( K
% 12.45/6.97 = ( semiri1314217659103216013at_int @ N4 ) )
% 12.45/6.97 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
% 12.45/6.97 => ~ ! [N4: nat] :
% 12.45/6.97 ( ( K
% 12.45/6.97 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 12.45/6.97 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6007_pi__less__4,axiom,
% 12.45/6.97 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6008_pi__ge__two,axiom,
% 12.45/6.97 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6009_zmult__zless__mono2__lemma,axiom,
% 12.45/6.97 ! [I: int,J: int,K: nat] :
% 12.45/6.97 ( ( ord_less_int @ I @ J )
% 12.45/6.97 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.97 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6010_not__zle__0__negative,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6011_negative__zless__0,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6012_negD,axiom,
% 12.45/6.97 ! [X4: int] :
% 12.45/6.97 ( ( ord_less_int @ X4 @ zero_zero_int )
% 12.45/6.97 => ? [N4: nat] :
% 12.45/6.97 ( X4
% 12.45/6.97 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6013_pi__half__neq__two,axiom,
% 12.45/6.97 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.45/6.97 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6014_pochhammer__rec,axiom,
% 12.45/6.97 ! [A: complex,N: nat] :
% 12.45/6.97 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6015_pochhammer__rec,axiom,
% 12.45/6.97 ! [A: real,N: nat] :
% 12.45/6.97 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6016_pochhammer__rec,axiom,
% 12.45/6.97 ! [A: rat,N: nat] :
% 12.45/6.97 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6017_pochhammer__rec,axiom,
% 12.45/6.97 ! [A: nat,N: nat] :
% 12.45/6.97 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6018_pochhammer__rec,axiom,
% 12.45/6.97 ! [A: int,N: nat] :
% 12.45/6.97 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6019_pochhammer__Suc,axiom,
% 12.45/6.97 ! [A: complex,N: nat] :
% 12.45/6.97 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6020_pochhammer__Suc,axiom,
% 12.45/6.97 ! [A: real,N: nat] :
% 12.45/6.97 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6021_pochhammer__Suc,axiom,
% 12.45/6.97 ! [A: rat,N: nat] :
% 12.45/6.97 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6022_pochhammer__Suc,axiom,
% 12.45/6.97 ! [A: nat,N: nat] :
% 12.45/6.97 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6023_pochhammer__Suc,axiom,
% 12.45/6.97 ! [A: int,N: nat] :
% 12.45/6.97 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6024_pochhammer__rec_H,axiom,
% 12.45/6.97 ! [Z: complex,N: nat] :
% 12.45/6.97 ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6025_pochhammer__rec_H,axiom,
% 12.45/6.97 ! [Z: real,N: nat] :
% 12.45/6.97 ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6026_pochhammer__rec_H,axiom,
% 12.45/6.97 ! [Z: rat,N: nat] :
% 12.45/6.97 ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6027_pochhammer__rec_H,axiom,
% 12.45/6.97 ! [Z: nat,N: nat] :
% 12.45/6.97 ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6028_pochhammer__rec_H,axiom,
% 12.45/6.97 ! [Z: int,N: nat] :
% 12.45/6.97 ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6029_pochhammer__of__nat__eq__0__lemma,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ord_less_nat @ N @ K )
% 12.45/6.97 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 12.45/6.97 = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6030_pochhammer__of__nat__eq__0__lemma,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ord_less_nat @ N @ K )
% 12.45/6.97 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 12.45/6.97 = zero_zero_complex ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6031_pochhammer__of__nat__eq__0__lemma,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ord_less_nat @ N @ K )
% 12.45/6.97 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 12.45/6.97 = zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6032_pochhammer__of__nat__eq__0__lemma,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ord_less_nat @ N @ K )
% 12.45/6.97 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 12.45/6.97 = zero_zero_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6033_pochhammer__of__nat__eq__0__lemma,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ord_less_nat @ N @ K )
% 12.45/6.97 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 12.45/6.97 = zero_zero_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6034_pochhammer__of__nat__eq__0__iff,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 12.45/6.97 = zero_z3403309356797280102nteger )
% 12.45/6.97 = ( ord_less_nat @ N @ K ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6035_pochhammer__of__nat__eq__0__iff,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 12.45/6.97 = zero_zero_complex )
% 12.45/6.97 = ( ord_less_nat @ N @ K ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6036_pochhammer__of__nat__eq__0__iff,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 = ( ord_less_nat @ N @ K ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6037_pochhammer__of__nat__eq__0__iff,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 12.45/6.97 = zero_zero_rat )
% 12.45/6.97 = ( ord_less_nat @ N @ K ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6038_pochhammer__of__nat__eq__0__iff,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 12.45/6.97 = zero_zero_int )
% 12.45/6.97 = ( ord_less_nat @ N @ K ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6039_pochhammer__eq__0__iff,axiom,
% 12.45/6.97 ! [A: complex,N: nat] :
% 12.45/6.97 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 12.45/6.97 = zero_zero_complex )
% 12.45/6.97 = ( ? [K3: nat] :
% 12.45/6.97 ( ( ord_less_nat @ K3 @ N )
% 12.45/6.97 & ( A
% 12.45/6.97 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6040_pochhammer__eq__0__iff,axiom,
% 12.45/6.97 ! [A: real,N: nat] :
% 12.45/6.97 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 = ( ? [K3: nat] :
% 12.45/6.97 ( ( ord_less_nat @ K3 @ N )
% 12.45/6.97 & ( A
% 12.45/6.97 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6041_pochhammer__eq__0__iff,axiom,
% 12.45/6.97 ! [A: rat,N: nat] :
% 12.45/6.97 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 12.45/6.97 = zero_zero_rat )
% 12.45/6.97 = ( ? [K3: nat] :
% 12.45/6.97 ( ( ord_less_nat @ K3 @ N )
% 12.45/6.97 & ( A
% 12.45/6.97 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6042_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.97 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 12.45/6.97 != zero_z3403309356797280102nteger ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6043_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.97 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 12.45/6.97 != zero_zero_complex ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6044_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.97 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 12.45/6.97 != zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6045_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.97 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 12.45/6.97 != zero_zero_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6046_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.97 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 12.45/6.97 != zero_zero_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6047_int__ops_I6_J,axiom,
% 12.45/6.97 ! [A: nat,B2: nat] :
% 12.45/6.97 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
% 12.45/6.97 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
% 12.45/6.97 = zero_zero_int ) )
% 12.45/6.97 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
% 12.45/6.97 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
% 12.45/6.97 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6048_pochhammer__product_H,axiom,
% 12.45/6.97 ! [Z: complex,N: nat,M: nat] :
% 12.45/6.97 ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
% 12.45/6.97 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6049_pochhammer__product_H,axiom,
% 12.45/6.97 ! [Z: real,N: nat,M: nat] :
% 12.45/6.97 ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
% 12.45/6.97 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6050_pochhammer__product_H,axiom,
% 12.45/6.97 ! [Z: rat,N: nat,M: nat] :
% 12.45/6.97 ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
% 12.45/6.97 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6051_pochhammer__product_H,axiom,
% 12.45/6.97 ! [Z: nat,N: nat,M: nat] :
% 12.45/6.97 ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
% 12.45/6.97 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6052_pochhammer__product_H,axiom,
% 12.45/6.97 ! [Z: int,N: nat,M: nat] :
% 12.45/6.97 ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
% 12.45/6.97 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6053_floor__split,axiom,
% 12.45/6.97 ! [P: int > $o,T: real] :
% 12.45/6.97 ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 12.45/6.97 = ( ! [I2: int] :
% 12.45/6.97 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I2 ) @ T )
% 12.45/6.97 & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) ) )
% 12.45/6.97 => ( P @ I2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6054_floor__split,axiom,
% 12.45/6.97 ! [P: int > $o,T: rat] :
% 12.45/6.97 ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 12.45/6.97 = ( ! [I2: int] :
% 12.45/6.97 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I2 ) @ T )
% 12.45/6.97 & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) ) )
% 12.45/6.97 => ( P @ I2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6055_floor__eq__iff,axiom,
% 12.45/6.97 ! [X4: real,A: int] :
% 12.45/6.97 ( ( ( archim6058952711729229775r_real @ X4 )
% 12.45/6.97 = A )
% 12.45/6.97 = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X4 )
% 12.45/6.97 & ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6056_floor__eq__iff,axiom,
% 12.45/6.97 ! [X4: rat,A: int] :
% 12.45/6.97 ( ( ( archim3151403230148437115or_rat @ X4 )
% 12.45/6.97 = A )
% 12.45/6.97 = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X4 )
% 12.45/6.97 & ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6057_floor__unique,axiom,
% 12.45/6.97 ! [Z: int,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
% 12.45/6.97 => ( ( archim6058952711729229775r_real @ X4 )
% 12.45/6.97 = Z ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6058_floor__unique,axiom,
% 12.45/6.97 ! [Z: int,X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X4 )
% 12.45/6.97 => ( ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
% 12.45/6.97 => ( ( archim3151403230148437115or_rat @ X4 )
% 12.45/6.97 = Z ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6059_le__mult__floor,axiom,
% 12.45/6.97 ! [A: real,B2: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.97 => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6060_le__mult__floor,axiom,
% 12.45/6.97 ! [A: rat,B2: rat] :
% 12.45/6.97 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.97 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.97 => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6061_less__floor__iff,axiom,
% 12.45/6.97 ! [Z: int,X4: real] :
% 12.45/6.97 ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6062_less__floor__iff,axiom,
% 12.45/6.97 ! [Z: int,X4: rat] :
% 12.45/6.97 ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6063_floor__le__iff,axiom,
% 12.45/6.97 ! [X4: real,Z: int] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
% 12.45/6.97 = ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6064_floor__le__iff,axiom,
% 12.45/6.97 ! [X4: rat,Z: int] :
% 12.45/6.97 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
% 12.45/6.97 = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6065_binomial__mono,axiom,
% 12.45/6.97 ! [K: nat,K6: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ K6 )
% 12.45/6.97 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 12.45/6.97 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6066_binomial__maximum_H,axiom,
% 12.45/6.97 ! [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 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6067_binomial__maximum,axiom,
% 12.45/6.97 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6068_binomial__antimono,axiom,
% 12.45/6.97 ! [K: nat,K6: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ K6 )
% 12.45/6.97 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 12.45/6.97 => ( ( ord_less_eq_nat @ K6 @ N )
% 12.45/6.97 => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6069_floor__correct,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) @ X4 )
% 12.45/6.97 & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6070_floor__correct,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) @ X4 )
% 12.45/6.97 & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6071_ceiling__split,axiom,
% 12.45/6.97 ! [P: int > $o,T: real] :
% 12.45/6.97 ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 12.45/6.97 = ( ! [I2: int] :
% 12.45/6.97 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
% 12.45/6.97 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
% 12.45/6.97 => ( P @ I2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6072_ceiling__split,axiom,
% 12.45/6.97 ! [P: int > $o,T: rat] :
% 12.45/6.97 ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 12.45/6.97 = ( ! [I2: int] :
% 12.45/6.97 ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
% 12.45/6.97 & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
% 12.45/6.97 => ( P @ I2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6073_ceiling__eq__iff,axiom,
% 12.45/6.97 ! [X4: real,A: int] :
% 12.45/6.97 ( ( ( archim7802044766580827645g_real @ X4 )
% 12.45/6.97 = A )
% 12.45/6.97 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X4 )
% 12.45/6.97 & ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6074_ceiling__eq__iff,axiom,
% 12.45/6.97 ! [X4: rat,A: int] :
% 12.45/6.97 ( ( ( archim2889992004027027881ng_rat @ X4 )
% 12.45/6.97 = A )
% 12.45/6.97 = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X4 )
% 12.45/6.97 & ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6075_ceiling__unique,axiom,
% 12.45/6.97 ! [Z: int,X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z ) )
% 12.45/6.97 => ( ( archim7802044766580827645g_real @ X4 )
% 12.45/6.97 = Z ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6076_ceiling__unique,axiom,
% 12.45/6.97 ! [Z: int,X4: rat] :
% 12.45/6.97 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z ) )
% 12.45/6.97 => ( ( archim2889992004027027881ng_rat @ X4 )
% 12.45/6.97 = Z ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6077_ceiling__correct,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) @ one_one_real ) @ X4 )
% 12.45/6.97 & ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6078_ceiling__correct,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) @ one_one_rat ) @ X4 )
% 12.45/6.97 & ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6079_mult__ceiling__le,axiom,
% 12.45/6.97 ! [A: real,B2: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 12.45/6.97 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6080_mult__ceiling__le,axiom,
% 12.45/6.97 ! [A: rat,B2: rat] :
% 12.45/6.97 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.97 => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 12.45/6.97 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6081_floor__log__nat__eq__if,axiom,
% 12.45/6.97 ! [B2: nat,N: nat,K: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 12.45/6.97 => ( ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 12.45/6.97 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 12.45/6.97 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 12.45/6.97 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6082_ceiling__less__iff,axiom,
% 12.45/6.97 ! [X4: real,Z: int] :
% 12.45/6.97 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ Z )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6083_ceiling__less__iff,axiom,
% 12.45/6.97 ! [X4: rat,Z: int] :
% 12.45/6.97 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z )
% 12.45/6.97 = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6084_le__ceiling__iff,axiom,
% 12.45/6.97 ! [Z: int,X4: rat] :
% 12.45/6.97 ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X4 ) )
% 12.45/6.97 = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6085_le__ceiling__iff,axiom,
% 12.45/6.97 ! [Z: int,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X4 ) )
% 12.45/6.97 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6086_floor__eq2,axiom,
% 12.45/6.97 ! [N: int,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 12.45/6.97 => ( ( archim6058952711729229775r_real @ X4 )
% 12.45/6.97 = N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6087_floor__divide__real__eq__div,axiom,
% 12.45/6.97 ! [B2: int,A: real] :
% 12.45/6.97 ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/6.97 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B2 ) ) )
% 12.45/6.97 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6088_neg__int__cases,axiom,
% 12.45/6.97 ! [K: int] :
% 12.45/6.97 ( ( ord_less_int @ K @ zero_zero_int )
% 12.45/6.97 => ~ ! [N4: nat] :
% 12.45/6.97 ( ( K
% 12.45/6.97 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 12.45/6.97 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6089_pi__half__neq__zero,axiom,
% 12.45/6.97 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.45/6.97 != zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6090_pi__half__less__two,axiom,
% 12.45/6.97 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6091_pi__half__le__two,axiom,
% 12.45/6.97 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6092_zdiff__int__split,axiom,
% 12.45/6.97 ! [P: int > $o,X4: nat,Y3: nat] :
% 12.45/6.97 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X4 @ Y3 ) ) )
% 12.45/6.97 = ( ( ( ord_less_eq_nat @ Y3 @ X4 )
% 12.45/6.97 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
% 12.45/6.97 & ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.97 => ( P @ zero_zero_int ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6093_floor__log__nat__eq__powr__iff,axiom,
% 12.45/6.97 ! [B2: nat,K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 12.45/6.97 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.97 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 12.45/6.97 = ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.97 = ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 12.45/6.97 & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6094_pochhammer__product,axiom,
% 12.45/6.97 ! [M: nat,N: nat,Z: complex] :
% 12.45/6.97 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( ( comm_s2602460028002588243omplex @ Z @ N )
% 12.45/6.97 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6095_pochhammer__product,axiom,
% 12.45/6.97 ! [M: nat,N: nat,Z: real] :
% 12.45/6.97 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( ( comm_s7457072308508201937r_real @ Z @ N )
% 12.45/6.97 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6096_pochhammer__product,axiom,
% 12.45/6.97 ! [M: nat,N: nat,Z: rat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( ( comm_s4028243227959126397er_rat @ Z @ N )
% 12.45/6.97 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6097_pochhammer__product,axiom,
% 12.45/6.97 ! [M: nat,N: nat,Z: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( ( comm_s4663373288045622133er_nat @ Z @ N )
% 12.45/6.97 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6098_pochhammer__product,axiom,
% 12.45/6.97 ! [M: nat,N: nat,Z: int] :
% 12.45/6.97 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( ( comm_s4660882817536571857er_int @ Z @ N )
% 12.45/6.97 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6099_floor__divide__lower,axiom,
% 12.45/6.97 ! [Q2: real,P2: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 12.45/6.97 => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ Q2 ) @ P2 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6100_floor__divide__lower,axiom,
% 12.45/6.97 ! [Q2: rat,P2: rat] :
% 12.45/6.97 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 12.45/6.97 => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ Q2 ) @ P2 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6101_binomial__less__binomial__Suc,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6102_binomial__strict__mono,axiom,
% 12.45/6.97 ! [K: nat,K6: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_nat @ K @ K6 )
% 12.45/6.97 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 12.45/6.97 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6103_binomial__strict__antimono,axiom,
% 12.45/6.97 ! [K: nat,K6: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_nat @ K @ K6 )
% 12.45/6.97 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 12.45/6.97 => ( ( ord_less_eq_nat @ K6 @ N )
% 12.45/6.97 => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6104_central__binomial__odd,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.97 => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.97 = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6105_ceiling__divide__upper,axiom,
% 12.45/6.97 ! [Q2: real,P2: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 12.45/6.97 => ( ord_less_eq_real @ P2 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6106_ceiling__divide__upper,axiom,
% 12.45/6.97 ! [Q2: rat,P2: rat] :
% 12.45/6.97 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 12.45/6.97 => ( ord_less_eq_rat @ P2 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6107_bit__nat__def,axiom,
% 12.45/6.97 ( bit_se1148574629649215175it_nat
% 12.45/6.97 = ( ^ [M6: nat,N2: nat] :
% 12.45/6.97 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6108_ceiling__log__nat__eq__if,axiom,
% 12.45/6.97 ! [B2: nat,N: nat,K: nat] :
% 12.45/6.97 ( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 12.45/6.97 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 12.45/6.97 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 12.45/6.97 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 12.45/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6109_pi__half__gt__zero,axiom,
% 12.45/6.97 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6110_pi__half__ge__zero,axiom,
% 12.45/6.97 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6111_pochhammer__absorb__comp,axiom,
% 12.45/6.97 ! [R2: code_integer,K: nat] :
% 12.45/6.97 ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
% 12.45/6.97 = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6112_pochhammer__absorb__comp,axiom,
% 12.45/6.97 ! [R2: complex,K: nat] :
% 12.45/6.97 ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
% 12.45/6.97 = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6113_pochhammer__absorb__comp,axiom,
% 12.45/6.97 ! [R2: real,K: nat] :
% 12.45/6.97 ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
% 12.45/6.97 = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6114_pochhammer__absorb__comp,axiom,
% 12.45/6.97 ! [R2: rat,K: nat] :
% 12.45/6.97 ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
% 12.45/6.97 = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6115_pochhammer__absorb__comp,axiom,
% 12.45/6.97 ! [R2: int,K: nat] :
% 12.45/6.97 ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
% 12.45/6.97 = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6116_m2pi__less__pi,axiom,
% 12.45/6.97 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6117_ceiling__log__nat__eq__powr__iff,axiom,
% 12.45/6.97 ! [B2: nat,K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 12.45/6.97 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.97 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 12.45/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 12.45/6.97 = ( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 12.45/6.97 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6118_floor__divide__upper,axiom,
% 12.45/6.97 ! [Q2: real,P2: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 12.45/6.97 => ( ord_less_real @ P2 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6119_floor__divide__upper,axiom,
% 12.45/6.97 ! [Q2: rat,P2: rat] :
% 12.45/6.97 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 12.45/6.97 => ( ord_less_rat @ P2 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6120_arctan__ubound,axiom,
% 12.45/6.97 ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6121_arctan__one,axiom,
% 12.45/6.97 ( ( arctan @ one_one_real )
% 12.45/6.97 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6122_round__def,axiom,
% 12.45/6.97 ( archim8280529875227126926d_real
% 12.45/6.97 = ( ^ [X: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6123_round__def,axiom,
% 12.45/6.97 ( archim7778729529865785530nd_rat
% 12.45/6.97 = ( ^ [X: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6124_ceiling__divide__lower,axiom,
% 12.45/6.97 ! [Q2: real,P2: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 12.45/6.97 => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P2 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6125_ceiling__divide__lower,axiom,
% 12.45/6.97 ! [Q2: rat,P2: rat] :
% 12.45/6.97 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 12.45/6.97 => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P2 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6126_ceiling__eq,axiom,
% 12.45/6.97 ! [N: int,X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 12.45/6.97 => ( ( archim7802044766580827645g_real @ X4 )
% 12.45/6.97 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6127_ceiling__eq,axiom,
% 12.45/6.97 ! [N: int,X4: rat] :
% 12.45/6.97 ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 12.45/6.97 => ( ( archim2889992004027027881ng_rat @ X4 )
% 12.45/6.97 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6128_exp__bound__half,axiom,
% 12.45/6.97 ! [Z: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6129_exp__bound__half,axiom,
% 12.45/6.97 ! [Z: complex] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6130_minus__pi__half__less__zero,axiom,
% 12.45/6.97 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6131_pochhammer__minus,axiom,
% 12.45/6.97 ! [B2: code_integer,K: nat] :
% 12.45/6.97 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K )
% 12.45/6.97 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6132_pochhammer__minus,axiom,
% 12.45/6.97 ! [B2: complex,K: nat] :
% 12.45/6.97 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K )
% 12.45/6.97 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6133_pochhammer__minus,axiom,
% 12.45/6.97 ! [B2: real,K: nat] :
% 12.45/6.97 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K )
% 12.45/6.97 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6134_pochhammer__minus,axiom,
% 12.45/6.97 ! [B2: rat,K: nat] :
% 12.45/6.97 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K )
% 12.45/6.97 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6135_pochhammer__minus,axiom,
% 12.45/6.97 ! [B2: int,K: nat] :
% 12.45/6.97 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K )
% 12.45/6.97 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6136_pochhammer__minus_H,axiom,
% 12.45/6.97 ! [B2: code_integer,K: nat] :
% 12.45/6.97 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 12.45/6.97 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6137_pochhammer__minus_H,axiom,
% 12.45/6.97 ! [B2: complex,K: nat] :
% 12.45/6.97 ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 12.45/6.97 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6138_pochhammer__minus_H,axiom,
% 12.45/6.97 ! [B2: real,K: nat] :
% 12.45/6.97 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 12.45/6.97 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6139_pochhammer__minus_H,axiom,
% 12.45/6.97 ! [B2: rat,K: nat] :
% 12.45/6.97 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 12.45/6.97 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6140_pochhammer__minus_H,axiom,
% 12.45/6.97 ! [B2: int,K: nat] :
% 12.45/6.97 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 12.45/6.97 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6141_arctan__lbound,axiom,
% 12.45/6.97 ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6142_arctan__bounded,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 12.45/6.97 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6143_exp__bound__lemma,axiom,
% 12.45/6.97 ! [Z: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( 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 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6144_exp__bound__lemma,axiom,
% 12.45/6.97 ! [Z: complex] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( 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 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6145_machin__Euler,axiom,
% 12.45/6.97 ( ( 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 ) ) ) ) ) ) ) ) ) ) )
% 12.45/6.97 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6146_norm__divide__numeral,axiom,
% 12.45/6.97 ! [A: real,W: num] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.97 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6147_norm__divide__numeral,axiom,
% 12.45/6.97 ! [A: complex,W: num] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 12.45/6.97 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6148_norm__mult__numeral1,axiom,
% 12.45/6.97 ! [W: num,A: real] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 12.45/6.97 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6149_norm__mult__numeral1,axiom,
% 12.45/6.97 ! [W: num,A: complex] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 12.45/6.97 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6150_norm__mult__numeral2,axiom,
% 12.45/6.97 ! [A: real,W: num] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.97 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6151_norm__mult__numeral2,axiom,
% 12.45/6.97 ! [A: complex,W: num] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 12.45/6.97 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6152_zero__less__binomial__iff,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 12.45/6.97 = ( ord_less_eq_nat @ K @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6153_norm__neg__numeral,axiom,
% 12.45/6.97 ! [W: num] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.97 = ( numeral_numeral_real @ W ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6154_norm__neg__numeral,axiom,
% 12.45/6.97 ! [W: num] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.45/6.97 = ( numeral_numeral_real @ W ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6155_norm__le__zero__iff,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X4 ) @ zero_zero_real )
% 12.45/6.97 = ( X4 = zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6156_norm__le__zero__iff,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ zero_zero_real )
% 12.45/6.97 = ( X4 = zero_zero_complex ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6157_binomial__Suc__n,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( binomial @ ( suc @ N ) @ N )
% 12.45/6.97 = ( suc @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6158_binomial__n__n,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( binomial @ N @ N )
% 12.45/6.97 = one_one_nat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6159_norm__one,axiom,
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ one_one_real )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6160_norm__one,axiom,
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ one_one_complex )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6161_norm__numeral,axiom,
% 12.45/6.97 ! [W: num] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 12.45/6.97 = ( numeral_numeral_real @ W ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6162_norm__numeral,axiom,
% 12.45/6.97 ! [W: num] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.45/6.97 = ( numeral_numeral_real @ W ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6163_binomial__0__Suc,axiom,
% 12.45/6.97 ! [K: nat] :
% 12.45/6.97 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 12.45/6.97 = zero_zero_nat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6164_binomial__1,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 12.45/6.97 = N ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6165_binomial__eq__0__iff,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ( binomial @ N @ K )
% 12.45/6.97 = zero_zero_nat )
% 12.45/6.97 = ( ord_less_nat @ N @ K ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6166_binomial__Suc__Suc,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 12.45/6.97 = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6167_binomial__n__0,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( binomial @ N @ zero_zero_nat )
% 12.45/6.97 = one_one_nat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6168_zero__less__norm__iff,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X4 ) )
% 12.45/6.97 = ( X4 != zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6169_zero__less__norm__iff,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X4 ) )
% 12.45/6.97 = ( X4 != zero_zero_complex ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6170_norm__minus__commute,axiom,
% 12.45/6.97 ! [A: real,B2: real] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) )
% 12.45/6.97 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6171_norm__minus__commute,axiom,
% 12.45/6.97 ! [A: complex,B2: complex] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) )
% 12.45/6.97 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6172_choose__one,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( binomial @ N @ one_one_nat )
% 12.45/6.97 = N ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6173_norm__not__less__zero,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6174_norm__ge__zero,axiom,
% 12.45/6.97 ! [X4: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6175_norm__mult,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( times_times_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6176_norm__mult,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X4 @ Y3 ) )
% 12.45/6.97 = ( times_times_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6177_norm__divide,axiom,
% 12.45/6.97 ! [A: real,B2: real] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.97 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6178_norm__divide,axiom,
% 12.45/6.97 ! [A: complex,B2: complex] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.45/6.97 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6179_norm__power,axiom,
% 12.45/6.97 ! [X4: real,N: nat] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( power_power_real @ X4 @ N ) )
% 12.45/6.97 = ( power_power_real @ ( real_V7735802525324610683m_real @ X4 ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6180_norm__power,axiom,
% 12.45/6.97 ! [X4: complex,N: nat] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X4 @ N ) )
% 12.45/6.97 = ( power_power_real @ ( real_V1022390504157884413omplex @ X4 ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6181_binomial__eq__0,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ord_less_nat @ N @ K )
% 12.45/6.97 => ( ( binomial @ N @ K )
% 12.45/6.97 = zero_zero_nat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6182_Suc__times__binomial,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 12.45/6.97 = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6183_Suc__times__binomial__eq,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 12.45/6.97 = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6184_binomial__symmetric,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.97 => ( ( binomial @ N @ K )
% 12.45/6.97 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6185_choose__mult__lemma,axiom,
% 12.45/6.97 ! [M: nat,R2: nat,K: nat] :
% 12.45/6.97 ( ( 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 ) )
% 12.45/6.97 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6186_binomial__le__pow,axiom,
% 12.45/6.97 ! [R2: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ R2 @ N )
% 12.45/6.97 => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6187_norm__uminus__minus,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) )
% 12.45/6.97 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6188_norm__uminus__minus,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ Y3 ) )
% 12.45/6.97 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6189_nonzero__norm__divide,axiom,
% 12.45/6.97 ! [B2: real,A: real] :
% 12.45/6.97 ( ( B2 != zero_zero_real )
% 12.45/6.97 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.97 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6190_nonzero__norm__divide,axiom,
% 12.45/6.97 ! [B2: complex,A: complex] :
% 12.45/6.97 ( ( B2 != zero_zero_complex )
% 12.45/6.97 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.45/6.97 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6191_power__eq__imp__eq__norm,axiom,
% 12.45/6.97 ! [W: real,N: nat,Z: real] :
% 12.45/6.97 ( ( ( power_power_real @ W @ N )
% 12.45/6.97 = ( power_power_real @ Z @ N ) )
% 12.45/6.97 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.97 => ( ( real_V7735802525324610683m_real @ W )
% 12.45/6.97 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6192_power__eq__imp__eq__norm,axiom,
% 12.45/6.97 ! [W: complex,N: nat,Z: complex] :
% 12.45/6.97 ( ( ( power_power_complex @ W @ N )
% 12.45/6.97 = ( power_power_complex @ Z @ N ) )
% 12.45/6.97 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.97 => ( ( real_V1022390504157884413omplex @ W )
% 12.45/6.97 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6193_norm__mult__less,axiom,
% 12.45/6.97 ! [X4: real,R2: real,Y3: real,S: real] :
% 12.45/6.97 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ R2 )
% 12.45/6.97 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 12.45/6.97 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X4 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6194_norm__mult__less,axiom,
% 12.45/6.97 ! [X4: complex,R2: real,Y3: complex,S: real] :
% 12.45/6.97 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ R2 )
% 12.45/6.97 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 12.45/6.97 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X4 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6195_norm__mult__ineq,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X4 @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6196_norm__mult__ineq,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X4 @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6197_norm__triangle__lt,axiom,
% 12.45/6.97 ! [X4: real,Y3: real,E: real] :
% 12.45/6.97 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 12.45/6.97 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ E ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6198_norm__triangle__lt,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex,E: real] :
% 12.45/6.97 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 12.45/6.97 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ E ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6199_norm__add__less,axiom,
% 12.45/6.97 ! [X4: real,R2: real,Y3: real,S: real] :
% 12.45/6.97 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ R2 )
% 12.45/6.97 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 12.45/6.97 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6200_norm__add__less,axiom,
% 12.45/6.97 ! [X4: complex,R2: real,Y3: complex,S: real] :
% 12.45/6.97 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ R2 )
% 12.45/6.97 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 12.45/6.97 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6201_norm__power__ineq,axiom,
% 12.45/6.97 ! [X4: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X4 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X4 ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6202_norm__power__ineq,axiom,
% 12.45/6.97 ! [X4: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X4 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X4 ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6203_norm__triangle__mono,axiom,
% 12.45/6.97 ! [A: real,R2: real,B2: real,S: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ S )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6204_norm__triangle__mono,axiom,
% 12.45/6.97 ! [A: complex,R2: real,B2: complex,S: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ S )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6205_norm__triangle__ineq,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6206_norm__triangle__ineq,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6207_norm__triangle__le,axiom,
% 12.45/6.97 ! [X4: real,Y3: real,E: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ E ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6208_norm__triangle__le,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex,E: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ E ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6209_norm__add__leD,axiom,
% 12.45/6.97 ! [A: real,B2: real,C: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ C )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6210_norm__add__leD,axiom,
% 12.45/6.97 ! [A: complex,B2: complex,C: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ C )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6211_norm__diff__triangle__less,axiom,
% 12.45/6.97 ! [X4: real,Y3: real,E1: real,Z: real,E22: real] :
% 12.45/6.97 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ E1 )
% 12.45/6.97 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 12.45/6.97 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6212_norm__diff__triangle__less,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 12.45/6.97 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) @ E1 )
% 12.45/6.97 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 12.45/6.97 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6213_norm__triangle__sub,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6214_norm__triangle__sub,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6215_norm__triangle__ineq4,axiom,
% 12.45/6.97 ! [A: real,B2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6216_norm__triangle__ineq4,axiom,
% 12.45/6.97 ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6217_norm__diff__triangle__le,axiom,
% 12.45/6.97 ! [X4: real,Y3: real,E1: real,Z: real,E22: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ E1 )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6218_norm__diff__triangle__le,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) @ E1 )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6219_norm__triangle__le__diff,axiom,
% 12.45/6.97 ! [X4: real,Y3: real,E: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ E ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6220_norm__triangle__le__diff,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex,E: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 12.45/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) @ E ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6221_norm__diff__ineq,axiom,
% 12.45/6.97 ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6222_norm__diff__ineq,axiom,
% 12.45/6.97 ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6223_norm__triangle__ineq2,axiom,
% 12.45/6.97 ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6224_norm__triangle__ineq2,axiom,
% 12.45/6.97 ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6225_zero__less__binomial,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.97 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6226_Suc__times__binomial__add,axiom,
% 12.45/6.97 ! [A: nat,B2: nat] :
% 12.45/6.97 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ ( suc @ A ) ) )
% 12.45/6.97 = ( times_times_nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6227_binomial__Suc__Suc__eq__times,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 12.45/6.97 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6228_choose__mult,axiom,
% 12.45/6.97 ! [K: nat,M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ M )
% 12.45/6.97 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 12.45/6.97 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6229_binomial__absorb__comp,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 12.45/6.97 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6230_power__eq__1__iff,axiom,
% 12.45/6.97 ! [W: real,N: nat] :
% 12.45/6.97 ( ( ( power_power_real @ W @ N )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 => ( ( ( real_V7735802525324610683m_real @ W )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 | ( N = zero_zero_nat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6231_power__eq__1__iff,axiom,
% 12.45/6.97 ! [W: complex,N: nat] :
% 12.45/6.97 ( ( ( power_power_complex @ W @ N )
% 12.45/6.97 = one_one_complex )
% 12.45/6.97 => ( ( ( real_V1022390504157884413omplex @ W )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 | ( N = zero_zero_nat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6232_norm__diff__triangle__ineq,axiom,
% 12.45/6.97 ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6233_norm__diff__triangle__ineq,axiom,
% 12.45/6.97 ! [A: complex,B2: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ D ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6234_norm__triangle__ineq3,axiom,
% 12.45/6.97 ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6235_norm__triangle__ineq3,axiom,
% 12.45/6.97 ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6236_binomial__absorption,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 12.45/6.97 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6237_binomial__ge__n__over__k__pow__k,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.97 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6238_binomial__ge__n__over__k__pow__k,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.97 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6239_binomial__le__pow2,axiom,
% 12.45/6.97 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6240_choose__reduce__nat,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.97 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.97 => ( ( binomial @ N @ K )
% 12.45/6.97 = ( 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 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6241_times__binomial__minus1__eq,axiom,
% 12.45/6.97 ! [K: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.97 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 12.45/6.97 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6242_square__norm__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 => ( ( real_V7735802525324610683m_real @ X4 )
% 12.45/6.97 = one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6243_square__norm__one,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = one_one_complex )
% 12.45/6.97 => ( ( real_V1022390504157884413omplex @ X4 )
% 12.45/6.97 = one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6244_norm__power__diff,axiom,
% 12.45/6.97 ! [Z: real,W: real,M: nat] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 12.45/6.97 => ( 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 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6245_norm__power__diff,axiom,
% 12.45/6.97 ! [Z: complex,W: complex,M: nat] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 12.45/6.97 => ( 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 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6246_binomial__addition__formula,axiom,
% 12.45/6.97 ! [N: nat,K: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.97 => ( ( binomial @ N @ ( suc @ K ) )
% 12.45/6.97 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6247_choose__two,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6248_sin__cos__npi,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( 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 ) ) ) )
% 12.45/6.97 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6249_cos__pi__eq__zero,axiom,
% 12.45/6.97 ! [M: nat] :
% 12.45/6.97 ( ( 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 ) ) ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6250_round__altdef,axiom,
% 12.45/6.97 ( archim8280529875227126926d_real
% 12.45/6.97 = ( ^ [X: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X ) ) @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6251_round__altdef,axiom,
% 12.45/6.97 ( archim7778729529865785530nd_rat
% 12.45/6.97 = ( ^ [X: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X ) ) @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6252_ceiling__log__eq__powr__iff,axiom,
% 12.45/6.97 ! [X4: real,B2: real,K: nat] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.97 => ( ( ( archim7802044766580827645g_real @ ( log @ B2 @ X4 ) )
% 12.45/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 12.45/6.97 = ( ( ord_less_real @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ X4 )
% 12.45/6.97 & ( ord_less_eq_real @ X4 @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6253_cos__zero__iff,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( cos_real @ X4 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 = ( ? [N2: nat] :
% 12.45/6.97 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 12.45/6.97 & ( X4
% 12.45/6.97 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.97 | ? [N2: nat] :
% 12.45/6.97 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 12.45/6.97 & ( X4
% 12.45/6.97 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6254_cot__less__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 12.45/6.97 => ( ord_less_real @ ( cot_real @ X4 ) @ zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6255_powr__one__eq__one,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( powr_real @ one_one_real @ A )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6256_cos__zero,axiom,
% 12.45/6.97 ( ( cos_complex @ zero_zero_complex )
% 12.45/6.97 = one_one_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6257_cos__zero,axiom,
% 12.45/6.97 ( ( cos_real @ zero_zero_real )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6258_powr__zero__eq__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( X4 = zero_zero_real )
% 12.45/6.97 => ( ( powr_real @ X4 @ zero_zero_real )
% 12.45/6.97 = zero_zero_real ) )
% 12.45/6.97 & ( ( X4 != zero_zero_real )
% 12.45/6.97 => ( ( powr_real @ X4 @ zero_zero_real )
% 12.45/6.97 = one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6259_powr__gt__zero,axiom,
% 12.45/6.97 ! [X4: real,A: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X4 @ A ) )
% 12.45/6.97 = ( X4 != zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6260_powr__nonneg__iff,axiom,
% 12.45/6.97 ! [A: real,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( powr_real @ A @ X4 ) @ zero_zero_real )
% 12.45/6.97 = ( A = zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6261_powr__less__cancel__iff,axiom,
% 12.45/6.97 ! [X4: real,A: real,B2: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) )
% 12.45/6.97 = ( ord_less_real @ A @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6262_sin__pi__minus,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( sin_real @ ( minus_minus_real @ pi @ X4 ) )
% 12.45/6.97 = ( sin_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6263_powr__eq__one__iff,axiom,
% 12.45/6.97 ! [A: real,X4: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ A )
% 12.45/6.97 => ( ( ( powr_real @ A @ X4 )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 = ( X4 = zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6264_powr__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( powr_real @ X4 @ one_one_real )
% 12.45/6.97 = X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6265_powr__one__gt__zero__iff,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( powr_real @ X4 @ one_one_real )
% 12.45/6.97 = X4 )
% 12.45/6.97 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6266_powr__le__cancel__iff,axiom,
% 12.45/6.97 ! [X4: real,A: real,B2: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) )
% 12.45/6.97 = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6267_numeral__powr__numeral__real,axiom,
% 12.45/6.97 ! [M: num,N: num] :
% 12.45/6.97 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 12.45/6.97 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6268_cos__pi,axiom,
% 12.45/6.97 ( ( cos_real @ pi )
% 12.45/6.97 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6269_cos__periodic__pi2,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( cos_real @ ( plus_plus_real @ pi @ X4 ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6270_cos__periodic__pi,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( cos_real @ ( plus_plus_real @ X4 @ pi ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6271_sin__periodic__pi2,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( sin_real @ ( plus_plus_real @ pi @ X4 ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6272_sin__periodic__pi,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( sin_real @ ( plus_plus_real @ X4 @ pi ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6273_cos__pi__minus,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( cos_real @ ( minus_minus_real @ pi @ X4 ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6274_cos__minus__pi,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( cos_real @ ( minus_minus_real @ X4 @ pi ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( cos_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6275_sin__minus__pi,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( sin_real @ ( minus_minus_real @ X4 @ pi ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6276_sin__cos__squared__add3,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ X4 ) ) @ ( times_times_complex @ ( sin_complex @ X4 ) @ ( sin_complex @ X4 ) ) )
% 12.45/6.97 = one_one_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6277_sin__cos__squared__add3,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ X4 ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ X4 ) ) )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6278_log__powr__cancel,axiom,
% 12.45/6.97 ! [A: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ( A != one_one_real )
% 12.45/6.97 => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
% 12.45/6.97 = Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6279_powr__log__cancel,axiom,
% 12.45/6.97 ! [A: real,X4: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ( A != one_one_real )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( powr_real @ A @ ( log @ A @ X4 ) )
% 12.45/6.97 = X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6280_sin__npi2,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6281_sin__npi,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6282_sin__npi__int,axiom,
% 12.45/6.97 ! [N: int] :
% 12.45/6.97 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6283_cot__npi,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6284_powr__numeral,axiom,
% 12.45/6.97 ! [X4: real,N: num] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( powr_real @ X4 @ ( numeral_numeral_real @ N ) )
% 12.45/6.97 = ( power_power_real @ X4 @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6285_cos__pi__half,axiom,
% 12.45/6.97 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6286_sin__two__pi,axiom,
% 12.45/6.97 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6287_sin__pi__half,axiom,
% 12.45/6.97 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6288_cos__two__pi,axiom,
% 12.45/6.97 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6289_cos__periodic,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( cos_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 12.45/6.97 = ( cos_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6290_sin__periodic,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( sin_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 12.45/6.97 = ( sin_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6291_cos__2pi__minus,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X4 ) )
% 12.45/6.97 = ( cos_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6292_cos__npi,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 12.45/6.97 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6293_cos__npi2,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.45/6.97 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6294_cot__periodic,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( cot_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 12.45/6.97 = ( cot_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6295_sin__cos__squared__add2,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6296_sin__cos__squared__add2,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = one_one_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6297_sin__cos__squared__add,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6298_sin__cos__squared__add,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = one_one_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6299_sin__2npi,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6300_cos__2npi,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6301_sin__2pi__minus,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X4 ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6302_sin__int__2pin,axiom,
% 12.45/6.97 ! [N: int] :
% 12.45/6.97 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6303_cos__int__2pin,axiom,
% 12.45/6.97 ! [N: int] :
% 12.45/6.97 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6304_cos__3over2__pi,axiom,
% 12.45/6.97 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6305_square__powr__half,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( powr_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = ( abs_abs_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6306_sin__3over2__pi,axiom,
% 12.45/6.97 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 12.45/6.97 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6307_cos__npi__int,axiom,
% 12.45/6.97 ! [N: int] :
% 12.45/6.97 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.45/6.97 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 12.45/6.97 = one_one_real ) )
% 12.45/6.97 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.45/6.97 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 12.45/6.97 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6308_sin__add,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( sin_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6309_sin__diff,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( sin_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6310_cot__def,axiom,
% 12.45/6.97 ( cot_complex
% 12.45/6.97 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X ) @ ( sin_complex @ X ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6311_cot__def,axiom,
% 12.45/6.97 ( cot_real
% 12.45/6.97 = ( ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6312_polar__Ex,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ? [R3: real,A5: real] :
% 12.45/6.97 ( ( X4
% 12.45/6.97 = ( times_times_real @ R3 @ ( cos_real @ A5 ) ) )
% 12.45/6.97 & ( Y3
% 12.45/6.97 = ( times_times_real @ R3 @ ( sin_real @ A5 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6313_cos__one__sin__zero,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( ( cos_complex @ X4 )
% 12.45/6.97 = one_one_complex )
% 12.45/6.97 => ( ( sin_complex @ X4 )
% 12.45/6.97 = zero_zero_complex ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6314_cos__one__sin__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( cos_real @ X4 )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 => ( ( sin_real @ X4 )
% 12.45/6.97 = zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6315_powr__powr,axiom,
% 12.45/6.97 ! [X4: real,A: real,B2: real] :
% 12.45/6.97 ( ( powr_real @ ( powr_real @ X4 @ A ) @ B2 )
% 12.45/6.97 = ( powr_real @ X4 @ ( times_times_real @ A @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6316_cos__add,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( cos_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6317_cos__diff,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( cos_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6318_sin__zero__norm__cos__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( sin_real @ X4 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X4 ) )
% 12.45/6.97 = one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6319_sin__zero__norm__cos__one,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( ( sin_complex @ X4 )
% 12.45/6.97 = zero_zero_complex )
% 12.45/6.97 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X4 ) )
% 12.45/6.97 = one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6320_sin__zero__abs__cos__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( sin_real @ X4 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 => ( ( abs_abs_real @ ( cos_real @ X4 ) )
% 12.45/6.97 = one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6321_sin__double,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.97 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X4 ) ) @ ( cos_complex @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6322_sin__double,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.97 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X4 ) ) @ ( cos_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6323_sincos__principal__value,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ? [Y4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 12.45/6.97 & ( ord_less_eq_real @ Y4 @ pi )
% 12.45/6.97 & ( ( sin_real @ Y4 )
% 12.45/6.97 = ( sin_real @ X4 ) )
% 12.45/6.97 & ( ( cos_real @ Y4 )
% 12.45/6.97 = ( cos_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6324_powr__less__mono2__neg,axiom,
% 12.45/6.97 ! [A: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.97 => ( ord_less_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X4 @ A ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6325_powr__non__neg,axiom,
% 12.45/6.97 ! [A: real,X4: real] :
% 12.45/6.97 ~ ( ord_less_real @ ( powr_real @ A @ X4 ) @ zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6326_powr__mono2,axiom,
% 12.45/6.97 ! [A: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.97 => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6327_powr__ge__pzero,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X4 @ Y3 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6328_powr__less__mono,axiom,
% 12.45/6.97 ! [A: real,B2: real,X4: real] :
% 12.45/6.97 ( ( ord_less_real @ A @ B2 )
% 12.45/6.97 => ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/6.97 => ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6329_powr__less__cancel,axiom,
% 12.45/6.97 ! [X4: real,A: real,B2: real] :
% 12.45/6.97 ( ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) )
% 12.45/6.97 => ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/6.97 => ( ord_less_real @ A @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6330_powr__mono,axiom,
% 12.45/6.97 ! [A: real,B2: real,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.97 => ( ( ord_less_eq_real @ one_one_real @ X4 )
% 12.45/6.97 => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6331_sin__x__le__x,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ord_less_eq_real @ ( sin_real @ X4 ) @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6332_sin__le__one,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_real @ ( sin_real @ X4 ) @ one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6333_cos__le__one,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_real @ ( cos_real @ X4 ) @ one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6334_abs__sin__x__le__abs__x,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X4 ) ) @ ( abs_abs_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6335_sin__cos__le1,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6336_frac__ge__0,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6337_frac__ge__0,axiom,
% 12.45/6.97 ! [X4: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6338_sin__squared__eq,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6339_sin__squared__eq,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6340_cos__squared__eq,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6341_cos__squared__eq,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6342_frac__lt__1,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X4 ) @ one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6343_frac__lt__1,axiom,
% 12.45/6.97 ! [X4: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X4 ) @ one_one_rat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6344_frac__1__eq,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ one_one_real ) )
% 12.45/6.97 = ( archim2898591450579166408c_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6345_frac__1__eq,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) )
% 12.45/6.97 = ( archimedean_frac_rat @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6346_powr__mono2_H,axiom,
% 12.45/6.97 ! [A: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.97 => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X4 @ A ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6347_powr__less__mono2,axiom,
% 12.45/6.97 ! [A: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.97 => ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6348_gr__one__powr,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ( ord_less_real @ one_one_real @ ( powr_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6349_powr__inj,axiom,
% 12.45/6.97 ! [A: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ( A != one_one_real )
% 12.45/6.97 => ( ( ( powr_real @ A @ X4 )
% 12.45/6.97 = ( powr_real @ A @ Y3 ) )
% 12.45/6.97 = ( X4 = Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6350_powr__le1,axiom,
% 12.45/6.97 ! [A: real,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.97 => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ one_one_real ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6351_powr__mono__both,axiom,
% 12.45/6.97 ! [A: real,B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.97 => ( ( ord_less_eq_real @ one_one_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.97 => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ B2 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6352_ge__one__powr__ge__zero,axiom,
% 12.45/6.97 ! [X4: real,A: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ one_one_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X4 @ A ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6353_powr__divide,axiom,
% 12.45/6.97 ! [X4: real,Y3: real,A: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ( ( powr_real @ ( divide_divide_real @ X4 @ Y3 ) @ A )
% 12.45/6.97 = ( divide_divide_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6354_powr__mult,axiom,
% 12.45/6.97 ! [X4: real,Y3: real,A: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ( ( powr_real @ ( times_times_real @ X4 @ Y3 ) @ A )
% 12.45/6.97 = ( times_times_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6355_divide__powr__uminus,axiom,
% 12.45/6.97 ! [A: real,B2: real,C: real] :
% 12.45/6.97 ( ( divide_divide_real @ A @ ( powr_real @ B2 @ C ) )
% 12.45/6.97 = ( times_times_real @ A @ ( powr_real @ B2 @ ( uminus_uminus_real @ C ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6356_sin__gt__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ pi )
% 12.45/6.97 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6357_sin__x__ge__neg__x,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ ( sin_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6358_sin__ge__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ pi )
% 12.45/6.97 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6359_log__base__powr,axiom,
% 12.45/6.97 ! [A: real,B2: real,X4: real] :
% 12.45/6.97 ( ( A != zero_zero_real )
% 12.45/6.97 => ( ( log @ ( powr_real @ A @ B2 ) @ X4 )
% 12.45/6.97 = ( divide_divide_real @ ( log @ A @ X4 ) @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6360_ln__powr,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( X4 != zero_zero_real )
% 12.45/6.97 => ( ( ln_ln_real @ ( powr_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( times_times_real @ Y3 @ ( ln_ln_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6361_log__powr,axiom,
% 12.45/6.97 ! [X4: real,B2: real,Y3: real] :
% 12.45/6.97 ( ( X4 != zero_zero_real )
% 12.45/6.97 => ( ( log @ B2 @ ( powr_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( times_times_real @ Y3 @ ( log @ B2 @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6362_sin__ge__minus__one,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6363_cos__monotone__0__pi__le,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ pi )
% 12.45/6.97 => ( ord_less_eq_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6364_cos__mono__le__eq,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ pi )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ pi )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) )
% 12.45/6.97 = ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6365_cos__inj__pi,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ pi )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ pi )
% 12.45/6.97 => ( ( ( cos_real @ X4 )
% 12.45/6.97 = ( cos_real @ Y3 ) )
% 12.45/6.97 => ( X4 = Y3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6366_cos__ge__minus__one,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6367_abs__sin__le__one,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X4 ) ) @ one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6368_abs__cos__le__one,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X4 ) ) @ one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6369_cos__diff__cos,axiom,
% 12.45/6.97 ! [W: complex,Z: complex] :
% 12.45/6.97 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6370_cos__diff__cos,axiom,
% 12.45/6.97 ! [W: real,Z: real] :
% 12.45/6.97 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6371_sin__diff__sin,axiom,
% 12.45/6.97 ! [W: complex,Z: complex] :
% 12.45/6.97 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6372_sin__diff__sin,axiom,
% 12.45/6.97 ! [W: real,Z: real] :
% 12.45/6.97 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6373_sin__plus__sin,axiom,
% 12.45/6.97 ! [W: complex,Z: complex] :
% 12.45/6.97 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6374_sin__plus__sin,axiom,
% 12.45/6.97 ! [W: real,Z: real] :
% 12.45/6.97 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6375_cos__times__sin,axiom,
% 12.45/6.97 ! [W: complex,Z: complex] :
% 12.45/6.97 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6376_cos__times__sin,axiom,
% 12.45/6.97 ! [W: real,Z: real] :
% 12.45/6.97 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6377_sin__times__cos,axiom,
% 12.45/6.97 ! [W: complex,Z: complex] :
% 12.45/6.97 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6378_sin__times__cos,axiom,
% 12.45/6.97 ! [W: real,Z: real] :
% 12.45/6.97 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6379_sin__times__sin,axiom,
% 12.45/6.97 ! [W: complex,Z: complex] :
% 12.45/6.97 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6380_sin__times__sin,axiom,
% 12.45/6.97 ! [W: real,Z: real] :
% 12.45/6.97 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6381_cos__double,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.97 = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6382_cos__double,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.97 = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6383_powr__add,axiom,
% 12.45/6.97 ! [X4: real,A: real,B2: real] :
% 12.45/6.97 ( ( powr_real @ X4 @ ( plus_plus_real @ A @ B2 ) )
% 12.45/6.97 = ( times_times_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6384_powr__diff,axiom,
% 12.45/6.97 ! [W: real,Z1: real,Z22: real] :
% 12.45/6.97 ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
% 12.45/6.97 = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6385_cos__double__sin,axiom,
% 12.45/6.97 ! [W: complex] :
% 12.45/6.97 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 12.45/6.97 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6386_cos__double__sin,axiom,
% 12.45/6.97 ! [W: real] :
% 12.45/6.97 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 12.45/6.97 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6387_powr__realpow,axiom,
% 12.45/6.97 ! [X4: real,N: nat] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( powr_real @ X4 @ ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.97 = ( power_power_real @ X4 @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6388_cos__two__neq__zero,axiom,
% 12.45/6.97 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.45/6.97 != zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6389_less__log__iff,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ Y3 @ ( log @ B2 @ X4 ) )
% 12.45/6.97 = ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6390_log__less__iff,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ ( log @ B2 @ X4 ) @ Y3 )
% 12.45/6.97 = ( ord_less_real @ X4 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6391_less__powr__iff,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( powr_real @ B2 @ Y3 ) )
% 12.45/6.97 = ( ord_less_real @ ( log @ B2 @ X4 ) @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6392_powr__less__iff,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X4 )
% 12.45/6.97 = ( ord_less_real @ Y3 @ ( log @ B2 @ X4 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6393_cos__monotone__0__pi,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ( ( ord_less_real @ Y3 @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ pi )
% 12.45/6.97 => ( ord_less_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6394_cos__mono__less__eq,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ pi )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ pi )
% 12.45/6.97 => ( ( ord_less_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) )
% 12.45/6.97 = ( ord_less_real @ Y3 @ X4 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6395_sin__eq__0__pi,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ pi )
% 12.45/6.97 => ( ( ( sin_real @ X4 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 => ( X4 = zero_zero_real ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6396_sin__zero__pi__iff,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ pi )
% 12.45/6.97 => ( ( ( sin_real @ X4 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 = ( X4 = zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6397_cos__monotone__minus__pi__0_H,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.97 => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X4 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6398_sin__zero__iff__int2,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( sin_real @ X4 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 = ( ? [I2: int] :
% 12.45/6.97 ( X4
% 12.45/6.97 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6399_frac__def,axiom,
% 12.45/6.97 ( archim2898591450579166408c_real
% 12.45/6.97 = ( ^ [X: real] : ( minus_minus_real @ X @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6400_frac__def,axiom,
% 12.45/6.97 ( archimedean_frac_rat
% 12.45/6.97 = ( ^ [X: rat] : ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6401_sincos__total__pi,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 => ? [T2: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ T2 )
% 12.45/6.97 & ( ord_less_eq_real @ T2 @ pi )
% 12.45/6.97 & ( X4
% 12.45/6.97 = ( cos_real @ T2 ) )
% 12.45/6.97 & ( Y3
% 12.45/6.97 = ( sin_real @ T2 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6402_sin__cos__sqrt,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X4 ) )
% 12.45/6.97 => ( ( sin_real @ X4 )
% 12.45/6.97 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6403_sin__expansion__lemma,axiom,
% 12.45/6.97 ! [X4: real,M: nat] :
% 12.45/6.97 ( ( sin_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.45/6.97 = ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6404_powr__minus__divide,axiom,
% 12.45/6.97 ! [X4: real,A: real] :
% 12.45/6.97 ( ( powr_real @ X4 @ ( uminus_uminus_real @ A ) )
% 12.45/6.97 = ( divide_divide_real @ one_one_real @ ( powr_real @ X4 @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6405_cos__expansion__lemma,axiom,
% 12.45/6.97 ! [X4: real,M: nat] :
% 12.45/6.97 ( ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6406_powr__neg__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( powr_real @ X4 @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.97 = ( divide_divide_real @ one_one_real @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6407_sin__gt__zero__02,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.45/6.97 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6408_powr__mult__base,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( times_times_real @ X4 @ ( powr_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( powr_real @ X4 @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6409_powr__le__iff,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X4 )
% 12.45/6.97 = ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X4 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6410_le__powr__iff,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ ( powr_real @ B2 @ Y3 ) )
% 12.45/6.97 = ( ord_less_eq_real @ ( log @ B2 @ X4 ) @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6411_log__le__iff,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( log @ B2 @ X4 ) @ Y3 )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6412_le__log__iff,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6413_cos__two__less__zero,axiom,
% 12.45/6.97 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6414_cos__two__le__zero,axiom,
% 12.45/6.97 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6415_cos__is__zero,axiom,
% 12.45/6.97 ? [X3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 12.45/6.97 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.45/6.97 & ( ( cos_real @ X3 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 & ! [Y5: real] :
% 12.45/6.97 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 12.45/6.97 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.45/6.97 & ( ( cos_real @ Y5 )
% 12.45/6.97 = zero_zero_real ) )
% 12.45/6.97 => ( Y5 = X3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6416_cos__monotone__minus__pi__0,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_real @ Y3 @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.97 => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X4 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6417_cos__total,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.97 => ? [X3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 12.45/6.97 & ( ord_less_eq_real @ X3 @ pi )
% 12.45/6.97 & ( ( cos_real @ X3 )
% 12.45/6.97 = Y3 )
% 12.45/6.97 & ! [Y5: real] :
% 12.45/6.97 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 12.45/6.97 & ( ord_less_eq_real @ Y5 @ pi )
% 12.45/6.97 & ( ( cos_real @ Y5 )
% 12.45/6.97 = Y3 ) )
% 12.45/6.97 => ( Y5 = X3 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6418_frac__eq,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( archim2898591450579166408c_real @ X4 )
% 12.45/6.97 = X4 )
% 12.45/6.97 = ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 & ( ord_less_real @ X4 @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6419_frac__eq,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( ( archimedean_frac_rat @ X4 )
% 12.45/6.97 = X4 )
% 12.45/6.97 = ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.45/6.97 & ( ord_less_rat @ X4 @ one_one_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6420_frac__add,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 12.45/6.97 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) ) )
% 12.45/6.97 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 12.45/6.97 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6421_frac__add,axiom,
% 12.45/6.97 ! [X4: rat,Y3: rat] :
% 12.45/6.97 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 12.45/6.97 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
% 12.45/6.97 = ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) ) )
% 12.45/6.97 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 12.45/6.97 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
% 12.45/6.97 = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6422_sincos__total__pi__half,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 => ? [T2: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ T2 )
% 12.45/6.97 & ( ord_less_eq_real @ T2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 & ( X4
% 12.45/6.97 = ( cos_real @ T2 ) )
% 12.45/6.97 & ( Y3
% 12.45/6.97 = ( sin_real @ T2 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6423_sincos__total__2pi__le,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 => ? [T2: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ T2 )
% 12.45/6.97 & ( ord_less_eq_real @ T2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.45/6.97 & ( X4
% 12.45/6.97 = ( cos_real @ T2 ) )
% 12.45/6.97 & ( Y3
% 12.45/6.97 = ( sin_real @ T2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6424_sincos__total__2pi,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 => ~ ! [T2: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ T2 )
% 12.45/6.97 => ( ( ord_less_real @ T2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.45/6.97 => ( ( X4
% 12.45/6.97 = ( cos_real @ T2 ) )
% 12.45/6.97 => ( Y3
% 12.45/6.97 != ( sin_real @ T2 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6425_ln__powr__bound,axiom,
% 12.45/6.97 ! [X4: real,A: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ one_one_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( divide_divide_real @ ( powr_real @ X4 @ A ) @ A ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6426_ln__powr__bound2,axiom,
% 12.45/6.97 ! [X4: real,A: real] :
% 12.45/6.97 ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X4 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6427_log__add__eq__powr,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.97 => ( ( B2 != one_one_real )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( plus_plus_real @ ( log @ B2 @ X4 ) @ Y3 )
% 12.45/6.97 = ( log @ B2 @ ( times_times_real @ X4 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6428_add__log__eq__powr,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.97 => ( ( B2 != one_one_real )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( plus_plus_real @ Y3 @ ( log @ B2 @ X4 ) )
% 12.45/6.97 = ( log @ B2 @ ( times_times_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6429_minus__log__eq__powr,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.97 => ( ( B2 != one_one_real )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( minus_minus_real @ Y3 @ ( log @ B2 @ X4 ) )
% 12.45/6.97 = ( log @ B2 @ ( divide_divide_real @ ( powr_real @ B2 @ Y3 ) @ X4 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6430_sin__pi__divide__n__ge__0,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( N != zero_zero_nat )
% 12.45/6.97 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6431_sin__45,axiom,
% 12.45/6.97 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.45/6.97 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6432_cos__45,axiom,
% 12.45/6.97 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.45/6.97 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6433_powr__def,axiom,
% 12.45/6.97 ( powr_real
% 12.45/6.97 = ( ^ [X: real,A4: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A4 @ ( ln_ln_real @ X ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6434_cos__plus__cos,axiom,
% 12.45/6.97 ! [W: complex,Z: complex] :
% 12.45/6.97 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6435_cos__plus__cos,axiom,
% 12.45/6.97 ! [W: real,Z: real] :
% 12.45/6.97 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6436_cos__times__cos,axiom,
% 12.45/6.97 ! [W: complex,Z: complex] :
% 12.45/6.97 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6437_cos__times__cos,axiom,
% 12.45/6.97 ! [W: real,Z: real] :
% 12.45/6.97 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 12.45/6.97 = ( 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 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6438_sin__gt__zero2,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6439_sin__lt__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ pi @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.45/6.97 => ( ord_less_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6440_cos__double__less__one,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.45/6.97 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6441_sin__30,axiom,
% 12.45/6.97 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 12.45/6.97 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6442_cos__gt__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6443_sin__monotone__2pi__le,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X4 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6444_sin__mono__le__eq,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6445_sin__inj__pi,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ( sin_real @ X4 )
% 12.45/6.97 = ( sin_real @ Y3 ) )
% 12.45/6.97 => ( X4 = Y3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6446_log__minus__eq__powr,axiom,
% 12.45/6.97 ! [B2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.97 => ( ( B2 != one_one_real )
% 12.45/6.97 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( minus_minus_real @ ( log @ B2 @ X4 ) @ Y3 )
% 12.45/6.97 = ( log @ B2 @ ( times_times_real @ X4 @ ( powr_real @ B2 @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6447_cos__60,axiom,
% 12.45/6.97 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 12.45/6.97 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6448_sin__60,axiom,
% 12.45/6.97 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 12.45/6.97 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6449_cos__30,axiom,
% 12.45/6.97 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 12.45/6.97 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6450_cos__one__2pi__int,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( cos_real @ X4 )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 = ( ? [X: int] :
% 12.45/6.97 ( X4
% 12.45/6.97 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6451_cos__double__cos,axiom,
% 12.45/6.97 ! [W: complex] :
% 12.45/6.97 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 12.45/6.97 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6452_cos__double__cos,axiom,
% 12.45/6.97 ! [W: real] :
% 12.45/6.97 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 12.45/6.97 = ( 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 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6453_cos__treble__cos,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X4 ) )
% 12.45/6.97 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6454_cos__treble__cos,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X4 ) )
% 12.45/6.97 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6455_powr__half__sqrt,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( powr_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = ( sqrt @ X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6456_powr__neg__numeral,axiom,
% 12.45/6.97 ! [X4: real,N: num] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( powr_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.45/6.97 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6457_sin__le__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ pi @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.45/6.97 => ( ord_less_eq_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6458_sin__less__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 12.45/6.97 => ( ord_less_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6459_sin__monotone__2pi,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_real @ Y3 @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X4 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6460_sin__mono__less__eq,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_real @ ( sin_real @ X4 ) @ ( sin_real @ Y3 ) )
% 12.45/6.97 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6461_sin__total,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.97 => ? [X3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 12.45/6.97 & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 & ( ( sin_real @ X3 )
% 12.45/6.97 = Y3 )
% 12.45/6.97 & ! [Y5: real] :
% 12.45/6.97 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 12.45/6.97 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 & ( ( sin_real @ Y5 )
% 12.45/6.97 = Y3 ) )
% 12.45/6.97 => ( Y5 = X3 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6462_cos__gt__zero__pi,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6463_cos__ge__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6464_cos__one__2pi,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( cos_real @ X4 )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 = ( ? [X: nat] :
% 12.45/6.97 ( X4
% 12.45/6.97 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 12.45/6.97 | ? [X: nat] :
% 12.45/6.97 ( X4
% 12.45/6.97 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6465_floor__add,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 12.45/6.97 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) )
% 12.45/6.97 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 12.45/6.97 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6466_floor__add,axiom,
% 12.45/6.97 ! [X4: rat,Y3: rat] :
% 12.45/6.97 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 12.45/6.97 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
% 12.45/6.97 = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) )
% 12.45/6.97 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 12.45/6.97 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
% 12.45/6.97 = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6467_sin__pi__divide__n__gt__0,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/6.97 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6468_sin__arctan,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( sin_real @ ( arctan @ X4 ) )
% 12.45/6.97 = ( divide_divide_real @ X4 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6469_floor__log__eq__powr__iff,axiom,
% 12.45/6.97 ! [X4: real,B2: real,K: int] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/6.97 => ( ( ( archim6058952711729229775r_real @ ( log @ B2 @ X4 ) )
% 12.45/6.97 = K )
% 12.45/6.97 = ( ( ord_less_eq_real @ ( powr_real @ B2 @ ( ring_1_of_int_real @ K ) ) @ X4 )
% 12.45/6.97 & ( ord_less_real @ X4 @ ( powr_real @ B2 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6470_cos__arctan,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( cos_real @ ( arctan @ X4 ) )
% 12.45/6.97 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6471_cot__gt__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6472_sin__zero__iff__int,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( sin_real @ X4 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 = ( ? [I2: int] :
% 12.45/6.97 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
% 12.45/6.97 & ( X4
% 12.45/6.97 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6473_cos__zero__iff__int,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( cos_real @ X4 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 = ( ? [I2: int] :
% 12.45/6.97 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
% 12.45/6.97 & ( X4
% 12.45/6.97 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6474_sin__zero__lemma,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ( sin_real @ X4 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 => ? [N4: nat] :
% 12.45/6.97 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 12.45/6.97 & ( X4
% 12.45/6.97 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6475_sin__zero__iff,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( sin_real @ X4 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 = ( ? [N2: nat] :
% 12.45/6.97 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 12.45/6.97 & ( X4
% 12.45/6.97 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.97 | ? [N2: nat] :
% 12.45/6.97 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 12.45/6.97 & ( X4
% 12.45/6.97 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6476_cos__zero__lemma,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ( cos_real @ X4 )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 => ? [N4: nat] :
% 12.45/6.97 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 12.45/6.97 & ( X4
% 12.45/6.97 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6477_tan__double,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( ( cos_complex @ X4 )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X4 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6478_tan__double,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( cos_real @ X4 )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.97 = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X4 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6479_sin__tan,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( sin_real @ X4 )
% 12.45/6.97 = ( divide_divide_real @ ( tan_real @ X4 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6480_cos__tan,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( cos_real @ X4 )
% 12.45/6.97 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6481_arcosh__def,axiom,
% 12.45/6.97 ( arcosh_real
% 12.45/6.97 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6482_complex__unimodular__polar,axiom,
% 12.45/6.97 ! [Z: complex] :
% 12.45/6.97 ( ( ( real_V1022390504157884413omplex @ Z )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 => ~ ! [T2: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ T2 )
% 12.45/6.97 => ( ( ord_less_real @ T2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.45/6.97 => ( Z
% 12.45/6.97 != ( complex2 @ ( cos_real @ T2 ) @ ( sin_real @ T2 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6483_cos__arcsin,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.97 => ( ( cos_real @ ( arcsin @ X4 ) )
% 12.45/6.97 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6484_of__real__1,axiom,
% 12.45/6.97 ( ( real_V1803761363581548252l_real @ one_one_real )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6485_of__real__1,axiom,
% 12.45/6.97 ( ( real_V4546457046886955230omplex @ one_one_real )
% 12.45/6.97 = one_one_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6486_of__real__eq__1__iff,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( real_V1803761363581548252l_real @ X4 )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 = ( X4 = one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6487_of__real__eq__1__iff,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ( real_V4546457046886955230omplex @ X4 )
% 12.45/6.97 = one_one_complex )
% 12.45/6.97 = ( X4 = one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6488_of__real__mult,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( real_V1803761363581548252l_real @ ( times_times_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( times_times_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6489_of__real__mult,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( real_V4546457046886955230omplex @ ( times_times_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( times_times_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6490_of__real__numeral,axiom,
% 12.45/6.97 ! [W: num] :
% 12.45/6.97 ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
% 12.45/6.97 = ( numeral_numeral_real @ W ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6491_of__real__numeral,axiom,
% 12.45/6.97 ! [W: num] :
% 12.45/6.97 ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
% 12.45/6.97 = ( numera6690914467698888265omplex @ W ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6492_of__real__divide,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6493_of__real__divide,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6494_of__real__power,axiom,
% 12.45/6.97 ! [X4: real,N: nat] :
% 12.45/6.97 ( ( real_V1803761363581548252l_real @ ( power_power_real @ X4 @ N ) )
% 12.45/6.97 = ( power_power_real @ ( real_V1803761363581548252l_real @ X4 ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6495_of__real__power,axiom,
% 12.45/6.97 ! [X4: real,N: nat] :
% 12.45/6.97 ( ( real_V4546457046886955230omplex @ ( power_power_real @ X4 @ N ) )
% 12.45/6.97 = ( power_power_complex @ ( real_V4546457046886955230omplex @ X4 ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6496_of__real__add,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6497_of__real__add,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6498_of__real__diff,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6499_of__real__diff,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6500_tan__periodic__pi,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( tan_real @ ( plus_plus_real @ X4 @ pi ) )
% 12.45/6.97 = ( tan_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6501_of__real__neg__numeral,axiom,
% 12.45/6.97 ! [W: num] :
% 12.45/6.97 ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6502_of__real__neg__numeral,axiom,
% 12.45/6.97 ! [W: num] :
% 12.45/6.97 ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.97 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6503_cos__of__real__pi,axiom,
% 12.45/6.97 ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
% 12.45/6.97 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6504_cos__of__real__pi,axiom,
% 12.45/6.97 ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 12.45/6.97 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6505_tan__npi,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6506_tan__periodic__n,axiom,
% 12.45/6.97 ! [X4: real,N: num] :
% 12.45/6.97 ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 12.45/6.97 = ( tan_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6507_sin__arcsin,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.97 => ( ( sin_real @ ( arcsin @ Y3 ) )
% 12.45/6.97 = Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6508_tan__periodic__nat,axiom,
% 12.45/6.97 ! [X4: real,N: nat] :
% 12.45/6.97 ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 12.45/6.97 = ( tan_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6509_tan__periodic__int,axiom,
% 12.45/6.97 ! [X4: real,I: int] :
% 12.45/6.97 ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 12.45/6.97 = ( tan_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6510_norm__cos__sin,axiom,
% 12.45/6.97 ! [T: real] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6511_norm__of__real__add1,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ one_one_real ) )
% 12.45/6.97 = ( abs_abs_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6512_norm__of__real__add1,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ one_one_complex ) )
% 12.45/6.97 = ( abs_abs_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6513_norm__of__real__addn,axiom,
% 12.45/6.97 ! [X4: real,B2: num] :
% 12.45/6.97 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( numeral_numeral_real @ B2 ) ) )
% 12.45/6.97 = ( abs_abs_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6514_norm__of__real__addn,axiom,
% 12.45/6.97 ! [X4: real,B2: num] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( numera6690914467698888265omplex @ B2 ) ) )
% 12.45/6.97 = ( abs_abs_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6515_arcsin__1,axiom,
% 12.45/6.97 ( ( arcsin @ one_one_real )
% 12.45/6.97 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6516_tan__periodic,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 12.45/6.97 = ( tan_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6517_cos__of__real__pi__half,axiom,
% 12.45/6.97 ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6518_cos__of__real__pi__half,axiom,
% 12.45/6.97 ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = zero_zero_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6519_sin__of__real__pi__half,axiom,
% 12.45/6.97 ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6520_sin__of__real__pi__half,axiom,
% 12.45/6.97 ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 12.45/6.97 = one_one_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6521_arcsin__minus__1,axiom,
% 12.45/6.97 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6522_complex__of__real__mult__Complex,axiom,
% 12.45/6.97 ! [R2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X4 @ Y3 ) )
% 12.45/6.97 = ( complex2 @ ( times_times_real @ R2 @ X4 ) @ ( times_times_real @ R2 @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6523_Complex__mult__complex__of__real,axiom,
% 12.45/6.97 ! [X4: real,Y3: real,R2: real] :
% 12.45/6.97 ( ( times_times_complex @ ( complex2 @ X4 @ Y3 ) @ ( real_V4546457046886955230omplex @ R2 ) )
% 12.45/6.97 = ( complex2 @ ( times_times_real @ X4 @ R2 ) @ ( times_times_real @ Y3 @ R2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6524_complex__of__real__add__Complex,axiom,
% 12.45/6.97 ! [R2: real,X4: real,Y3: real] :
% 12.45/6.97 ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X4 @ Y3 ) )
% 12.45/6.97 = ( complex2 @ ( plus_plus_real @ R2 @ X4 ) @ Y3 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6525_Complex__add__complex__of__real,axiom,
% 12.45/6.97 ! [X4: real,Y3: real,R2: real] :
% 12.45/6.97 ( ( plus_plus_complex @ ( complex2 @ X4 @ Y3 ) @ ( real_V4546457046886955230omplex @ R2 ) )
% 12.45/6.97 = ( complex2 @ ( plus_plus_real @ X4 @ R2 ) @ Y3 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6526_complex__diff,axiom,
% 12.45/6.97 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.97 ( ( minus_minus_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
% 12.45/6.97 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6527_Complex__eq__1,axiom,
% 12.45/6.97 ! [A: real,B2: real] :
% 12.45/6.97 ( ( ( complex2 @ A @ B2 )
% 12.45/6.97 = one_one_complex )
% 12.45/6.97 = ( ( A = one_one_real )
% 12.45/6.97 & ( B2 = zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6528_one__complex_Ocode,axiom,
% 12.45/6.97 ( one_one_complex
% 12.45/6.97 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6529_Complex__eq__numeral,axiom,
% 12.45/6.97 ! [A: real,B2: real,W: num] :
% 12.45/6.97 ( ( ( complex2 @ A @ B2 )
% 12.45/6.97 = ( numera6690914467698888265omplex @ W ) )
% 12.45/6.97 = ( ( A
% 12.45/6.97 = ( numeral_numeral_real @ W ) )
% 12.45/6.97 & ( B2 = zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6530_complex__add,axiom,
% 12.45/6.97 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.97 ( ( plus_plus_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
% 12.45/6.97 = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6531_nonzero__of__real__divide,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( Y3 != zero_zero_real )
% 12.45/6.97 => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6532_nonzero__of__real__divide,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( Y3 != zero_zero_real )
% 12.45/6.97 => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6533_Complex__eq__neg__1,axiom,
% 12.45/6.97 ! [A: real,B2: real] :
% 12.45/6.97 ( ( ( complex2 @ A @ B2 )
% 12.45/6.97 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.97 = ( ( A
% 12.45/6.97 = ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.97 & ( B2 = zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6534_Complex__eq__neg__numeral,axiom,
% 12.45/6.97 ! [A: real,B2: real,W: num] :
% 12.45/6.97 ( ( ( complex2 @ A @ B2 )
% 12.45/6.97 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.45/6.97 = ( ( A
% 12.45/6.97 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.97 & ( B2 = zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6535_complex__mult,axiom,
% 12.45/6.97 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.97 ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
% 12.45/6.97 = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6536_arcsin__le__arcsin,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.97 => ( ord_less_eq_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6537_arcsin__minus,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.97 => ( ( arcsin @ ( uminus_uminus_real @ X4 ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( arcsin @ X4 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6538_arcsin__le__mono,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6539_arcsin__eq__iff,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 12.45/6.97 => ( ( ( arcsin @ X4 )
% 12.45/6.97 = ( arcsin @ Y3 ) )
% 12.45/6.97 = ( X4 = Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6540_norm__less__p1,axiom,
% 12.45/6.97 ! [X4: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X4 ) ) @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6541_norm__less__p1,axiom,
% 12.45/6.97 ! [X4: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X4 ) ) @ one_one_complex ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6542_tan__def,axiom,
% 12.45/6.97 ( tan_complex
% 12.45/6.97 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X ) @ ( cos_complex @ X ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6543_tan__def,axiom,
% 12.45/6.97 ( tan_real
% 12.45/6.97 = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ X ) @ ( cos_real @ X ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6544_arcsin__less__arcsin,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.97 => ( ord_less_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6545_arcsin__less__mono,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 12.45/6.97 => ( ( ord_less_real @ ( arcsin @ X4 ) @ ( arcsin @ Y3 ) )
% 12.45/6.97 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6546_norm__of__real__diff,axiom,
% 12.45/6.97 ! [B2: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B2 ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6547_norm__of__real__diff,axiom,
% 12.45/6.97 ! [B2: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B2 ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6548_tan__45,axiom,
% 12.45/6.97 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6549_tan__60,axiom,
% 12.45/6.97 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 12.45/6.97 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6550_cos__int__times__real,axiom,
% 12.45/6.97 ! [M: int,X4: real] :
% 12.45/6.97 ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X4 ) ) )
% 12.45/6.97 = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6551_cos__int__times__real,axiom,
% 12.45/6.97 ! [M: int,X4: real] :
% 12.45/6.97 ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X4 ) ) )
% 12.45/6.97 = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6552_sin__int__times__real,axiom,
% 12.45/6.97 ! [M: int,X4: real] :
% 12.45/6.97 ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X4 ) ) )
% 12.45/6.97 = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6553_sin__int__times__real,axiom,
% 12.45/6.97 ! [M: int,X4: real] :
% 12.45/6.97 ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X4 ) ) )
% 12.45/6.97 = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6554_cos__arcsin__nonzero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/6.97 => ( ( cos_real @ ( arcsin @ X4 ) )
% 12.45/6.97 != zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6555_lemma__tan__total,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ? [X3: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X3 )
% 12.45/6.97 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 & ( ord_less_real @ Y3 @ ( tan_real @ X3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6556_tan__gt__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6557_tan__total,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ? [X3: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 12.45/6.97 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 & ( ( tan_real @ X3 )
% 12.45/6.97 = Y3 )
% 12.45/6.97 & ! [Y5: real] :
% 12.45/6.97 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 12.45/6.97 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 & ( ( tan_real @ Y5 )
% 12.45/6.97 = Y3 ) )
% 12.45/6.97 => ( Y5 = X3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6558_tan__monotone,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_real @ Y3 @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X4 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6559_tan__monotone_H,axiom,
% 12.45/6.97 ! [Y3: real,X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_real @ Y3 @ X4 )
% 12.45/6.97 = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X4 ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6560_tan__mono__lt__eq,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) )
% 12.45/6.97 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6561_lemma__tan__total1,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ? [X3: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 12.45/6.97 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 & ( ( tan_real @ X3 )
% 12.45/6.97 = Y3 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6562_tan__minus__45,axiom,
% 12.45/6.97 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.97 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6563_tan__inverse,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y3 ) )
% 12.45/6.97 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6564_complex__norm,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( real_V1022390504157884413omplex @ ( complex2 @ X4 @ Y3 ) )
% 12.45/6.97 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6565_add__tan__eq,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex] :
% 12.45/6.97 ( ( ( cos_complex @ X4 )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( ( cos_complex @ Y3 )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( plus_plus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6566_add__tan__eq,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ( cos_real @ X4 )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( ( cos_real @ Y3 )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( plus_plus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) )
% 12.45/6.97 = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6567_tan__cot_H,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) )
% 12.45/6.97 = ( cot_real @ X4 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6568_sin__cos__eq,axiom,
% 12.45/6.97 ( sin_real
% 12.45/6.97 = ( ^ [X: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6569_sin__cos__eq,axiom,
% 12.45/6.97 ( sin_complex
% 12.45/6.97 = ( ^ [X: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6570_cos__sin__eq,axiom,
% 12.45/6.97 ( cos_real
% 12.45/6.97 = ( ^ [X: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6571_cos__sin__eq,axiom,
% 12.45/6.97 ( cos_complex
% 12.45/6.97 = ( ^ [X: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6572_tan__pos__pi2__le,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6573_tan__total__pos,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.97 => ? [X3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 12.45/6.97 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 & ( ( tan_real @ X3 )
% 12.45/6.97 = Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6574_tan__less__zero,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 12.45/6.97 => ( ord_less_real @ ( tan_real @ X4 ) @ zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6575_tan__mono__le,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.97 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ord_less_eq_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6576_tan__mono__le__eq,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6577_tan__bound__pi2,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.45/6.97 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X4 ) ) @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6578_tan__30,axiom,
% 12.45/6.97 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 12.45/6.97 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6579_arctan,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 12.45/6.97 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 & ( ( tan_real @ ( arctan @ Y3 ) )
% 12.45/6.97 = Y3 ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6580_arctan__tan,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( arctan @ ( tan_real @ X4 ) )
% 12.45/6.97 = X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6581_arctan__unique,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ( tan_real @ X4 )
% 12.45/6.97 = Y3 )
% 12.45/6.97 => ( ( arctan @ Y3 )
% 12.45/6.97 = X4 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6582_lemma__tan__add1,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex] :
% 12.45/6.97 ( ( ( cos_complex @ X4 )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( ( cos_complex @ Y3 )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X4 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6583_lemma__tan__add1,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ( cos_real @ X4 )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( ( cos_real @ Y3 )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) )
% 12.45/6.97 = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X4 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6584_tan__diff,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex] :
% 12.45/6.97 ( ( ( cos_complex @ X4 )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( ( cos_complex @ Y3 )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( ( cos_complex @ ( minus_minus_complex @ X4 @ Y3 ) )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( tan_complex @ ( minus_minus_complex @ X4 @ Y3 ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6585_tan__diff,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ( cos_real @ X4 )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( ( cos_real @ Y3 )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( ( cos_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( tan_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6586_tan__add,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex] :
% 12.45/6.97 ( ( ( cos_complex @ X4 )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( ( cos_complex @ Y3 )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( ( cos_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
% 12.45/6.97 != zero_zero_complex )
% 12.45/6.97 => ( ( tan_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6587_tan__add,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ( cos_real @ X4 )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( ( cos_real @ Y3 )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( ( cos_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( tan_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6588_minus__sin__cos__eq,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( uminus_uminus_real @ ( sin_real @ X4 ) )
% 12.45/6.97 = ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6589_minus__sin__cos__eq,axiom,
% 12.45/6.97 ! [X4: complex] :
% 12.45/6.97 ( ( uminus1482373934393186551omplex @ ( sin_complex @ X4 ) )
% 12.45/6.97 = ( cos_complex @ ( plus_plus_complex @ X4 @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6590_tan__total__pi4,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.97 => ? [Z2: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
% 12.45/6.97 & ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.45/6.97 & ( ( tan_real @ Z2 )
% 12.45/6.97 = X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6591_arcsin__lt__bounded,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_real @ Y3 @ one_one_real )
% 12.45/6.97 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 12.45/6.97 & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6592_arcsin__lbound,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.97 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6593_arcsin__ubound,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.97 => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6594_arcsin__bounded,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 12.45/6.97 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6595_arcsin__sin,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( arcsin @ ( sin_real @ X4 ) )
% 12.45/6.97 = X4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6596_tan__half,axiom,
% 12.45/6.97 ( tan_complex
% 12.45/6.97 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ one_one_complex ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6597_tan__half,axiom,
% 12.45/6.97 ( tan_real
% 12.45/6.97 = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6598_le__arcsin__iff,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X4 ) )
% 12.45/6.97 = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X4 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6599_arcsin__le__iff,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( arcsin @ X4 ) @ Y3 )
% 12.45/6.97 = ( ord_less_eq_real @ X4 @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6600_arcsin__pi,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 12.45/6.97 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
% 12.45/6.97 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 12.45/6.97 = Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6601_arcsin,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.97 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 12.45/6.97 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.97 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 12.45/6.97 = Y3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6602_arsinh__def,axiom,
% 12.45/6.97 ( arsinh_real
% 12.45/6.97 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6603_sin__arccos__abs,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 12.45/6.97 => ( ( sin_real @ ( arccos @ Y3 ) )
% 12.45/6.97 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6604_sin__arccos,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.97 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.97 => ( ( sin_real @ ( arccos @ X4 ) )
% 12.45/6.97 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6605_arccos__cos__eq__abs__2pi,axiom,
% 12.45/6.97 ! [Theta: real] :
% 12.45/6.97 ~ ! [K2: int] :
% 12.45/6.97 ( ( arccos @ ( cos_real @ Theta ) )
% 12.45/6.97 != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6606_fact__double,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.97 = ( 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 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6607_fact__double,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.97 = ( 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 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6608_fact__double,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.97 = ( 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 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6609_modulo__int__unfold,axiom,
% 12.45/6.97 ! [L2: int,K: int,N: nat,M: nat] :
% 12.45/6.97 ( ( ( ( ( sgn_sgn_int @ L2 )
% 12.45/6.97 = zero_zero_int )
% 12.45/6.97 | ( ( sgn_sgn_int @ K )
% 12.45/6.97 = zero_zero_int )
% 12.45/6.97 | ( N = zero_zero_nat ) )
% 12.45/6.97 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.45/6.97 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 12.45/6.97 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 12.45/6.97 = zero_zero_int )
% 12.45/6.97 | ( ( sgn_sgn_int @ K )
% 12.45/6.97 = zero_zero_int )
% 12.45/6.97 | ( N = zero_zero_nat ) )
% 12.45/6.97 => ( ( ( ( sgn_sgn_int @ K )
% 12.45/6.97 = ( sgn_sgn_int @ L2 ) )
% 12.45/6.97 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.45/6.97 = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 12.45/6.97 & ( ( ( sgn_sgn_int @ K )
% 12.45/6.97 != ( sgn_sgn_int @ L2 ) )
% 12.45/6.97 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.45/6.97 = ( times_times_int @ ( sgn_sgn_int @ L2 )
% 12.45/6.97 @ ( minus_minus_int
% 12.45/6.97 @ ( semiri1314217659103216013at_int
% 12.45/6.97 @ ( times_times_nat @ N
% 12.45/6.97 @ ( zero_n2687167440665602831ol_nat
% 12.45/6.97 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 12.45/6.97 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6610_sgn__sgn,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
% 12.45/6.97 = ( sgn_sgn_int @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6611_sgn__sgn,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
% 12.45/6.97 = ( sgn_sgn_real @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6612_sgn__sgn,axiom,
% 12.45/6.97 ! [A: complex] :
% 12.45/6.97 ( ( sgn_sgn_complex @ ( sgn_sgn_complex @ A ) )
% 12.45/6.97 = ( sgn_sgn_complex @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6613_sgn__sgn,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 12.45/6.97 = ( sgn_sgn_Code_integer @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6614_sgn__sgn,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
% 12.45/6.97 = ( sgn_sgn_rat @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6615_sgn__0,axiom,
% 12.45/6.97 ( ( sgn_sgn_complex @ zero_zero_complex )
% 12.45/6.97 = zero_zero_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6616_sgn__0,axiom,
% 12.45/6.97 ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 12.45/6.97 = zero_z3403309356797280102nteger ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6617_sgn__0,axiom,
% 12.45/6.97 ( ( sgn_sgn_real @ zero_zero_real )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6618_sgn__0,axiom,
% 12.45/6.97 ( ( sgn_sgn_rat @ zero_zero_rat )
% 12.45/6.97 = zero_zero_rat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6619_sgn__0,axiom,
% 12.45/6.97 ( ( sgn_sgn_int @ zero_zero_int )
% 12.45/6.97 = zero_zero_int ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6620_sgn__1,axiom,
% 12.45/6.97 ( ( sgn_sgn_int @ one_one_int )
% 12.45/6.97 = one_one_int ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6621_sgn__1,axiom,
% 12.45/6.97 ( ( sgn_sgn_real @ one_one_real )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6622_sgn__1,axiom,
% 12.45/6.97 ( ( sgn_sgn_complex @ one_one_complex )
% 12.45/6.97 = one_one_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6623_sgn__1,axiom,
% 12.45/6.97 ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
% 12.45/6.97 = one_one_Code_integer ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6624_sgn__1,axiom,
% 12.45/6.97 ( ( sgn_sgn_rat @ one_one_rat )
% 12.45/6.97 = one_one_rat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6625_sgn__one,axiom,
% 12.45/6.97 ( ( sgn_sgn_real @ one_one_real )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6626_sgn__one,axiom,
% 12.45/6.97 ( ( sgn_sgn_complex @ one_one_complex )
% 12.45/6.97 = one_one_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6627_sgn__divide,axiom,
% 12.45/6.97 ! [A: complex,B2: complex] :
% 12.45/6.97 ( ( sgn_sgn_complex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.45/6.97 = ( divide1717551699836669952omplex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6628_sgn__divide,axiom,
% 12.45/6.97 ! [A: real,B2: real] :
% 12.45/6.97 ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.97 = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6629_sgn__divide,axiom,
% 12.45/6.97 ! [A: rat,B2: rat] :
% 12.45/6.97 ( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B2 ) )
% 12.45/6.97 = ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6630_idom__abs__sgn__class_Osgn__minus,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
% 12.45/6.97 = ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6631_idom__abs__sgn__class_Osgn__minus,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
% 12.45/6.97 = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6632_idom__abs__sgn__class_Osgn__minus,axiom,
% 12.45/6.97 ! [A: complex] :
% 12.45/6.97 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ A ) )
% 12.45/6.97 = ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6633_idom__abs__sgn__class_Osgn__minus,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.97 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6634_idom__abs__sgn__class_Osgn__minus,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
% 12.45/6.97 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6635_power__sgn,axiom,
% 12.45/6.97 ! [A: code_integer,N: nat] :
% 12.45/6.97 ( ( sgn_sgn_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 12.45/6.97 = ( power_8256067586552552935nteger @ ( sgn_sgn_Code_integer @ A ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6636_power__sgn,axiom,
% 12.45/6.97 ! [A: rat,N: nat] :
% 12.45/6.97 ( ( sgn_sgn_rat @ ( power_power_rat @ A @ N ) )
% 12.45/6.97 = ( power_power_rat @ ( sgn_sgn_rat @ A ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6637_power__sgn,axiom,
% 12.45/6.97 ! [A: real,N: nat] :
% 12.45/6.97 ( ( sgn_sgn_real @ ( power_power_real @ A @ N ) )
% 12.45/6.97 = ( power_power_real @ ( sgn_sgn_real @ A ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6638_power__sgn,axiom,
% 12.45/6.97 ! [A: int,N: nat] :
% 12.45/6.97 ( ( sgn_sgn_int @ ( power_power_int @ A @ N ) )
% 12.45/6.97 = ( power_power_int @ ( sgn_sgn_int @ A ) @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6639_sgn__greater,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
% 12.45/6.97 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6640_sgn__greater,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 12.45/6.97 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6641_sgn__greater,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
% 12.45/6.97 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6642_sgn__greater,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
% 12.45/6.97 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6643_sgn__less,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 12.45/6.97 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6644_sgn__less,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 12.45/6.97 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6645_sgn__less,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
% 12.45/6.97 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6646_sgn__less,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
% 12.45/6.97 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6647_divide__sgn,axiom,
% 12.45/6.97 ! [A: real,B2: real] :
% 12.45/6.97 ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B2 ) )
% 12.45/6.97 = ( times_times_real @ A @ ( sgn_sgn_real @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6648_divide__sgn,axiom,
% 12.45/6.97 ! [A: rat,B2: rat] :
% 12.45/6.97 ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B2 ) )
% 12.45/6.97 = ( times_times_rat @ A @ ( sgn_sgn_rat @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6649_fact__0,axiom,
% 12.45/6.97 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 12.45/6.97 = one_one_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6650_fact__0,axiom,
% 12.45/6.97 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 12.45/6.97 = one_one_rat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6651_fact__0,axiom,
% 12.45/6.97 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 12.45/6.97 = one_one_int ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6652_fact__0,axiom,
% 12.45/6.97 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 12.45/6.97 = one_one_nat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6653_fact__0,axiom,
% 12.45/6.97 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6654_fact__1,axiom,
% 12.45/6.97 ( ( semiri5044797733671781792omplex @ one_one_nat )
% 12.45/6.97 = one_one_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6655_fact__1,axiom,
% 12.45/6.97 ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 12.45/6.97 = one_one_rat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6656_fact__1,axiom,
% 12.45/6.97 ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 12.45/6.97 = one_one_int ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6657_fact__1,axiom,
% 12.45/6.97 ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 12.45/6.97 = one_one_nat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6658_fact__1,axiom,
% 12.45/6.97 ( ( semiri2265585572941072030t_real @ one_one_nat )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6659_arccos__1,axiom,
% 12.45/6.97 ( ( arccos @ one_one_real )
% 12.45/6.97 = zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6660_sgn__pos,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 12.45/6.97 => ( ( sgn_sgn_Code_integer @ A )
% 12.45/6.97 = one_one_Code_integer ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6661_sgn__pos,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.97 => ( ( sgn_sgn_real @ A )
% 12.45/6.97 = one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6662_sgn__pos,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.97 => ( ( sgn_sgn_rat @ A )
% 12.45/6.97 = one_one_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6663_sgn__pos,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/6.97 => ( ( sgn_sgn_int @ A )
% 12.45/6.97 = one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6664_abs__sgn__eq__1,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( A != zero_z3403309356797280102nteger )
% 12.45/6.97 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 12.45/6.97 = one_one_Code_integer ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6665_abs__sgn__eq__1,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( A != zero_zero_real )
% 12.45/6.97 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 12.45/6.97 = one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6666_abs__sgn__eq__1,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( A != zero_zero_rat )
% 12.45/6.97 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 12.45/6.97 = one_one_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6667_abs__sgn__eq__1,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( A != zero_zero_int )
% 12.45/6.97 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 12.45/6.97 = one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6668_fact__Suc__0,axiom,
% 12.45/6.97 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 12.45/6.97 = one_one_complex ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6669_fact__Suc__0,axiom,
% 12.45/6.97 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 12.45/6.97 = one_one_rat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6670_fact__Suc__0,axiom,
% 12.45/6.97 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 12.45/6.97 = one_one_int ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6671_fact__Suc__0,axiom,
% 12.45/6.97 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 12.45/6.97 = one_one_nat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6672_fact__Suc__0,axiom,
% 12.45/6.97 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 12.45/6.97 = one_one_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6673_fact__Suc,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6674_fact__Suc,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6675_fact__Suc,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6676_fact__Suc,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6677_fact__Suc,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 12.45/6.97 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6678_sgn__mult__self__eq,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
% 12.45/6.97 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6679_sgn__mult__self__eq,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 12.45/6.97 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6680_sgn__mult__self__eq,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
% 12.45/6.97 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6681_sgn__mult__self__eq,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 12.45/6.97 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6682_idom__abs__sgn__class_Oabs__sgn,axiom,
% 12.45/6.97 ! [A: complex] :
% 12.45/6.97 ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
% 12.45/6.97 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6683_idom__abs__sgn__class_Oabs__sgn,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
% 12.45/6.97 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6684_idom__abs__sgn__class_Oabs__sgn,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
% 12.45/6.97 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6685_idom__abs__sgn__class_Oabs__sgn,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
% 12.45/6.97 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6686_idom__abs__sgn__class_Oabs__sgn,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 12.45/6.97 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6687_sgn__abs,axiom,
% 12.45/6.97 ! [A: complex] :
% 12.45/6.97 ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
% 12.45/6.97 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6688_sgn__abs,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 12.45/6.97 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6689_sgn__abs,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 12.45/6.97 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6690_sgn__abs,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 12.45/6.97 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6691_sgn__abs,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 12.45/6.97 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6692_arccos__minus__1,axiom,
% 12.45/6.97 ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.97 = pi ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6693_sgn__mult__dvd__iff,axiom,
% 12.45/6.97 ! [R2: int,L2: int,K: int] :
% 12.45/6.97 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L2 ) @ K )
% 12.45/6.97 = ( ( dvd_dvd_int @ L2 @ K )
% 12.45/6.97 & ( ( R2 = zero_zero_int )
% 12.45/6.97 => ( K = zero_zero_int ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6694_mult__sgn__dvd__iff,axiom,
% 12.45/6.97 ! [L2: int,R2: int,K: int] :
% 12.45/6.97 ( ( dvd_dvd_int @ ( times_times_int @ L2 @ ( sgn_sgn_int @ R2 ) ) @ K )
% 12.45/6.97 = ( ( dvd_dvd_int @ L2 @ K )
% 12.45/6.97 & ( ( R2 = zero_zero_int )
% 12.45/6.97 => ( K = zero_zero_int ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6695_dvd__sgn__mult__iff,axiom,
% 12.45/6.97 ! [L2: int,R2: int,K: int] :
% 12.45/6.97 ( ( dvd_dvd_int @ L2 @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
% 12.45/6.97 = ( ( dvd_dvd_int @ L2 @ K )
% 12.45/6.97 | ( R2 = zero_zero_int ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6696_dvd__mult__sgn__iff,axiom,
% 12.45/6.97 ! [L2: int,K: int,R2: int] :
% 12.45/6.97 ( ( dvd_dvd_int @ L2 @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
% 12.45/6.97 = ( ( dvd_dvd_int @ L2 @ K )
% 12.45/6.97 | ( R2 = zero_zero_int ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6697_sgn__neg,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.97 => ( ( sgn_sgn_real @ A )
% 12.45/6.97 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6698_sgn__neg,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( ord_less_int @ A @ zero_zero_int )
% 12.45/6.97 => ( ( sgn_sgn_int @ A )
% 12.45/6.97 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6699_sgn__neg,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 12.45/6.97 => ( ( sgn_sgn_Code_integer @ A )
% 12.45/6.97 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6700_sgn__neg,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.97 => ( ( sgn_sgn_rat @ A )
% 12.45/6.97 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6701_fact__2,axiom,
% 12.45/6.97 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6702_fact__2,axiom,
% 12.45/6.97 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6703_fact__2,axiom,
% 12.45/6.97 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6704_fact__2,axiom,
% 12.45/6.97 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6705_fact__2,axiom,
% 12.45/6.97 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.97 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6706_sgn__of__nat,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.45/6.97 = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6707_sgn__of__nat,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
% 12.45/6.97 = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6708_sgn__of__nat,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
% 12.45/6.97 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6709_sgn__of__nat,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 12.45/6.97 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6710_cos__arccos,axiom,
% 12.45/6.97 ! [Y3: real] :
% 12.45/6.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.97 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.97 => ( ( cos_real @ ( arccos @ Y3 ) )
% 12.45/6.97 = Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6711_arccos__0,axiom,
% 12.45/6.97 ( ( arccos @ zero_zero_real )
% 12.45/6.97 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6712_complex__exp__exists,axiom,
% 12.45/6.97 ! [Z: complex] :
% 12.45/6.97 ? [A5: complex,R3: real] :
% 12.45/6.97 ( Z
% 12.45/6.97 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( exp_complex @ A5 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6713_fact__mono__nat,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6714_fact__ge__self,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6715_sgn__eq__0__iff,axiom,
% 12.45/6.97 ! [A: complex] :
% 12.45/6.97 ( ( ( sgn_sgn_complex @ A )
% 12.45/6.97 = zero_zero_complex )
% 12.45/6.97 = ( A = zero_zero_complex ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6716_sgn__eq__0__iff,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( ( sgn_sgn_Code_integer @ A )
% 12.45/6.97 = zero_z3403309356797280102nteger )
% 12.45/6.97 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6717_sgn__eq__0__iff,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( ( sgn_sgn_real @ A )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 = ( A = zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6718_sgn__eq__0__iff,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( ( sgn_sgn_rat @ A )
% 12.45/6.97 = zero_zero_rat )
% 12.45/6.97 = ( A = zero_zero_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6719_sgn__eq__0__iff,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( ( sgn_sgn_int @ A )
% 12.45/6.97 = zero_zero_int )
% 12.45/6.97 = ( A = zero_zero_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6720_sgn__0__0,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( ( sgn_sgn_Code_integer @ A )
% 12.45/6.97 = zero_z3403309356797280102nteger )
% 12.45/6.97 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6721_sgn__0__0,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( ( sgn_sgn_real @ A )
% 12.45/6.97 = zero_zero_real )
% 12.45/6.97 = ( A = zero_zero_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6722_sgn__0__0,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( ( sgn_sgn_rat @ A )
% 12.45/6.97 = zero_zero_rat )
% 12.45/6.97 = ( A = zero_zero_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6723_sgn__0__0,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( ( sgn_sgn_int @ A )
% 12.45/6.97 = zero_zero_int )
% 12.45/6.97 = ( A = zero_zero_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6724_Real__Vector__Spaces_Osgn__mult,axiom,
% 12.45/6.97 ! [X4: complex,Y3: complex] :
% 12.45/6.97 ( ( sgn_sgn_complex @ ( times_times_complex @ X4 @ Y3 ) )
% 12.45/6.97 = ( times_times_complex @ ( sgn_sgn_complex @ X4 ) @ ( sgn_sgn_complex @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6725_Real__Vector__Spaces_Osgn__mult,axiom,
% 12.45/6.97 ! [X4: real,Y3: real] :
% 12.45/6.97 ( ( sgn_sgn_real @ ( times_times_real @ X4 @ Y3 ) )
% 12.45/6.97 = ( times_times_real @ ( sgn_sgn_real @ X4 ) @ ( sgn_sgn_real @ Y3 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6726_sgn__mult,axiom,
% 12.45/6.97 ! [A: complex,B2: complex] :
% 12.45/6.97 ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B2 ) )
% 12.45/6.97 = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6727_sgn__mult,axiom,
% 12.45/6.97 ! [A: code_integer,B2: code_integer] :
% 12.45/6.97 ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 12.45/6.97 = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6728_sgn__mult,axiom,
% 12.45/6.97 ! [A: real,B2: real] :
% 12.45/6.97 ( ( sgn_sgn_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.97 = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6729_sgn__mult,axiom,
% 12.45/6.97 ! [A: rat,B2: rat] :
% 12.45/6.97 ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.97 = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6730_sgn__mult,axiom,
% 12.45/6.97 ! [A: int,B2: int] :
% 12.45/6.97 ( ( sgn_sgn_int @ ( times_times_int @ A @ B2 ) )
% 12.45/6.97 = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B2 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6731_same__sgn__sgn__add,axiom,
% 12.45/6.97 ! [B2: code_integer,A: code_integer] :
% 12.45/6.97 ( ( ( sgn_sgn_Code_integer @ B2 )
% 12.45/6.97 = ( sgn_sgn_Code_integer @ A ) )
% 12.45/6.97 => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 12.45/6.97 = ( sgn_sgn_Code_integer @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6732_same__sgn__sgn__add,axiom,
% 12.45/6.97 ! [B2: real,A: real] :
% 12.45/6.97 ( ( ( sgn_sgn_real @ B2 )
% 12.45/6.97 = ( sgn_sgn_real @ A ) )
% 12.45/6.97 => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B2 ) )
% 12.45/6.97 = ( sgn_sgn_real @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6733_same__sgn__sgn__add,axiom,
% 12.45/6.97 ! [B2: rat,A: rat] :
% 12.45/6.97 ( ( ( sgn_sgn_rat @ B2 )
% 12.45/6.97 = ( sgn_sgn_rat @ A ) )
% 12.45/6.97 => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B2 ) )
% 12.45/6.97 = ( sgn_sgn_rat @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6734_same__sgn__sgn__add,axiom,
% 12.45/6.97 ! [B2: int,A: int] :
% 12.45/6.97 ( ( ( sgn_sgn_int @ B2 )
% 12.45/6.97 = ( sgn_sgn_int @ A ) )
% 12.45/6.97 => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B2 ) )
% 12.45/6.97 = ( sgn_sgn_int @ A ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6735_fact__less__mono__nat,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.97 => ( ( ord_less_nat @ M @ N )
% 12.45/6.97 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6736_sgn__not__eq__imp,axiom,
% 12.45/6.97 ! [B2: real,A: real] :
% 12.45/6.97 ( ( ( sgn_sgn_real @ B2 )
% 12.45/6.97 != ( sgn_sgn_real @ A ) )
% 12.45/6.97 => ( ( ( sgn_sgn_real @ A )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( ( sgn_sgn_real @ B2 )
% 12.45/6.97 != zero_zero_real )
% 12.45/6.97 => ( ( sgn_sgn_real @ A )
% 12.45/6.97 = ( uminus_uminus_real @ ( sgn_sgn_real @ B2 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6737_sgn__not__eq__imp,axiom,
% 12.45/6.97 ! [B2: int,A: int] :
% 12.45/6.97 ( ( ( sgn_sgn_int @ B2 )
% 12.45/6.97 != ( sgn_sgn_int @ A ) )
% 12.45/6.97 => ( ( ( sgn_sgn_int @ A )
% 12.45/6.97 != zero_zero_int )
% 12.45/6.97 => ( ( ( sgn_sgn_int @ B2 )
% 12.45/6.97 != zero_zero_int )
% 12.45/6.97 => ( ( sgn_sgn_int @ A )
% 12.45/6.97 = ( uminus_uminus_int @ ( sgn_sgn_int @ B2 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6738_sgn__not__eq__imp,axiom,
% 12.45/6.97 ! [B2: code_integer,A: code_integer] :
% 12.45/6.97 ( ( ( sgn_sgn_Code_integer @ B2 )
% 12.45/6.97 != ( sgn_sgn_Code_integer @ A ) )
% 12.45/6.97 => ( ( ( sgn_sgn_Code_integer @ A )
% 12.45/6.97 != zero_z3403309356797280102nteger )
% 12.45/6.97 => ( ( ( sgn_sgn_Code_integer @ B2 )
% 12.45/6.97 != zero_z3403309356797280102nteger )
% 12.45/6.97 => ( ( sgn_sgn_Code_integer @ A )
% 12.45/6.97 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B2 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6739_sgn__not__eq__imp,axiom,
% 12.45/6.97 ! [B2: rat,A: rat] :
% 12.45/6.97 ( ( ( sgn_sgn_rat @ B2 )
% 12.45/6.97 != ( sgn_sgn_rat @ A ) )
% 12.45/6.97 => ( ( ( sgn_sgn_rat @ A )
% 12.45/6.97 != zero_zero_rat )
% 12.45/6.97 => ( ( ( sgn_sgn_rat @ B2 )
% 12.45/6.97 != zero_zero_rat )
% 12.45/6.97 => ( ( sgn_sgn_rat @ A )
% 12.45/6.97 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B2 ) ) ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6740_sgn__minus__1,axiom,
% 12.45/6.97 ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.97 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6741_sgn__minus__1,axiom,
% 12.45/6.97 ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.97 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6742_sgn__minus__1,axiom,
% 12.45/6.97 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.45/6.97 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6743_sgn__minus__1,axiom,
% 12.45/6.97 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.97 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6744_sgn__minus__1,axiom,
% 12.45/6.97 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.97 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6745_fact__ge__zero,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6746_fact__ge__zero,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6747_fact__ge__zero,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6748_fact__ge__zero,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6749_linordered__idom__class_Oabs__sgn,axiom,
% 12.45/6.97 ( abs_abs_Code_integer
% 12.45/6.97 = ( ^ [K3: code_integer] : ( times_3573771949741848930nteger @ K3 @ ( sgn_sgn_Code_integer @ K3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6750_linordered__idom__class_Oabs__sgn,axiom,
% 12.45/6.97 ( abs_abs_real
% 12.45/6.97 = ( ^ [K3: real] : ( times_times_real @ K3 @ ( sgn_sgn_real @ K3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6751_linordered__idom__class_Oabs__sgn,axiom,
% 12.45/6.97 ( abs_abs_rat
% 12.45/6.97 = ( ^ [K3: rat] : ( times_times_rat @ K3 @ ( sgn_sgn_rat @ K3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6752_linordered__idom__class_Oabs__sgn,axiom,
% 12.45/6.97 ( abs_abs_int
% 12.45/6.97 = ( ^ [K3: int] : ( times_times_int @ K3 @ ( sgn_sgn_int @ K3 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6753_abs__mult__sgn,axiom,
% 12.45/6.97 ! [A: complex] :
% 12.45/6.97 ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
% 12.45/6.97 = A ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6754_abs__mult__sgn,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 12.45/6.97 = A ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6755_abs__mult__sgn,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
% 12.45/6.97 = A ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6756_abs__mult__sgn,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 12.45/6.97 = A ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6757_abs__mult__sgn,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
% 12.45/6.97 = A ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6758_sgn__mult__abs,axiom,
% 12.45/6.97 ! [A: complex] :
% 12.45/6.97 ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
% 12.45/6.97 = A ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6759_sgn__mult__abs,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 12.45/6.97 = A ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6760_sgn__mult__abs,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
% 12.45/6.97 = A ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6761_sgn__mult__abs,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
% 12.45/6.97 = A ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6762_sgn__mult__abs,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
% 12.45/6.97 = A ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6763_mult__sgn__abs,axiom,
% 12.45/6.97 ! [X4: code_integer] :
% 12.45/6.97 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X4 ) @ ( abs_abs_Code_integer @ X4 ) )
% 12.45/6.97 = X4 ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6764_mult__sgn__abs,axiom,
% 12.45/6.97 ! [X4: real] :
% 12.45/6.97 ( ( times_times_real @ ( sgn_sgn_real @ X4 ) @ ( abs_abs_real @ X4 ) )
% 12.45/6.97 = X4 ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6765_mult__sgn__abs,axiom,
% 12.45/6.97 ! [X4: rat] :
% 12.45/6.97 ( ( times_times_rat @ ( sgn_sgn_rat @ X4 ) @ ( abs_abs_rat @ X4 ) )
% 12.45/6.97 = X4 ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6766_mult__sgn__abs,axiom,
% 12.45/6.97 ! [X4: int] :
% 12.45/6.97 ( ( times_times_int @ ( sgn_sgn_int @ X4 ) @ ( abs_abs_int @ X4 ) )
% 12.45/6.97 = X4 ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6767_fact__not__neg,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6768_fact__not__neg,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6769_fact__not__neg,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6770_fact__not__neg,axiom,
% 12.45/6.97 ! [N: nat] :
% 12.45/6.97 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6771_fact__gt__zero,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6772_fact__gt__zero,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6773_fact__gt__zero,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6774_fact__gt__zero,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6775_int__sgnE,axiom,
% 12.45/6.97 ! [K: int] :
% 12.45/6.97 ~ ! [N4: nat,L3: int] :
% 12.45/6.97 ( K
% 12.45/6.97 != ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6776_same__sgn__abs__add,axiom,
% 12.45/6.97 ! [B2: code_integer,A: code_integer] :
% 12.45/6.97 ( ( ( sgn_sgn_Code_integer @ B2 )
% 12.45/6.97 = ( sgn_sgn_Code_integer @ A ) )
% 12.45/6.97 => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 12.45/6.97 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6777_same__sgn__abs__add,axiom,
% 12.45/6.97 ! [B2: real,A: real] :
% 12.45/6.97 ( ( ( sgn_sgn_real @ B2 )
% 12.45/6.97 = ( sgn_sgn_real @ A ) )
% 12.45/6.97 => ( ( abs_abs_real @ ( plus_plus_real @ A @ B2 ) )
% 12.45/6.97 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6778_same__sgn__abs__add,axiom,
% 12.45/6.97 ! [B2: rat,A: rat] :
% 12.45/6.97 ( ( ( sgn_sgn_rat @ B2 )
% 12.45/6.97 = ( sgn_sgn_rat @ A ) )
% 12.45/6.97 => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B2 ) )
% 12.45/6.97 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6779_same__sgn__abs__add,axiom,
% 12.45/6.97 ! [B2: int,A: int] :
% 12.45/6.97 ( ( ( sgn_sgn_int @ B2 )
% 12.45/6.97 = ( sgn_sgn_int @ A ) )
% 12.45/6.97 => ( ( abs_abs_int @ ( plus_plus_int @ A @ B2 ) )
% 12.45/6.97 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6780_fact__ge__1,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6781_fact__ge__1,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6782_fact__ge__1,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6783_fact__ge__1,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6784_fact__mono,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6785_fact__mono,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6786_fact__mono,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6787_fact__mono,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6788_div__eq__sgn__abs,axiom,
% 12.45/6.97 ! [K: int,L2: int] :
% 12.45/6.97 ( ( ( sgn_sgn_int @ K )
% 12.45/6.97 = ( sgn_sgn_int @ L2 ) )
% 12.45/6.97 => ( ( divide_divide_int @ K @ L2 )
% 12.45/6.97 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6789_fact__dvd,axiom,
% 12.45/6.97 ! [N: nat,M: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.97 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6790_fact__dvd,axiom,
% 12.45/6.97 ! [N: nat,M: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.97 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6791_fact__dvd,axiom,
% 12.45/6.97 ! [N: nat,M: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.97 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6792_fact__dvd,axiom,
% 12.45/6.97 ! [N: nat,M: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.97 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6793_pochhammer__fact,axiom,
% 12.45/6.97 ( semiri5044797733671781792omplex
% 12.45/6.97 = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6794_pochhammer__fact,axiom,
% 12.45/6.97 ( semiri773545260158071498ct_rat
% 12.45/6.97 = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6795_pochhammer__fact,axiom,
% 12.45/6.97 ( semiri1406184849735516958ct_int
% 12.45/6.97 = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6796_pochhammer__fact,axiom,
% 12.45/6.97 ( semiri1408675320244567234ct_nat
% 12.45/6.97 = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6797_pochhammer__fact,axiom,
% 12.45/6.97 ( semiri2265585572941072030t_real
% 12.45/6.97 = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6798_fact__ge__Suc__0__nat,axiom,
% 12.45/6.97 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6799_sgn__1__pos,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( ( sgn_sgn_Code_integer @ A )
% 12.45/6.97 = one_one_Code_integer )
% 12.45/6.97 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6800_sgn__1__pos,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( ( sgn_sgn_real @ A )
% 12.45/6.97 = one_one_real )
% 12.45/6.97 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6801_sgn__1__pos,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( ( sgn_sgn_rat @ A )
% 12.45/6.97 = one_one_rat )
% 12.45/6.97 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6802_sgn__1__pos,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( ( sgn_sgn_int @ A )
% 12.45/6.97 = one_one_int )
% 12.45/6.97 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6803_abs__sgn__eq,axiom,
% 12.45/6.97 ! [A: code_integer] :
% 12.45/6.97 ( ( ( A = zero_z3403309356797280102nteger )
% 12.45/6.97 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 12.45/6.97 = zero_z3403309356797280102nteger ) )
% 12.45/6.97 & ( ( A != zero_z3403309356797280102nteger )
% 12.45/6.97 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 12.45/6.97 = one_one_Code_integer ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6804_abs__sgn__eq,axiom,
% 12.45/6.97 ! [A: real] :
% 12.45/6.97 ( ( ( A = zero_zero_real )
% 12.45/6.97 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 12.45/6.97 = zero_zero_real ) )
% 12.45/6.97 & ( ( A != zero_zero_real )
% 12.45/6.97 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 12.45/6.97 = one_one_real ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6805_abs__sgn__eq,axiom,
% 12.45/6.97 ! [A: rat] :
% 12.45/6.97 ( ( ( A = zero_zero_rat )
% 12.45/6.97 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 12.45/6.97 = zero_zero_rat ) )
% 12.45/6.97 & ( ( A != zero_zero_rat )
% 12.45/6.97 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 12.45/6.97 = one_one_rat ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6806_abs__sgn__eq,axiom,
% 12.45/6.97 ! [A: int] :
% 12.45/6.97 ( ( ( A = zero_zero_int )
% 12.45/6.97 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 12.45/6.97 = zero_zero_int ) )
% 12.45/6.97 & ( ( A != zero_zero_int )
% 12.45/6.97 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 12.45/6.97 = one_one_int ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6807_dvd__fact,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 12.45/6.97 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.97 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6808_fact__less__mono,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.97 => ( ( ord_less_nat @ M @ N )
% 12.45/6.97 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6809_fact__less__mono,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.97 => ( ( ord_less_nat @ M @ N )
% 12.45/6.97 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6810_fact__less__mono,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.97 => ( ( ord_less_nat @ M @ N )
% 12.45/6.97 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6811_fact__less__mono,axiom,
% 12.45/6.97 ! [M: nat,N: nat] :
% 12.45/6.97 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/6.97 => ( ( ord_less_nat @ M @ N )
% 12.45/6.97 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 12.45/6.97
% 12.45/6.97 thf(fact_6812_sgn__mod,axiom,
% 12.45/6.97 ! [L2: int,K: int] :
% 12.45/6.97 ( ( L2 != zero_zero_int )
% 12.45/6.97 => ( ~ ( dvd_dvd_int @ L2 @ K )
% 12.45/6.97 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
% 12.45/6.97 = ( sgn_sgn_int @ L2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6813_fact__mod,axiom,
% 12.45/6.98 ! [M: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.98 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 12.45/6.98 = zero_zero_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6814_fact__mod,axiom,
% 12.45/6.98 ! [M: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.98 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 12.45/6.98 = zero_z3403309356797280102nteger ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6815_fact__mod,axiom,
% 12.45/6.98 ! [M: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.98 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 12.45/6.98 = zero_zero_nat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6816_fact__fact__dvd__fact,axiom,
% 12.45/6.98 ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6817_fact__fact__dvd__fact,axiom,
% 12.45/6.98 ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6818_fact__fact__dvd__fact,axiom,
% 12.45/6.98 ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6819_fact__fact__dvd__fact,axiom,
% 12.45/6.98 ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6820_fact__fact__dvd__fact,axiom,
% 12.45/6.98 ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6821_fact__le__power,axiom,
% 12.45/6.98 ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6822_fact__le__power,axiom,
% 12.45/6.98 ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6823_fact__le__power,axiom,
% 12.45/6.98 ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6824_fact__le__power,axiom,
% 12.45/6.98 ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6825_arccos__le__arccos,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.98 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X4 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6826_arccos__eq__iff,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.98 & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
% 12.45/6.98 => ( ( ( arccos @ X4 )
% 12.45/6.98 = ( arccos @ Y3 ) )
% 12.45/6.98 = ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6827_arccos__le__mono,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.98 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 12.45/6.98 => ( ( ord_less_eq_real @ ( arccos @ X4 ) @ ( arccos @ Y3 ) )
% 12.45/6.98 = ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6828_fact__diff__Suc,axiom,
% 12.45/6.98 ! [N: nat,M: nat] :
% 12.45/6.98 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 12.45/6.98 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 12.45/6.98 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6829_sgn__if,axiom,
% 12.45/6.98 ( sgn_sgn_real
% 12.45/6.98 = ( ^ [X: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6830_sgn__if,axiom,
% 12.45/6.98 ( sgn_sgn_int
% 12.45/6.98 = ( ^ [X: int] : ( if_int @ ( X = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6831_sgn__if,axiom,
% 12.45/6.98 ( sgn_sgn_Code_integer
% 12.45/6.98 = ( ^ [X: code_integer] : ( if_Code_integer @ ( X = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6832_sgn__if,axiom,
% 12.45/6.98 ( sgn_sgn_rat
% 12.45/6.98 = ( ^ [X: rat] : ( if_rat @ ( X = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6833_sgn__1__neg,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ( sgn_sgn_real @ A )
% 12.45/6.98 = ( uminus_uminus_real @ one_one_real ) )
% 12.45/6.98 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6834_sgn__1__neg,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ( ( ( sgn_sgn_int @ A )
% 12.45/6.98 = ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.98 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6835_sgn__1__neg,axiom,
% 12.45/6.98 ! [A: code_integer] :
% 12.45/6.98 ( ( ( sgn_sgn_Code_integer @ A )
% 12.45/6.98 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.98 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6836_sgn__1__neg,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ( sgn_sgn_rat @ A )
% 12.45/6.98 = ( uminus_uminus_rat @ one_one_rat ) )
% 12.45/6.98 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6837_zsgn__def,axiom,
% 12.45/6.98 ( sgn_sgn_int
% 12.45/6.98 = ( ^ [I2: int] : ( if_int @ ( I2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6838_fact__div__fact__le__pow,axiom,
% 12.45/6.98 ! [R2: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ R2 @ N )
% 12.45/6.98 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6839_binomial__fact__lemma,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 12.45/6.98 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6840_norm__sgn,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ( X4 = zero_zero_real )
% 12.45/6.98 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X4 ) )
% 12.45/6.98 = zero_zero_real ) )
% 12.45/6.98 & ( ( X4 != zero_zero_real )
% 12.45/6.98 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X4 ) )
% 12.45/6.98 = one_one_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6841_norm__sgn,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( ( X4 = zero_zero_complex )
% 12.45/6.98 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X4 ) )
% 12.45/6.98 = zero_zero_real ) )
% 12.45/6.98 & ( ( X4 != zero_zero_complex )
% 12.45/6.98 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X4 ) )
% 12.45/6.98 = one_one_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6842_div__sgn__abs__cancel,axiom,
% 12.45/6.98 ! [V: int,K: int,L2: int] :
% 12.45/6.98 ( ( V != zero_zero_int )
% 12.45/6.98 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L2 ) ) )
% 12.45/6.98 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6843_div__dvd__sgn__abs,axiom,
% 12.45/6.98 ! [L2: int,K: int] :
% 12.45/6.98 ( ( dvd_dvd_int @ L2 @ K )
% 12.45/6.98 => ( ( divide_divide_int @ K @ L2 )
% 12.45/6.98 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6844_choose__dvd,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6845_choose__dvd,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6846_choose__dvd,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6847_choose__dvd,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6848_choose__dvd,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6849_fact__numeral,axiom,
% 12.45/6.98 ! [K: num] :
% 12.45/6.98 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 12.45/6.98 = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6850_fact__numeral,axiom,
% 12.45/6.98 ! [K: num] :
% 12.45/6.98 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 12.45/6.98 = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6851_fact__numeral,axiom,
% 12.45/6.98 ! [K: num] :
% 12.45/6.98 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 12.45/6.98 = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6852_fact__numeral,axiom,
% 12.45/6.98 ! [K: num] :
% 12.45/6.98 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 12.45/6.98 = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6853_fact__numeral,axiom,
% 12.45/6.98 ! [K: num] :
% 12.45/6.98 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 12.45/6.98 = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6854_arccos__lbound,axiom,
% 12.45/6.98 ! [Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.98 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6855_arccos__less__arccos,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.98 => ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.98 => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X4 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6856_arccos__less__mono,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.98 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 12.45/6.98 => ( ( ord_less_real @ ( arccos @ X4 ) @ ( arccos @ Y3 ) )
% 12.45/6.98 = ( ord_less_real @ Y3 @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6857_arccos__ubound,axiom,
% 12.45/6.98 ! [Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.98 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6858_arccos__cos,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_real @ X4 @ pi )
% 12.45/6.98 => ( ( arccos @ ( cos_real @ X4 ) )
% 12.45/6.98 = X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6859_cos__arccos__abs,axiom,
% 12.45/6.98 ! [Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 12.45/6.98 => ( ( cos_real @ ( arccos @ Y3 ) )
% 12.45/6.98 = Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6860_arccos__cos__eq__abs,axiom,
% 12.45/6.98 ! [Theta: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 12.45/6.98 => ( ( arccos @ ( cos_real @ Theta ) )
% 12.45/6.98 = ( abs_abs_real @ Theta ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6861_binomial__altdef__nat,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( ( binomial @ N @ K )
% 12.45/6.98 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6862_square__fact__le__2__fact,axiom,
% 12.45/6.98 ! [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 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6863_arccos__lt__bounded,axiom,
% 12.45/6.98 ! [Y3: real] :
% 12.45/6.98 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.98 => ( ( ord_less_real @ Y3 @ one_one_real )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 12.45/6.98 & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6864_arccos__bounded,axiom,
% 12.45/6.98 ! [Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.98 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 12.45/6.98 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6865_sin__arccos__nonzero,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.98 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/6.98 => ( ( sin_real @ ( arccos @ X4 ) )
% 12.45/6.98 != zero_zero_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6866_arccos__cos2,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.98 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X4 )
% 12.45/6.98 => ( ( arccos @ ( cos_real @ X4 ) )
% 12.45/6.98 = ( uminus_uminus_real @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6867_arccos__minus,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.98 => ( ( arccos @ ( uminus_uminus_real @ X4 ) )
% 12.45/6.98 = ( minus_minus_real @ pi @ ( arccos @ X4 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6868_fact__num__eq__if,axiom,
% 12.45/6.98 ( semiri5044797733671781792omplex
% 12.45/6.98 = ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6869_fact__num__eq__if,axiom,
% 12.45/6.98 ( semiri773545260158071498ct_rat
% 12.45/6.98 = ( ^ [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 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6870_fact__num__eq__if,axiom,
% 12.45/6.98 ( semiri1406184849735516958ct_int
% 12.45/6.98 = ( ^ [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 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6871_fact__num__eq__if,axiom,
% 12.45/6.98 ( semiri1408675320244567234ct_nat
% 12.45/6.98 = ( ^ [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 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6872_fact__num__eq__if,axiom,
% 12.45/6.98 ( semiri2265585572941072030t_real
% 12.45/6.98 = ( ^ [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 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6873_fact__reduce,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.98 => ( ( semiri5044797733671781792omplex @ N )
% 12.45/6.98 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6874_fact__reduce,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.98 => ( ( semiri773545260158071498ct_rat @ N )
% 12.45/6.98 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6875_fact__reduce,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.98 => ( ( semiri1406184849735516958ct_int @ N )
% 12.45/6.98 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6876_fact__reduce,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.98 => ( ( semiri1408675320244567234ct_nat @ N )
% 12.45/6.98 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6877_fact__reduce,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.98 => ( ( semiri2265585572941072030t_real @ N )
% 12.45/6.98 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6878_pochhammer__same,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
% 12.45/6.98 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6879_pochhammer__same,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
% 12.45/6.98 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6880_pochhammer__same,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
% 12.45/6.98 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6881_pochhammer__same,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
% 12.45/6.98 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6882_pochhammer__same,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
% 12.45/6.98 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6883_binomial__fact,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 12.45/6.98 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6884_binomial__fact,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 12.45/6.98 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6885_binomial__fact,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 12.45/6.98 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6886_fact__binomial,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 12.45/6.98 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6887_fact__binomial,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 12.45/6.98 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6888_fact__binomial,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 12.45/6.98 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6889_arccos,axiom,
% 12.45/6.98 ! [Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.98 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 12.45/6.98 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
% 12.45/6.98 & ( ( cos_real @ ( arccos @ Y3 ) )
% 12.45/6.98 = Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6890_arccos__minus__abs,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.98 => ( ( arccos @ ( uminus_uminus_real @ X4 ) )
% 12.45/6.98 = ( minus_minus_real @ pi @ ( arccos @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6891_div__noneq__sgn__abs,axiom,
% 12.45/6.98 ! [L2: int,K: int] :
% 12.45/6.98 ( ( L2 != zero_zero_int )
% 12.45/6.98 => ( ( ( sgn_sgn_int @ K )
% 12.45/6.98 != ( sgn_sgn_int @ L2 ) )
% 12.45/6.98 => ( ( divide_divide_int @ K @ L2 )
% 12.45/6.98 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) )
% 12.45/6.98 @ ( zero_n2684676970156552555ol_int
% 12.45/6.98 @ ~ ( dvd_dvd_int @ L2 @ K ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6892_arccos__le__pi2,axiom,
% 12.45/6.98 ! [Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 12.45/6.98 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6893_divide__int__unfold,axiom,
% 12.45/6.98 ! [L2: int,K: int,N: nat,M: nat] :
% 12.45/6.98 ( ( ( ( ( sgn_sgn_int @ L2 )
% 12.45/6.98 = zero_zero_int )
% 12.45/6.98 | ( ( sgn_sgn_int @ K )
% 12.45/6.98 = zero_zero_int )
% 12.45/6.98 | ( N = zero_zero_nat ) )
% 12.45/6.98 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.45/6.98 = zero_zero_int ) )
% 12.45/6.98 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 12.45/6.98 = zero_zero_int )
% 12.45/6.98 | ( ( sgn_sgn_int @ K )
% 12.45/6.98 = zero_zero_int )
% 12.45/6.98 | ( N = zero_zero_nat ) )
% 12.45/6.98 => ( ( ( ( sgn_sgn_int @ K )
% 12.45/6.98 = ( sgn_sgn_int @ L2 ) )
% 12.45/6.98 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.45/6.98 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 12.45/6.98 & ( ( ( sgn_sgn_int @ K )
% 12.45/6.98 != ( sgn_sgn_int @ L2 ) )
% 12.45/6.98 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.45/6.98 = ( uminus_uminus_int
% 12.45/6.98 @ ( semiri1314217659103216013at_int
% 12.45/6.98 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 12.45/6.98 @ ( zero_n2687167440665602831ol_nat
% 12.45/6.98 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6894_sin__coeff__def,axiom,
% 12.45/6.98 ( sin_coeff
% 12.45/6.98 = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6895_binomial__code,axiom,
% 12.45/6.98 ( binomial
% 12.45/6.98 = ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6896_cos__coeff__def,axiom,
% 12.45/6.98 ( cos_coeff
% 12.45/6.98 = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6897_exp__two__pi__i,axiom,
% 12.45/6.98 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 12.45/6.98 = one_one_complex ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6898_exp__two__pi__i_H,axiom,
% 12.45/6.98 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 12.45/6.98 = one_one_complex ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6899_zero__le__sgn__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X4 ) )
% 12.45/6.98 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6900_sgn__le__0__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X4 ) @ zero_zero_real )
% 12.45/6.98 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6901_norm__ii,axiom,
% 12.45/6.98 ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 12.45/6.98 = one_one_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6902_cos__coeff__0,axiom,
% 12.45/6.98 ( ( cos_coeff @ zero_zero_nat )
% 12.45/6.98 = one_one_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6903_divide__i,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( divide1717551699836669952omplex @ X4 @ imaginary_unit )
% 12.45/6.98 = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6904_complex__i__mult__minus,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X4 ) )
% 12.45/6.98 = ( uminus1482373934393186551omplex @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6905_i__squared,axiom,
% 12.45/6.98 ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 12.45/6.98 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6906_divide__numeral__i,axiom,
% 12.45/6.98 ! [Z: complex,N: num] :
% 12.45/6.98 ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 12.45/6.98 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6907_power2__i,axiom,
% 12.45/6.98 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.98 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6908_i__even__power,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.98 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6909_exp__pi__i_H,axiom,
% 12.45/6.98 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 12.45/6.98 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6910_exp__pi__i,axiom,
% 12.45/6.98 ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 12.45/6.98 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6911_real__sgn__eq,axiom,
% 12.45/6.98 ( sgn_sgn_real
% 12.45/6.98 = ( ^ [X: real] : ( divide_divide_real @ X @ ( abs_abs_real @ X ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6912_sgn__eq,axiom,
% 12.45/6.98 ( sgn_sgn_complex
% 12.45/6.98 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ Z5 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z5 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6913_sin__coeff__Suc,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( sin_coeff @ ( suc @ N ) )
% 12.45/6.98 = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6914_i__times__eq__iff,axiom,
% 12.45/6.98 ! [W: complex,Z: complex] :
% 12.45/6.98 ( ( ( times_times_complex @ imaginary_unit @ W )
% 12.45/6.98 = Z )
% 12.45/6.98 = ( W
% 12.45/6.98 = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6915_cos__coeff__Suc,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( cos_coeff @ ( suc @ N ) )
% 12.45/6.98 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6916_imaginary__unit_Ocode,axiom,
% 12.45/6.98 ( imaginary_unit
% 12.45/6.98 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6917_Complex__eq__i,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ( complex2 @ X4 @ Y3 )
% 12.45/6.98 = imaginary_unit )
% 12.45/6.98 = ( ( X4 = zero_zero_real )
% 12.45/6.98 & ( Y3 = one_one_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6918_i__mult__Complex,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B2 ) )
% 12.45/6.98 = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6919_Complex__mult__i,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ imaginary_unit )
% 12.45/6.98 = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6920_sgn__real__def,axiom,
% 12.45/6.98 ( sgn_sgn_real
% 12.45/6.98 = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6921_sgn__power__injE,axiom,
% 12.45/6.98 ! [A: real,N: nat,X4: real,B2: real] :
% 12.45/6.98 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 12.45/6.98 = X4 )
% 12.45/6.98 => ( ( X4
% 12.45/6.98 = ( times_times_real @ ( sgn_sgn_real @ B2 ) @ ( power_power_real @ ( abs_abs_real @ B2 ) @ N ) ) )
% 12.45/6.98 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.98 => ( A = B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6922_fold__atLeastAtMost__nat_Osimps,axiom,
% 12.45/6.98 ( set_fo2584398358068434914at_nat
% 12.45/6.98 = ( ^ [F2: nat > nat > nat,A4: nat,B3: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B3 @ A4 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B3 @ ( F2 @ A4 @ Acc ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6923_fold__atLeastAtMost__nat_Oelims,axiom,
% 12.45/6.98 ! [X4: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y3: nat] :
% 12.45/6.98 ( ( ( set_fo2584398358068434914at_nat @ X4 @ Xa @ Xb @ Xc )
% 12.45/6.98 = Y3 )
% 12.45/6.98 => ( ( ( ord_less_nat @ Xb @ Xa )
% 12.45/6.98 => ( Y3 = Xc ) )
% 12.45/6.98 & ( ~ ( ord_less_nat @ Xb @ Xa )
% 12.45/6.98 => ( Y3
% 12.45/6.98 = ( set_fo2584398358068434914at_nat @ X4 @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X4 @ Xa @ Xc ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6924_complex__of__real__i,axiom,
% 12.45/6.98 ! [R2: real] :
% 12.45/6.98 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
% 12.45/6.98 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6925_i__complex__of__real,axiom,
% 12.45/6.98 ! [R2: real] :
% 12.45/6.98 ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
% 12.45/6.98 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6926_Complex__eq,axiom,
% 12.45/6.98 ( complex2
% 12.45/6.98 = ( ^ [A4: real,B3: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A4 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6927_complex__split__polar,axiom,
% 12.45/6.98 ! [Z: complex] :
% 12.45/6.98 ? [R3: real,A5: real] :
% 12.45/6.98 ( Z
% 12.45/6.98 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A5 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A5 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6928_cmod__unit__one,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 12.45/6.98 = one_one_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6929_cmod__complex__polar,axiom,
% 12.45/6.98 ! [R2: real,A: real] :
% 12.45/6.98 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
% 12.45/6.98 = ( abs_abs_real @ R2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6930_arctan__inverse,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( X4 != zero_zero_real )
% 12.45/6.98 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X4 ) )
% 12.45/6.98 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X4 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6931_fact__code,axiom,
% 12.45/6.98 ( semiri5044797733671781792omplex
% 12.45/6.98 = ( ^ [N2: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6932_fact__code,axiom,
% 12.45/6.98 ( semiri773545260158071498ct_rat
% 12.45/6.98 = ( ^ [N2: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6933_fact__code,axiom,
% 12.45/6.98 ( semiri1406184849735516958ct_int
% 12.45/6.98 = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6934_fact__code,axiom,
% 12.45/6.98 ( semiri1408675320244567234ct_nat
% 12.45/6.98 = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6935_fact__code,axiom,
% 12.45/6.98 ( semiri2265585572941072030t_real
% 12.45/6.98 = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6936_Arg__minus__ii,axiom,
% 12.45/6.98 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 12.45/6.98 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6937_csqrt__ii,axiom,
% 12.45/6.98 ( ( csqrt @ imaginary_unit )
% 12.45/6.98 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6938_Arg__ii,axiom,
% 12.45/6.98 ( ( arg @ imaginary_unit )
% 12.45/6.98 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6939_cis__minus__pi__half,axiom,
% 12.45/6.98 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.45/6.98 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6940_modulo__int__def,axiom,
% 12.45/6.98 ( modulo_modulo_int
% 12.45/6.98 = ( ^ [K3: int,L: int] :
% 12.45/6.98 ( if_int @ ( L = zero_zero_int ) @ K3
% 12.45/6.98 @ ( if_int
% 12.45/6.98 @ ( ( sgn_sgn_int @ K3 )
% 12.45/6.98 = ( sgn_sgn_int @ L ) )
% 12.45/6.98 @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
% 12.45/6.98 @ ( times_times_int @ ( sgn_sgn_int @ L )
% 12.45/6.98 @ ( minus_minus_int
% 12.45/6.98 @ ( times_times_int @ ( abs_abs_int @ L )
% 12.45/6.98 @ ( zero_n2684676970156552555ol_int
% 12.45/6.98 @ ~ ( dvd_dvd_int @ L @ K3 ) ) )
% 12.45/6.98 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6941_nat__numeral,axiom,
% 12.45/6.98 ! [K: num] :
% 12.45/6.98 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 12.45/6.98 = ( numeral_numeral_nat @ K ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6942_norm__cis,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 12.45/6.98 = one_one_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6943_nat__1,axiom,
% 12.45/6.98 ( ( nat2 @ one_one_int )
% 12.45/6.98 = ( suc @ zero_zero_nat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6944_zless__nat__conj,axiom,
% 12.45/6.98 ! [W: int,Z: int] :
% 12.45/6.98 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 12.45/6.98 = ( ( ord_less_int @ zero_zero_int @ Z )
% 12.45/6.98 & ( ord_less_int @ W @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6945_of__nat__nat__take__bit__eq,axiom,
% 12.45/6.98 ! [N: nat,K: int] :
% 12.45/6.98 ( ( semiri8010041392384452111omplex @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 12.45/6.98 = ( ring_17405671764205052669omplex @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6946_of__nat__nat__take__bit__eq,axiom,
% 12.45/6.98 ! [N: nat,K: int] :
% 12.45/6.98 ( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 12.45/6.98 = ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6947_of__nat__nat__take__bit__eq,axiom,
% 12.45/6.98 ! [N: nat,K: int] :
% 12.45/6.98 ( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 12.45/6.98 = ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6948_of__nat__nat__take__bit__eq,axiom,
% 12.45/6.98 ! [N: nat,K: int] :
% 12.45/6.98 ( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 12.45/6.98 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6949_zero__less__nat__eq,axiom,
% 12.45/6.98 ! [Z: int] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 12.45/6.98 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6950_diff__nat__numeral,axiom,
% 12.45/6.98 ! [V: num,V2: num] :
% 12.45/6.98 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V2 ) )
% 12.45/6.98 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6951_numeral__power__eq__nat__cancel__iff,axiom,
% 12.45/6.98 ! [X4: num,N: nat,Y3: int] :
% 12.45/6.98 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N )
% 12.45/6.98 = ( nat2 @ Y3 ) )
% 12.45/6.98 = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N )
% 12.45/6.98 = Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6952_nat__eq__numeral__power__cancel__iff,axiom,
% 12.45/6.98 ! [Y3: int,X4: num,N: nat] :
% 12.45/6.98 ( ( ( nat2 @ Y3 )
% 12.45/6.98 = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
% 12.45/6.98 = ( Y3
% 12.45/6.98 = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6953_power2__csqrt,axiom,
% 12.45/6.98 ! [Z: complex] :
% 12.45/6.98 ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.98 = Z ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6954_nat__ceiling__le__eq,axiom,
% 12.45/6.98 ! [X4: real,A: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X4 ) ) @ A )
% 12.45/6.98 = ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6955_one__less__nat__eq,axiom,
% 12.45/6.98 ! [Z: int] :
% 12.45/6.98 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 12.45/6.98 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6956_nat__numeral__diff__1,axiom,
% 12.45/6.98 ! [V: num] :
% 12.45/6.98 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 12.45/6.98 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6957_numeral__power__less__nat__cancel__iff,axiom,
% 12.45/6.98 ! [X4: num,N: nat,A: int] :
% 12.45/6.98 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) @ ( nat2 @ A ) )
% 12.45/6.98 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6958_nat__less__numeral__power__cancel__iff,axiom,
% 12.45/6.98 ! [A: int,X4: num,N: nat] :
% 12.45/6.98 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
% 12.45/6.98 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6959_numeral__power__le__nat__cancel__iff,axiom,
% 12.45/6.98 ! [X4: num,N: nat,A: int] :
% 12.45/6.98 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) @ ( nat2 @ A ) )
% 12.45/6.98 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6960_nat__le__numeral__power__cancel__iff,axiom,
% 12.45/6.98 ! [A: int,X4: num,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
% 12.45/6.98 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6961_cis__pi__half,axiom,
% 12.45/6.98 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.98 = imaginary_unit ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6962_cis__2pi,axiom,
% 12.45/6.98 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.45/6.98 = one_one_complex ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6963_cis__divide,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B2 ) )
% 12.45/6.98 = ( cis @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6964_nat__zero__as__int,axiom,
% 12.45/6.98 ( zero_zero_nat
% 12.45/6.98 = ( nat2 @ zero_zero_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6965_nat__numeral__as__int,axiom,
% 12.45/6.98 ( numeral_numeral_nat
% 12.45/6.98 = ( ^ [I2: num] : ( nat2 @ ( numeral_numeral_int @ I2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6966_nat__mono,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.98 => ( ord_less_eq_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6967_nat__one__as__int,axiom,
% 12.45/6.98 ( one_one_nat
% 12.45/6.98 = ( nat2 @ one_one_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6968_unset__bit__nat__def,axiom,
% 12.45/6.98 ( bit_se4205575877204974255it_nat
% 12.45/6.98 = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M6 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6969_DeMoivre,axiom,
% 12.45/6.98 ! [A: real,N: nat] :
% 12.45/6.98 ( ( power_power_complex @ ( cis @ A ) @ N )
% 12.45/6.98 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6970_nat__mask__eq,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 12.45/6.98 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6971_Arg__correct,axiom,
% 12.45/6.98 ! [Z: complex] :
% 12.45/6.98 ( ( Z != zero_zero_complex )
% 12.45/6.98 => ( ( ( sgn_sgn_complex @ Z )
% 12.45/6.98 = ( cis @ ( arg @ Z ) ) )
% 12.45/6.98 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 12.45/6.98 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6972_cis__Arg__unique,axiom,
% 12.45/6.98 ! [Z: complex,X4: real] :
% 12.45/6.98 ( ( ( sgn_sgn_complex @ Z )
% 12.45/6.98 = ( cis @ X4 ) )
% 12.45/6.98 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_real @ X4 @ pi )
% 12.45/6.98 => ( ( arg @ Z )
% 12.45/6.98 = X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6973_cis__mult,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B2 ) )
% 12.45/6.98 = ( cis @ ( plus_plus_real @ A @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6974_nat__mono__iff,axiom,
% 12.45/6.98 ! [Z: int,W: int] :
% 12.45/6.98 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.45/6.98 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 12.45/6.98 = ( ord_less_int @ W @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6975_of__nat__ceiling,axiom,
% 12.45/6.98 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6976_of__nat__ceiling,axiom,
% 12.45/6.98 ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6977_zless__nat__eq__int__zless,axiom,
% 12.45/6.98 ! [M: nat,Z: int] :
% 12.45/6.98 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 12.45/6.98 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6978_nat__le__iff,axiom,
% 12.45/6.98 ! [X4: int,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ ( nat2 @ X4 ) @ N )
% 12.45/6.98 = ( ord_less_eq_int @ X4 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6979_nat__int__add,axiom,
% 12.45/6.98 ! [A: nat,B2: nat] :
% 12.45/6.98 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
% 12.45/6.98 = ( plus_plus_nat @ A @ B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6980_int__minus,axiom,
% 12.45/6.98 ! [N: nat,M: nat] :
% 12.45/6.98 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 12.45/6.98 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6981_nat__abs__mult__distrib,axiom,
% 12.45/6.98 ! [W: int,Z: int] :
% 12.45/6.98 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 12.45/6.98 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6982_and__nat__def,axiom,
% 12.45/6.98 ( bit_se727722235901077358nd_nat
% 12.45/6.98 = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6983_real__nat__ceiling__ge,axiom,
% 12.45/6.98 ! [X4: real] : ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6984_of__nat__floor,axiom,
% 12.45/6.98 ! [R2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 12.45/6.98 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6985_of__nat__floor,axiom,
% 12.45/6.98 ! [R2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6986_nat__less__eq__zless,axiom,
% 12.45/6.98 ! [W: int,Z: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 12.45/6.98 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 12.45/6.98 = ( ord_less_int @ W @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6987_nat__le__eq__zle,axiom,
% 12.45/6.98 ! [W: int,Z: int] :
% 12.45/6.98 ( ( ( ord_less_int @ zero_zero_int @ W )
% 12.45/6.98 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 12.45/6.98 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 12.45/6.98 = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6988_split__nat,axiom,
% 12.45/6.98 ! [P: nat > $o,I: int] :
% 12.45/6.98 ( ( P @ ( nat2 @ I ) )
% 12.45/6.98 = ( ! [N2: nat] :
% 12.45/6.98 ( ( I
% 12.45/6.98 = ( semiri1314217659103216013at_int @ N2 ) )
% 12.45/6.98 => ( P @ N2 ) )
% 12.45/6.98 & ( ( ord_less_int @ I @ zero_zero_int )
% 12.45/6.98 => ( P @ zero_zero_nat ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6989_le__mult__nat__floor,axiom,
% 12.45/6.98 ! [A: real,B2: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6990_le__mult__nat__floor,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6991_le__nat__iff,axiom,
% 12.45/6.98 ! [K: int,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.98 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 12.45/6.98 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6992_nat__add__distrib,axiom,
% 12.45/6.98 ! [Z: int,Z6: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.45/6.98 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 12.45/6.98 => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
% 12.45/6.98 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6993_nat__mult__distrib,axiom,
% 12.45/6.98 ! [Z: int,Z6: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.45/6.98 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 12.45/6.98 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6994_Suc__as__int,axiom,
% 12.45/6.98 ( suc
% 12.45/6.98 = ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6995_nat__diff__distrib,axiom,
% 12.45/6.98 ! [Z6: int,Z: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 12.45/6.98 => ( ( ord_less_eq_int @ Z6 @ Z )
% 12.45/6.98 => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
% 12.45/6.98 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6996_nat__diff__distrib_H,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.98 => ( ( nat2 @ ( minus_minus_int @ X4 @ Y3 ) )
% 12.45/6.98 = ( minus_minus_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6997_nat__abs__triangle__ineq,axiom,
% 12.45/6.98 ! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6998_nat__div__distrib,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.98 => ( ( nat2 @ ( divide_divide_int @ X4 @ Y3 ) )
% 12.45/6.98 = ( divide_divide_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_6999_nat__div__distrib_H,axiom,
% 12.45/6.98 ! [Y3: int,X4: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.98 => ( ( nat2 @ ( divide_divide_int @ X4 @ Y3 ) )
% 12.45/6.98 = ( divide_divide_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7000_nat__power__eq,axiom,
% 12.45/6.98 ! [Z: int,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.45/6.98 => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 12.45/6.98 = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7001_nat__floor__neg,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.98 => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.98 = zero_zero_nat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7002_nat__mod__distrib,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.98 => ( ( nat2 @ ( modulo_modulo_int @ X4 @ Y3 ) )
% 12.45/6.98 = ( modulo_modulo_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7003_div__abs__eq__div__nat,axiom,
% 12.45/6.98 ! [K: int,L2: int] :
% 12.45/6.98 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
% 12.45/6.98 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7004_floor__eq3,axiom,
% 12.45/6.98 ! [N: nat,X4: real] :
% 12.45/6.98 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X4 )
% 12.45/6.98 => ( ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 12.45/6.98 => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.98 = N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7005_le__nat__floor,axiom,
% 12.45/6.98 ! [X4: nat,A: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ A )
% 12.45/6.98 => ( ord_less_eq_nat @ X4 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7006_mod__abs__eq__div__nat,axiom,
% 12.45/6.98 ! [K: int,L2: int] :
% 12.45/6.98 ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
% 12.45/6.98 = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7007_nat__take__bit__eq,axiom,
% 12.45/6.98 ! [K: int,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.98 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 12.45/6.98 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7008_take__bit__nat__eq,axiom,
% 12.45/6.98 ! [K: int,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.98 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 12.45/6.98 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7009_bit__nat__iff,axiom,
% 12.45/6.98 ! [K: int,N: nat] :
% 12.45/6.98 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 12.45/6.98 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.98 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7010_nat__2,axiom,
% 12.45/6.98 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.98 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7011_Suc__nat__eq__nat__zadd1,axiom,
% 12.45/6.98 ! [Z: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.45/6.98 => ( ( suc @ ( nat2 @ Z ) )
% 12.45/6.98 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7012_nat__less__iff,axiom,
% 12.45/6.98 ! [W: int,M: nat] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 12.45/6.98 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 12.45/6.98 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7013_nat__mult__distrib__neg,axiom,
% 12.45/6.98 ! [Z: int,Z6: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 12.45/6.98 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 12.45/6.98 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7014_nat__abs__int__diff,axiom,
% 12.45/6.98 ! [A: nat,B2: nat] :
% 12.45/6.98 ( ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.98 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
% 12.45/6.98 = ( minus_minus_nat @ B2 @ A ) ) )
% 12.45/6.98 & ( ~ ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.98 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
% 12.45/6.98 = ( minus_minus_nat @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7015_floor__eq4,axiom,
% 12.45/6.98 ! [N: nat,X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X4 )
% 12.45/6.98 => ( ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 12.45/6.98 => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
% 12.45/6.98 = N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7016_cis__conv__exp,axiom,
% 12.45/6.98 ( cis
% 12.45/6.98 = ( ^ [B3: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7017_of__int__of__nat,axiom,
% 12.45/6.98 ( ring_18347121197199848620nteger
% 12.45/6.98 = ( ^ [K3: int] : ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7018_of__int__of__nat,axiom,
% 12.45/6.98 ( ring_17405671764205052669omplex
% 12.45/6.98 = ( ^ [K3: int] : ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7019_of__int__of__nat,axiom,
% 12.45/6.98 ( ring_1_of_int_real
% 12.45/6.98 = ( ^ [K3: int] : ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7020_of__int__of__nat,axiom,
% 12.45/6.98 ( ring_1_of_int_rat
% 12.45/6.98 = ( ^ [K3: int] : ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7021_of__int__of__nat,axiom,
% 12.45/6.98 ( ring_1_of_int_int
% 12.45/6.98 = ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7022_of__real__sqrt,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X4 ) )
% 12.45/6.98 = ( csqrt @ ( real_V4546457046886955230omplex @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7023_Arg__bounded,axiom,
% 12.45/6.98 ! [Z: complex] :
% 12.45/6.98 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 12.45/6.98 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7024_even__nat__iff,axiom,
% 12.45/6.98 ! [K: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.98 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 12.45/6.98 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7025_powr__int,axiom,
% 12.45/6.98 ! [X4: real,I: int] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 12.45/6.98 => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ I ) )
% 12.45/6.98 = ( power_power_real @ X4 @ ( nat2 @ I ) ) ) )
% 12.45/6.98 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 12.45/6.98 => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ I ) )
% 12.45/6.98 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X4 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7026_divide__int__def,axiom,
% 12.45/6.98 ( divide_divide_int
% 12.45/6.98 = ( ^ [K3: int,L: int] :
% 12.45/6.98 ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
% 12.45/6.98 @ ( if_int
% 12.45/6.98 @ ( ( sgn_sgn_int @ K3 )
% 12.45/6.98 = ( sgn_sgn_int @ L ) )
% 12.45/6.98 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
% 12.45/6.98 @ ( uminus_uminus_int
% 12.45/6.98 @ ( semiri1314217659103216013at_int
% 12.45/6.98 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
% 12.45/6.98 @ ( zero_n2687167440665602831ol_nat
% 12.45/6.98 @ ~ ( dvd_dvd_int @ L @ K3 ) ) ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7027_cis__multiple__2pi,axiom,
% 12.45/6.98 ! [N: real] :
% 12.45/6.98 ( ( member_real @ N @ ring_1_Ints_real )
% 12.45/6.98 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 12.45/6.98 = one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7028_powr__real__of__int,axiom,
% 12.45/6.98 ! [X4: real,N: int] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 12.45/6.98 => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N ) )
% 12.45/6.98 = ( power_power_real @ X4 @ ( nat2 @ N ) ) ) )
% 12.45/6.98 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 12.45/6.98 => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N ) )
% 12.45/6.98 = ( inverse_inverse_real @ ( power_power_real @ X4 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7029_Suc__0__xor__eq,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.45/6.98 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/6.98 @ ( zero_n2687167440665602831ol_nat
% 12.45/6.98 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7030_xor__Suc__0__eq,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.45/6.98 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/6.98 @ ( zero_n2687167440665602831ol_nat
% 12.45/6.98 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7031_gbinomial__absorption_H,axiom,
% 12.45/6.98 ! [K: nat,A: complex] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.98 => ( ( gbinomial_complex @ A @ K )
% 12.45/6.98 = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7032_gbinomial__absorption_H,axiom,
% 12.45/6.98 ! [K: nat,A: real] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.98 => ( ( gbinomial_real @ A @ K )
% 12.45/6.98 = ( 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 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7033_gbinomial__absorption_H,axiom,
% 12.45/6.98 ! [K: nat,A: rat] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.98 => ( ( gbinomial_rat @ A @ K )
% 12.45/6.98 = ( 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 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7034_inverse__inverse__eq,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 12.45/6.98 = A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7035_inverse__inverse__eq,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 12.45/6.98 = A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7036_inverse__inverse__eq,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 12.45/6.98 = A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7037_inverse__eq__iff__eq,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ( inverse_inverse_real @ A )
% 12.45/6.98 = ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 = ( A = B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7038_inverse__eq__iff__eq,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( ( invers8013647133539491842omplex @ A )
% 12.45/6.98 = ( invers8013647133539491842omplex @ B2 ) )
% 12.45/6.98 = ( A = B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7039_inverse__eq__iff__eq,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ( inverse_inverse_rat @ A )
% 12.45/6.98 = ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 = ( A = B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7040_bit_Oxor__left__self,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) )
% 12.45/6.98 = Y3 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7041_inverse__nonzero__iff__nonzero,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ( inverse_inverse_real @ A )
% 12.45/6.98 = zero_zero_real )
% 12.45/6.98 = ( A = zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7042_inverse__nonzero__iff__nonzero,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( ( invers8013647133539491842omplex @ A )
% 12.45/6.98 = zero_zero_complex )
% 12.45/6.98 = ( A = zero_zero_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7043_inverse__nonzero__iff__nonzero,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ( inverse_inverse_rat @ A )
% 12.45/6.98 = zero_zero_rat )
% 12.45/6.98 = ( A = zero_zero_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7044_inverse__zero,axiom,
% 12.45/6.98 ( ( inverse_inverse_real @ zero_zero_real )
% 12.45/6.98 = zero_zero_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7045_inverse__zero,axiom,
% 12.45/6.98 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 12.45/6.98 = zero_zero_complex ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7046_inverse__zero,axiom,
% 12.45/6.98 ( ( inverse_inverse_rat @ zero_zero_rat )
% 12.45/6.98 = zero_zero_rat ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7047_inverse__mult__distrib,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( inverse_inverse_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.98 = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7048_inverse__mult__distrib,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B2 ) )
% 12.45/6.98 = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7049_inverse__mult__distrib,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.98 = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7050_inverse__eq__1__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ( inverse_inverse_real @ X4 )
% 12.45/6.98 = one_one_real )
% 12.45/6.98 = ( X4 = one_one_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7051_inverse__eq__1__iff,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( ( invers8013647133539491842omplex @ X4 )
% 12.45/6.98 = one_one_complex )
% 12.45/6.98 = ( X4 = one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7052_inverse__eq__1__iff,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ( ( ( inverse_inverse_rat @ X4 )
% 12.45/6.98 = one_one_rat )
% 12.45/6.98 = ( X4 = one_one_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7053_inverse__1,axiom,
% 12.45/6.98 ( ( inverse_inverse_real @ one_one_real )
% 12.45/6.98 = one_one_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7054_inverse__1,axiom,
% 12.45/6.98 ( ( invers8013647133539491842omplex @ one_one_complex )
% 12.45/6.98 = one_one_complex ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7055_inverse__1,axiom,
% 12.45/6.98 ( ( inverse_inverse_rat @ one_one_rat )
% 12.45/6.98 = one_one_rat ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7056_inverse__divide,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B2 ) )
% 12.45/6.98 = ( divide_divide_real @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7057_inverse__divide,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.45/6.98 = ( divide1717551699836669952omplex @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7058_inverse__divide,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B2 ) )
% 12.45/6.98 = ( divide_divide_rat @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7059_inverse__minus__eq,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 12.45/6.98 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7060_inverse__minus__eq,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 12.45/6.98 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7061_inverse__minus__eq,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 12.45/6.98 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7062_xor_Oright__neutral,axiom,
% 12.45/6.98 ! [A: nat] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 12.45/6.98 = A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7063_xor_Oright__neutral,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 12.45/6.98 = A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7064_xor_Oleft__neutral,axiom,
% 12.45/6.98 ! [A: nat] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 12.45/6.98 = A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7065_xor_Oleft__neutral,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 12.45/6.98 = A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7066_xor__self__eq,axiom,
% 12.45/6.98 ! [A: nat] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ A @ A )
% 12.45/6.98 = zero_zero_nat ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7067_xor__self__eq,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ A @ A )
% 12.45/6.98 = zero_zero_int ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7068_bit_Oxor__self,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ X4 @ X4 )
% 12.45/6.98 = zero_zero_int ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7069_abs__inverse,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 12.45/6.98 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7070_abs__inverse,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( abs_abs_complex @ ( invers8013647133539491842omplex @ A ) )
% 12.45/6.98 = ( invers8013647133539491842omplex @ ( abs_abs_complex @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7071_abs__inverse,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 12.45/6.98 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7072_inverse__sgn,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( inverse_inverse_real @ ( sgn_sgn_real @ A ) )
% 12.45/6.98 = ( sgn_sgn_real @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7073_inverse__sgn,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) )
% 12.45/6.98 = ( sgn_sgn_rat @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7074_sgn__inverse,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( sgn_sgn_real @ ( inverse_inverse_real @ A ) )
% 12.45/6.98 = ( inverse_inverse_real @ ( sgn_sgn_real @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7075_sgn__inverse,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( sgn_sgn_complex @ ( invers8013647133539491842omplex @ A ) )
% 12.45/6.98 = ( invers8013647133539491842omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7076_sgn__inverse,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( sgn_sgn_rat @ ( inverse_inverse_rat @ A ) )
% 12.45/6.98 = ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7077_take__bit__xor,axiom,
% 12.45/6.98 ! [N: nat,A: int,B2: int] :
% 12.45/6.98 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B2 ) )
% 12.45/6.98 = ( bit_se6526347334894502574or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7078_take__bit__xor,axiom,
% 12.45/6.98 ! [N: nat,A: nat,B2: nat] :
% 12.45/6.98 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B2 ) )
% 12.45/6.98 = ( bit_se6528837805403552850or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7079_inverse__nonpositive__iff__nonpositive,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 12.45/6.98 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7080_inverse__nonpositive__iff__nonpositive,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 12.45/6.98 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7081_inverse__nonnegative__iff__nonnegative,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 12.45/6.98 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7082_inverse__nonnegative__iff__nonnegative,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 12.45/6.98 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7083_inverse__positive__iff__positive,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 12.45/6.98 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7084_inverse__positive__iff__positive,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 12.45/6.98 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7085_inverse__negative__iff__negative,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 12.45/6.98 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7086_inverse__negative__iff__negative,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 12.45/6.98 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7087_inverse__less__iff__less__neg,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 12.45/6.98 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 = ( ord_less_real @ B2 @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7088_inverse__less__iff__less__neg,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 12.45/6.98 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 = ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7089_inverse__less__iff__less,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.98 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 = ( ord_less_real @ B2 @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7090_inverse__less__iff__less,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 12.45/6.98 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 = ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7091_gbinomial__0_I2_J,axiom,
% 12.45/6.98 ! [K: nat] :
% 12.45/6.98 ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 12.45/6.98 = zero_zero_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7092_gbinomial__0_I2_J,axiom,
% 12.45/6.98 ! [K: nat] :
% 12.45/6.98 ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 12.45/6.98 = zero_zero_rat ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7093_gbinomial__0_I2_J,axiom,
% 12.45/6.98 ! [K: nat] :
% 12.45/6.98 ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 12.45/6.98 = zero_zero_nat ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7094_gbinomial__0_I2_J,axiom,
% 12.45/6.98 ! [K: nat] :
% 12.45/6.98 ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 12.45/6.98 = zero_zero_int ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7095_gbinomial__0_I1_J,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( gbinomial_complex @ A @ zero_zero_nat )
% 12.45/6.98 = one_one_complex ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7096_gbinomial__0_I1_J,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( gbinomial_real @ A @ zero_zero_nat )
% 12.45/6.98 = one_one_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7097_gbinomial__0_I1_J,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( gbinomial_rat @ A @ zero_zero_nat )
% 12.45/6.98 = one_one_rat ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7098_gbinomial__0_I1_J,axiom,
% 12.45/6.98 ! [A: nat] :
% 12.45/6.98 ( ( gbinomial_nat @ A @ zero_zero_nat )
% 12.45/6.98 = one_one_nat ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7099_gbinomial__0_I1_J,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ( ( gbinomial_int @ A @ zero_zero_nat )
% 12.45/6.98 = one_one_int ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7100_inverse__le__iff__le,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 = ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7101_inverse__le__iff__le,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 = ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7102_inverse__le__iff__le__neg,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 12.45/6.98 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 = ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7103_inverse__le__iff__le__neg,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 12.45/6.98 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 = ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7104_right__inverse,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
% 12.45/6.98 = one_one_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7105_right__inverse,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
% 12.45/6.98 = one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7106_right__inverse,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
% 12.45/6.98 = one_one_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7107_left__inverse,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 12.45/6.98 = one_one_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7108_left__inverse,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 12.45/6.98 = one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7109_left__inverse,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 12.45/6.98 = one_one_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7110_inverse__eq__divide__numeral,axiom,
% 12.45/6.98 ! [W: num] :
% 12.45/6.98 ( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
% 12.45/6.98 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7111_inverse__eq__divide__numeral,axiom,
% 12.45/6.98 ! [W: num] :
% 12.45/6.98 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.45/6.98 = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7112_inverse__eq__divide__numeral,axiom,
% 12.45/6.98 ! [W: num] :
% 12.45/6.98 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
% 12.45/6.98 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7113_floor__add2,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ( member_real @ X4 @ ring_1_Ints_real )
% 12.45/6.98 | ( member_real @ Y3 @ ring_1_Ints_real ) )
% 12.45/6.98 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.98 = ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7114_floor__add2,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( ( member_rat @ X4 @ ring_1_Ints_rat )
% 12.45/6.98 | ( member_rat @ Y3 @ ring_1_Ints_rat ) )
% 12.45/6.98 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y3 ) )
% 12.45/6.98 = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7115_frac__gt__0__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X4 ) )
% 12.45/6.98 = ( ~ ( member_real @ X4 @ ring_1_Ints_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7116_frac__gt__0__iff,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X4 ) )
% 12.45/6.98 = ( ~ ( member_rat @ X4 @ ring_1_Ints_rat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7117_inverse__eq__divide__neg__numeral,axiom,
% 12.45/6.98 ! [W: num] :
% 12.45/6.98 ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.45/6.98 = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7118_inverse__eq__divide__neg__numeral,axiom,
% 12.45/6.98 ! [W: num] :
% 12.45/6.98 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.45/6.98 = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7119_inverse__eq__divide__neg__numeral,axiom,
% 12.45/6.98 ! [W: num] :
% 12.45/6.98 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.45/6.98 = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7120_xor__numerals_I3_J,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 12.45/6.98 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7121_xor__numerals_I3_J,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 12.45/6.98 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7122_xor__numerals_I1_J,axiom,
% 12.45/6.98 ! [Y3: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 12.45/6.98 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7123_xor__numerals_I1_J,axiom,
% 12.45/6.98 ! [Y3: num] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 12.45/6.98 = ( numeral_numeral_int @ ( bit1 @ Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7124_xor__numerals_I2_J,axiom,
% 12.45/6.98 ! [Y3: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 12.45/6.98 = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7125_xor__numerals_I2_J,axiom,
% 12.45/6.98 ! [Y3: num] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 12.45/6.98 = ( numeral_numeral_int @ ( bit0 @ Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7126_xor__numerals_I5_J,axiom,
% 12.45/6.98 ! [X4: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ one_one_nat )
% 12.45/6.98 = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7127_xor__numerals_I5_J,axiom,
% 12.45/6.98 ! [X4: num] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ one_one_int )
% 12.45/6.98 = ( numeral_numeral_int @ ( bit1 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7128_xor__numerals_I8_J,axiom,
% 12.45/6.98 ! [X4: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ one_one_nat )
% 12.45/6.98 = ( numeral_numeral_nat @ ( bit0 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7129_xor__numerals_I8_J,axiom,
% 12.45/6.98 ! [X4: num] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ one_one_int )
% 12.45/6.98 = ( numeral_numeral_int @ ( bit0 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7130_xor__numerals_I7_J,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 12.45/6.98 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7131_xor__numerals_I7_J,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 12.45/6.98 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7132_xor__nat__numerals_I1_J,axiom,
% 12.45/6.98 ! [Y3: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 12.45/6.98 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7133_xor__nat__numerals_I2_J,axiom,
% 12.45/6.98 ! [Y3: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 12.45/6.98 = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7134_xor__nat__numerals_I3_J,axiom,
% 12.45/6.98 ! [X4: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 12.45/6.98 = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7135_xor__nat__numerals_I4_J,axiom,
% 12.45/6.98 ! [X4: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 12.45/6.98 = ( numeral_numeral_nat @ ( bit0 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7136_xor__numerals_I4_J,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 12.45/6.98 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7137_xor__numerals_I4_J,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 12.45/6.98 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7138_xor__numerals_I6_J,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 12.45/6.98 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7139_xor__numerals_I6_J,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 12.45/6.98 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7140_power__inverse,axiom,
% 12.45/6.98 ! [A: real,N: nat] :
% 12.45/6.98 ( ( power_power_real @ ( inverse_inverse_real @ A ) @ N )
% 12.45/6.98 = ( inverse_inverse_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7141_power__inverse,axiom,
% 12.45/6.98 ! [A: complex,N: nat] :
% 12.45/6.98 ( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N )
% 12.45/6.98 = ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7142_power__inverse,axiom,
% 12.45/6.98 ! [A: rat,N: nat] :
% 12.45/6.98 ( ( power_power_rat @ ( inverse_inverse_rat @ A ) @ N )
% 12.45/6.98 = ( inverse_inverse_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7143_mult__commute__imp__mult__inverse__commute,axiom,
% 12.45/6.98 ! [Y3: real,X4: real] :
% 12.45/6.98 ( ( ( times_times_real @ Y3 @ X4 )
% 12.45/6.98 = ( times_times_real @ X4 @ Y3 ) )
% 12.45/6.98 => ( ( times_times_real @ ( inverse_inverse_real @ Y3 ) @ X4 )
% 12.45/6.98 = ( times_times_real @ X4 @ ( inverse_inverse_real @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7144_mult__commute__imp__mult__inverse__commute,axiom,
% 12.45/6.98 ! [Y3: complex,X4: complex] :
% 12.45/6.98 ( ( ( times_times_complex @ Y3 @ X4 )
% 12.45/6.98 = ( times_times_complex @ X4 @ Y3 ) )
% 12.45/6.98 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y3 ) @ X4 )
% 12.45/6.98 = ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7145_mult__commute__imp__mult__inverse__commute,axiom,
% 12.45/6.98 ! [Y3: rat,X4: rat] :
% 12.45/6.98 ( ( ( times_times_rat @ Y3 @ X4 )
% 12.45/6.98 = ( times_times_rat @ X4 @ Y3 ) )
% 12.45/6.98 => ( ( times_times_rat @ ( inverse_inverse_rat @ Y3 ) @ X4 )
% 12.45/6.98 = ( times_times_rat @ X4 @ ( inverse_inverse_rat @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7146_of__nat__xor__eq,axiom,
% 12.45/6.98 ! [M: nat,N: nat] :
% 12.45/6.98 ( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 12.45/6.98 = ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7147_of__nat__xor__eq,axiom,
% 12.45/6.98 ! [M: nat,N: nat] :
% 12.45/6.98 ( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 12.45/6.98 = ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7148_of__int__xor__eq,axiom,
% 12.45/6.98 ! [K: int,L2: int] :
% 12.45/6.98 ( ( ring_1_of_int_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
% 12.45/6.98 = ( bit_se6526347334894502574or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7149_xor_Oleft__commute,axiom,
% 12.45/6.98 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ B2 @ ( bit_se6528837805403552850or_nat @ A @ C ) )
% 12.45/6.98 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B2 @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7150_xor_Oleft__commute,axiom,
% 12.45/6.98 ! [B2: int,A: int,C: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ B2 @ ( bit_se6526347334894502574or_int @ A @ C ) )
% 12.45/6.98 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B2 @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7151_xor_Ocommute,axiom,
% 12.45/6.98 ( bit_se6528837805403552850or_nat
% 12.45/6.98 = ( ^ [A4: nat,B3: nat] : ( bit_se6528837805403552850or_nat @ B3 @ A4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7152_xor_Ocommute,axiom,
% 12.45/6.98 ( bit_se6526347334894502574or_int
% 12.45/6.98 = ( ^ [A4: int,B3: int] : ( bit_se6526347334894502574or_int @ B3 @ A4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7153_xor_Oassoc,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ ( bit_se6528837805403552850or_nat @ A @ B2 ) @ C )
% 12.45/6.98 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B2 @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7154_xor_Oassoc,axiom,
% 12.45/6.98 ! [A: int,B2: int,C: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ ( bit_se6526347334894502574or_int @ A @ B2 ) @ C )
% 12.45/6.98 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B2 @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7155_inverse__eq__imp__eq,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ( inverse_inverse_real @ A )
% 12.45/6.98 = ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 => ( A = B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7156_inverse__eq__imp__eq,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( ( invers8013647133539491842omplex @ A )
% 12.45/6.98 = ( invers8013647133539491842omplex @ B2 ) )
% 12.45/6.98 => ( A = B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7157_inverse__eq__imp__eq,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ( inverse_inverse_rat @ A )
% 12.45/6.98 = ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 => ( A = B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7158_field__class_Ofield__inverse__zero,axiom,
% 12.45/6.98 ( ( inverse_inverse_real @ zero_zero_real )
% 12.45/6.98 = zero_zero_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7159_field__class_Ofield__inverse__zero,axiom,
% 12.45/6.98 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 12.45/6.98 = zero_zero_complex ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7160_field__class_Ofield__inverse__zero,axiom,
% 12.45/6.98 ( ( inverse_inverse_rat @ zero_zero_rat )
% 12.45/6.98 = zero_zero_rat ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7161_inverse__zero__imp__zero,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ( inverse_inverse_real @ A )
% 12.45/6.98 = zero_zero_real )
% 12.45/6.98 => ( A = zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7162_inverse__zero__imp__zero,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( ( invers8013647133539491842omplex @ A )
% 12.45/6.98 = zero_zero_complex )
% 12.45/6.98 => ( A = zero_zero_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7163_inverse__zero__imp__zero,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ( inverse_inverse_rat @ A )
% 12.45/6.98 = zero_zero_rat )
% 12.45/6.98 => ( A = zero_zero_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7164_nonzero__inverse__eq__imp__eq,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ( inverse_inverse_real @ A )
% 12.45/6.98 = ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 => ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( B2 != zero_zero_real )
% 12.45/6.98 => ( A = B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7165_nonzero__inverse__eq__imp__eq,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( ( invers8013647133539491842omplex @ A )
% 12.45/6.98 = ( invers8013647133539491842omplex @ B2 ) )
% 12.45/6.98 => ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( B2 != zero_zero_complex )
% 12.45/6.98 => ( A = B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7166_nonzero__inverse__eq__imp__eq,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ( inverse_inverse_rat @ A )
% 12.45/6.98 = ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 => ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( B2 != zero_zero_rat )
% 12.45/6.98 => ( A = B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7167_nonzero__inverse__inverse__eq,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 12.45/6.98 = A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7168_nonzero__inverse__inverse__eq,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 12.45/6.98 = A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7169_nonzero__inverse__inverse__eq,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 12.45/6.98 = A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7170_nonzero__imp__inverse__nonzero,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( inverse_inverse_real @ A )
% 12.45/6.98 != zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7171_nonzero__imp__inverse__nonzero,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( invers8013647133539491842omplex @ A )
% 12.45/6.98 != zero_zero_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7172_nonzero__imp__inverse__nonzero,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( inverse_inverse_rat @ A )
% 12.45/6.98 != zero_zero_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7173_Ints__numeral,axiom,
% 12.45/6.98 ! [N: num] : ( member_complex @ ( numera6690914467698888265omplex @ N ) @ ring_1_Ints_complex ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7174_Ints__numeral,axiom,
% 12.45/6.98 ! [N: num] : ( member_real @ ( numeral_numeral_real @ N ) @ ring_1_Ints_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7175_Ints__numeral,axiom,
% 12.45/6.98 ! [N: num] : ( member_rat @ ( numeral_numeral_rat @ N ) @ ring_1_Ints_rat ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7176_Ints__numeral,axiom,
% 12.45/6.98 ! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7177_Ints__mult,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.45/6.98 => ( ( member_complex @ B2 @ ring_1_Ints_complex )
% 12.45/6.98 => ( member_complex @ ( times_times_complex @ A @ B2 ) @ ring_1_Ints_complex ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7178_Ints__mult,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ( member_real @ B2 @ ring_1_Ints_real )
% 12.45/6.98 => ( member_real @ ( times_times_real @ A @ B2 ) @ ring_1_Ints_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7179_Ints__mult,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( member_rat @ B2 @ ring_1_Ints_rat )
% 12.45/6.98 => ( member_rat @ ( times_times_rat @ A @ B2 ) @ ring_1_Ints_rat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7180_Ints__mult,axiom,
% 12.45/6.98 ! [A: int,B2: int] :
% 12.45/6.98 ( ( member_int @ A @ ring_1_Ints_int )
% 12.45/6.98 => ( ( member_int @ B2 @ ring_1_Ints_int )
% 12.45/6.98 => ( member_int @ ( times_times_int @ A @ B2 ) @ ring_1_Ints_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7181_Ints__1,axiom,
% 12.45/6.98 member_complex @ one_one_complex @ ring_1_Ints_complex ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7182_Ints__1,axiom,
% 12.45/6.98 member_rat @ one_one_rat @ ring_1_Ints_rat ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7183_Ints__1,axiom,
% 12.45/6.98 member_int @ one_one_int @ ring_1_Ints_int ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7184_Ints__1,axiom,
% 12.45/6.98 member_real @ one_one_real @ ring_1_Ints_real ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7185_Ints__add,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.45/6.98 => ( ( member_complex @ B2 @ ring_1_Ints_complex )
% 12.45/6.98 => ( member_complex @ ( plus_plus_complex @ A @ B2 ) @ ring_1_Ints_complex ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7186_Ints__add,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ( member_real @ B2 @ ring_1_Ints_real )
% 12.45/6.98 => ( member_real @ ( plus_plus_real @ A @ B2 ) @ ring_1_Ints_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7187_Ints__add,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( member_rat @ B2 @ ring_1_Ints_rat )
% 12.45/6.98 => ( member_rat @ ( plus_plus_rat @ A @ B2 ) @ ring_1_Ints_rat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7188_Ints__add,axiom,
% 12.45/6.98 ! [A: int,B2: int] :
% 12.45/6.98 ( ( member_int @ A @ ring_1_Ints_int )
% 12.45/6.98 => ( ( member_int @ B2 @ ring_1_Ints_int )
% 12.45/6.98 => ( member_int @ ( plus_plus_int @ A @ B2 ) @ ring_1_Ints_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7189_real__sqrt__inverse,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( sqrt @ ( inverse_inverse_real @ X4 ) )
% 12.45/6.98 = ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7190_Ints__diff,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.45/6.98 => ( ( member_complex @ B2 @ ring_1_Ints_complex )
% 12.45/6.98 => ( member_complex @ ( minus_minus_complex @ A @ B2 ) @ ring_1_Ints_complex ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7191_Ints__diff,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ( member_real @ B2 @ ring_1_Ints_real )
% 12.45/6.98 => ( member_real @ ( minus_minus_real @ A @ B2 ) @ ring_1_Ints_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7192_Ints__diff,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( member_rat @ B2 @ ring_1_Ints_rat )
% 12.45/6.98 => ( member_rat @ ( minus_minus_rat @ A @ B2 ) @ ring_1_Ints_rat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7193_Ints__diff,axiom,
% 12.45/6.98 ! [A: int,B2: int] :
% 12.45/6.98 ( ( member_int @ A @ ring_1_Ints_int )
% 12.45/6.98 => ( ( member_int @ B2 @ ring_1_Ints_int )
% 12.45/6.98 => ( member_int @ ( minus_minus_int @ A @ B2 ) @ ring_1_Ints_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7194_Ints__power,axiom,
% 12.45/6.98 ! [A: real,N: nat] :
% 12.45/6.98 ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7195_Ints__power,axiom,
% 12.45/6.98 ! [A: int,N: nat] :
% 12.45/6.98 ( ( member_int @ A @ ring_1_Ints_int )
% 12.45/6.98 => ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7196_Ints__power,axiom,
% 12.45/6.98 ! [A: complex,N: nat] :
% 12.45/6.98 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.45/6.98 => ( member_complex @ ( power_power_complex @ A @ N ) @ ring_1_Ints_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7197_bit_Oconj__xor__distrib2,axiom,
% 12.45/6.98 ! [Y3: int,Z: int,X4: int] :
% 12.45/6.98 ( ( bit_se725231765392027082nd_int @ ( bit_se6526347334894502574or_int @ Y3 @ Z ) @ X4 )
% 12.45/6.98 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ Y3 @ X4 ) @ ( bit_se725231765392027082nd_int @ Z @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7198_bit_Oconj__xor__distrib,axiom,
% 12.45/6.98 ! [X4: int,Y3: int,Z: int] :
% 12.45/6.98 ( ( bit_se725231765392027082nd_int @ X4 @ ( bit_se6526347334894502574or_int @ Y3 @ Z ) )
% 12.45/6.98 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ ( bit_se725231765392027082nd_int @ X4 @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7199_bit__xor__iff,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,N: nat] :
% 12.45/6.98 ( ( bit_se1148574629649215175it_nat @ ( bit_se6528837805403552850or_nat @ A @ B2 ) @ N )
% 12.45/6.98 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.45/6.98 != ( bit_se1148574629649215175it_nat @ B2 @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7200_bit__xor__iff,axiom,
% 12.45/6.98 ! [A: int,B2: int,N: nat] :
% 12.45/6.98 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ A @ B2 ) @ N )
% 12.45/6.98 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.45/6.98 != ( bit_se1146084159140164899it_int @ B2 @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7201_norm__inverse__le__norm,axiom,
% 12.45/6.98 ! [R2: real,X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X4 ) )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ R2 )
% 12.45/6.98 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X4 ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7202_norm__inverse__le__norm,axiom,
% 12.45/6.98 ! [R2: real,X4: complex] :
% 12.45/6.98 ( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X4 ) )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ R2 )
% 12.45/6.98 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X4 ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7203_positive__imp__inverse__positive,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7204_positive__imp__inverse__positive,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7205_negative__imp__inverse__negative,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ zero_zero_real )
% 12.45/6.98 => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7206_negative__imp__inverse__negative,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.45/6.98 => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7207_inverse__positive__imp__positive,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 12.45/6.98 => ( ( A != zero_zero_real )
% 12.45/6.98 => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7208_inverse__positive__imp__positive,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 12.45/6.98 => ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7209_inverse__negative__imp__negative,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 12.45/6.98 => ( ( A != zero_zero_real )
% 12.45/6.98 => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7210_inverse__negative__imp__negative,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 12.45/6.98 => ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7211_less__imp__inverse__less__neg,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 12.45/6.98 => ( ord_less_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7212_less__imp__inverse__less__neg,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 12.45/6.98 => ( ord_less_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7213_inverse__less__imp__less__neg,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 12.45/6.98 => ( ord_less_real @ B2 @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7214_inverse__less__imp__less__neg,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 12.45/6.98 => ( ord_less_rat @ B2 @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7215_less__imp__inverse__less,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ord_less_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7216_less__imp__inverse__less,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ord_less_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7217_inverse__less__imp__less,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ord_less_real @ B2 @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7218_inverse__less__imp__less,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ord_less_rat @ B2 @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7219_nonzero__inverse__mult__distrib,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( B2 != zero_zero_real )
% 12.45/6.98 => ( ( inverse_inverse_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.98 = ( times_times_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7220_nonzero__inverse__mult__distrib,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( B2 != zero_zero_complex )
% 12.45/6.98 => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B2 ) )
% 12.45/6.98 = ( times_times_complex @ ( invers8013647133539491842omplex @ B2 ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7221_nonzero__inverse__mult__distrib,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( B2 != zero_zero_rat )
% 12.45/6.98 => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.98 = ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7222_nonzero__inverse__minus__eq,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 12.45/6.98 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7223_nonzero__inverse__minus__eq,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 12.45/6.98 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7224_nonzero__inverse__minus__eq,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 12.45/6.98 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7225_inverse__numeral__1,axiom,
% 12.45/6.98 ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
% 12.45/6.98 = ( numeral_numeral_real @ one ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7226_inverse__numeral__1,axiom,
% 12.45/6.98 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
% 12.45/6.98 = ( numera6690914467698888265omplex @ one ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7227_inverse__numeral__1,axiom,
% 12.45/6.98 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
% 12.45/6.98 = ( numeral_numeral_rat @ one ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7228_inverse__unique,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ( times_times_real @ A @ B2 )
% 12.45/6.98 = one_one_real )
% 12.45/6.98 => ( ( inverse_inverse_real @ A )
% 12.45/6.98 = B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7229_inverse__unique,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( ( times_times_complex @ A @ B2 )
% 12.45/6.98 = one_one_complex )
% 12.45/6.98 => ( ( invers8013647133539491842omplex @ A )
% 12.45/6.98 = B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7230_inverse__unique,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ( times_times_rat @ A @ B2 )
% 12.45/6.98 = one_one_rat )
% 12.45/6.98 => ( ( inverse_inverse_rat @ A )
% 12.45/6.98 = B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7231_divide__inverse__commute,axiom,
% 12.45/6.98 ( divide_divide_real
% 12.45/6.98 = ( ^ [A4: real,B3: real] : ( times_times_real @ ( inverse_inverse_real @ B3 ) @ A4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7232_divide__inverse__commute,axiom,
% 12.45/6.98 ( divide1717551699836669952omplex
% 12.45/6.98 = ( ^ [A4: complex,B3: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B3 ) @ A4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7233_divide__inverse__commute,axiom,
% 12.45/6.98 ( divide_divide_rat
% 12.45/6.98 = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B3 ) @ A4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7234_divide__inverse,axiom,
% 12.45/6.98 ( divide_divide_real
% 12.45/6.98 = ( ^ [A4: real,B3: real] : ( times_times_real @ A4 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7235_divide__inverse,axiom,
% 12.45/6.98 ( divide1717551699836669952omplex
% 12.45/6.98 = ( ^ [A4: complex,B3: complex] : ( times_times_complex @ A4 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7236_divide__inverse,axiom,
% 12.45/6.98 ( divide_divide_rat
% 12.45/6.98 = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ A4 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7237_field__class_Ofield__divide__inverse,axiom,
% 12.45/6.98 ( divide_divide_real
% 12.45/6.98 = ( ^ [A4: real,B3: real] : ( times_times_real @ A4 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7238_field__class_Ofield__divide__inverse,axiom,
% 12.45/6.98 ( divide1717551699836669952omplex
% 12.45/6.98 = ( ^ [A4: complex,B3: complex] : ( times_times_complex @ A4 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7239_field__class_Ofield__divide__inverse,axiom,
% 12.45/6.98 ( divide_divide_rat
% 12.45/6.98 = ( ^ [A4: rat,B3: rat] : ( times_times_rat @ A4 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7240_inverse__eq__divide,axiom,
% 12.45/6.98 ( inverse_inverse_real
% 12.45/6.98 = ( divide_divide_real @ one_one_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7241_inverse__eq__divide,axiom,
% 12.45/6.98 ( invers8013647133539491842omplex
% 12.45/6.98 = ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7242_inverse__eq__divide,axiom,
% 12.45/6.98 ( inverse_inverse_rat
% 12.45/6.98 = ( divide_divide_rat @ one_one_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7243_power__mult__power__inverse__commute,axiom,
% 12.45/6.98 ! [X4: real,M: nat,N: nat] :
% 12.45/6.98 ( ( times_times_real @ ( power_power_real @ X4 @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ N ) )
% 12.45/6.98 = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ N ) @ ( power_power_real @ X4 @ M ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7244_power__mult__power__inverse__commute,axiom,
% 12.45/6.98 ! [X4: complex,M: nat,N: nat] :
% 12.45/6.98 ( ( times_times_complex @ ( power_power_complex @ X4 @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ N ) )
% 12.45/6.98 = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ N ) @ ( power_power_complex @ X4 @ M ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7245_power__mult__power__inverse__commute,axiom,
% 12.45/6.98 ! [X4: rat,M: nat,N: nat] :
% 12.45/6.98 ( ( times_times_rat @ ( power_power_rat @ X4 @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X4 ) @ N ) )
% 12.45/6.98 = ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X4 ) @ N ) @ ( power_power_rat @ X4 @ M ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7246_power__mult__inverse__distrib,axiom,
% 12.45/6.98 ! [X4: real,M: nat] :
% 12.45/6.98 ( ( times_times_real @ ( power_power_real @ X4 @ M ) @ ( inverse_inverse_real @ X4 ) )
% 12.45/6.98 = ( times_times_real @ ( inverse_inverse_real @ X4 ) @ ( power_power_real @ X4 @ M ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7247_power__mult__inverse__distrib,axiom,
% 12.45/6.98 ! [X4: complex,M: nat] :
% 12.45/6.98 ( ( times_times_complex @ ( power_power_complex @ X4 @ M ) @ ( invers8013647133539491842omplex @ X4 ) )
% 12.45/6.98 = ( times_times_complex @ ( invers8013647133539491842omplex @ X4 ) @ ( power_power_complex @ X4 @ M ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7248_power__mult__inverse__distrib,axiom,
% 12.45/6.98 ! [X4: rat,M: nat] :
% 12.45/6.98 ( ( times_times_rat @ ( power_power_rat @ X4 @ M ) @ ( inverse_inverse_rat @ X4 ) )
% 12.45/6.98 = ( times_times_rat @ ( inverse_inverse_rat @ X4 ) @ ( power_power_rat @ X4 @ M ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7249_mult__inverse__of__nat__commute,axiom,
% 12.45/6.98 ! [Xa: nat,X4: real] :
% 12.45/6.98 ( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X4 )
% 12.45/6.98 = ( times_times_real @ X4 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7250_mult__inverse__of__nat__commute,axiom,
% 12.45/6.98 ! [Xa: nat,X4: complex] :
% 12.45/6.98 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X4 )
% 12.45/6.98 = ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7251_mult__inverse__of__nat__commute,axiom,
% 12.45/6.98 ! [Xa: nat,X4: rat] :
% 12.45/6.98 ( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) @ X4 )
% 12.45/6.98 = ( times_times_rat @ X4 @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7252_nonzero__abs__inverse,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 12.45/6.98 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7253_nonzero__abs__inverse,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 12.45/6.98 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7254_mult__inverse__of__int__commute,axiom,
% 12.45/6.98 ! [Xa: int,X4: real] :
% 12.45/6.98 ( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) @ X4 )
% 12.45/6.98 = ( times_times_real @ X4 @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7255_mult__inverse__of__int__commute,axiom,
% 12.45/6.98 ! [Xa: int,X4: complex] :
% 12.45/6.98 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) @ X4 )
% 12.45/6.98 = ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7256_mult__inverse__of__int__commute,axiom,
% 12.45/6.98 ! [Xa: int,X4: rat] :
% 12.45/6.98 ( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) @ X4 )
% 12.45/6.98 = ( times_times_rat @ X4 @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7257_divide__real__def,axiom,
% 12.45/6.98 ( divide_divide_real
% 12.45/6.98 = ( ^ [X: real,Y: real] : ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7258_Ints__double__eq__0__iff,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.45/6.98 => ( ( ( plus_plus_complex @ A @ A )
% 12.45/6.98 = zero_zero_complex )
% 12.45/6.98 = ( A = zero_zero_complex ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7259_Ints__double__eq__0__iff,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ( ( plus_plus_real @ A @ A )
% 12.45/6.98 = zero_zero_real )
% 12.45/6.98 = ( A = zero_zero_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7260_Ints__double__eq__0__iff,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( ( plus_plus_rat @ A @ A )
% 12.45/6.98 = zero_zero_rat )
% 12.45/6.98 = ( A = zero_zero_rat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7261_Ints__double__eq__0__iff,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ( ( member_int @ A @ ring_1_Ints_int )
% 12.45/6.98 => ( ( ( plus_plus_int @ A @ A )
% 12.45/6.98 = zero_zero_int )
% 12.45/6.98 = ( A = zero_zero_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7262_inverse__le__imp__le,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7263_inverse__le__imp__le,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7264_le__imp__inverse__le,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ord_less_eq_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7265_le__imp__inverse__le,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7266_inverse__le__imp__le__neg,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 12.45/6.98 => ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7267_inverse__le__imp__le__neg,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 12.45/6.98 => ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7268_le__imp__inverse__le__neg,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ zero_zero_real )
% 12.45/6.98 => ( ord_less_eq_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7269_le__imp__inverse__le__neg,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 12.45/6.98 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7270_inverse__le__1__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( inverse_inverse_real @ X4 ) @ one_one_real )
% 12.45/6.98 = ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.98 | ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7271_inverse__le__1__iff,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X4 ) @ one_one_rat )
% 12.45/6.98 = ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 12.45/6.98 | ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7272_one__less__inverse__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X4 ) )
% 12.45/6.98 = ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 & ( ord_less_real @ X4 @ one_one_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7273_one__less__inverse__iff,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X4 ) )
% 12.45/6.98 = ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 12.45/6.98 & ( ord_less_rat @ X4 @ one_one_rat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7274_one__less__inverse,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ( ord_less_real @ A @ one_one_real )
% 12.45/6.98 => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7275_one__less__inverse,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ( ord_less_rat @ A @ one_one_rat )
% 12.45/6.98 => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7276_field__class_Ofield__inverse,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 12.45/6.98 = one_one_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7277_field__class_Ofield__inverse,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 12.45/6.98 = one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7278_field__class_Ofield__inverse,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 12.45/6.98 = one_one_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7279_division__ring__inverse__add,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( B2 != zero_zero_real )
% 12.45/6.98 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B2 ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7280_division__ring__inverse__add,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( B2 != zero_zero_complex )
% 12.45/6.98 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
% 12.45/6.98 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B2 ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7281_division__ring__inverse__add,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( B2 != zero_zero_rat )
% 12.45/6.98 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B2 ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7282_inverse__add,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( B2 != zero_zero_real )
% 12.45/6.98 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7283_inverse__add,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( B2 != zero_zero_complex )
% 12.45/6.98 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
% 12.45/6.98 = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7284_inverse__add,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( B2 != zero_zero_rat )
% 12.45/6.98 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7285_division__ring__inverse__diff,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( B2 != zero_zero_real )
% 12.45/6.98 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B2 @ A ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7286_division__ring__inverse__diff,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( B2 != zero_zero_complex )
% 12.45/6.98 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
% 12.45/6.98 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B2 @ A ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7287_division__ring__inverse__diff,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( B2 != zero_zero_rat )
% 12.45/6.98 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B2 @ A ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7288_nonzero__inverse__eq__divide,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( inverse_inverse_real @ A )
% 12.45/6.98 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7289_nonzero__inverse__eq__divide,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( invers8013647133539491842omplex @ A )
% 12.45/6.98 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7290_nonzero__inverse__eq__divide,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( inverse_inverse_rat @ A )
% 12.45/6.98 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7291_gbinomial__Suc__Suc,axiom,
% 12.45/6.98 ! [A: complex,K: nat] :
% 12.45/6.98 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 12.45/6.98 = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7292_gbinomial__Suc__Suc,axiom,
% 12.45/6.98 ! [A: real,K: nat] :
% 12.45/6.98 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 12.45/6.98 = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7293_gbinomial__Suc__Suc,axiom,
% 12.45/6.98 ! [A: rat,K: nat] :
% 12.45/6.98 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 12.45/6.98 = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7294_gbinomial__of__nat__symmetric,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K )
% 12.45/6.98 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7295_gbinomial__of__nat__symmetric,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 12.45/6.98 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7296_gbinomial__of__nat__symmetric,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.45/6.98 => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
% 12.45/6.98 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7297_inverse__powr,axiom,
% 12.45/6.98 ! [Y3: real,A: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.98 => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A )
% 12.45/6.98 = ( inverse_inverse_real @ ( powr_real @ Y3 @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7298_Ints__odd__nonzero,axiom,
% 12.45/6.98 ! [A: complex] :
% 12.45/6.98 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.45/6.98 => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
% 12.45/6.98 != zero_zero_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7299_Ints__odd__nonzero,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
% 12.45/6.98 != zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7300_Ints__odd__nonzero,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
% 12.45/6.98 != zero_zero_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7301_Ints__odd__nonzero,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ( ( member_int @ A @ ring_1_Ints_int )
% 12.45/6.98 => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
% 12.45/6.98 != zero_zero_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7302_of__int__divide__in__Ints,axiom,
% 12.45/6.98 ! [B2: int,A: int] :
% 12.45/6.98 ( ( dvd_dvd_int @ B2 @ A )
% 12.45/6.98 => ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B2 ) ) @ ring_1_Ints_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7303_of__int__divide__in__Ints,axiom,
% 12.45/6.98 ! [B2: int,A: int] :
% 12.45/6.98 ( ( dvd_dvd_int @ B2 @ A )
% 12.45/6.98 => ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B2 ) ) @ ring_1_Ints_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7304_of__int__divide__in__Ints,axiom,
% 12.45/6.98 ! [B2: int,A: int] :
% 12.45/6.98 ( ( dvd_dvd_int @ B2 @ A )
% 12.45/6.98 => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B2 ) ) @ ring_1_Ints_rat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7305_of__int__divide__in__Ints,axiom,
% 12.45/6.98 ! [B2: int,A: int] :
% 12.45/6.98 ( ( dvd_dvd_int @ B2 @ A )
% 12.45/6.98 => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B2 ) ) @ ring_1_Ints_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7306_inverse__le__iff,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.98 => ( ord_less_eq_real @ B2 @ A ) )
% 12.45/6.98 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 12.45/6.98 => ( ord_less_eq_real @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7307_inverse__le__iff,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.98 => ( ord_less_eq_rat @ B2 @ A ) )
% 12.45/6.98 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 12.45/6.98 => ( ord_less_eq_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7308_inverse__less__iff,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 12.45/6.98 => ( ord_less_real @ B2 @ A ) )
% 12.45/6.98 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 12.45/6.98 => ( ord_less_real @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7309_inverse__less__iff,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 12.45/6.98 => ( ord_less_rat @ B2 @ A ) )
% 12.45/6.98 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 12.45/6.98 => ( ord_less_rat @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7310_one__le__inverse,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ( ord_less_eq_real @ A @ one_one_real )
% 12.45/6.98 => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7311_one__le__inverse,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 12.45/6.98 => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7312_inverse__less__1__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ ( inverse_inverse_real @ X4 ) @ one_one_real )
% 12.45/6.98 = ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.98 | ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7313_inverse__less__1__iff,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ ( inverse_inverse_rat @ X4 ) @ one_one_rat )
% 12.45/6.98 = ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
% 12.45/6.98 | ( ord_less_rat @ one_one_rat @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7314_one__le__inverse__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X4 ) )
% 12.45/6.98 = ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 & ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7315_one__le__inverse__iff,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X4 ) )
% 12.45/6.98 = ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 12.45/6.98 & ( ord_less_eq_rat @ X4 @ one_one_rat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7316_inverse__diff__inverse,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( A != zero_zero_real )
% 12.45/6.98 => ( ( B2 != zero_zero_real )
% 12.45/6.98 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 12.45/6.98 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B2 ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7317_inverse__diff__inverse,axiom,
% 12.45/6.98 ! [A: complex,B2: complex] :
% 12.45/6.98 ( ( A != zero_zero_complex )
% 12.45/6.98 => ( ( B2 != zero_zero_complex )
% 12.45/6.98 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) )
% 12.45/6.98 = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B2 ) ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7318_inverse__diff__inverse,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( A != zero_zero_rat )
% 12.45/6.98 => ( ( B2 != zero_zero_rat )
% 12.45/6.98 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 12.45/6.98 = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B2 ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7319_reals__Archimedean,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ? [N4: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7320_reals__Archimedean,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 12.45/6.98 => ? [N4: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7321_gbinomial__addition__formula,axiom,
% 12.45/6.98 ! [A: complex,K: nat] :
% 12.45/6.98 ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 12.45/6.98 = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7322_gbinomial__addition__formula,axiom,
% 12.45/6.98 ! [A: real,K: nat] :
% 12.45/6.98 ( ( gbinomial_real @ A @ ( suc @ K ) )
% 12.45/6.98 = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7323_gbinomial__addition__formula,axiom,
% 12.45/6.98 ! [A: rat,K: nat] :
% 12.45/6.98 ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 12.45/6.98 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7324_gbinomial__absorb__comp,axiom,
% 12.45/6.98 ! [A: complex,K: nat] :
% 12.45/6.98 ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
% 12.45/6.98 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7325_gbinomial__absorb__comp,axiom,
% 12.45/6.98 ! [A: real,K: nat] :
% 12.45/6.98 ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
% 12.45/6.98 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7326_gbinomial__absorb__comp,axiom,
% 12.45/6.98 ! [A: rat,K: nat] :
% 12.45/6.98 ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
% 12.45/6.98 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7327_gbinomial__ge__n__over__k__pow__k,axiom,
% 12.45/6.98 ! [K: nat,A: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 12.45/6.98 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7328_gbinomial__ge__n__over__k__pow__k,axiom,
% 12.45/6.98 ! [K: nat,A: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 12.45/6.98 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7329_gbinomial__mult__1_H,axiom,
% 12.45/6.98 ! [A: complex,K: nat] :
% 12.45/6.98 ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
% 12.45/6.98 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7330_gbinomial__mult__1_H,axiom,
% 12.45/6.98 ! [A: real,K: nat] :
% 12.45/6.98 ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 12.45/6.98 = ( 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 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7331_gbinomial__mult__1_H,axiom,
% 12.45/6.98 ! [A: rat,K: nat] :
% 12.45/6.98 ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 12.45/6.98 = ( 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 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7332_gbinomial__mult__1,axiom,
% 12.45/6.98 ! [A: complex,K: nat] :
% 12.45/6.98 ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
% 12.45/6.98 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7333_gbinomial__mult__1,axiom,
% 12.45/6.98 ! [A: real,K: nat] :
% 12.45/6.98 ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 12.45/6.98 = ( 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 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7334_gbinomial__mult__1,axiom,
% 12.45/6.98 ! [A: rat,K: nat] :
% 12.45/6.98 ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 12.45/6.98 = ( 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 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7335_forall__pos__mono__1,axiom,
% 12.45/6.98 ! [P: real > $o,E: real] :
% 12.45/6.98 ( ! [D3: real,E2: real] :
% 12.45/6.98 ( ( ord_less_real @ D3 @ E2 )
% 12.45/6.98 => ( ( P @ D3 )
% 12.45/6.98 => ( P @ E2 ) ) )
% 12.45/6.98 => ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ E )
% 12.45/6.98 => ( P @ E ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7336_forall__pos__mono,axiom,
% 12.45/6.98 ! [P: real > $o,E: real] :
% 12.45/6.98 ( ! [D3: real,E2: real] :
% 12.45/6.98 ( ( ord_less_real @ D3 @ E2 )
% 12.45/6.98 => ( ( P @ D3 )
% 12.45/6.98 => ( P @ E2 ) ) )
% 12.45/6.98 => ( ! [N4: nat] :
% 12.45/6.98 ( ( N4 != zero_zero_nat )
% 12.45/6.98 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ E )
% 12.45/6.98 => ( P @ E ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7337_real__arch__inverse,axiom,
% 12.45/6.98 ! [E: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ E )
% 12.45/6.98 = ( ? [N2: nat] :
% 12.45/6.98 ( ( N2 != zero_zero_nat )
% 12.45/6.98 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 12.45/6.98 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7338_even__xor__iff,axiom,
% 12.45/6.98 ! [A: code_integer,B2: code_integer] :
% 12.45/6.98 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ A @ B2 ) )
% 12.45/6.98 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.45/6.98 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7339_even__xor__iff,axiom,
% 12.45/6.98 ! [A: nat,B2: nat] :
% 12.45/6.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B2 ) )
% 12.45/6.98 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.45/6.98 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7340_even__xor__iff,axiom,
% 12.45/6.98 ! [A: int,B2: int] :
% 12.45/6.98 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B2 ) )
% 12.45/6.98 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.45/6.98 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7341_sqrt__divide__self__eq,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( divide_divide_real @ ( sqrt @ X4 ) @ X4 )
% 12.45/6.98 = ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7342_ln__inverse,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( ln_ln_real @ ( inverse_inverse_real @ X4 ) )
% 12.45/6.98 = ( uminus_uminus_real @ ( ln_ln_real @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7343_Ints__odd__less__0,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
% 12.45/6.98 = ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7344_Ints__odd__less__0,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
% 12.45/6.98 = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7345_Ints__odd__less__0,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ( ( member_int @ A @ ring_1_Ints_int )
% 12.45/6.98 => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
% 12.45/6.98 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7346_Ints__nonzero__abs__ge1,axiom,
% 12.45/6.98 ! [X4: code_integer] :
% 12.45/6.98 ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
% 12.45/6.98 => ( ( X4 != zero_z3403309356797280102nteger )
% 12.45/6.98 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7347_Ints__nonzero__abs__ge1,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( member_real @ X4 @ ring_1_Ints_real )
% 12.45/6.98 => ( ( X4 != zero_zero_real )
% 12.45/6.98 => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7348_Ints__nonzero__abs__ge1,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ( ( member_rat @ X4 @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( X4 != zero_zero_rat )
% 12.45/6.98 => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7349_Ints__nonzero__abs__ge1,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ( ( member_int @ X4 @ ring_1_Ints_int )
% 12.45/6.98 => ( ( X4 != zero_zero_int )
% 12.45/6.98 => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7350_Ints__nonzero__abs__less1,axiom,
% 12.45/6.98 ! [X4: code_integer] :
% 12.45/6.98 ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
% 12.45/6.98 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer )
% 12.45/6.98 => ( X4 = zero_z3403309356797280102nteger ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7351_Ints__nonzero__abs__less1,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( member_real @ X4 @ ring_1_Ints_real )
% 12.45/6.98 => ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.98 => ( X4 = zero_zero_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7352_Ints__nonzero__abs__less1,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ( ( member_rat @ X4 @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( ord_less_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat )
% 12.45/6.98 => ( X4 = zero_zero_rat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7353_Ints__nonzero__abs__less1,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ( ( member_int @ X4 @ ring_1_Ints_int )
% 12.45/6.98 => ( ( ord_less_int @ ( abs_abs_int @ X4 ) @ one_one_int )
% 12.45/6.98 => ( X4 = zero_zero_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7354_Ints__eq__abs__less1,axiom,
% 12.45/6.98 ! [X4: code_integer,Y3: code_integer] :
% 12.45/6.98 ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
% 12.45/6.98 => ( ( member_Code_integer @ Y3 @ ring_11222124179247155820nteger )
% 12.45/6.98 => ( ( X4 = Y3 )
% 12.45/6.98 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ Y3 ) ) @ one_one_Code_integer ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7355_Ints__eq__abs__less1,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( member_real @ X4 @ ring_1_Ints_real )
% 12.45/6.98 => ( ( member_real @ Y3 @ ring_1_Ints_real )
% 12.45/6.98 => ( ( X4 = Y3 )
% 12.45/6.98 = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ one_one_real ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7356_Ints__eq__abs__less1,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( member_rat @ X4 @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( member_rat @ Y3 @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( X4 = Y3 )
% 12.45/6.98 = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ Y3 ) ) @ one_one_rat ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7357_Ints__eq__abs__less1,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( member_int @ X4 @ ring_1_Ints_int )
% 12.45/6.98 => ( ( member_int @ Y3 @ ring_1_Ints_int )
% 12.45/6.98 => ( ( X4 = Y3 )
% 12.45/6.98 = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7358_sin__times__pi__eq__0,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ( sin_real @ ( times_times_real @ X4 @ pi ) )
% 12.45/6.98 = zero_zero_real )
% 12.45/6.98 = ( member_real @ X4 @ ring_1_Ints_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7359_ex__inverse__of__nat__less,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ? [N4: nat] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 12.45/6.98 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7360_ex__inverse__of__nat__less,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ zero_zero_rat @ X4 )
% 12.45/6.98 => ? [N4: nat] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 12.45/6.98 & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N4 ) ) @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7361_power__diff__conv__inverse,axiom,
% 12.45/6.98 ! [X4: real,M: nat,N: nat] :
% 12.45/6.98 ( ( X4 != zero_zero_real )
% 12.45/6.98 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.98 => ( ( power_power_real @ X4 @ ( minus_minus_nat @ N @ M ) )
% 12.45/6.98 = ( times_times_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ M ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7362_power__diff__conv__inverse,axiom,
% 12.45/6.98 ! [X4: complex,M: nat,N: nat] :
% 12.45/6.98 ( ( X4 != zero_zero_complex )
% 12.45/6.98 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.98 => ( ( power_power_complex @ X4 @ ( minus_minus_nat @ N @ M ) )
% 12.45/6.98 = ( times_times_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ M ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7363_power__diff__conv__inverse,axiom,
% 12.45/6.98 ! [X4: rat,M: nat,N: nat] :
% 12.45/6.98 ( ( X4 != zero_zero_rat )
% 12.45/6.98 => ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.98 => ( ( power_power_rat @ X4 @ ( minus_minus_nat @ N @ M ) )
% 12.45/6.98 = ( times_times_rat @ ( power_power_rat @ X4 @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X4 ) @ M ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7364_Suc__times__gbinomial,axiom,
% 12.45/6.98 ! [K: nat,A: complex] :
% 12.45/6.98 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 12.45/6.98 = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7365_Suc__times__gbinomial,axiom,
% 12.45/6.98 ! [K: nat,A: real] :
% 12.45/6.98 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 12.45/6.98 = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7366_Suc__times__gbinomial,axiom,
% 12.45/6.98 ! [K: nat,A: rat] :
% 12.45/6.98 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 12.45/6.98 = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7367_gbinomial__absorption,axiom,
% 12.45/6.98 ! [K: nat,A: complex] :
% 12.45/6.98 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 12.45/6.98 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7368_gbinomial__absorption,axiom,
% 12.45/6.98 ! [K: nat,A: real] :
% 12.45/6.98 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 12.45/6.98 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7369_gbinomial__absorption,axiom,
% 12.45/6.98 ! [K: nat,A: rat] :
% 12.45/6.98 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 12.45/6.98 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7370_gbinomial__trinomial__revision,axiom,
% 12.45/6.98 ! [K: nat,M: nat,A: complex] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ M )
% 12.45/6.98 => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 12.45/6.98 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7371_gbinomial__trinomial__revision,axiom,
% 12.45/6.98 ! [K: nat,M: nat,A: real] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ M )
% 12.45/6.98 => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 12.45/6.98 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7372_gbinomial__trinomial__revision,axiom,
% 12.45/6.98 ! [K: nat,M: nat,A: rat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ K @ M )
% 12.45/6.98 => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 12.45/6.98 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7373_log__inverse,axiom,
% 12.45/6.98 ! [A: real,X4: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ( A != one_one_real )
% 12.45/6.98 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( log @ A @ ( inverse_inverse_real @ X4 ) )
% 12.45/6.98 = ( uminus_uminus_real @ ( log @ A @ X4 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7374_frac__neg,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ( member_real @ X4 @ ring_1_Ints_real )
% 12.45/6.98 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X4 ) )
% 12.45/6.98 = zero_zero_real ) )
% 12.45/6.98 & ( ~ ( member_real @ X4 @ ring_1_Ints_real )
% 12.45/6.98 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X4 ) )
% 12.45/6.98 = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X4 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7375_frac__neg,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ( ( ( member_rat @ X4 @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X4 ) )
% 12.45/6.98 = zero_zero_rat ) )
% 12.45/6.98 & ( ~ ( member_rat @ X4 @ ring_1_Ints_rat )
% 12.45/6.98 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X4 ) )
% 12.45/6.98 = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X4 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7376_gbinomial__factors,axiom,
% 12.45/6.98 ! [A: complex,K: nat] :
% 12.45/6.98 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 12.45/6.98 = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7377_gbinomial__factors,axiom,
% 12.45/6.98 ! [A: real,K: nat] :
% 12.45/6.98 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 12.45/6.98 = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7378_gbinomial__factors,axiom,
% 12.45/6.98 ! [A: rat,K: nat] :
% 12.45/6.98 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 12.45/6.98 = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7379_gbinomial__rec,axiom,
% 12.45/6.98 ! [A: complex,K: nat] :
% 12.45/6.98 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 12.45/6.98 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7380_gbinomial__rec,axiom,
% 12.45/6.98 ! [A: real,K: nat] :
% 12.45/6.98 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 12.45/6.98 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7381_gbinomial__rec,axiom,
% 12.45/6.98 ! [A: rat,K: nat] :
% 12.45/6.98 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 12.45/6.98 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7382_gbinomial__negated__upper,axiom,
% 12.45/6.98 ( gbinomial_complex
% 12.45/6.98 = ( ^ [A4: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A4 ) @ one_one_complex ) @ K3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7383_gbinomial__negated__upper,axiom,
% 12.45/6.98 ( gbinomial_real
% 12.45/6.98 = ( ^ [A4: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A4 ) @ one_one_real ) @ K3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7384_gbinomial__negated__upper,axiom,
% 12.45/6.98 ( gbinomial_rat
% 12.45/6.98 = ( ^ [A4: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A4 ) @ one_one_rat ) @ K3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7385_gbinomial__index__swap,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
% 12.45/6.98 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7386_gbinomial__index__swap,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( 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 ) )
% 12.45/6.98 = ( 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 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7387_gbinomial__index__swap,axiom,
% 12.45/6.98 ! [K: nat,N: nat] :
% 12.45/6.98 ( ( 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 ) )
% 12.45/6.98 = ( 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 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7388_exp__plus__inverse__exp,axiom,
% 12.45/6.98 ! [X4: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7389_le__mult__floor__Ints,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7390_le__mult__floor__Ints,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7391_le__mult__floor__Ints,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7392_le__mult__floor__Ints,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7393_le__mult__floor__Ints,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7394_le__mult__floor__Ints,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7395_frac__unique__iff,axiom,
% 12.45/6.98 ! [X4: real,A: real] :
% 12.45/6.98 ( ( ( archim2898591450579166408c_real @ X4 )
% 12.45/6.98 = A )
% 12.45/6.98 = ( ( member_real @ ( minus_minus_real @ X4 @ A ) @ ring_1_Ints_real )
% 12.45/6.98 & ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.98 & ( ord_less_real @ A @ one_one_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7396_frac__unique__iff,axiom,
% 12.45/6.98 ! [X4: rat,A: rat] :
% 12.45/6.98 ( ( ( archimedean_frac_rat @ X4 )
% 12.45/6.98 = A )
% 12.45/6.98 = ( ( member_rat @ ( minus_minus_rat @ X4 @ A ) @ ring_1_Ints_rat )
% 12.45/6.98 & ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.98 & ( ord_less_rat @ A @ one_one_rat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7397_mult__ceiling__le__Ints,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7398_mult__ceiling__le__Ints,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7399_mult__ceiling__le__Ints,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.45/6.98 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.45/6.98 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7400_mult__ceiling__le__Ints,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7401_mult__ceiling__le__Ints,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7402_mult__ceiling__le__Ints,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.45/6.98 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.45/6.98 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B2 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7403_gbinomial__minus,axiom,
% 12.45/6.98 ! [A: complex,K: nat] :
% 12.45/6.98 ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 12.45/6.98 = ( 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 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7404_gbinomial__minus,axiom,
% 12.45/6.98 ! [A: real,K: nat] :
% 12.45/6.98 ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 12.45/6.98 = ( 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 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7405_gbinomial__minus,axiom,
% 12.45/6.98 ! [A: rat,K: nat] :
% 12.45/6.98 ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 12.45/6.98 = ( 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 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7406_plus__inverse__ge__2,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7407_real__inv__sqrt__pow2,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.98 = ( inverse_inverse_real @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7408_gbinomial__reduce__nat,axiom,
% 12.45/6.98 ! [K: nat,A: complex] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.98 => ( ( gbinomial_complex @ A @ K )
% 12.45/6.98 = ( 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 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7409_gbinomial__reduce__nat,axiom,
% 12.45/6.98 ! [K: nat,A: real] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.98 => ( ( gbinomial_real @ A @ K )
% 12.45/6.98 = ( 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 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7410_gbinomial__reduce__nat,axiom,
% 12.45/6.98 ! [K: nat,A: rat] :
% 12.45/6.98 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.45/6.98 => ( ( gbinomial_rat @ A @ K )
% 12.45/6.98 = ( 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 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7411_gbinomial__pochhammer,axiom,
% 12.45/6.98 ( gbinomial_complex
% 12.45/6.98 = ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A4 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7412_gbinomial__pochhammer,axiom,
% 12.45/6.98 ( gbinomial_rat
% 12.45/6.98 = ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A4 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7413_gbinomial__pochhammer,axiom,
% 12.45/6.98 ( gbinomial_real
% 12.45/6.98 = ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A4 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7414_gbinomial__pochhammer_H,axiom,
% 12.45/6.98 ( gbinomial_complex
% 12.45/6.98 = ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7415_gbinomial__pochhammer_H,axiom,
% 12.45/6.98 ( gbinomial_rat
% 12.45/6.98 = ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7416_gbinomial__pochhammer_H,axiom,
% 12.45/6.98 ( gbinomial_real
% 12.45/6.98 = ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7417_tan__cot,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) )
% 12.45/6.98 = ( inverse_inverse_real @ ( tan_real @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7418_sin__integer__2pi,axiom,
% 12.45/6.98 ! [N: real] :
% 12.45/6.98 ( ( member_real @ N @ ring_1_Ints_real )
% 12.45/6.98 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 12.45/6.98 = zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7419_cos__integer__2pi,axiom,
% 12.45/6.98 ! [N: real] :
% 12.45/6.98 ( ( member_real @ N @ ring_1_Ints_real )
% 12.45/6.98 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 12.45/6.98 = one_one_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7420_xor__nat__unfold,axiom,
% 12.45/6.98 ( bit_se6528837805403552850or_nat
% 12.45/6.98 = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = 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 @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7421_xor__nat__rec,axiom,
% 12.45/6.98 ( bit_se6528837805403552850or_nat
% 12.45/6.98 = ( ^ [M6: nat,N2: nat] :
% 12.45/6.98 ( plus_plus_nat
% 12.45/6.98 @ ( zero_n2687167440665602831ol_nat
% 12.45/6.98 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
% 12.45/6.98 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 12.45/6.98 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7422_real__le__x__sinh,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ord_less_eq_real @ X4 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7423_xor__one__eq,axiom,
% 12.45/6.98 ! [A: code_integer] :
% 12.45/6.98 ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
% 12.45/6.98 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 12.45/6.98 @ ( zero_n356916108424825756nteger
% 12.45/6.98 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7424_xor__one__eq,axiom,
% 12.45/6.98 ! [A: nat] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
% 12.45/6.98 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 12.45/6.98 @ ( zero_n2687167440665602831ol_nat
% 12.45/6.98 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7425_xor__one__eq,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
% 12.45/6.98 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 12.45/6.98 @ ( zero_n2684676970156552555ol_int
% 12.45/6.98 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7426_one__xor__eq,axiom,
% 12.45/6.98 ! [A: code_integer] :
% 12.45/6.98 ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
% 12.45/6.98 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 12.45/6.98 @ ( zero_n356916108424825756nteger
% 12.45/6.98 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7427_one__xor__eq,axiom,
% 12.45/6.98 ! [A: nat] :
% 12.45/6.98 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
% 12.45/6.98 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 12.45/6.98 @ ( zero_n2687167440665602831ol_nat
% 12.45/6.98 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7428_one__xor__eq,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
% 12.45/6.98 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 12.45/6.98 @ ( zero_n2684676970156552555ol_int
% 12.45/6.98 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7429_real__le__abs__sinh,axiom,
% 12.45/6.98 ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7430_tan__sec,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ( cos_real @ X4 )
% 12.45/6.98 != zero_zero_real )
% 12.45/6.98 => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.98 = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7431_tan__sec,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( ( cos_complex @ X4 )
% 12.45/6.98 != zero_zero_complex )
% 12.45/6.98 => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.98 = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7432_sinh__ln__real,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( sinh_real @ ( ln_ln_real @ X4 ) )
% 12.45/6.98 = ( divide_divide_real @ ( minus_minus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7433_cosh__ln__real,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( cosh_real @ ( ln_ln_real @ X4 ) )
% 12.45/6.98 = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7434_cosh__zero__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ( cosh_real @ X4 )
% 12.45/6.98 = zero_zero_real )
% 12.45/6.98 = ( ( power_power_real @ ( exp_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.98 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7435_cosh__zero__iff,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( ( cosh_complex @ X4 )
% 12.45/6.98 = zero_zero_complex )
% 12.45/6.98 = ( ( power_power_complex @ ( exp_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.98 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7436_psubsetI,axiom,
% 12.45/6.98 ! [A3: set_nat,B4: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 12.45/6.98 => ( ( A3 != B4 )
% 12.45/6.98 => ( ord_less_set_nat @ A3 @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7437_push__bit__numeral__minus__1,axiom,
% 12.45/6.98 ! [N: num] :
% 12.45/6.98 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.98 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7438_push__bit__numeral__minus__1,axiom,
% 12.45/6.98 ! [N: num] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.98 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7439_push__bit__nonnegative__int__iff,axiom,
% 12.45/6.98 ! [N: nat,K: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 12.45/6.98 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7440_push__bit__negative__int__iff,axiom,
% 12.45/6.98 ! [N: nat,K: int] :
% 12.45/6.98 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 12.45/6.98 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7441_push__bit__of__0,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
% 12.45/6.98 = zero_zero_int ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7442_push__bit__of__0,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
% 12.45/6.98 = zero_zero_nat ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7443_push__bit__eq__0__iff,axiom,
% 12.45/6.98 ! [N: nat,A: int] :
% 12.45/6.98 ( ( ( bit_se545348938243370406it_int @ N @ A )
% 12.45/6.98 = zero_zero_int )
% 12.45/6.98 = ( A = zero_zero_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7444_push__bit__eq__0__iff,axiom,
% 12.45/6.98 ! [N: nat,A: nat] :
% 12.45/6.98 ( ( ( bit_se547839408752420682it_nat @ N @ A )
% 12.45/6.98 = zero_zero_nat )
% 12.45/6.98 = ( A = zero_zero_nat ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7445_push__bit__push__bit,axiom,
% 12.45/6.98 ! [M: nat,N: nat,A: int] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 12.45/6.98 = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7446_push__bit__push__bit,axiom,
% 12.45/6.98 ! [M: nat,N: nat,A: nat] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 12.45/6.98 = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7447_push__bit__and,axiom,
% 12.45/6.98 ! [N: nat,A: int,B2: int] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B2 ) )
% 12.45/6.98 = ( bit_se725231765392027082nd_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7448_push__bit__and,axiom,
% 12.45/6.98 ! [N: nat,A: nat,B2: nat] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B2 ) )
% 12.45/6.98 = ( bit_se727722235901077358nd_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7449_push__bit__xor,axiom,
% 12.45/6.98 ! [N: nat,A: int,B2: int] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B2 ) )
% 12.45/6.98 = ( bit_se6526347334894502574or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7450_push__bit__xor,axiom,
% 12.45/6.98 ! [N: nat,A: nat,B2: nat] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B2 ) )
% 12.45/6.98 = ( bit_se6528837805403552850or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7451_concat__bit__of__zero__1,axiom,
% 12.45/6.98 ! [N: nat,L2: int] :
% 12.45/6.98 ( ( bit_concat_bit @ N @ zero_zero_int @ L2 )
% 12.45/6.98 = ( bit_se545348938243370406it_int @ N @ L2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7452_sinh__real__less__iff,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) )
% 12.45/6.98 = ( ord_less_real @ X4 @ Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7453_sinh__real__le__iff,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) )
% 12.45/6.98 = ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7454_cosh__0,axiom,
% 12.45/6.98 ( ( cosh_complex @ zero_zero_complex )
% 12.45/6.98 = one_one_complex ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7455_cosh__0,axiom,
% 12.45/6.98 ( ( cosh_real @ zero_zero_real )
% 12.45/6.98 = one_one_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7456_sinh__real__pos__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X4 ) )
% 12.45/6.98 = ( ord_less_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7457_sinh__real__neg__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_real @ ( sinh_real @ X4 ) @ zero_zero_real )
% 12.45/6.98 = ( ord_less_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7458_sinh__real__nonneg__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X4 ) )
% 12.45/6.98 = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7459_sinh__real__nonpos__iff,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ ( sinh_real @ X4 ) @ zero_zero_real )
% 12.45/6.98 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7460_xor__nonnegative__int__iff,axiom,
% 12.45/6.98 ! [K: int,L2: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
% 12.45/6.98 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/6.98 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7461_xor__negative__int__iff,axiom,
% 12.45/6.98 ! [K: int,L2: int] :
% 12.45/6.98 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ zero_zero_int )
% 12.45/6.98 = ( ( ord_less_int @ K @ zero_zero_int )
% 12.45/6.98 != ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7462_push__bit__Suc__numeral,axiom,
% 12.45/6.98 ! [N: nat,K: num] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( numeral_numeral_int @ K ) )
% 12.45/6.98 = ( bit_se545348938243370406it_int @ N @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7463_push__bit__Suc__numeral,axiom,
% 12.45/6.98 ! [N: nat,K: num] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.45/6.98 = ( bit_se547839408752420682it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7464_push__bit__Suc__minus__numeral,axiom,
% 12.45/6.98 ! [N: nat,K: num] :
% 12.45/6.98 ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 12.45/6.98 = ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7465_push__bit__Suc__minus__numeral,axiom,
% 12.45/6.98 ! [N: nat,K: num] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/6.98 = ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7466_push__bit__numeral,axiom,
% 12.45/6.98 ! [L2: num,K: num] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ K ) )
% 12.45/6.98 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7467_push__bit__numeral,axiom,
% 12.45/6.98 ! [L2: num,K: num] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ K ) )
% 12.45/6.98 = ( bit_se547839408752420682it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7468_push__bit__of__Suc__0,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.45/6.98 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7469_push__bit__Suc,axiom,
% 12.45/6.98 ! [N: nat,A: int] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
% 12.45/6.98 = ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7470_push__bit__Suc,axiom,
% 12.45/6.98 ! [N: nat,A: nat] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
% 12.45/6.98 = ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7471_push__bit__of__1,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ N @ one_one_int )
% 12.45/6.98 = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7472_push__bit__of__1,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
% 12.45/6.98 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7473_even__push__bit__iff,axiom,
% 12.45/6.98 ! [N: nat,A: code_integer] :
% 12.45/6.98 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 12.45/6.98 = ( ( N != zero_zero_nat )
% 12.45/6.98 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7474_even__push__bit__iff,axiom,
% 12.45/6.98 ! [N: nat,A: int] :
% 12.45/6.98 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
% 12.45/6.98 = ( ( N != zero_zero_nat )
% 12.45/6.98 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7475_even__push__bit__iff,axiom,
% 12.45/6.98 ! [N: nat,A: nat] :
% 12.45/6.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 12.45/6.98 = ( ( N != zero_zero_nat )
% 12.45/6.98 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7476_push__bit__minus__numeral,axiom,
% 12.45/6.98 ! [L2: num,K: num] :
% 12.45/6.98 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 12.45/6.98 = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7477_push__bit__minus__numeral,axiom,
% 12.45/6.98 ! [L2: num,K: num] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/6.98 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7478_flip__bit__int__def,axiom,
% 12.45/6.98 ( bit_se2159334234014336723it_int
% 12.45/6.98 = ( ^ [N2: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7479_push__bit__add,axiom,
% 12.45/6.98 ! [N: nat,A: int,B2: int] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ N @ ( plus_plus_int @ A @ B2 ) )
% 12.45/6.98 = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7480_push__bit__add,axiom,
% 12.45/6.98 ! [N: nat,A: nat,B2: nat] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ N @ ( plus_plus_nat @ A @ B2 ) )
% 12.45/6.98 = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7481_cosh__plus__sinh,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( plus_plus_complex @ ( cosh_complex @ X4 ) @ ( sinh_complex @ X4 ) )
% 12.45/6.98 = ( exp_complex @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7482_cosh__plus__sinh,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( plus_plus_real @ ( cosh_real @ X4 ) @ ( sinh_real @ X4 ) )
% 12.45/6.98 = ( exp_real @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7483_sinh__plus__cosh,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( plus_plus_complex @ ( sinh_complex @ X4 ) @ ( cosh_complex @ X4 ) )
% 12.45/6.98 = ( exp_complex @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7484_sinh__plus__cosh,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( plus_plus_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) )
% 12.45/6.98 = ( exp_real @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7485_psubsetD,axiom,
% 12.45/6.98 ! [A3: set_Extended_enat,B4: set_Extended_enat,C: extended_enat] :
% 12.45/6.98 ( ( ord_le2529575680413868914d_enat @ A3 @ B4 )
% 12.45/6.98 => ( ( member_Extended_enat @ C @ A3 )
% 12.45/6.98 => ( member_Extended_enat @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7486_psubsetD,axiom,
% 12.45/6.98 ! [A3: set_complex,B4: set_complex,C: complex] :
% 12.45/6.98 ( ( ord_less_set_complex @ A3 @ B4 )
% 12.45/6.98 => ( ( member_complex @ C @ A3 )
% 12.45/6.98 => ( member_complex @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7487_psubsetD,axiom,
% 12.45/6.98 ! [A3: set_real,B4: set_real,C: real] :
% 12.45/6.98 ( ( ord_less_set_real @ A3 @ B4 )
% 12.45/6.98 => ( ( member_real @ C @ A3 )
% 12.45/6.98 => ( member_real @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7488_psubsetD,axiom,
% 12.45/6.98 ! [A3: set_nat,B4: set_nat,C: nat] :
% 12.45/6.98 ( ( ord_less_set_nat @ A3 @ B4 )
% 12.45/6.98 => ( ( member_nat @ C @ A3 )
% 12.45/6.98 => ( member_nat @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7489_psubsetD,axiom,
% 12.45/6.98 ! [A3: set_int,B4: set_int,C: int] :
% 12.45/6.98 ( ( ord_less_set_int @ A3 @ B4 )
% 12.45/6.98 => ( ( member_int @ C @ A3 )
% 12.45/6.98 => ( member_int @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7490_sinh__le__cosh__real,axiom,
% 12.45/6.98 ! [X4: real] : ( ord_less_eq_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7491_sinh__less__cosh__real,axiom,
% 12.45/6.98 ! [X4: real] : ( ord_less_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7492_push__bit__of__int,axiom,
% 12.45/6.98 ! [N: nat,K: int] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ N @ ( ring_1_of_int_int @ K ) )
% 12.45/6.98 = ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7493_of__nat__push__bit,axiom,
% 12.45/6.98 ! [M: nat,N: nat] :
% 12.45/6.98 ( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 12.45/6.98 = ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7494_of__nat__push__bit,axiom,
% 12.45/6.98 ! [M: nat,N: nat] :
% 12.45/6.98 ( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 12.45/6.98 = ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7495_push__bit__of__nat,axiom,
% 12.45/6.98 ! [N: nat,M: nat] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 12.45/6.98 = ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7496_push__bit__of__nat,axiom,
% 12.45/6.98 ! [N: nat,M: nat] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 12.45/6.98 = ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7497_psubset__imp__ex__mem,axiom,
% 12.45/6.98 ! [A3: set_Extended_enat,B4: set_Extended_enat] :
% 12.45/6.98 ( ( ord_le2529575680413868914d_enat @ A3 @ B4 )
% 12.45/6.98 => ? [B5: extended_enat] : ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ B4 @ A3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7498_psubset__imp__ex__mem,axiom,
% 12.45/6.98 ! [A3: set_complex,B4: set_complex] :
% 12.45/6.98 ( ( ord_less_set_complex @ A3 @ B4 )
% 12.45/6.98 => ? [B5: complex] : ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B4 @ A3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7499_psubset__imp__ex__mem,axiom,
% 12.45/6.98 ! [A3: set_real,B4: set_real] :
% 12.45/6.98 ( ( ord_less_set_real @ A3 @ B4 )
% 12.45/6.98 => ? [B5: real] : ( member_real @ B5 @ ( minus_minus_set_real @ B4 @ A3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7500_psubset__imp__ex__mem,axiom,
% 12.45/6.98 ! [A3: set_int,B4: set_int] :
% 12.45/6.98 ( ( ord_less_set_int @ A3 @ B4 )
% 12.45/6.98 => ? [B5: int] : ( member_int @ B5 @ ( minus_minus_set_int @ B4 @ A3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7501_psubset__imp__ex__mem,axiom,
% 12.45/6.98 ! [A3: set_nat,B4: set_nat] :
% 12.45/6.98 ( ( ord_less_set_nat @ A3 @ B4 )
% 12.45/6.98 => ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B4 @ A3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7502_sinh__diff,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( sinh_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 12.45/6.98 = ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( cosh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7503_cosh__diff,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( cosh_real @ ( minus_minus_real @ X4 @ Y3 ) )
% 12.45/6.98 = ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7504_sinh__add,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( sinh_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.98 = ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( cosh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7505_cosh__add,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( cosh_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.98 = ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) ) @ ( times_times_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7506_push__bit__nat__eq,axiom,
% 12.45/6.98 ! [N: nat,K: int] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 12.45/6.98 = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7507_push__bit__minus,axiom,
% 12.45/6.98 ! [N: nat,A: code_integer] :
% 12.45/6.98 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ A ) )
% 12.45/6.98 = ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7508_push__bit__minus,axiom,
% 12.45/6.98 ! [N: nat,A: int] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ A ) )
% 12.45/6.98 = ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7509_tanh__def,axiom,
% 12.45/6.98 ( tanh_complex
% 12.45/6.98 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7510_tanh__def,axiom,
% 12.45/6.98 ( tanh_real
% 12.45/6.98 = ( ^ [X: real] : ( divide_divide_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7511_bit__xor__int__iff,axiom,
% 12.45/6.98 ! [K: int,L2: int,N: nat] :
% 12.45/6.98 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N )
% 12.45/6.98 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 12.45/6.98 != ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7512_sinh__minus__cosh,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( minus_minus_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) )
% 12.45/6.98 = ( uminus_uminus_real @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7513_sinh__minus__cosh,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( minus_minus_complex @ ( sinh_complex @ X4 ) @ ( cosh_complex @ X4 ) )
% 12.45/6.98 = ( uminus1482373934393186551omplex @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7514_cosh__minus__sinh,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( minus_minus_real @ ( cosh_real @ X4 ) @ ( sinh_real @ X4 ) )
% 12.45/6.98 = ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7515_cosh__minus__sinh,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( minus_minus_complex @ ( cosh_complex @ X4 ) @ ( sinh_complex @ X4 ) )
% 12.45/6.98 = ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7516_XOR__lower,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/6.98 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7517_cosh__real__pos,axiom,
% 12.45/6.98 ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7518_cosh__real__nonneg,axiom,
% 12.45/6.98 ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7519_cosh__real__nonneg__le__iff,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
% 12.45/6.98 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7520_cosh__real__nonpos__le__iff,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.98 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 12.45/6.98 => ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
% 12.45/6.98 = ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7521_cosh__real__ge__1,axiom,
% 12.45/6.98 ! [X4: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7522_sinh__double,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.98 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X4 ) ) @ ( cosh_complex @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7523_sinh__double,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.98 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X4 ) ) @ ( cosh_real @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7524_push__bit__take__bit,axiom,
% 12.45/6.98 ! [M: nat,N: nat,A: int] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 12.45/6.98 = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7525_push__bit__take__bit,axiom,
% 12.45/6.98 ! [M: nat,N: nat,A: nat] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 12.45/6.98 = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7526_take__bit__push__bit,axiom,
% 12.45/6.98 ! [M: nat,N: nat,A: int] :
% 12.45/6.98 ( ( bit_se2923211474154528505it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 12.45/6.98 = ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7527_take__bit__push__bit,axiom,
% 12.45/6.98 ! [M: nat,N: nat,A: nat] :
% 12.45/6.98 ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 12.45/6.98 = ( bit_se547839408752420682it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7528_divide__complex__def,axiom,
% 12.45/6.98 ( divide1717551699836669952omplex
% 12.45/6.98 = ( ^ [X: complex,Y: complex] : ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7529_flip__bit__nat__def,axiom,
% 12.45/6.98 ( bit_se2161824704523386999it_nat
% 12.45/6.98 = ( ^ [M6: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7530_cosh__real__strict__mono,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 => ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7531_cosh__real__nonneg__less__iff,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/6.98 => ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
% 12.45/6.98 = ( ord_less_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7532_cosh__real__nonpos__less__iff,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/6.98 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 12.45/6.98 => ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y3 ) )
% 12.45/6.98 = ( ord_less_real @ Y3 @ X4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7533_bit__push__bit__iff__int,axiom,
% 12.45/6.98 ! [M: nat,K: int,N: nat] :
% 12.45/6.98 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 12.45/6.98 = ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.98 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7534_cosh__square__eq,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.98 = ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7535_cosh__square__eq,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.98 = ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7536_sinh__square__eq,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.98 = ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7537_sinh__square__eq,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/6.98 = ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7538_hyperbolic__pythagoras,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.98 = one_one_complex ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7539_hyperbolic__pythagoras,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.98 = one_one_real ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7540_xor__nat__def,axiom,
% 12.45/6.98 ( bit_se6528837805403552850or_nat
% 12.45/6.98 = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7541_psubsetE,axiom,
% 12.45/6.98 ! [A3: set_nat,B4: set_nat] :
% 12.45/6.98 ( ( ord_less_set_nat @ A3 @ B4 )
% 12.45/6.98 => ~ ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 12.45/6.98 => ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7542_psubset__eq,axiom,
% 12.45/6.98 ( ord_less_set_nat
% 12.45/6.98 = ( ^ [A6: set_nat,B7: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ A6 @ B7 )
% 12.45/6.98 & ( A6 != B7 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7543_psubset__imp__subset,axiom,
% 12.45/6.98 ! [A3: set_nat,B4: set_nat] :
% 12.45/6.98 ( ( ord_less_set_nat @ A3 @ B4 )
% 12.45/6.98 => ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7544_psubset__subset__trans,axiom,
% 12.45/6.98 ! [A3: set_nat,B4: set_nat,C5: set_nat] :
% 12.45/6.98 ( ( ord_less_set_nat @ A3 @ B4 )
% 12.45/6.98 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 12.45/6.98 => ( ord_less_set_nat @ A3 @ C5 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7545_subset__not__subset__eq,axiom,
% 12.45/6.98 ( ord_less_set_nat
% 12.45/6.98 = ( ^ [A6: set_nat,B7: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ A6 @ B7 )
% 12.45/6.98 & ~ ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7546_subset__psubset__trans,axiom,
% 12.45/6.98 ! [A3: set_nat,B4: set_nat,C5: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 12.45/6.98 => ( ( ord_less_set_nat @ B4 @ C5 )
% 12.45/6.98 => ( ord_less_set_nat @ A3 @ C5 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7547_subset__iff__psubset__eq,axiom,
% 12.45/6.98 ( ord_less_eq_set_nat
% 12.45/6.98 = ( ^ [A6: set_nat,B7: set_nat] :
% 12.45/6.98 ( ( ord_less_set_nat @ A6 @ B7 )
% 12.45/6.98 | ( A6 = B7 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7548_double__diff,axiom,
% 12.45/6.98 ! [A3: set_nat,B4: set_nat,C5: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ A3 @ B4 )
% 12.45/6.98 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 12.45/6.98 => ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C5 @ A3 ) )
% 12.45/6.98 = A3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7549_Diff__subset,axiom,
% 12.45/6.98 ! [A3: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ A3 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7550_Diff__mono,axiom,
% 12.45/6.98 ! [A3: set_nat,C5: set_nat,D4: set_nat,B4: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ A3 @ C5 )
% 12.45/6.98 => ( ( ord_less_eq_set_nat @ D4 @ B4 )
% 12.45/6.98 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ ( minus_minus_set_nat @ C5 @ D4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7551_bit__push__bit__iff__nat,axiom,
% 12.45/6.98 ! [M: nat,Q2: nat,N: nat] :
% 12.45/6.98 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
% 12.45/6.98 = ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.98 & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7552_concat__bit__eq,axiom,
% 12.45/6.98 ( bit_concat_bit
% 12.45/6.98 = ( ^ [N2: nat,K3: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7553_arcosh__cosh__real,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.98 => ( ( arcosh_real @ ( cosh_real @ X4 ) )
% 12.45/6.98 = X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7554_flip__bit__eq__xor,axiom,
% 12.45/6.98 ( bit_se2159334234014336723it_int
% 12.45/6.98 = ( ^ [N2: nat,A4: int] : ( bit_se6526347334894502574or_int @ A4 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7555_flip__bit__eq__xor,axiom,
% 12.45/6.98 ( bit_se2161824704523386999it_nat
% 12.45/6.98 = ( ^ [N2: nat,A4: nat] : ( bit_se6528837805403552850or_nat @ A4 @ ( bit_se547839408752420682it_nat @ N2 @ one_one_nat ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7556_cosh__double,axiom,
% 12.45/6.98 ! [X4: complex] :
% 12.45/6.98 ( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.98 = ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7557_cosh__double,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
% 12.45/6.98 = ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7558_push__bit__double,axiom,
% 12.45/6.98 ! [N: nat,A: int] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 12.45/6.98 = ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7559_push__bit__double,axiom,
% 12.45/6.98 ! [N: nat,A: nat] :
% 12.45/6.98 ( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.98 = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7560_bit__iff__and__push__bit__not__eq__0,axiom,
% 12.45/6.98 ( bit_se1146084159140164899it_int
% 12.45/6.98 = ( ^ [A4: int,N2: nat] :
% 12.45/6.98 ( ( bit_se725231765392027082nd_int @ A4 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) )
% 12.45/6.98 != zero_zero_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7561_bit__iff__and__push__bit__not__eq__0,axiom,
% 12.45/6.98 ( bit_se1148574629649215175it_nat
% 12.45/6.98 = ( ^ [A4: nat,N2: nat] :
% 12.45/6.98 ( ( bit_se727722235901077358nd_nat @ A4 @ ( bit_se547839408752420682it_nat @ N2 @ one_one_nat ) )
% 12.45/6.98 != zero_zero_nat ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7562_push__bit__int__def,axiom,
% 12.45/6.98 ( bit_se545348938243370406it_int
% 12.45/6.98 = ( ^ [N2: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7563_push__bit__nat__def,axiom,
% 12.45/6.98 ( bit_se547839408752420682it_nat
% 12.45/6.98 = ( ^ [N2: nat,M6: nat] : ( times_times_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7564_push__bit__eq__mult,axiom,
% 12.45/6.98 ( bit_se545348938243370406it_int
% 12.45/6.98 = ( ^ [N2: nat,A4: int] : ( times_times_int @ A4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7565_push__bit__eq__mult,axiom,
% 12.45/6.98 ( bit_se547839408752420682it_nat
% 12.45/6.98 = ( ^ [N2: nat,A4: nat] : ( times_times_nat @ A4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7566_exp__dvdE,axiom,
% 12.45/6.98 ! [N: nat,A: code_integer] :
% 12.45/6.98 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A )
% 12.45/6.98 => ~ ! [B5: code_integer] :
% 12.45/6.98 ( A
% 12.45/6.98 != ( bit_se7788150548672797655nteger @ N @ B5 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7567_exp__dvdE,axiom,
% 12.45/6.98 ! [N: nat,A: int] :
% 12.45/6.98 ( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A )
% 12.45/6.98 => ~ ! [B5: int] :
% 12.45/6.98 ( A
% 12.45/6.98 != ( bit_se545348938243370406it_int @ N @ B5 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7568_exp__dvdE,axiom,
% 12.45/6.98 ! [N: nat,A: nat] :
% 12.45/6.98 ( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A )
% 12.45/6.98 => ~ ! [B5: nat] :
% 12.45/6.98 ( A
% 12.45/6.98 != ( bit_se547839408752420682it_nat @ N @ B5 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7569_push__bit__minus__one,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.98 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7570_tanh__add,axiom,
% 12.45/6.98 ! [X4: complex,Y3: complex] :
% 12.45/6.98 ( ( ( cosh_complex @ X4 )
% 12.45/6.98 != zero_zero_complex )
% 12.45/6.98 => ( ( ( cosh_complex @ Y3 )
% 12.45/6.98 != zero_zero_complex )
% 12.45/6.98 => ( ( tanh_complex @ ( plus_plus_complex @ X4 @ Y3 ) )
% 12.45/6.98 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X4 ) @ ( tanh_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X4 ) @ ( tanh_complex @ Y3 ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7571_tanh__add,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ( cosh_real @ X4 )
% 12.45/6.98 != zero_zero_real )
% 12.45/6.98 => ( ( ( cosh_real @ Y3 )
% 12.45/6.98 != zero_zero_real )
% 12.45/6.98 => ( ( tanh_real @ ( plus_plus_real @ X4 @ Y3 ) )
% 12.45/6.98 = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y3 ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7572_XOR__upper,axiom,
% 12.45/6.98 ! [X4: int,N: nat,Y3: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/6.98 => ( ( ord_less_int @ X4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.98 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.98 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7573_cosh__field__def,axiom,
% 12.45/6.98 ( cosh_real
% 12.45/6.98 = ( ^ [Z5: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7574_cosh__field__def,axiom,
% 12.45/6.98 ( cosh_complex
% 12.45/6.98 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7575_complex__inverse,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B2 ) )
% 12.45/6.98 = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B2 ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7576_xor__int__rec,axiom,
% 12.45/6.98 ( bit_se6526347334894502574or_int
% 12.45/6.98 = ( ^ [K3: int,L: int] :
% 12.45/6.98 ( plus_plus_int
% 12.45/6.98 @ ( zero_n2684676970156552555ol_int
% 12.45/6.98 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
% 12.45/6.98 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 12.45/6.98 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7577_sinh__field__def,axiom,
% 12.45/6.98 ( sinh_real
% 12.45/6.98 = ( ^ [Z5: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7578_sinh__field__def,axiom,
% 12.45/6.98 ( sinh_complex
% 12.45/6.98 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7579_xor__int__unfold,axiom,
% 12.45/6.98 ( bit_se6526347334894502574or_int
% 12.45/6.98 = ( ^ [K3: int,L: int] :
% 12.45/6.98 ( if_int
% 12.45/6.98 @ ( K3
% 12.45/6.98 = ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.98 @ ( bit_ri7919022796975470100ot_int @ L )
% 12.45/6.98 @ ( if_int
% 12.45/6.98 @ ( L
% 12.45/6.98 = ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.98 @ ( bit_ri7919022796975470100ot_int @ K3 )
% 12.45/6.98 @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7580_True,axiom,
% 12.45/6.98 ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7581_Cauchy__iff2,axiom,
% 12.45/6.98 ( topolo4055970368930404560y_real
% 12.45/6.98 = ( ^ [X6: nat > real] :
% 12.45/6.98 ! [J3: nat] :
% 12.45/6.98 ? [M8: nat] :
% 12.45/6.98 ! [M6: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ M8 @ M6 )
% 12.45/6.98 => ! [N2: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ M8 @ N2 )
% 12.45/6.98 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7582_dual__order_Orefl,axiom,
% 12.45/6.98 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7583_dual__order_Orefl,axiom,
% 12.45/6.98 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7584_dual__order_Orefl,axiom,
% 12.45/6.98 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7585_dual__order_Orefl,axiom,
% 12.45/6.98 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7586_dual__order_Orefl,axiom,
% 12.45/6.98 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7587_order__refl,axiom,
% 12.45/6.98 ! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7588_order__refl,axiom,
% 12.45/6.98 ! [X4: rat] : ( ord_less_eq_rat @ X4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7589_order__refl,axiom,
% 12.45/6.98 ! [X4: num] : ( ord_less_eq_num @ X4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7590_order__refl,axiom,
% 12.45/6.98 ! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7591_order__refl,axiom,
% 12.45/6.98 ! [X4: int] : ( ord_less_eq_int @ X4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7592__C5_Ohyps_C_I4_J,axiom,
% 12.45/6.98 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 12.45/6.98 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7593_DiffI,axiom,
% 12.45/6.98 ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
% 12.45/6.98 ( ( member_Extended_enat @ C @ A3 )
% 12.45/6.98 => ( ~ ( member_Extended_enat @ C @ B4 )
% 12.45/6.98 => ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7594_DiffI,axiom,
% 12.45/6.98 ! [C: complex,A3: set_complex,B4: set_complex] :
% 12.45/6.98 ( ( member_complex @ C @ A3 )
% 12.45/6.98 => ( ~ ( member_complex @ C @ B4 )
% 12.45/6.98 => ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7595_DiffI,axiom,
% 12.45/6.98 ! [C: real,A3: set_real,B4: set_real] :
% 12.45/6.98 ( ( member_real @ C @ A3 )
% 12.45/6.98 => ( ~ ( member_real @ C @ B4 )
% 12.45/6.98 => ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7596_DiffI,axiom,
% 12.45/6.98 ! [C: int,A3: set_int,B4: set_int] :
% 12.45/6.98 ( ( member_int @ C @ A3 )
% 12.45/6.98 => ( ~ ( member_int @ C @ B4 )
% 12.45/6.98 => ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7597_DiffI,axiom,
% 12.45/6.98 ! [C: nat,A3: set_nat,B4: set_nat] :
% 12.45/6.98 ( ( member_nat @ C @ A3 )
% 12.45/6.98 => ( ~ ( member_nat @ C @ B4 )
% 12.45/6.98 => ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7598_Diff__iff,axiom,
% 12.45/6.98 ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
% 12.45/6.98 ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
% 12.45/6.98 = ( ( member_Extended_enat @ C @ A3 )
% 12.45/6.98 & ~ ( member_Extended_enat @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7599_Diff__iff,axiom,
% 12.45/6.98 ! [C: complex,A3: set_complex,B4: set_complex] :
% 12.45/6.98 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
% 12.45/6.98 = ( ( member_complex @ C @ A3 )
% 12.45/6.98 & ~ ( member_complex @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7600_Diff__iff,axiom,
% 12.45/6.98 ! [C: real,A3: set_real,B4: set_real] :
% 12.45/6.98 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
% 12.45/6.98 = ( ( member_real @ C @ A3 )
% 12.45/6.98 & ~ ( member_real @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7601_Diff__iff,axiom,
% 12.45/6.98 ! [C: int,A3: set_int,B4: set_int] :
% 12.45/6.98 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
% 12.45/6.98 = ( ( member_int @ C @ A3 )
% 12.45/6.98 & ~ ( member_int @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7602_Diff__iff,axiom,
% 12.45/6.98 ! [C: nat,A3: set_nat,B4: set_nat] :
% 12.45/6.98 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
% 12.45/6.98 = ( ( member_nat @ C @ A3 )
% 12.45/6.98 & ~ ( member_nat @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7603_Diff__idemp,axiom,
% 12.45/6.98 ! [A3: set_nat,B4: set_nat] :
% 12.45/6.98 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ B4 )
% 12.45/6.98 = ( minus_minus_set_nat @ A3 @ B4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7604_bit_Ocompl__eq__compl__iff,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ( bit_ri7919022796975470100ot_int @ X4 )
% 12.45/6.98 = ( bit_ri7919022796975470100ot_int @ Y3 ) )
% 12.45/6.98 = ( X4 = Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7605_bit_Odouble__compl,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ( ( bit_ri7919022796975470100ot_int @ ( bit_ri7919022796975470100ot_int @ X4 ) )
% 12.45/6.98 = X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7606_bit_Oxor__compl__right,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_ri7919022796975470100ot_int @ Y3 ) )
% 12.45/6.98 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7607_bit_Oxor__compl__left,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ Y3 )
% 12.45/6.98 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X4 @ Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7608_bit_Oconj__cancel__right,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ( ( bit_se725231765392027082nd_int @ X4 @ ( bit_ri7919022796975470100ot_int @ X4 ) )
% 12.45/6.98 = zero_zero_int ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7609_bit_Oconj__cancel__left,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ X4 )
% 12.45/6.98 = zero_zero_int ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7610_bit_Ocompl__zero,axiom,
% 12.45/6.98 ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
% 12.45/6.98 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7611_bit_Ocompl__zero,axiom,
% 12.45/6.98 ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
% 12.45/6.98 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7612_bit_Ocompl__one,axiom,
% 12.45/6.98 ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.98 = zero_z3403309356797280102nteger ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7613_bit_Ocompl__one,axiom,
% 12.45/6.98 ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.98 = zero_zero_int ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7614_bit_Oxor__one__left,axiom,
% 12.45/6.98 ! [X4: code_integer] :
% 12.45/6.98 ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X4 )
% 12.45/6.98 = ( bit_ri7632146776885996613nteger @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7615_bit_Oxor__one__left,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X4 )
% 12.45/6.98 = ( bit_ri7919022796975470100ot_int @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7616_bit_Oxor__one__right,axiom,
% 12.45/6.98 ! [X4: code_integer] :
% 12.45/6.98 ( ( bit_se3222712562003087583nteger @ X4 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.98 = ( bit_ri7632146776885996613nteger @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7617_bit_Oxor__one__right,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ X4 @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.98 = ( bit_ri7919022796975470100ot_int @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7618_bit_Oxor__cancel__left,axiom,
% 12.45/6.98 ! [X4: code_integer] :
% 12.45/6.98 ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X4 ) @ X4 )
% 12.45/6.98 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7619_bit_Oxor__cancel__left,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ X4 )
% 12.45/6.98 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7620_bit_Oxor__cancel__right,axiom,
% 12.45/6.98 ! [X4: code_integer] :
% 12.45/6.98 ( ( bit_se3222712562003087583nteger @ X4 @ ( bit_ri7632146776885996613nteger @ X4 ) )
% 12.45/6.98 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7621_bit_Oxor__cancel__right,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_ri7919022796975470100ot_int @ X4 ) )
% 12.45/6.98 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7622_not__negative__int__iff,axiom,
% 12.45/6.98 ! [K: int] :
% 12.45/6.98 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 12.45/6.98 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7623_not__nonnegative__int__iff,axiom,
% 12.45/6.98 ! [K: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 12.45/6.98 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7624_minus__not__numeral__eq,axiom,
% 12.45/6.98 ! [N: num] :
% 12.45/6.98 ( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.45/6.98 = ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7625_minus__not__numeral__eq,axiom,
% 12.45/6.98 ! [N: num] :
% 12.45/6.98 ( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/6.98 = ( numeral_numeral_int @ ( inc @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7626_even__not__iff,axiom,
% 12.45/6.98 ! [A: code_integer] :
% 12.45/6.98 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri7632146776885996613nteger @ A ) )
% 12.45/6.98 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7627_even__not__iff,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ A ) )
% 12.45/6.98 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7628_push__bit__minus__one__eq__not__mask,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.45/6.98 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7629_push__bit__minus__one__eq__not__mask,axiom,
% 12.45/6.98 ! [N: nat] :
% 12.45/6.98 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/6.98 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7630_not__one__eq,axiom,
% 12.45/6.98 ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
% 12.45/6.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7631_not__one__eq,axiom,
% 12.45/6.98 ( ( bit_ri7919022796975470100ot_int @ one_one_int )
% 12.45/6.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7632_length__induct,axiom,
% 12.45/6.98 ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
% 12.45/6.98 ( ! [Xs3: list_VEBT_VEBT] :
% 12.45/6.98 ( ! [Ys: list_VEBT_VEBT] :
% 12.45/6.98 ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 12.45/6.98 => ( P @ Ys ) )
% 12.45/6.98 => ( P @ Xs3 ) )
% 12.45/6.98 => ( P @ Xs2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7633_length__induct,axiom,
% 12.45/6.98 ! [P: list_o > $o,Xs2: list_o] :
% 12.45/6.98 ( ! [Xs3: list_o] :
% 12.45/6.98 ( ! [Ys: list_o] :
% 12.45/6.98 ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs3 ) )
% 12.45/6.98 => ( P @ Ys ) )
% 12.45/6.98 => ( P @ Xs3 ) )
% 12.45/6.98 => ( P @ Xs2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7634_length__induct,axiom,
% 12.45/6.98 ! [P: list_nat > $o,Xs2: list_nat] :
% 12.45/6.98 ( ! [Xs3: list_nat] :
% 12.45/6.98 ( ! [Ys: list_nat] :
% 12.45/6.98 ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs3 ) )
% 12.45/6.98 => ( P @ Ys ) )
% 12.45/6.98 => ( P @ Xs3 ) )
% 12.45/6.98 => ( P @ Xs2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7635_length__induct,axiom,
% 12.45/6.98 ! [P: list_int > $o,Xs2: list_int] :
% 12.45/6.98 ( ! [Xs3: list_int] :
% 12.45/6.98 ( ! [Ys: list_int] :
% 12.45/6.98 ( ( ord_less_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs3 ) )
% 12.45/6.98 => ( P @ Ys ) )
% 12.45/6.98 => ( P @ Xs3 ) )
% 12.45/6.98 => ( P @ Xs2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7636_DiffE,axiom,
% 12.45/6.98 ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
% 12.45/6.98 ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
% 12.45/6.98 => ~ ( ( member_Extended_enat @ C @ A3 )
% 12.45/6.98 => ( member_Extended_enat @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7637_DiffE,axiom,
% 12.45/6.98 ! [C: complex,A3: set_complex,B4: set_complex] :
% 12.45/6.98 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
% 12.45/6.98 => ~ ( ( member_complex @ C @ A3 )
% 12.45/6.98 => ( member_complex @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7638_DiffE,axiom,
% 12.45/6.98 ! [C: real,A3: set_real,B4: set_real] :
% 12.45/6.98 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
% 12.45/6.98 => ~ ( ( member_real @ C @ A3 )
% 12.45/6.98 => ( member_real @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7639_DiffE,axiom,
% 12.45/6.98 ! [C: int,A3: set_int,B4: set_int] :
% 12.45/6.98 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
% 12.45/6.98 => ~ ( ( member_int @ C @ A3 )
% 12.45/6.98 => ( member_int @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7640_DiffE,axiom,
% 12.45/6.98 ! [C: nat,A3: set_nat,B4: set_nat] :
% 12.45/6.98 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
% 12.45/6.98 => ~ ( ( member_nat @ C @ A3 )
% 12.45/6.98 => ( member_nat @ C @ B4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7641_DiffD1,axiom,
% 12.45/6.98 ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
% 12.45/6.98 ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
% 12.45/6.98 => ( member_Extended_enat @ C @ A3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7642_DiffD1,axiom,
% 12.45/6.98 ! [C: complex,A3: set_complex,B4: set_complex] :
% 12.45/6.98 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
% 12.45/6.98 => ( member_complex @ C @ A3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7643_DiffD1,axiom,
% 12.45/6.98 ! [C: real,A3: set_real,B4: set_real] :
% 12.45/6.98 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
% 12.45/6.98 => ( member_real @ C @ A3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7644_DiffD1,axiom,
% 12.45/6.98 ! [C: int,A3: set_int,B4: set_int] :
% 12.45/6.98 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
% 12.45/6.98 => ( member_int @ C @ A3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7645_DiffD1,axiom,
% 12.45/6.98 ! [C: nat,A3: set_nat,B4: set_nat] :
% 12.45/6.98 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
% 12.45/6.98 => ( member_nat @ C @ A3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7646_DiffD2,axiom,
% 12.45/6.98 ! [C: extended_enat,A3: set_Extended_enat,B4: set_Extended_enat] :
% 12.45/6.98 ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A3 @ B4 ) )
% 12.45/6.98 => ~ ( member_Extended_enat @ C @ B4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7647_DiffD2,axiom,
% 12.45/6.98 ! [C: complex,A3: set_complex,B4: set_complex] :
% 12.45/6.98 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A3 @ B4 ) )
% 12.45/6.98 => ~ ( member_complex @ C @ B4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7648_DiffD2,axiom,
% 12.45/6.98 ! [C: real,A3: set_real,B4: set_real] :
% 12.45/6.98 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B4 ) )
% 12.45/6.98 => ~ ( member_real @ C @ B4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7649_DiffD2,axiom,
% 12.45/6.98 ! [C: int,A3: set_int,B4: set_int] :
% 12.45/6.98 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B4 ) )
% 12.45/6.98 => ~ ( member_int @ C @ B4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7650_DiffD2,axiom,
% 12.45/6.98 ! [C: nat,A3: set_nat,B4: set_nat] :
% 12.45/6.98 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
% 12.45/6.98 => ~ ( member_nat @ C @ B4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7651_take__bit__not__iff,axiom,
% 12.45/6.98 ! [N: nat,A: int,B2: int] :
% 12.45/6.98 ( ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 12.45/6.98 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ B2 ) ) )
% 12.45/6.98 = ( ( bit_se2923211474154528505it_int @ N @ A )
% 12.45/6.98 = ( bit_se2923211474154528505it_int @ N @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7652_take__bit__not__take__bit,axiom,
% 12.45/6.98 ! [N: nat,A: int] :
% 12.45/6.98 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 12.45/6.98 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7653_bit__not__int__iff,axiom,
% 12.45/6.98 ! [K: int,N: nat] :
% 12.45/6.98 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 12.45/6.98 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7654_of__int__not__eq,axiom,
% 12.45/6.98 ! [K: int] :
% 12.45/6.98 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 12.45/6.98 = ( bit_ri7919022796975470100ot_int @ ( ring_1_of_int_int @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7655_size__neq__size__imp__neq,axiom,
% 12.45/6.98 ! [X4: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 12.45/6.98 ( ( ( size_s6755466524823107622T_VEBT @ X4 )
% 12.45/6.98 != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 12.45/6.98 => ( X4 != Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7656_size__neq__size__imp__neq,axiom,
% 12.45/6.98 ! [X4: list_o,Y3: list_o] :
% 12.45/6.98 ( ( ( size_size_list_o @ X4 )
% 12.45/6.98 != ( size_size_list_o @ Y3 ) )
% 12.45/6.98 => ( X4 != Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7657_size__neq__size__imp__neq,axiom,
% 12.45/6.98 ! [X4: list_nat,Y3: list_nat] :
% 12.45/6.98 ( ( ( size_size_list_nat @ X4 )
% 12.45/6.98 != ( size_size_list_nat @ Y3 ) )
% 12.45/6.98 => ( X4 != Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7658_size__neq__size__imp__neq,axiom,
% 12.45/6.98 ! [X4: list_int,Y3: list_int] :
% 12.45/6.98 ( ( ( size_size_list_int @ X4 )
% 12.45/6.98 != ( size_size_list_int @ Y3 ) )
% 12.45/6.98 => ( X4 != Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7659_size__neq__size__imp__neq,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( ( size_size_num @ X4 )
% 12.45/6.98 != ( size_size_num @ Y3 ) )
% 12.45/6.98 => ( X4 != Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7660_of__int__not__numeral,axiom,
% 12.45/6.98 ! [K: num] :
% 12.45/6.98 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/6.98 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7661_not__add__distrib,axiom,
% 12.45/6.98 ! [A: int,B2: int] :
% 12.45/6.98 ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B2 ) )
% 12.45/6.98 = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7662_not__diff__distrib,axiom,
% 12.45/6.98 ! [A: int,B2: int] :
% 12.45/6.98 ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B2 ) )
% 12.45/6.98 = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7663_minus__eq__not__plus__1,axiom,
% 12.45/6.98 ( uminus1351360451143612070nteger
% 12.45/6.98 = ( ^ [A4: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A4 ) @ one_one_Code_integer ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7664_minus__eq__not__plus__1,axiom,
% 12.45/6.98 ( uminus_uminus_int
% 12.45/6.98 = ( ^ [A4: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A4 ) @ one_one_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7665_not__eq__complement,axiom,
% 12.45/6.98 ( bit_ri7632146776885996613nteger
% 12.45/6.98 = ( ^ [A4: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A4 ) @ one_one_Code_integer ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7666_not__eq__complement,axiom,
% 12.45/6.98 ( bit_ri7919022796975470100ot_int
% 12.45/6.98 = ( ^ [A4: int] : ( minus_minus_int @ ( uminus_uminus_int @ A4 ) @ one_one_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7667_minus__eq__not__minus__1,axiom,
% 12.45/6.98 ( uminus1351360451143612070nteger
% 12.45/6.98 = ( ^ [A4: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A4 @ one_one_Code_integer ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7668_minus__eq__not__minus__1,axiom,
% 12.45/6.98 ( uminus_uminus_int
% 12.45/6.98 = ( ^ [A4: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A4 @ one_one_int ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7669_not__int__def,axiom,
% 12.45/6.98 ( bit_ri7919022796975470100ot_int
% 12.45/6.98 = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7670_and__not__numerals_I1_J,axiom,
% 12.45/6.98 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.45/6.98 = zero_zero_int ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7671_disjunctive__diff,axiom,
% 12.45/6.98 ! [B2: int,A: int] :
% 12.45/6.98 ( ! [N4: nat] :
% 12.45/6.98 ( ( bit_se1146084159140164899it_int @ B2 @ N4 )
% 12.45/6.98 => ( bit_se1146084159140164899it_int @ A @ N4 ) )
% 12.45/6.98 => ( ( minus_minus_int @ A @ B2 )
% 12.45/6.98 = ( bit_se725231765392027082nd_int @ A @ ( bit_ri7919022796975470100ot_int @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7672_nle__le,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ~ ( ord_less_eq_rat @ A @ B2 ) )
% 12.45/6.98 = ( ( ord_less_eq_rat @ B2 @ A )
% 12.45/6.98 & ( B2 != A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7673_nle__le,axiom,
% 12.45/6.98 ! [A: num,B2: num] :
% 12.45/6.98 ( ( ~ ( ord_less_eq_num @ A @ B2 ) )
% 12.45/6.98 = ( ( ord_less_eq_num @ B2 @ A )
% 12.45/6.98 & ( B2 != A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7674_nle__le,axiom,
% 12.45/6.98 ! [A: nat,B2: nat] :
% 12.45/6.98 ( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
% 12.45/6.98 = ( ( ord_less_eq_nat @ B2 @ A )
% 12.45/6.98 & ( B2 != A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7675_nle__le,axiom,
% 12.45/6.98 ! [A: int,B2: int] :
% 12.45/6.98 ( ( ~ ( ord_less_eq_int @ A @ B2 ) )
% 12.45/6.98 = ( ( ord_less_eq_int @ B2 @ A )
% 12.45/6.98 & ( B2 != A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7676_le__cases3,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat,Z: rat] :
% 12.45/6.98 ( ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
% 12.45/6.98 => ( ( ( ord_less_eq_rat @ Y3 @ X4 )
% 12.45/6.98 => ~ ( ord_less_eq_rat @ X4 @ Z ) )
% 12.45/6.98 => ( ( ( ord_less_eq_rat @ X4 @ Z )
% 12.45/6.98 => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
% 12.45/6.98 => ( ( ( ord_less_eq_rat @ Z @ Y3 )
% 12.45/6.98 => ~ ( ord_less_eq_rat @ Y3 @ X4 ) )
% 12.45/6.98 => ( ( ( ord_less_eq_rat @ Y3 @ Z )
% 12.45/6.98 => ~ ( ord_less_eq_rat @ Z @ X4 ) )
% 12.45/6.98 => ~ ( ( ord_less_eq_rat @ Z @ X4 )
% 12.45/6.98 => ~ ( ord_less_eq_rat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7677_le__cases3,axiom,
% 12.45/6.98 ! [X4: num,Y3: num,Z: num] :
% 12.45/6.98 ( ( ( ord_less_eq_num @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_eq_num @ Y3 @ Z ) )
% 12.45/6.98 => ( ( ( ord_less_eq_num @ Y3 @ X4 )
% 12.45/6.98 => ~ ( ord_less_eq_num @ X4 @ Z ) )
% 12.45/6.98 => ( ( ( ord_less_eq_num @ X4 @ Z )
% 12.45/6.98 => ~ ( ord_less_eq_num @ Z @ Y3 ) )
% 12.45/6.98 => ( ( ( ord_less_eq_num @ Z @ Y3 )
% 12.45/6.98 => ~ ( ord_less_eq_num @ Y3 @ X4 ) )
% 12.45/6.98 => ( ( ( ord_less_eq_num @ Y3 @ Z )
% 12.45/6.98 => ~ ( ord_less_eq_num @ Z @ X4 ) )
% 12.45/6.98 => ~ ( ( ord_less_eq_num @ Z @ X4 )
% 12.45/6.98 => ~ ( ord_less_eq_num @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7678_le__cases3,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat,Z: nat] :
% 12.45/6.98 ( ( ( ord_less_eq_nat @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 12.45/6.98 => ( ( ( ord_less_eq_nat @ Y3 @ X4 )
% 12.45/6.98 => ~ ( ord_less_eq_nat @ X4 @ Z ) )
% 12.45/6.98 => ( ( ( ord_less_eq_nat @ X4 @ Z )
% 12.45/6.98 => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 12.45/6.98 => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 12.45/6.98 => ~ ( ord_less_eq_nat @ Y3 @ X4 ) )
% 12.45/6.98 => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 12.45/6.98 => ~ ( ord_less_eq_nat @ Z @ X4 ) )
% 12.45/6.98 => ~ ( ( ord_less_eq_nat @ Z @ X4 )
% 12.45/6.98 => ~ ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7679_le__cases3,axiom,
% 12.45/6.98 ! [X4: int,Y3: int,Z: int] :
% 12.45/6.98 ( ( ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_eq_int @ Y3 @ Z ) )
% 12.45/6.98 => ( ( ( ord_less_eq_int @ Y3 @ X4 )
% 12.45/6.98 => ~ ( ord_less_eq_int @ X4 @ Z ) )
% 12.45/6.98 => ( ( ( ord_less_eq_int @ X4 @ Z )
% 12.45/6.98 => ~ ( ord_less_eq_int @ Z @ Y3 ) )
% 12.45/6.98 => ( ( ( ord_less_eq_int @ Z @ Y3 )
% 12.45/6.98 => ~ ( ord_less_eq_int @ Y3 @ X4 ) )
% 12.45/6.98 => ( ( ( ord_less_eq_int @ Y3 @ Z )
% 12.45/6.98 => ~ ( ord_less_eq_int @ Z @ X4 ) )
% 12.45/6.98 => ~ ( ( ord_less_eq_int @ Z @ X4 )
% 12.45/6.98 => ~ ( ord_less_eq_int @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7680_order__class_Oorder__eq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: set_nat,Z4: set_nat] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [X: set_nat,Y: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ X @ Y )
% 12.45/6.98 & ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7681_order__class_Oorder__eq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [X: rat,Y: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X @ Y )
% 12.45/6.98 & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7682_order__class_Oorder__eq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [X: num,Y: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X @ Y )
% 12.45/6.98 & ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7683_order__class_Oorder__eq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [X: nat,Y: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X @ Y )
% 12.45/6.98 & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7684_order__class_Oorder__eq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [X: int,Y: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ X @ Y )
% 12.45/6.98 & ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7685_ord__eq__le__trans,axiom,
% 12.45/6.98 ! [A: set_nat,B2: set_nat,C: set_nat] :
% 12.45/6.98 ( ( A = B2 )
% 12.45/6.98 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 12.45/6.98 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7686_ord__eq__le__trans,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.98 ( ( A = B2 )
% 12.45/6.98 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.98 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7687_ord__eq__le__trans,axiom,
% 12.45/6.98 ! [A: num,B2: num,C: num] :
% 12.45/6.98 ( ( A = B2 )
% 12.45/6.98 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.98 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7688_ord__eq__le__trans,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.98 ( ( A = B2 )
% 12.45/6.98 => ( ( ord_less_eq_nat @ B2 @ C )
% 12.45/6.98 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7689_ord__eq__le__trans,axiom,
% 12.45/6.98 ! [A: int,B2: int,C: int] :
% 12.45/6.98 ( ( A = B2 )
% 12.45/6.98 => ( ( ord_less_eq_int @ B2 @ C )
% 12.45/6.98 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7690_ord__le__eq__trans,axiom,
% 12.45/6.98 ! [A: set_nat,B2: set_nat,C: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ A @ B2 )
% 12.45/6.98 => ( ( B2 = C )
% 12.45/6.98 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7691_ord__le__eq__trans,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( B2 = C )
% 12.45/6.98 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7692_ord__le__eq__trans,axiom,
% 12.45/6.98 ! [A: num,B2: num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( ( B2 = C )
% 12.45/6.98 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7693_ord__le__eq__trans,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.98 => ( ( B2 = C )
% 12.45/6.98 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7694_ord__le__eq__trans,axiom,
% 12.45/6.98 ! [A: int,B2: int,C: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.98 => ( ( B2 = C )
% 12.45/6.98 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7695_order__antisym,axiom,
% 12.45/6.98 ! [X4: set_nat,Y3: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_set_nat @ Y3 @ X4 )
% 12.45/6.98 => ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7696_order__antisym,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_rat @ Y3 @ X4 )
% 12.45/6.98 => ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7697_order__antisym,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_num @ Y3 @ X4 )
% 12.45/6.98 => ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7698_order__antisym,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_nat @ Y3 @ X4 )
% 12.45/6.98 => ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7699_order__antisym,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_int @ Y3 @ X4 )
% 12.45/6.98 => ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7700_order_Otrans,axiom,
% 12.45/6.98 ! [A: set_nat,B2: set_nat,C: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 12.45/6.98 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7701_order_Otrans,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.98 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7702_order_Otrans,axiom,
% 12.45/6.98 ! [A: num,B2: num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.98 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7703_order_Otrans,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_nat @ B2 @ C )
% 12.45/6.98 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7704_order_Otrans,axiom,
% 12.45/6.98 ! [A: int,B2: int,C: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_int @ B2 @ C )
% 12.45/6.98 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7705_order__trans,axiom,
% 12.45/6.98 ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_set_nat @ Y3 @ Z )
% 12.45/6.98 => ( ord_less_eq_set_nat @ X4 @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7706_order__trans,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat,Z: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 12.45/6.98 => ( ord_less_eq_rat @ X4 @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7707_order__trans,axiom,
% 12.45/6.98 ! [X4: num,Y3: num,Z: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_num @ Y3 @ Z )
% 12.45/6.98 => ( ord_less_eq_num @ X4 @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7708_order__trans,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat,Z: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 12.45/6.98 => ( ord_less_eq_nat @ X4 @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7709_order__trans,axiom,
% 12.45/6.98 ! [X4: int,Y3: int,Z: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_eq_int @ Y3 @ Z )
% 12.45/6.98 => ( ord_less_eq_int @ X4 @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7710_linorder__wlog,axiom,
% 12.45/6.98 ! [P: rat > rat > $o,A: rat,B2: rat] :
% 12.45/6.98 ( ! [A5: rat,B5: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A5 @ B5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( ! [A5: rat,B5: rat] :
% 12.45/6.98 ( ( P @ B5 @ A5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( P @ A @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7711_linorder__wlog,axiom,
% 12.45/6.98 ! [P: num > num > $o,A: num,B2: num] :
% 12.45/6.98 ( ! [A5: num,B5: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ A5 @ B5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( ! [A5: num,B5: num] :
% 12.45/6.98 ( ( P @ B5 @ A5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( P @ A @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7712_linorder__wlog,axiom,
% 12.45/6.98 ! [P: nat > nat > $o,A: nat,B2: nat] :
% 12.45/6.98 ( ! [A5: nat,B5: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ A5 @ B5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( ! [A5: nat,B5: nat] :
% 12.45/6.98 ( ( P @ B5 @ A5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( P @ A @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7713_linorder__wlog,axiom,
% 12.45/6.98 ! [P: int > int > $o,A: int,B2: int] :
% 12.45/6.98 ( ! [A5: int,B5: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ A5 @ B5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( ! [A5: int,B5: int] :
% 12.45/6.98 ( ( P @ B5 @ A5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( P @ A @ B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7714_dual__order_Oeq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: set_nat,Z4: set_nat] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [A4: set_nat,B3: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ B3 @ A4 )
% 12.45/6.98 & ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7715_dual__order_Oeq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [A4: rat,B3: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ B3 @ A4 )
% 12.45/6.98 & ( ord_less_eq_rat @ A4 @ B3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7716_dual__order_Oeq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [A4: num,B3: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ B3 @ A4 )
% 12.45/6.98 & ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7717_dual__order_Oeq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [A4: nat,B3: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ B3 @ A4 )
% 12.45/6.98 & ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7718_dual__order_Oeq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [A4: int,B3: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ B3 @ A4 )
% 12.45/6.98 & ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7719_dual__order_Oantisym,axiom,
% 12.45/6.98 ! [B2: set_nat,A: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_eq_set_nat @ A @ B2 )
% 12.45/6.98 => ( A = B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7720_dual__order_Oantisym,axiom,
% 12.45/6.98 ! [B2: rat,A: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( A = B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7721_dual__order_Oantisym,axiom,
% 12.45/6.98 ! [B2: num,A: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( A = B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7722_dual__order_Oantisym,axiom,
% 12.45/6.98 ! [B2: nat,A: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.98 => ( A = B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7723_dual__order_Oantisym,axiom,
% 12.45/6.98 ! [B2: int,A: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.98 => ( A = B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7724_dual__order_Otrans,axiom,
% 12.45/6.98 ! [B2: set_nat,A: set_nat,C: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_eq_set_nat @ C @ B2 )
% 12.45/6.98 => ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7725_dual__order_Otrans,axiom,
% 12.45/6.98 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_eq_rat @ C @ B2 )
% 12.45/6.98 => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7726_dual__order_Otrans,axiom,
% 12.45/6.98 ! [B2: num,A: num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_eq_num @ C @ B2 )
% 12.45/6.98 => ( ord_less_eq_num @ C @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7727_dual__order_Otrans,axiom,
% 12.45/6.98 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_eq_nat @ C @ B2 )
% 12.45/6.98 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7728_dual__order_Otrans,axiom,
% 12.45/6.98 ! [B2: int,A: int,C: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_eq_int @ C @ B2 )
% 12.45/6.98 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7729_antisym,axiom,
% 12.45/6.98 ! [A: set_nat,B2: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_set_nat @ B2 @ A )
% 12.45/6.98 => ( A = B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7730_antisym,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_rat @ B2 @ A )
% 12.45/6.98 => ( A = B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7731_antisym,axiom,
% 12.45/6.98 ! [A: num,B2: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_num @ B2 @ A )
% 12.45/6.98 => ( A = B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7732_antisym,axiom,
% 12.45/6.98 ! [A: nat,B2: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_nat @ B2 @ A )
% 12.45/6.98 => ( A = B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7733_antisym,axiom,
% 12.45/6.98 ! [A: int,B2: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_int @ B2 @ A )
% 12.45/6.98 => ( A = B2 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7734_Orderings_Oorder__eq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: set_nat,Z4: set_nat] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [A4: set_nat,B3: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ A4 @ B3 )
% 12.45/6.98 & ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7735_Orderings_Oorder__eq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [A4: rat,B3: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A4 @ B3 )
% 12.45/6.98 & ( ord_less_eq_rat @ B3 @ A4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7736_Orderings_Oorder__eq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [A4: num,B3: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ A4 @ B3 )
% 12.45/6.98 & ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7737_Orderings_Oorder__eq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [A4: nat,B3: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ A4 @ B3 )
% 12.45/6.98 & ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7738_Orderings_Oorder__eq__iff,axiom,
% 12.45/6.98 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 12.45/6.98 = ( ^ [A4: int,B3: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ A4 @ B3 )
% 12.45/6.98 & ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7739_order__subst1,axiom,
% 12.45/6.98 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7740_order__subst1,axiom,
% 12.45/6.98 ! [A: rat,F: num > rat,B2: num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7741_order__subst1,axiom,
% 12.45/6.98 ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_nat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7742_order__subst1,axiom,
% 12.45/6.98 ! [A: rat,F: int > rat,B2: int,C: int] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_int @ B2 @ C )
% 12.45/6.98 => ( ! [X3: int,Y4: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7743_order__subst1,axiom,
% 12.45/6.98 ! [A: num,F: rat > num,B2: rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7744_order__subst1,axiom,
% 12.45/6.98 ! [A: num,F: num > num,B2: num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7745_order__subst1,axiom,
% 12.45/6.98 ! [A: num,F: nat > num,B2: nat,C: nat] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_nat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7746_order__subst1,axiom,
% 12.45/6.98 ! [A: num,F: int > num,B2: int,C: int] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_int @ B2 @ C )
% 12.45/6.98 => ( ! [X3: int,Y4: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7747_order__subst1,axiom,
% 12.45/6.98 ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7748_order__subst1,axiom,
% 12.45/6.98 ! [A: nat,F: num > nat,B2: num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7749_order__subst2,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7750_order__subst2,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7751_order__subst2,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7752_order__subst2,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > int,C: int] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7753_order__subst2,axiom,
% 12.45/6.98 ! [A: num,B2: num,F: num > rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7754_order__subst2,axiom,
% 12.45/6.98 ! [A: num,B2: num,F: num > num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7755_order__subst2,axiom,
% 12.45/6.98 ! [A: num,B2: num,F: num > nat,C: nat] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7756_order__subst2,axiom,
% 12.45/6.98 ! [A: num,B2: num,F: num > int,C: int] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7757_order__subst2,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7758_order__subst2,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,F: nat > num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7759_order__eq__refl,axiom,
% 12.45/6.98 ! [X4: set_nat,Y3: set_nat] :
% 12.45/6.98 ( ( X4 = Y3 )
% 12.45/6.98 => ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7760_order__eq__refl,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( X4 = Y3 )
% 12.45/6.98 => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7761_order__eq__refl,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( X4 = Y3 )
% 12.45/6.98 => ( ord_less_eq_num @ X4 @ Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7762_order__eq__refl,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ( X4 = Y3 )
% 12.45/6.98 => ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7763_order__eq__refl,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( X4 = Y3 )
% 12.45/6.98 => ( ord_less_eq_int @ X4 @ Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7764_linorder__linear,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.98 | ( ord_less_eq_rat @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7765_linorder__linear,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X4 @ Y3 )
% 12.45/6.98 | ( ord_less_eq_num @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7766_linorder__linear,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 12.45/6.98 | ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7767_linorder__linear,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.98 | ( ord_less_eq_int @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7768_ord__eq__le__subst,axiom,
% 12.45/6.98 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7769_ord__eq__le__subst,axiom,
% 12.45/6.98 ! [A: num,F: rat > num,B2: rat,C: rat] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7770_ord__eq__le__subst,axiom,
% 12.45/6.98 ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7771_ord__eq__le__subst,axiom,
% 12.45/6.98 ! [A: int,F: rat > int,B2: rat,C: rat] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7772_ord__eq__le__subst,axiom,
% 12.45/6.98 ! [A: rat,F: num > rat,B2: num,C: num] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7773_ord__eq__le__subst,axiom,
% 12.45/6.98 ! [A: num,F: num > num,B2: num,C: num] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7774_ord__eq__le__subst,axiom,
% 12.45/6.98 ! [A: nat,F: num > nat,B2: num,C: num] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7775_ord__eq__le__subst,axiom,
% 12.45/6.98 ! [A: int,F: num > int,B2: num,C: num] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7776_ord__eq__le__subst,axiom,
% 12.45/6.98 ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_nat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7777_ord__eq__le__subst,axiom,
% 12.45/6.98 ! [A: num,F: nat > num,B2: nat,C: nat] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_eq_nat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7778_ord__le__eq__subst,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7779_ord__le__eq__subst,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7780_ord__le__eq__subst,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7781_ord__le__eq__subst,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > int,C: int] :
% 12.45/6.98 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7782_ord__le__eq__subst,axiom,
% 12.45/6.98 ! [A: num,B2: num,F: num > rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7783_ord__le__eq__subst,axiom,
% 12.45/6.98 ! [A: num,B2: num,F: num > num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7784_ord__le__eq__subst,axiom,
% 12.45/6.98 ! [A: num,B2: num,F: num > nat,C: nat] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7785_ord__le__eq__subst,axiom,
% 12.45/6.98 ! [A: num,B2: num,F: num > int,C: int] :
% 12.45/6.98 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7786_ord__le__eq__subst,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7787_ord__le__eq__subst,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,F: nat > num,C: num] :
% 12.45/6.98 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7788_linorder__le__cases,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ~ ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.98 => ( ord_less_eq_rat @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7789_linorder__le__cases,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ~ ( ord_less_eq_num @ X4 @ Y3 )
% 12.45/6.98 => ( ord_less_eq_num @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7790_linorder__le__cases,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ~ ( ord_less_eq_nat @ X4 @ Y3 )
% 12.45/6.98 => ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7791_linorder__le__cases,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ~ ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.98 => ( ord_less_eq_int @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7792_order__antisym__conv,axiom,
% 12.45/6.98 ! [Y3: set_nat,X4: set_nat] :
% 12.45/6.98 ( ( ord_less_eq_set_nat @ Y3 @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 12.45/6.98 = ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7793_order__antisym__conv,axiom,
% 12.45/6.98 ! [Y3: rat,X4: rat] :
% 12.45/6.98 ( ( ord_less_eq_rat @ Y3 @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.98 = ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7794_order__antisym__conv,axiom,
% 12.45/6.98 ! [Y3: num,X4: num] :
% 12.45/6.98 ( ( ord_less_eq_num @ Y3 @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_num @ X4 @ Y3 )
% 12.45/6.98 = ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7795_order__antisym__conv,axiom,
% 12.45/6.98 ! [Y3: nat,X4: nat] :
% 12.45/6.98 ( ( ord_less_eq_nat @ Y3 @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_nat @ X4 @ Y3 )
% 12.45/6.98 = ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7796_order__antisym__conv,axiom,
% 12.45/6.98 ! [Y3: int,X4: int] :
% 12.45/6.98 ( ( ord_less_eq_int @ Y3 @ X4 )
% 12.45/6.98 => ( ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.98 = ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7797_lt__ex,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ? [Y4: real] : ( ord_less_real @ Y4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7798_lt__ex,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7799_lt__ex,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ? [Y4: int] : ( ord_less_int @ Y4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7800_gt__ex,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7801_gt__ex,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7802_gt__ex,axiom,
% 12.45/6.98 ! [X4: nat] :
% 12.45/6.98 ? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7803_gt__ex,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ? [X_1: int] : ( ord_less_int @ X4 @ X_1 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7804_dense,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 => ? [Z2: real] :
% 12.45/6.98 ( ( ord_less_real @ X4 @ Z2 )
% 12.45/6.98 & ( ord_less_real @ Z2 @ Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7805_dense,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.98 => ? [Z2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X4 @ Z2 )
% 12.45/6.98 & ( ord_less_rat @ Z2 @ Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7806_less__imp__neq,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 => ( X4 != Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7807_less__imp__neq,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.98 => ( X4 != Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7808_less__imp__neq,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.98 => ( X4 != Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7809_less__imp__neq,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.98 => ( X4 != Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7810_less__imp__neq,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.98 => ( X4 != Y3 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7811_order_Oasym,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ~ ( ord_less_real @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7812_order_Oasym,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ~ ( ord_less_rat @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7813_order_Oasym,axiom,
% 12.45/6.98 ! [A: num,B2: num] :
% 12.45/6.98 ( ( ord_less_num @ A @ B2 )
% 12.45/6.98 => ~ ( ord_less_num @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7814_order_Oasym,axiom,
% 12.45/6.98 ! [A: nat,B2: nat] :
% 12.45/6.98 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.98 => ~ ( ord_less_nat @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7815_order_Oasym,axiom,
% 12.45/6.98 ! [A: int,B2: int] :
% 12.45/6.98 ( ( ord_less_int @ A @ B2 )
% 12.45/6.98 => ~ ( ord_less_int @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7816_ord__eq__less__trans,axiom,
% 12.45/6.98 ! [A: real,B2: real,C: real] :
% 12.45/6.98 ( ( A = B2 )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.98 => ( ord_less_real @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7817_ord__eq__less__trans,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.98 ( ( A = B2 )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.98 => ( ord_less_rat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7818_ord__eq__less__trans,axiom,
% 12.45/6.98 ! [A: num,B2: num,C: num] :
% 12.45/6.98 ( ( A = B2 )
% 12.45/6.98 => ( ( ord_less_num @ B2 @ C )
% 12.45/6.98 => ( ord_less_num @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7819_ord__eq__less__trans,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.98 ( ( A = B2 )
% 12.45/6.98 => ( ( ord_less_nat @ B2 @ C )
% 12.45/6.98 => ( ord_less_nat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7820_ord__eq__less__trans,axiom,
% 12.45/6.98 ! [A: int,B2: int,C: int] :
% 12.45/6.98 ( ( A = B2 )
% 12.45/6.98 => ( ( ord_less_int @ B2 @ C )
% 12.45/6.98 => ( ord_less_int @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7821_ord__less__eq__trans,axiom,
% 12.45/6.98 ! [A: real,B2: real,C: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( B2 = C )
% 12.45/6.98 => ( ord_less_real @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7822_ord__less__eq__trans,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( B2 = C )
% 12.45/6.98 => ( ord_less_rat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7823_ord__less__eq__trans,axiom,
% 12.45/6.98 ! [A: num,B2: num,C: num] :
% 12.45/6.98 ( ( ord_less_num @ A @ B2 )
% 12.45/6.98 => ( ( B2 = C )
% 12.45/6.98 => ( ord_less_num @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7824_ord__less__eq__trans,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.98 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.98 => ( ( B2 = C )
% 12.45/6.98 => ( ord_less_nat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7825_ord__less__eq__trans,axiom,
% 12.45/6.98 ! [A: int,B2: int,C: int] :
% 12.45/6.98 ( ( ord_less_int @ A @ B2 )
% 12.45/6.98 => ( ( B2 = C )
% 12.45/6.98 => ( ord_less_int @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7826_less__induct,axiom,
% 12.45/6.98 ! [P: nat > $o,A: nat] :
% 12.45/6.98 ( ! [X3: nat] :
% 12.45/6.98 ( ! [Y5: nat] :
% 12.45/6.98 ( ( ord_less_nat @ Y5 @ X3 )
% 12.45/6.98 => ( P @ Y5 ) )
% 12.45/6.98 => ( P @ X3 ) )
% 12.45/6.98 => ( P @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7827_antisym__conv3,axiom,
% 12.45/6.98 ! [Y3: real,X4: real] :
% 12.45/6.98 ( ~ ( ord_less_real @ Y3 @ X4 )
% 12.45/6.98 => ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
% 12.45/6.98 = ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7828_antisym__conv3,axiom,
% 12.45/6.98 ! [Y3: rat,X4: rat] :
% 12.45/6.98 ( ~ ( ord_less_rat @ Y3 @ X4 )
% 12.45/6.98 => ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
% 12.45/6.98 = ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7829_antisym__conv3,axiom,
% 12.45/6.98 ! [Y3: num,X4: num] :
% 12.45/6.98 ( ~ ( ord_less_num @ Y3 @ X4 )
% 12.45/6.98 => ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
% 12.45/6.98 = ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7830_antisym__conv3,axiom,
% 12.45/6.98 ! [Y3: nat,X4: nat] :
% 12.45/6.98 ( ~ ( ord_less_nat @ Y3 @ X4 )
% 12.45/6.98 => ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 12.45/6.98 = ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7831_antisym__conv3,axiom,
% 12.45/6.98 ! [Y3: int,X4: int] :
% 12.45/6.98 ( ~ ( ord_less_int @ Y3 @ X4 )
% 12.45/6.98 => ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
% 12.45/6.98 = ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7832_linorder__cases,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ~ ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 => ( ( X4 != Y3 )
% 12.45/6.98 => ( ord_less_real @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7833_linorder__cases,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ~ ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.98 => ( ( X4 != Y3 )
% 12.45/6.98 => ( ord_less_rat @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7834_linorder__cases,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ~ ( ord_less_num @ X4 @ Y3 )
% 12.45/6.98 => ( ( X4 != Y3 )
% 12.45/6.98 => ( ord_less_num @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7835_linorder__cases,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ~ ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.98 => ( ( X4 != Y3 )
% 12.45/6.98 => ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7836_linorder__cases,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ~ ( ord_less_int @ X4 @ Y3 )
% 12.45/6.98 => ( ( X4 != Y3 )
% 12.45/6.98 => ( ord_less_int @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7837_dual__order_Oasym,axiom,
% 12.45/6.98 ! [B2: real,A: real] :
% 12.45/6.98 ( ( ord_less_real @ B2 @ A )
% 12.45/6.98 => ~ ( ord_less_real @ A @ B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7838_dual__order_Oasym,axiom,
% 12.45/6.98 ! [B2: rat,A: rat] :
% 12.45/6.98 ( ( ord_less_rat @ B2 @ A )
% 12.45/6.98 => ~ ( ord_less_rat @ A @ B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7839_dual__order_Oasym,axiom,
% 12.45/6.98 ! [B2: num,A: num] :
% 12.45/6.98 ( ( ord_less_num @ B2 @ A )
% 12.45/6.98 => ~ ( ord_less_num @ A @ B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7840_dual__order_Oasym,axiom,
% 12.45/6.98 ! [B2: nat,A: nat] :
% 12.45/6.98 ( ( ord_less_nat @ B2 @ A )
% 12.45/6.98 => ~ ( ord_less_nat @ A @ B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7841_dual__order_Oasym,axiom,
% 12.45/6.98 ! [B2: int,A: int] :
% 12.45/6.98 ( ( ord_less_int @ B2 @ A )
% 12.45/6.98 => ~ ( ord_less_int @ A @ B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7842_dual__order_Oirrefl,axiom,
% 12.45/6.98 ! [A: real] :
% 12.45/6.98 ~ ( ord_less_real @ A @ A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7843_dual__order_Oirrefl,axiom,
% 12.45/6.98 ! [A: rat] :
% 12.45/6.98 ~ ( ord_less_rat @ A @ A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7844_dual__order_Oirrefl,axiom,
% 12.45/6.98 ! [A: num] :
% 12.45/6.98 ~ ( ord_less_num @ A @ A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7845_dual__order_Oirrefl,axiom,
% 12.45/6.98 ! [A: nat] :
% 12.45/6.98 ~ ( ord_less_nat @ A @ A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7846_dual__order_Oirrefl,axiom,
% 12.45/6.98 ! [A: int] :
% 12.45/6.98 ~ ( ord_less_int @ A @ A ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7847_exists__least__iff,axiom,
% 12.45/6.98 ( ( ^ [P5: nat > $o] :
% 12.45/6.98 ? [X7: nat] : ( P5 @ X7 ) )
% 12.45/6.98 = ( ^ [P6: nat > $o] :
% 12.45/6.98 ? [N2: nat] :
% 12.45/6.98 ( ( P6 @ N2 )
% 12.45/6.98 & ! [M6: nat] :
% 12.45/6.98 ( ( ord_less_nat @ M6 @ N2 )
% 12.45/6.98 => ~ ( P6 @ M6 ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7848_linorder__less__wlog,axiom,
% 12.45/6.98 ! [P: real > real > $o,A: real,B2: real] :
% 12.45/6.98 ( ! [A5: real,B5: real] :
% 12.45/6.98 ( ( ord_less_real @ A5 @ B5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( ! [A5: real] : ( P @ A5 @ A5 )
% 12.45/6.98 => ( ! [A5: real,B5: real] :
% 12.45/6.98 ( ( P @ B5 @ A5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( P @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7849_linorder__less__wlog,axiom,
% 12.45/6.98 ! [P: rat > rat > $o,A: rat,B2: rat] :
% 12.45/6.98 ( ! [A5: rat,B5: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A5 @ B5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( ! [A5: rat] : ( P @ A5 @ A5 )
% 12.45/6.98 => ( ! [A5: rat,B5: rat] :
% 12.45/6.98 ( ( P @ B5 @ A5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( P @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7850_linorder__less__wlog,axiom,
% 12.45/6.98 ! [P: num > num > $o,A: num,B2: num] :
% 12.45/6.98 ( ! [A5: num,B5: num] :
% 12.45/6.98 ( ( ord_less_num @ A5 @ B5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( ! [A5: num] : ( P @ A5 @ A5 )
% 12.45/6.98 => ( ! [A5: num,B5: num] :
% 12.45/6.98 ( ( P @ B5 @ A5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( P @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7851_linorder__less__wlog,axiom,
% 12.45/6.98 ! [P: nat > nat > $o,A: nat,B2: nat] :
% 12.45/6.98 ( ! [A5: nat,B5: nat] :
% 12.45/6.98 ( ( ord_less_nat @ A5 @ B5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( ! [A5: nat] : ( P @ A5 @ A5 )
% 12.45/6.98 => ( ! [A5: nat,B5: nat] :
% 12.45/6.98 ( ( P @ B5 @ A5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( P @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7852_linorder__less__wlog,axiom,
% 12.45/6.98 ! [P: int > int > $o,A: int,B2: int] :
% 12.45/6.98 ( ! [A5: int,B5: int] :
% 12.45/6.98 ( ( ord_less_int @ A5 @ B5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( ! [A5: int] : ( P @ A5 @ A5 )
% 12.45/6.98 => ( ! [A5: int,B5: int] :
% 12.45/6.98 ( ( P @ B5 @ A5 )
% 12.45/6.98 => ( P @ A5 @ B5 ) )
% 12.45/6.98 => ( P @ A @ B2 ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7853_order_Ostrict__trans,axiom,
% 12.45/6.98 ! [A: real,B2: real,C: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.98 => ( ord_less_real @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7854_order_Ostrict__trans,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.98 => ( ord_less_rat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7855_order_Ostrict__trans,axiom,
% 12.45/6.98 ! [A: num,B2: num,C: num] :
% 12.45/6.98 ( ( ord_less_num @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_num @ B2 @ C )
% 12.45/6.98 => ( ord_less_num @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7856_order_Ostrict__trans,axiom,
% 12.45/6.98 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.98 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_nat @ B2 @ C )
% 12.45/6.98 => ( ord_less_nat @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7857_order_Ostrict__trans,axiom,
% 12.45/6.98 ! [A: int,B2: int,C: int] :
% 12.45/6.98 ( ( ord_less_int @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_int @ B2 @ C )
% 12.45/6.98 => ( ord_less_int @ A @ C ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7858_not__less__iff__gr__or__eq,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
% 12.45/6.98 = ( ( ord_less_real @ Y3 @ X4 )
% 12.45/6.98 | ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7859_not__less__iff__gr__or__eq,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
% 12.45/6.98 = ( ( ord_less_rat @ Y3 @ X4 )
% 12.45/6.98 | ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7860_not__less__iff__gr__or__eq,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
% 12.45/6.98 = ( ( ord_less_num @ Y3 @ X4 )
% 12.45/6.98 | ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7861_not__less__iff__gr__or__eq,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 12.45/6.98 = ( ( ord_less_nat @ Y3 @ X4 )
% 12.45/6.98 | ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7862_not__less__iff__gr__or__eq,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
% 12.45/6.98 = ( ( ord_less_int @ Y3 @ X4 )
% 12.45/6.98 | ( X4 = Y3 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7863_dual__order_Ostrict__trans,axiom,
% 12.45/6.98 ! [B2: real,A: real,C: real] :
% 12.45/6.98 ( ( ord_less_real @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_real @ C @ B2 )
% 12.45/6.98 => ( ord_less_real @ C @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7864_dual__order_Ostrict__trans,axiom,
% 12.45/6.98 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.98 ( ( ord_less_rat @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_rat @ C @ B2 )
% 12.45/6.98 => ( ord_less_rat @ C @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7865_dual__order_Ostrict__trans,axiom,
% 12.45/6.98 ! [B2: num,A: num,C: num] :
% 12.45/6.98 ( ( ord_less_num @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_num @ C @ B2 )
% 12.45/6.98 => ( ord_less_num @ C @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7866_dual__order_Ostrict__trans,axiom,
% 12.45/6.98 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.98 ( ( ord_less_nat @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_nat @ C @ B2 )
% 12.45/6.98 => ( ord_less_nat @ C @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7867_dual__order_Ostrict__trans,axiom,
% 12.45/6.98 ! [B2: int,A: int,C: int] :
% 12.45/6.98 ( ( ord_less_int @ B2 @ A )
% 12.45/6.98 => ( ( ord_less_int @ C @ B2 )
% 12.45/6.98 => ( ord_less_int @ C @ A ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7868_order_Ostrict__implies__not__eq,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( A != B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7869_order_Ostrict__implies__not__eq,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( A != B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7870_order_Ostrict__implies__not__eq,axiom,
% 12.45/6.98 ! [A: num,B2: num] :
% 12.45/6.98 ( ( ord_less_num @ A @ B2 )
% 12.45/6.98 => ( A != B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7871_order_Ostrict__implies__not__eq,axiom,
% 12.45/6.98 ! [A: nat,B2: nat] :
% 12.45/6.98 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.98 => ( A != B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7872_order_Ostrict__implies__not__eq,axiom,
% 12.45/6.98 ! [A: int,B2: int] :
% 12.45/6.98 ( ( ord_less_int @ A @ B2 )
% 12.45/6.98 => ( A != B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7873_dual__order_Ostrict__implies__not__eq,axiom,
% 12.45/6.98 ! [B2: real,A: real] :
% 12.45/6.98 ( ( ord_less_real @ B2 @ A )
% 12.45/6.98 => ( A != B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7874_dual__order_Ostrict__implies__not__eq,axiom,
% 12.45/6.98 ! [B2: rat,A: rat] :
% 12.45/6.98 ( ( ord_less_rat @ B2 @ A )
% 12.45/6.98 => ( A != B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7875_dual__order_Ostrict__implies__not__eq,axiom,
% 12.45/6.98 ! [B2: num,A: num] :
% 12.45/6.98 ( ( ord_less_num @ B2 @ A )
% 12.45/6.98 => ( A != B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7876_dual__order_Ostrict__implies__not__eq,axiom,
% 12.45/6.98 ! [B2: nat,A: nat] :
% 12.45/6.98 ( ( ord_less_nat @ B2 @ A )
% 12.45/6.98 => ( A != B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7877_dual__order_Ostrict__implies__not__eq,axiom,
% 12.45/6.98 ! [B2: int,A: int] :
% 12.45/6.98 ( ( ord_less_int @ B2 @ A )
% 12.45/6.98 => ( A != B2 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7878_linorder__neqE,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( X4 != Y3 )
% 12.45/6.98 => ( ~ ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 => ( ord_less_real @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7879_linorder__neqE,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( X4 != Y3 )
% 12.45/6.98 => ( ~ ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.98 => ( ord_less_rat @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7880_linorder__neqE,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( X4 != Y3 )
% 12.45/6.98 => ( ~ ( ord_less_num @ X4 @ Y3 )
% 12.45/6.98 => ( ord_less_num @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7881_linorder__neqE,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ( X4 != Y3 )
% 12.45/6.98 => ( ~ ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.98 => ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7882_linorder__neqE,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( X4 != Y3 )
% 12.45/6.98 => ( ~ ( ord_less_int @ X4 @ Y3 )
% 12.45/6.98 => ( ord_less_int @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7883_order__less__asym,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_real @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7884_order__less__asym,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_rat @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7885_order__less__asym,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_num @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7886_order__less__asym,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7887_order__less__asym,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_int @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7888_linorder__neq__iff,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( X4 != Y3 )
% 12.45/6.98 = ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 | ( ord_less_real @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7889_linorder__neq__iff,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( X4 != Y3 )
% 12.45/6.98 = ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.98 | ( ord_less_rat @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7890_linorder__neq__iff,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( X4 != Y3 )
% 12.45/6.98 = ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.98 | ( ord_less_num @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7891_linorder__neq__iff,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ( X4 != Y3 )
% 12.45/6.98 = ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.98 | ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7892_linorder__neq__iff,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( X4 != Y3 )
% 12.45/6.98 = ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.98 | ( ord_less_int @ Y3 @ X4 ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7893_order__less__asym_H,axiom,
% 12.45/6.98 ! [A: real,B2: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ~ ( ord_less_real @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7894_order__less__asym_H,axiom,
% 12.45/6.98 ! [A: rat,B2: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ~ ( ord_less_rat @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7895_order__less__asym_H,axiom,
% 12.45/6.98 ! [A: num,B2: num] :
% 12.45/6.98 ( ( ord_less_num @ A @ B2 )
% 12.45/6.98 => ~ ( ord_less_num @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7896_order__less__asym_H,axiom,
% 12.45/6.98 ! [A: nat,B2: nat] :
% 12.45/6.98 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.98 => ~ ( ord_less_nat @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7897_order__less__asym_H,axiom,
% 12.45/6.98 ! [A: int,B2: int] :
% 12.45/6.98 ( ( ord_less_int @ A @ B2 )
% 12.45/6.98 => ~ ( ord_less_int @ B2 @ A ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7898_order__less__trans,axiom,
% 12.45/6.98 ! [X4: real,Y3: real,Z: real] :
% 12.45/6.98 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_real @ Y3 @ Z )
% 12.45/6.98 => ( ord_less_real @ X4 @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7899_order__less__trans,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat,Z: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_rat @ Y3 @ Z )
% 12.45/6.98 => ( ord_less_rat @ X4 @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7900_order__less__trans,axiom,
% 12.45/6.98 ! [X4: num,Y3: num,Z: num] :
% 12.45/6.98 ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_num @ Y3 @ Z )
% 12.45/6.98 => ( ord_less_num @ X4 @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7901_order__less__trans,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat,Z: nat] :
% 12.45/6.98 ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_nat @ Y3 @ Z )
% 12.45/6.98 => ( ord_less_nat @ X4 @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7902_order__less__trans,axiom,
% 12.45/6.98 ! [X4: int,Y3: int,Z: int] :
% 12.45/6.98 ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_int @ Y3 @ Z )
% 12.45/6.98 => ( ord_less_int @ X4 @ Z ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7903_ord__eq__less__subst,axiom,
% 12.45/6.98 ! [A: real,F: real > real,B2: real,C: real] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7904_ord__eq__less__subst,axiom,
% 12.45/6.98 ! [A: rat,F: real > rat,B2: real,C: real] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7905_ord__eq__less__subst,axiom,
% 12.45/6.98 ! [A: num,F: real > num,B2: real,C: real] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7906_ord__eq__less__subst,axiom,
% 12.45/6.98 ! [A: nat,F: real > nat,B2: real,C: real] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7907_ord__eq__less__subst,axiom,
% 12.45/6.98 ! [A: int,F: real > int,B2: real,C: real] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7908_ord__eq__less__subst,axiom,
% 12.45/6.98 ! [A: real,F: rat > real,B2: rat,C: rat] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7909_ord__eq__less__subst,axiom,
% 12.45/6.98 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7910_ord__eq__less__subst,axiom,
% 12.45/6.98 ! [A: num,F: rat > num,B2: rat,C: rat] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7911_ord__eq__less__subst,axiom,
% 12.45/6.98 ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7912_ord__eq__less__subst,axiom,
% 12.45/6.98 ! [A: int,F: rat > int,B2: rat,C: rat] :
% 12.45/6.98 ( ( A
% 12.45/6.98 = ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7913_ord__less__eq__subst,axiom,
% 12.45/6.98 ! [A: real,B2: real,F: real > real,C: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7914_ord__less__eq__subst,axiom,
% 12.45/6.98 ! [A: real,B2: real,F: real > rat,C: rat] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7915_ord__less__eq__subst,axiom,
% 12.45/6.98 ! [A: real,B2: real,F: real > num,C: num] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7916_ord__less__eq__subst,axiom,
% 12.45/6.98 ! [A: real,B2: real,F: real > nat,C: nat] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7917_ord__less__eq__subst,axiom,
% 12.45/6.98 ! [A: real,B2: real,F: real > int,C: int] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7918_ord__less__eq__subst,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > real,C: real] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7919_ord__less__eq__subst,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7920_ord__less__eq__subst,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > num,C: num] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7921_ord__less__eq__subst,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7922_ord__less__eq__subst,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > int,C: int] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ( F @ B2 )
% 12.45/6.98 = C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7923_order__less__irrefl,axiom,
% 12.45/6.98 ! [X4: real] :
% 12.45/6.98 ~ ( ord_less_real @ X4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7924_order__less__irrefl,axiom,
% 12.45/6.98 ! [X4: rat] :
% 12.45/6.98 ~ ( ord_less_rat @ X4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7925_order__less__irrefl,axiom,
% 12.45/6.98 ! [X4: num] :
% 12.45/6.98 ~ ( ord_less_num @ X4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7926_order__less__irrefl,axiom,
% 12.45/6.98 ! [X4: nat] :
% 12.45/6.98 ~ ( ord_less_nat @ X4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7927_order__less__irrefl,axiom,
% 12.45/6.98 ! [X4: int] :
% 12.45/6.98 ~ ( ord_less_int @ X4 @ X4 ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7928_order__less__subst1,axiom,
% 12.45/6.98 ! [A: real,F: real > real,B2: real,C: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7929_order__less__subst1,axiom,
% 12.45/6.98 ! [A: real,F: rat > real,B2: rat,C: rat] :
% 12.45/6.98 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7930_order__less__subst1,axiom,
% 12.45/6.98 ! [A: real,F: num > real,B2: num,C: num] :
% 12.45/6.98 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_num @ B2 @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7931_order__less__subst1,axiom,
% 12.45/6.98 ! [A: real,F: nat > real,B2: nat,C: nat] :
% 12.45/6.98 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_nat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.98 ( ( ord_less_nat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7932_order__less__subst1,axiom,
% 12.45/6.98 ! [A: real,F: int > real,B2: int,C: int] :
% 12.45/6.98 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_int @ B2 @ C )
% 12.45/6.98 => ( ! [X3: int,Y4: int] :
% 12.45/6.98 ( ( ord_less_int @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7933_order__less__subst1,axiom,
% 12.45/6.98 ! [A: rat,F: real > rat,B2: real,C: real] :
% 12.45/6.98 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7934_order__less__subst1,axiom,
% 12.45/6.98 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7935_order__less__subst1,axiom,
% 12.45/6.98 ! [A: rat,F: num > rat,B2: num,C: num] :
% 12.45/6.98 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_num @ B2 @ C )
% 12.45/6.98 => ( ! [X3: num,Y4: num] :
% 12.45/6.98 ( ( ord_less_num @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7936_order__less__subst1,axiom,
% 12.45/6.98 ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_nat @ B2 @ C )
% 12.45/6.98 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.98 ( ( ord_less_nat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7937_order__less__subst1,axiom,
% 12.45/6.98 ! [A: rat,F: int > rat,B2: int,C: int] :
% 12.45/6.98 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 12.45/6.98 => ( ( ord_less_int @ B2 @ C )
% 12.45/6.98 => ( ! [X3: int,Y4: int] :
% 12.45/6.98 ( ( ord_less_int @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7938_order__less__subst2,axiom,
% 12.45/6.98 ! [A: real,B2: real,F: real > real,C: real] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7939_order__less__subst2,axiom,
% 12.45/6.98 ! [A: real,B2: real,F: real > rat,C: rat] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7940_order__less__subst2,axiom,
% 12.45/6.98 ! [A: real,B2: real,F: real > num,C: num] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7941_order__less__subst2,axiom,
% 12.45/6.98 ! [A: real,B2: real,F: real > nat,C: nat] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7942_order__less__subst2,axiom,
% 12.45/6.98 ! [A: real,B2: real,F: real > int,C: int] :
% 12.45/6.98 ( ( ord_less_real @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: real,Y4: real] :
% 12.45/6.98 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7943_order__less__subst2,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > real,C: real] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7944_order__less__subst2,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7945_order__less__subst2,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > num,C: num] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7946_order__less__subst2,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7947_order__less__subst2,axiom,
% 12.45/6.98 ! [A: rat,B2: rat,F: rat > int,C: int] :
% 12.45/6.98 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.98 => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 12.45/6.98 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.98 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.98 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7948_order__less__not__sym,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_real @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7949_order__less__not__sym,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_rat @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7950_order__less__not__sym,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_num @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7951_order__less__not__sym,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7952_order__less__not__sym,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.98 => ~ ( ord_less_int @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7953_order__less__imp__triv,axiom,
% 12.45/6.98 ! [X4: real,Y3: real,P: $o] :
% 12.45/6.98 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_real @ Y3 @ X4 )
% 12.45/6.98 => P ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7954_order__less__imp__triv,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat,P: $o] :
% 12.45/6.98 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_rat @ Y3 @ X4 )
% 12.45/6.98 => P ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7955_order__less__imp__triv,axiom,
% 12.45/6.98 ! [X4: num,Y3: num,P: $o] :
% 12.45/6.98 ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_num @ Y3 @ X4 )
% 12.45/6.98 => P ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7956_order__less__imp__triv,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat,P: $o] :
% 12.45/6.98 ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_nat @ Y3 @ X4 )
% 12.45/6.98 => P ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7957_order__less__imp__triv,axiom,
% 12.45/6.98 ! [X4: int,Y3: int,P: $o] :
% 12.45/6.98 ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.98 => ( ( ord_less_int @ Y3 @ X4 )
% 12.45/6.98 => P ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7958_linorder__less__linear,axiom,
% 12.45/6.98 ! [X4: real,Y3: real] :
% 12.45/6.98 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.98 | ( X4 = Y3 )
% 12.45/6.98 | ( ord_less_real @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7959_linorder__less__linear,axiom,
% 12.45/6.98 ! [X4: rat,Y3: rat] :
% 12.45/6.98 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.98 | ( X4 = Y3 )
% 12.45/6.98 | ( ord_less_rat @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7960_linorder__less__linear,axiom,
% 12.45/6.98 ! [X4: num,Y3: num] :
% 12.45/6.98 ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.98 | ( X4 = Y3 )
% 12.45/6.98 | ( ord_less_num @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7961_linorder__less__linear,axiom,
% 12.45/6.98 ! [X4: nat,Y3: nat] :
% 12.45/6.98 ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.98 | ( X4 = Y3 )
% 12.45/6.98 | ( ord_less_nat @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7962_linorder__less__linear,axiom,
% 12.45/6.98 ! [X4: int,Y3: int] :
% 12.45/6.98 ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.98 | ( X4 = Y3 )
% 12.45/6.98 | ( ord_less_int @ Y3 @ X4 ) ) ).
% 12.45/6.98
% 12.45/6.98 thf(fact_7963_order__less__imp__not__eq,axiom,
% 12.45/6.99 ! [X4: real,Y3: real] :
% 12.45/6.99 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.99 => ( X4 != Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7964_order__less__imp__not__eq,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.99 => ( X4 != Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7965_order__less__imp__not__eq,axiom,
% 12.45/6.99 ! [X4: num,Y3: num] :
% 12.45/6.99 ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.99 => ( X4 != Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7966_order__less__imp__not__eq,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.99 => ( X4 != Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7967_order__less__imp__not__eq,axiom,
% 12.45/6.99 ! [X4: int,Y3: int] :
% 12.45/6.99 ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.99 => ( X4 != Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7968_order__less__imp__not__eq2,axiom,
% 12.45/6.99 ! [X4: real,Y3: real] :
% 12.45/6.99 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.99 => ( Y3 != X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7969_order__less__imp__not__eq2,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.99 => ( Y3 != X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7970_order__less__imp__not__eq2,axiom,
% 12.45/6.99 ! [X4: num,Y3: num] :
% 12.45/6.99 ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.99 => ( Y3 != X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7971_order__less__imp__not__eq2,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.99 => ( Y3 != X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7972_order__less__imp__not__eq2,axiom,
% 12.45/6.99 ! [X4: int,Y3: int] :
% 12.45/6.99 ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.99 => ( Y3 != X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7973_order__less__imp__not__less,axiom,
% 12.45/6.99 ! [X4: real,Y3: real] :
% 12.45/6.99 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.99 => ~ ( ord_less_real @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7974_order__less__imp__not__less,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.99 => ~ ( ord_less_rat @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7975_order__less__imp__not__less,axiom,
% 12.45/6.99 ! [X4: num,Y3: num] :
% 12.45/6.99 ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.99 => ~ ( ord_less_num @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7976_order__less__imp__not__less,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.99 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7977_order__less__imp__not__less,axiom,
% 12.45/6.99 ! [X4: int,Y3: int] :
% 12.45/6.99 ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.99 => ~ ( ord_less_int @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7978_take__bit__not__eq__mask__diff,axiom,
% 12.45/6.99 ! [N: nat,A: int] :
% 12.45/6.99 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 12.45/6.99 = ( minus_minus_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7979_minus__numeral__inc__eq,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
% 12.45/6.99 = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7980_minus__numeral__inc__eq,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
% 12.45/6.99 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7981_unset__bit__int__def,axiom,
% 12.45/6.99 ( bit_se4203085406695923979it_int
% 12.45/6.99 = ( ^ [N2: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7982_not__int__div__2,axiom,
% 12.45/6.99 ! [K: int] :
% 12.45/6.99 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.99 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7983_even__not__iff__int,axiom,
% 12.45/6.99 ! [K: int] :
% 12.45/6.99 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 12.45/6.99 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7984_not__numeral__Bit0__eq,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
% 12.45/6.99 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7985_not__numeral__Bit0__eq,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
% 12.45/6.99 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7986_and__not__numerals_I2_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.45/6.99 = one_one_int ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7987_and__not__numerals_I4_J,axiom,
% 12.45/6.99 ! [M: num] :
% 12.45/6.99 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.45/6.99 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7988_bit__minus__int__iff,axiom,
% 12.45/6.99 ! [K: int,N: nat] :
% 12.45/6.99 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 12.45/6.99 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7989_take__bit__not__mask__eq__0,axiom,
% 12.45/6.99 ! [M: nat,N: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.99 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) )
% 12.45/6.99 = zero_zero_int ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7990_push__bit__mask__eq,axiom,
% 12.45/6.99 ! [M: nat,N: nat] :
% 12.45/6.99 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N ) )
% 12.45/6.99 = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7991_unset__bit__eq__and__not,axiom,
% 12.45/6.99 ( bit_se4203085406695923979it_int
% 12.45/6.99 = ( ^ [N2: nat,A4: int] : ( bit_se725231765392027082nd_int @ A4 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7992_and__not__numerals_I5_J,axiom,
% 12.45/6.99 ! [M: num,N: num] :
% 12.45/6.99 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.45/6.99 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7993_and__not__numerals_I7_J,axiom,
% 12.45/6.99 ! [M: num] :
% 12.45/6.99 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.45/6.99 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7994_and__not__numerals_I3_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.45/6.99 = zero_zero_int ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7995_and__not__numerals_I9_J,axiom,
% 12.45/6.99 ! [M: num,N: num] :
% 12.45/6.99 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.45/6.99 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7996_and__not__numerals_I6_J,axiom,
% 12.45/6.99 ! [M: num,N: num] :
% 12.45/6.99 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.45/6.99 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7997_bit__not__iff__eq,axiom,
% 12.45/6.99 ! [A: int,N: nat] :
% 12.45/6.99 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N )
% 12.45/6.99 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.45/6.99 != zero_zero_int )
% 12.45/6.99 & ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7998_minus__exp__eq__not__mask,axiom,
% 12.45/6.99 ! [N: nat] :
% 12.45/6.99 ( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.99 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_7999_minus__exp__eq__not__mask,axiom,
% 12.45/6.99 ! [N: nat] :
% 12.45/6.99 ( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.99 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8000_CauchyD,axiom,
% 12.45/6.99 ! [X8: nat > complex,E: real] :
% 12.45/6.99 ( ( topolo6517432010174082258omplex @ X8 )
% 12.45/6.99 => ( ( ord_less_real @ zero_zero_real @ E )
% 12.45/6.99 => ? [M9: nat] :
% 12.45/6.99 ! [M5: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M9 @ M5 )
% 12.45/6.99 => ! [N6: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M9 @ N6 )
% 12.45/6.99 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ E ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8001_CauchyD,axiom,
% 12.45/6.99 ! [X8: nat > real,E: real] :
% 12.45/6.99 ( ( topolo4055970368930404560y_real @ X8 )
% 12.45/6.99 => ( ( ord_less_real @ zero_zero_real @ E )
% 12.45/6.99 => ? [M9: nat] :
% 12.45/6.99 ! [M5: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M9 @ M5 )
% 12.45/6.99 => ! [N6: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M9 @ N6 )
% 12.45/6.99 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ E ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8002_CauchyI,axiom,
% 12.45/6.99 ! [X8: nat > complex] :
% 12.45/6.99 ( ! [E2: real] :
% 12.45/6.99 ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.45/6.99 => ? [M10: nat] :
% 12.45/6.99 ! [M4: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M10 @ M4 )
% 12.45/6.99 => ! [N4: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M10 @ N4 )
% 12.45/6.99 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ E2 ) ) ) )
% 12.45/6.99 => ( topolo6517432010174082258omplex @ X8 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8003_CauchyI,axiom,
% 12.45/6.99 ! [X8: nat > real] :
% 12.45/6.99 ( ! [E2: real] :
% 12.45/6.99 ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.45/6.99 => ? [M10: nat] :
% 12.45/6.99 ! [M4: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M10 @ M4 )
% 12.45/6.99 => ! [N4: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M10 @ N4 )
% 12.45/6.99 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ E2 ) ) ) )
% 12.45/6.99 => ( topolo4055970368930404560y_real @ X8 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8004_Cauchy__iff,axiom,
% 12.45/6.99 ( topolo6517432010174082258omplex
% 12.45/6.99 = ( ^ [X6: nat > complex] :
% 12.45/6.99 ! [E3: real] :
% 12.45/6.99 ( ( ord_less_real @ zero_zero_real @ E3 )
% 12.45/6.99 => ? [M8: nat] :
% 12.45/6.99 ! [M6: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M8 @ M6 )
% 12.45/6.99 => ! [N2: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M8 @ N2 )
% 12.45/6.99 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8005_Cauchy__iff,axiom,
% 12.45/6.99 ( topolo4055970368930404560y_real
% 12.45/6.99 = ( ^ [X6: nat > real] :
% 12.45/6.99 ! [E3: real] :
% 12.45/6.99 ( ( ord_less_real @ zero_zero_real @ E3 )
% 12.45/6.99 => ? [M8: nat] :
% 12.45/6.99 ! [M6: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M8 @ M6 )
% 12.45/6.99 => ! [N2: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M8 @ N2 )
% 12.45/6.99 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8006_leD,axiom,
% 12.45/6.99 ! [Y3: real,X4: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ Y3 @ X4 )
% 12.45/6.99 => ~ ( ord_less_real @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8007_leD,axiom,
% 12.45/6.99 ! [Y3: set_nat,X4: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ Y3 @ X4 )
% 12.45/6.99 => ~ ( ord_less_set_nat @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8008_leD,axiom,
% 12.45/6.99 ! [Y3: rat,X4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ Y3 @ X4 )
% 12.45/6.99 => ~ ( ord_less_rat @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8009_leD,axiom,
% 12.45/6.99 ! [Y3: num,X4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ Y3 @ X4 )
% 12.45/6.99 => ~ ( ord_less_num @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8010_leD,axiom,
% 12.45/6.99 ! [Y3: nat,X4: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ Y3 @ X4 )
% 12.45/6.99 => ~ ( ord_less_nat @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8011_leD,axiom,
% 12.45/6.99 ! [Y3: int,X4: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ Y3 @ X4 )
% 12.45/6.99 => ~ ( ord_less_int @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8012_leI,axiom,
% 12.45/6.99 ! [X4: real,Y3: real] :
% 12.45/6.99 ( ~ ( ord_less_real @ X4 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_real @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8013_leI,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat] :
% 12.45/6.99 ( ~ ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_rat @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8014_leI,axiom,
% 12.45/6.99 ! [X4: num,Y3: num] :
% 12.45/6.99 ( ~ ( ord_less_num @ X4 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_num @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8015_leI,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat] :
% 12.45/6.99 ( ~ ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8016_leI,axiom,
% 12.45/6.99 ! [X4: int,Y3: int] :
% 12.45/6.99 ( ~ ( ord_less_int @ X4 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_int @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8017_nless__le,axiom,
% 12.45/6.99 ! [A: real,B2: real] :
% 12.45/6.99 ( ( ~ ( ord_less_real @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_real @ A @ B2 )
% 12.45/6.99 | ( A = B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8018_nless__le,axiom,
% 12.45/6.99 ! [A: set_nat,B2: set_nat] :
% 12.45/6.99 ( ( ~ ( ord_less_set_nat @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_set_nat @ A @ B2 )
% 12.45/6.99 | ( A = B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8019_nless__le,axiom,
% 12.45/6.99 ! [A: rat,B2: rat] :
% 12.45/6.99 ( ( ~ ( ord_less_rat @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.99 | ( A = B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8020_nless__le,axiom,
% 12.45/6.99 ! [A: num,B2: num] :
% 12.45/6.99 ( ( ~ ( ord_less_num @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_num @ A @ B2 )
% 12.45/6.99 | ( A = B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8021_nless__le,axiom,
% 12.45/6.99 ! [A: nat,B2: nat] :
% 12.45/6.99 ( ( ~ ( ord_less_nat @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.99 | ( A = B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8022_nless__le,axiom,
% 12.45/6.99 ! [A: int,B2: int] :
% 12.45/6.99 ( ( ~ ( ord_less_int @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_int @ A @ B2 )
% 12.45/6.99 | ( A = B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8023_antisym__conv1,axiom,
% 12.45/6.99 ! [X4: real,Y3: real] :
% 12.45/6.99 ( ~ ( ord_less_real @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8024_antisym__conv1,axiom,
% 12.45/6.99 ! [X4: set_nat,Y3: set_nat] :
% 12.45/6.99 ( ~ ( ord_less_set_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8025_antisym__conv1,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat] :
% 12.45/6.99 ( ~ ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8026_antisym__conv1,axiom,
% 12.45/6.99 ! [X4: num,Y3: num] :
% 12.45/6.99 ( ~ ( ord_less_num @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_num @ X4 @ Y3 )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8027_antisym__conv1,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat] :
% 12.45/6.99 ( ~ ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_nat @ X4 @ Y3 )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8028_antisym__conv1,axiom,
% 12.45/6.99 ! [X4: int,Y3: int] :
% 12.45/6.99 ( ~ ( ord_less_int @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8029_antisym__conv2,axiom,
% 12.45/6.99 ! [X4: real,Y3: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.99 => ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8030_antisym__conv2,axiom,
% 12.45/6.99 ! [X4: set_nat,Y3: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ~ ( ord_less_set_nat @ X4 @ Y3 ) )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8031_antisym__conv2,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8032_antisym__conv2,axiom,
% 12.45/6.99 ! [X4: num,Y3: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X4 @ Y3 )
% 12.45/6.99 => ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8033_antisym__conv2,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8034_antisym__conv2,axiom,
% 12.45/6.99 ! [X4: int,Y3: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.99 => ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
% 12.45/6.99 = ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8035_dense__ge,axiom,
% 12.45/6.99 ! [Z: real,Y3: real] :
% 12.45/6.99 ( ! [X3: real] :
% 12.45/6.99 ( ( ord_less_real @ Z @ X3 )
% 12.45/6.99 => ( ord_less_eq_real @ Y3 @ X3 ) )
% 12.45/6.99 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8036_dense__ge,axiom,
% 12.45/6.99 ! [Z: rat,Y3: rat] :
% 12.45/6.99 ( ! [X3: rat] :
% 12.45/6.99 ( ( ord_less_rat @ Z @ X3 )
% 12.45/6.99 => ( ord_less_eq_rat @ Y3 @ X3 ) )
% 12.45/6.99 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8037_dense__le,axiom,
% 12.45/6.99 ! [Y3: real,Z: real] :
% 12.45/6.99 ( ! [X3: real] :
% 12.45/6.99 ( ( ord_less_real @ X3 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_real @ X3 @ Z ) )
% 12.45/6.99 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8038_dense__le,axiom,
% 12.45/6.99 ! [Y3: rat,Z: rat] :
% 12.45/6.99 ( ! [X3: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X3 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_rat @ X3 @ Z ) )
% 12.45/6.99 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8039_less__le__not__le,axiom,
% 12.45/6.99 ( ord_less_real
% 12.45/6.99 = ( ^ [X: real,Y: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ X @ Y )
% 12.45/6.99 & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8040_less__le__not__le,axiom,
% 12.45/6.99 ( ord_less_set_nat
% 12.45/6.99 = ( ^ [X: set_nat,Y: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ X @ Y )
% 12.45/6.99 & ~ ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8041_less__le__not__le,axiom,
% 12.45/6.99 ( ord_less_rat
% 12.45/6.99 = ( ^ [X: rat,Y: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X @ Y )
% 12.45/6.99 & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8042_less__le__not__le,axiom,
% 12.45/6.99 ( ord_less_num
% 12.45/6.99 = ( ^ [X: num,Y: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X @ Y )
% 12.45/6.99 & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8043_less__le__not__le,axiom,
% 12.45/6.99 ( ord_less_nat
% 12.45/6.99 = ( ^ [X: nat,Y: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ X @ Y )
% 12.45/6.99 & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8044_less__le__not__le,axiom,
% 12.45/6.99 ( ord_less_int
% 12.45/6.99 = ( ^ [X: int,Y: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ X @ Y )
% 12.45/6.99 & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8045_not__le__imp__less,axiom,
% 12.45/6.99 ! [Y3: real,X4: real] :
% 12.45/6.99 ( ~ ( ord_less_eq_real @ Y3 @ X4 )
% 12.45/6.99 => ( ord_less_real @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8046_not__le__imp__less,axiom,
% 12.45/6.99 ! [Y3: rat,X4: rat] :
% 12.45/6.99 ( ~ ( ord_less_eq_rat @ Y3 @ X4 )
% 12.45/6.99 => ( ord_less_rat @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8047_not__le__imp__less,axiom,
% 12.45/6.99 ! [Y3: num,X4: num] :
% 12.45/6.99 ( ~ ( ord_less_eq_num @ Y3 @ X4 )
% 12.45/6.99 => ( ord_less_num @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8048_not__le__imp__less,axiom,
% 12.45/6.99 ! [Y3: nat,X4: nat] :
% 12.45/6.99 ( ~ ( ord_less_eq_nat @ Y3 @ X4 )
% 12.45/6.99 => ( ord_less_nat @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8049_not__le__imp__less,axiom,
% 12.45/6.99 ! [Y3: int,X4: int] :
% 12.45/6.99 ( ~ ( ord_less_eq_int @ Y3 @ X4 )
% 12.45/6.99 => ( ord_less_int @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8050_order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_real
% 12.45/6.99 = ( ^ [A4: real,B3: real] :
% 12.45/6.99 ( ( ord_less_real @ A4 @ B3 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8051_order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_set_nat
% 12.45/6.99 = ( ^ [A4: set_nat,B3: set_nat] :
% 12.45/6.99 ( ( ord_less_set_nat @ A4 @ B3 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8052_order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_rat
% 12.45/6.99 = ( ^ [A4: rat,B3: rat] :
% 12.45/6.99 ( ( ord_less_rat @ A4 @ B3 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8053_order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_num
% 12.45/6.99 = ( ^ [A4: num,B3: num] :
% 12.45/6.99 ( ( ord_less_num @ A4 @ B3 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8054_order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_nat
% 12.45/6.99 = ( ^ [A4: nat,B3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ A4 @ B3 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8055_order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_int
% 12.45/6.99 = ( ^ [A4: int,B3: int] :
% 12.45/6.99 ( ( ord_less_int @ A4 @ B3 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8056_order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_real
% 12.45/6.99 = ( ^ [A4: real,B3: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ A4 @ B3 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8057_order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_set_nat
% 12.45/6.99 = ( ^ [A4: set_nat,B3: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ A4 @ B3 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8058_order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_rat
% 12.45/6.99 = ( ^ [A4: rat,B3: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A4 @ B3 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8059_order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_num
% 12.45/6.99 = ( ^ [A4: num,B3: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ A4 @ B3 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8060_order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_nat
% 12.45/6.99 = ( ^ [A4: nat,B3: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ A4 @ B3 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8061_order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_int
% 12.45/6.99 = ( ^ [A4: int,B3: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ A4 @ B3 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8062_order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [A: real,B2: real,C: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.99 => ( ord_less_real @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8063_order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [A: set_nat,B2: set_nat,C: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_set_nat @ B2 @ C )
% 12.45/6.99 => ( ord_less_set_nat @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8064_order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.99 => ( ord_less_rat @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8065_order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [A: num,B2: num,C: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_num @ B2 @ C )
% 12.45/6.99 => ( ord_less_num @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8066_order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_nat @ B2 @ C )
% 12.45/6.99 => ( ord_less_nat @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8067_order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [A: int,B2: int,C: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_int @ B2 @ C )
% 12.45/6.99 => ( ord_less_int @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8068_order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [A: real,B2: real,C: real] :
% 12.45/6.99 ( ( ord_less_real @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_real @ B2 @ C )
% 12.45/6.99 => ( ord_less_real @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8069_order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [A: set_nat,B2: set_nat,C: set_nat] :
% 12.45/6.99 ( ( ord_less_set_nat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 12.45/6.99 => ( ord_less_set_nat @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8070_order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [A: rat,B2: rat,C: rat] :
% 12.45/6.99 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.99 => ( ord_less_rat @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8071_order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [A: num,B2: num,C: num] :
% 12.45/6.99 ( ( ord_less_num @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.99 => ( ord_less_num @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8072_order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [A: nat,B2: nat,C: nat] :
% 12.45/6.99 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_nat @ B2 @ C )
% 12.45/6.99 => ( ord_less_nat @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8073_order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [A: int,B2: int,C: int] :
% 12.45/6.99 ( ( ord_less_int @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_int @ B2 @ C )
% 12.45/6.99 => ( ord_less_int @ A @ C ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8074_order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_real
% 12.45/6.99 = ( ^ [A4: real,B3: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ A4 @ B3 )
% 12.45/6.99 & ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8075_order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_set_nat
% 12.45/6.99 = ( ^ [A4: set_nat,B3: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ A4 @ B3 )
% 12.45/6.99 & ~ ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8076_order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_rat
% 12.45/6.99 = ( ^ [A4: rat,B3: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A4 @ B3 )
% 12.45/6.99 & ~ ( ord_less_eq_rat @ B3 @ A4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8077_order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_num
% 12.45/6.99 = ( ^ [A4: num,B3: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ A4 @ B3 )
% 12.45/6.99 & ~ ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8078_order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_nat
% 12.45/6.99 = ( ^ [A4: nat,B3: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ A4 @ B3 )
% 12.45/6.99 & ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8079_order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_int
% 12.45/6.99 = ( ^ [A4: int,B3: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ A4 @ B3 )
% 12.45/6.99 & ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8080_dense__ge__bounded,axiom,
% 12.45/6.99 ! [Z: real,X4: real,Y3: real] :
% 12.45/6.99 ( ( ord_less_real @ Z @ X4 )
% 12.45/6.99 => ( ! [W3: real] :
% 12.45/6.99 ( ( ord_less_real @ Z @ W3 )
% 12.45/6.99 => ( ( ord_less_real @ W3 @ X4 )
% 12.45/6.99 => ( ord_less_eq_real @ Y3 @ W3 ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8081_dense__ge__bounded,axiom,
% 12.45/6.99 ! [Z: rat,X4: rat,Y3: rat] :
% 12.45/6.99 ( ( ord_less_rat @ Z @ X4 )
% 12.45/6.99 => ( ! [W3: rat] :
% 12.45/6.99 ( ( ord_less_rat @ Z @ W3 )
% 12.45/6.99 => ( ( ord_less_rat @ W3 @ X4 )
% 12.45/6.99 => ( ord_less_eq_rat @ Y3 @ W3 ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8082_dense__le__bounded,axiom,
% 12.45/6.99 ! [X4: real,Y3: real,Z: real] :
% 12.45/6.99 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.99 => ( ! [W3: real] :
% 12.45/6.99 ( ( ord_less_real @ X4 @ W3 )
% 12.45/6.99 => ( ( ord_less_real @ W3 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_real @ W3 @ Z ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8083_dense__le__bounded,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat,Z: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.99 => ( ! [W3: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X4 @ W3 )
% 12.45/6.99 => ( ( ord_less_rat @ W3 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_rat @ W3 @ Z ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8084_dual__order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_real
% 12.45/6.99 = ( ^ [B3: real,A4: real] :
% 12.45/6.99 ( ( ord_less_real @ B3 @ A4 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8085_dual__order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_set_nat
% 12.45/6.99 = ( ^ [B3: set_nat,A4: set_nat] :
% 12.45/6.99 ( ( ord_less_set_nat @ B3 @ A4 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8086_dual__order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_rat
% 12.45/6.99 = ( ^ [B3: rat,A4: rat] :
% 12.45/6.99 ( ( ord_less_rat @ B3 @ A4 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8087_dual__order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_num
% 12.45/6.99 = ( ^ [B3: num,A4: num] :
% 12.45/6.99 ( ( ord_less_num @ B3 @ A4 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8088_dual__order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_nat
% 12.45/6.99 = ( ^ [B3: nat,A4: nat] :
% 12.45/6.99 ( ( ord_less_nat @ B3 @ A4 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8089_dual__order_Oorder__iff__strict,axiom,
% 12.45/6.99 ( ord_less_eq_int
% 12.45/6.99 = ( ^ [B3: int,A4: int] :
% 12.45/6.99 ( ( ord_less_int @ B3 @ A4 )
% 12.45/6.99 | ( A4 = B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8090_dual__order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_real
% 12.45/6.99 = ( ^ [B3: real,A4: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ B3 @ A4 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8091_dual__order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_set_nat
% 12.45/6.99 = ( ^ [B3: set_nat,A4: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ B3 @ A4 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8092_dual__order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_rat
% 12.45/6.99 = ( ^ [B3: rat,A4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ B3 @ A4 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8093_dual__order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_num
% 12.45/6.99 = ( ^ [B3: num,A4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ B3 @ A4 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8094_dual__order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_nat
% 12.45/6.99 = ( ^ [B3: nat,A4: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ B3 @ A4 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8095_dual__order_Ostrict__iff__order,axiom,
% 12.45/6.99 ( ord_less_int
% 12.45/6.99 = ( ^ [B3: int,A4: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ B3 @ A4 )
% 12.45/6.99 & ( A4 != B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8096_dual__order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [B2: real,A: real,C: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_real @ C @ B2 )
% 12.45/6.99 => ( ord_less_real @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8097_dual__order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [B2: set_nat,A: set_nat,C: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_set_nat @ C @ B2 )
% 12.45/6.99 => ( ord_less_set_nat @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8098_dual__order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_rat @ C @ B2 )
% 12.45/6.99 => ( ord_less_rat @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8099_dual__order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [B2: num,A: num,C: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_num @ C @ B2 )
% 12.45/6.99 => ( ord_less_num @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8100_dual__order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_nat @ C @ B2 )
% 12.45/6.99 => ( ord_less_nat @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8101_dual__order_Ostrict__trans1,axiom,
% 12.45/6.99 ! [B2: int,A: int,C: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_int @ C @ B2 )
% 12.45/6.99 => ( ord_less_int @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8102_dual__order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [B2: real,A: real,C: real] :
% 12.45/6.99 ( ( ord_less_real @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_eq_real @ C @ B2 )
% 12.45/6.99 => ( ord_less_real @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8103_dual__order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [B2: set_nat,A: set_nat,C: set_nat] :
% 12.45/6.99 ( ( ord_less_set_nat @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_eq_set_nat @ C @ B2 )
% 12.45/6.99 => ( ord_less_set_nat @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8104_dual__order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [B2: rat,A: rat,C: rat] :
% 12.45/6.99 ( ( ord_less_rat @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_eq_rat @ C @ B2 )
% 12.45/6.99 => ( ord_less_rat @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8105_dual__order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [B2: num,A: num,C: num] :
% 12.45/6.99 ( ( ord_less_num @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_eq_num @ C @ B2 )
% 12.45/6.99 => ( ord_less_num @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8106_dual__order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [B2: nat,A: nat,C: nat] :
% 12.45/6.99 ( ( ord_less_nat @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_eq_nat @ C @ B2 )
% 12.45/6.99 => ( ord_less_nat @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8107_dual__order_Ostrict__trans2,axiom,
% 12.45/6.99 ! [B2: int,A: int,C: int] :
% 12.45/6.99 ( ( ord_less_int @ B2 @ A )
% 12.45/6.99 => ( ( ord_less_eq_int @ C @ B2 )
% 12.45/6.99 => ( ord_less_int @ C @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8108_dual__order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_real
% 12.45/6.99 = ( ^ [B3: real,A4: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ B3 @ A4 )
% 12.45/6.99 & ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8109_dual__order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_set_nat
% 12.45/6.99 = ( ^ [B3: set_nat,A4: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ B3 @ A4 )
% 12.45/6.99 & ~ ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8110_dual__order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_rat
% 12.45/6.99 = ( ^ [B3: rat,A4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ B3 @ A4 )
% 12.45/6.99 & ~ ( ord_less_eq_rat @ A4 @ B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8111_dual__order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_num
% 12.45/6.99 = ( ^ [B3: num,A4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ B3 @ A4 )
% 12.45/6.99 & ~ ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8112_dual__order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_nat
% 12.45/6.99 = ( ^ [B3: nat,A4: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ B3 @ A4 )
% 12.45/6.99 & ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8113_dual__order_Ostrict__iff__not,axiom,
% 12.45/6.99 ( ord_less_int
% 12.45/6.99 = ( ^ [B3: int,A4: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ B3 @ A4 )
% 12.45/6.99 & ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8114_order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [A: real,B2: real] :
% 12.45/6.99 ( ( ord_less_real @ A @ B2 )
% 12.45/6.99 => ( ord_less_eq_real @ A @ B2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8115_order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [A: set_nat,B2: set_nat] :
% 12.45/6.99 ( ( ord_less_set_nat @ A @ B2 )
% 12.45/6.99 => ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8116_order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [A: rat,B2: rat] :
% 12.45/6.99 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.99 => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8117_order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [A: num,B2: num] :
% 12.45/6.99 ( ( ord_less_num @ A @ B2 )
% 12.45/6.99 => ( ord_less_eq_num @ A @ B2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8118_order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [A: nat,B2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.99 => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8119_order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [A: int,B2: int] :
% 12.45/6.99 ( ( ord_less_int @ A @ B2 )
% 12.45/6.99 => ( ord_less_eq_int @ A @ B2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8120_dual__order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [B2: real,A: real] :
% 12.45/6.99 ( ( ord_less_real @ B2 @ A )
% 12.45/6.99 => ( ord_less_eq_real @ B2 @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8121_dual__order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [B2: set_nat,A: set_nat] :
% 12.45/6.99 ( ( ord_less_set_nat @ B2 @ A )
% 12.45/6.99 => ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8122_dual__order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [B2: rat,A: rat] :
% 12.45/6.99 ( ( ord_less_rat @ B2 @ A )
% 12.45/6.99 => ( ord_less_eq_rat @ B2 @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8123_dual__order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [B2: num,A: num] :
% 12.45/6.99 ( ( ord_less_num @ B2 @ A )
% 12.45/6.99 => ( ord_less_eq_num @ B2 @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8124_dual__order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [B2: nat,A: nat] :
% 12.45/6.99 ( ( ord_less_nat @ B2 @ A )
% 12.45/6.99 => ( ord_less_eq_nat @ B2 @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8125_dual__order_Ostrict__implies__order,axiom,
% 12.45/6.99 ! [B2: int,A: int] :
% 12.45/6.99 ( ( ord_less_int @ B2 @ A )
% 12.45/6.99 => ( ord_less_eq_int @ B2 @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8126_order__le__less,axiom,
% 12.45/6.99 ( ord_less_eq_real
% 12.45/6.99 = ( ^ [X: real,Y: real] :
% 12.45/6.99 ( ( ord_less_real @ X @ Y )
% 12.45/6.99 | ( X = Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8127_order__le__less,axiom,
% 12.45/6.99 ( ord_less_eq_set_nat
% 12.45/6.99 = ( ^ [X: set_nat,Y: set_nat] :
% 12.45/6.99 ( ( ord_less_set_nat @ X @ Y )
% 12.45/6.99 | ( X = Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8128_order__le__less,axiom,
% 12.45/6.99 ( ord_less_eq_rat
% 12.45/6.99 = ( ^ [X: rat,Y: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X @ Y )
% 12.45/6.99 | ( X = Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8129_order__le__less,axiom,
% 12.45/6.99 ( ord_less_eq_num
% 12.45/6.99 = ( ^ [X: num,Y: num] :
% 12.45/6.99 ( ( ord_less_num @ X @ Y )
% 12.45/6.99 | ( X = Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8130_order__le__less,axiom,
% 12.45/6.99 ( ord_less_eq_nat
% 12.45/6.99 = ( ^ [X: nat,Y: nat] :
% 12.45/6.99 ( ( ord_less_nat @ X @ Y )
% 12.45/6.99 | ( X = Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8131_order__le__less,axiom,
% 12.45/6.99 ( ord_less_eq_int
% 12.45/6.99 = ( ^ [X: int,Y: int] :
% 12.45/6.99 ( ( ord_less_int @ X @ Y )
% 12.45/6.99 | ( X = Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8132_order__less__le,axiom,
% 12.45/6.99 ( ord_less_real
% 12.45/6.99 = ( ^ [X: real,Y: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ X @ Y )
% 12.45/6.99 & ( X != Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8133_order__less__le,axiom,
% 12.45/6.99 ( ord_less_set_nat
% 12.45/6.99 = ( ^ [X: set_nat,Y: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ X @ Y )
% 12.45/6.99 & ( X != Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8134_order__less__le,axiom,
% 12.45/6.99 ( ord_less_rat
% 12.45/6.99 = ( ^ [X: rat,Y: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X @ Y )
% 12.45/6.99 & ( X != Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8135_order__less__le,axiom,
% 12.45/6.99 ( ord_less_num
% 12.45/6.99 = ( ^ [X: num,Y: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X @ Y )
% 12.45/6.99 & ( X != Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8136_order__less__le,axiom,
% 12.45/6.99 ( ord_less_nat
% 12.45/6.99 = ( ^ [X: nat,Y: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ X @ Y )
% 12.45/6.99 & ( X != Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8137_order__less__le,axiom,
% 12.45/6.99 ( ord_less_int
% 12.45/6.99 = ( ^ [X: int,Y: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ X @ Y )
% 12.45/6.99 & ( X != Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8138_linorder__not__le,axiom,
% 12.45/6.99 ! [X4: real,Y3: real] :
% 12.45/6.99 ( ( ~ ( ord_less_eq_real @ X4 @ Y3 ) )
% 12.45/6.99 = ( ord_less_real @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8139_linorder__not__le,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat] :
% 12.45/6.99 ( ( ~ ( ord_less_eq_rat @ X4 @ Y3 ) )
% 12.45/6.99 = ( ord_less_rat @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8140_linorder__not__le,axiom,
% 12.45/6.99 ! [X4: num,Y3: num] :
% 12.45/6.99 ( ( ~ ( ord_less_eq_num @ X4 @ Y3 ) )
% 12.45/6.99 = ( ord_less_num @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8141_linorder__not__le,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat] :
% 12.45/6.99 ( ( ~ ( ord_less_eq_nat @ X4 @ Y3 ) )
% 12.45/6.99 = ( ord_less_nat @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8142_linorder__not__le,axiom,
% 12.45/6.99 ! [X4: int,Y3: int] :
% 12.45/6.99 ( ( ~ ( ord_less_eq_int @ X4 @ Y3 ) )
% 12.45/6.99 = ( ord_less_int @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8143_linorder__not__less,axiom,
% 12.45/6.99 ! [X4: real,Y3: real] :
% 12.45/6.99 ( ( ~ ( ord_less_real @ X4 @ Y3 ) )
% 12.45/6.99 = ( ord_less_eq_real @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8144_linorder__not__less,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat] :
% 12.45/6.99 ( ( ~ ( ord_less_rat @ X4 @ Y3 ) )
% 12.45/6.99 = ( ord_less_eq_rat @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8145_linorder__not__less,axiom,
% 12.45/6.99 ! [X4: num,Y3: num] :
% 12.45/6.99 ( ( ~ ( ord_less_num @ X4 @ Y3 ) )
% 12.45/6.99 = ( ord_less_eq_num @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8146_linorder__not__less,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat] :
% 12.45/6.99 ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 12.45/6.99 = ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8147_linorder__not__less,axiom,
% 12.45/6.99 ! [X4: int,Y3: int] :
% 12.45/6.99 ( ( ~ ( ord_less_int @ X4 @ Y3 ) )
% 12.45/6.99 = ( ord_less_eq_int @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8148_order__less__imp__le,axiom,
% 12.45/6.99 ! [X4: real,Y3: real] :
% 12.45/6.99 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_real @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8149_order__less__imp__le,axiom,
% 12.45/6.99 ! [X4: set_nat,Y3: set_nat] :
% 12.45/6.99 ( ( ord_less_set_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8150_order__less__imp__le,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_rat @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8151_order__less__imp__le,axiom,
% 12.45/6.99 ! [X4: num,Y3: num] :
% 12.45/6.99 ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_num @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8152_order__less__imp__le,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8153_order__less__imp__le,axiom,
% 12.45/6.99 ! [X4: int,Y3: int] :
% 12.45/6.99 ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.99 => ( ord_less_eq_int @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8154_order__le__neq__trans,axiom,
% 12.45/6.99 ! [A: real,B2: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.99 => ( ( A != B2 )
% 12.45/6.99 => ( ord_less_real @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8155_order__le__neq__trans,axiom,
% 12.45/6.99 ! [A: set_nat,B2: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ A @ B2 )
% 12.45/6.99 => ( ( A != B2 )
% 12.45/6.99 => ( ord_less_set_nat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8156_order__le__neq__trans,axiom,
% 12.45/6.99 ! [A: rat,B2: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.99 => ( ( A != B2 )
% 12.45/6.99 => ( ord_less_rat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8157_order__le__neq__trans,axiom,
% 12.45/6.99 ! [A: num,B2: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.99 => ( ( A != B2 )
% 12.45/6.99 => ( ord_less_num @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8158_order__le__neq__trans,axiom,
% 12.45/6.99 ! [A: nat,B2: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.99 => ( ( A != B2 )
% 12.45/6.99 => ( ord_less_nat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8159_order__le__neq__trans,axiom,
% 12.45/6.99 ! [A: int,B2: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.99 => ( ( A != B2 )
% 12.45/6.99 => ( ord_less_int @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8160_order__neq__le__trans,axiom,
% 12.45/6.99 ! [A: real,B2: real] :
% 12.45/6.99 ( ( A != B2 )
% 12.45/6.99 => ( ( ord_less_eq_real @ A @ B2 )
% 12.45/6.99 => ( ord_less_real @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8161_order__neq__le__trans,axiom,
% 12.45/6.99 ! [A: set_nat,B2: set_nat] :
% 12.45/6.99 ( ( A != B2 )
% 12.45/6.99 => ( ( ord_less_eq_set_nat @ A @ B2 )
% 12.45/6.99 => ( ord_less_set_nat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8162_order__neq__le__trans,axiom,
% 12.45/6.99 ! [A: rat,B2: rat] :
% 12.45/6.99 ( ( A != B2 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.99 => ( ord_less_rat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8163_order__neq__le__trans,axiom,
% 12.45/6.99 ! [A: num,B2: num] :
% 12.45/6.99 ( ( A != B2 )
% 12.45/6.99 => ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.99 => ( ord_less_num @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8164_order__neq__le__trans,axiom,
% 12.45/6.99 ! [A: nat,B2: nat] :
% 12.45/6.99 ( ( A != B2 )
% 12.45/6.99 => ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.99 => ( ord_less_nat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8165_order__neq__le__trans,axiom,
% 12.45/6.99 ! [A: int,B2: int] :
% 12.45/6.99 ( ( A != B2 )
% 12.45/6.99 => ( ( ord_less_eq_int @ A @ B2 )
% 12.45/6.99 => ( ord_less_int @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8166_order__le__less__trans,axiom,
% 12.45/6.99 ! [X4: real,Y3: real,Z: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_real @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_real @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8167_order__le__less__trans,axiom,
% 12.45/6.99 ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_set_nat @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_set_nat @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8168_order__le__less__trans,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat,Z: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_rat @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_rat @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8169_order__le__less__trans,axiom,
% 12.45/6.99 ! [X4: num,Y3: num,Z: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_num @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_num @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8170_order__le__less__trans,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat,Z: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_nat @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_nat @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8171_order__le__less__trans,axiom,
% 12.45/6.99 ! [X4: int,Y3: int,Z: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_int @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_int @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8172_order__less__le__trans,axiom,
% 12.45/6.99 ! [X4: real,Y3: real,Z: real] :
% 12.45/6.99 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_real @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8173_order__less__le__trans,axiom,
% 12.45/6.99 ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
% 12.45/6.99 ( ( ord_less_set_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_set_nat @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_set_nat @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8174_order__less__le__trans,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat,Z: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_rat @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8175_order__less__le__trans,axiom,
% 12.45/6.99 ! [X4: num,Y3: num,Z: num] :
% 12.45/6.99 ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_num @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_num @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8176_order__less__le__trans,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat,Z: nat] :
% 12.45/6.99 ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_nat @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8177_order__less__le__trans,axiom,
% 12.45/6.99 ! [X4: int,Y3: int,Z: int] :
% 12.45/6.99 ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_eq_int @ Y3 @ Z )
% 12.45/6.99 => ( ord_less_int @ X4 @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8178_order__le__less__subst1,axiom,
% 12.45/6.99 ! [A: real,F: real > real,B2: real,C: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.99 => ( ! [X3: real,Y4: real] :
% 12.45/6.99 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8179_order__le__less__subst1,axiom,
% 12.45/6.99 ! [A: real,F: rat > real,B2: rat,C: rat] :
% 12.45/6.99 ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8180_order__le__less__subst1,axiom,
% 12.45/6.99 ! [A: real,F: num > real,B2: num,C: num] :
% 12.45/6.99 ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_num @ B2 @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8181_order__le__less__subst1,axiom,
% 12.45/6.99 ! [A: real,F: nat > real,B2: nat,C: nat] :
% 12.45/6.99 ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_nat @ B2 @ C )
% 12.45/6.99 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.99 ( ( ord_less_nat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8182_order__le__less__subst1,axiom,
% 12.45/6.99 ! [A: real,F: int > real,B2: int,C: int] :
% 12.45/6.99 ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_int @ B2 @ C )
% 12.45/6.99 => ( ! [X3: int,Y4: int] :
% 12.45/6.99 ( ( ord_less_int @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8183_order__le__less__subst1,axiom,
% 12.45/6.99 ! [A: rat,F: real > rat,B2: real,C: real] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_real @ B2 @ C )
% 12.45/6.99 => ( ! [X3: real,Y4: real] :
% 12.45/6.99 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8184_order__le__less__subst1,axiom,
% 12.45/6.99 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_rat @ B2 @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8185_order__le__less__subst1,axiom,
% 12.45/6.99 ! [A: rat,F: num > rat,B2: num,C: num] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_num @ B2 @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8186_order__le__less__subst1,axiom,
% 12.45/6.99 ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_nat @ B2 @ C )
% 12.45/6.99 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.99 ( ( ord_less_nat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8187_order__le__less__subst1,axiom,
% 12.45/6.99 ! [A: rat,F: int > rat,B2: int,C: int] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_int @ B2 @ C )
% 12.45/6.99 => ( ! [X3: int,Y4: int] :
% 12.45/6.99 ( ( ord_less_int @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8188_order__le__less__subst2,axiom,
% 12.45/6.99 ! [A: rat,B2: rat,F: rat > real,C: real] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8189_order__le__less__subst2,axiom,
% 12.45/6.99 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8190_order__le__less__subst2,axiom,
% 12.45/6.99 ! [A: rat,B2: rat,F: rat > num,C: num] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8191_order__le__less__subst2,axiom,
% 12.45/6.99 ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8192_order__le__less__subst2,axiom,
% 12.45/6.99 ! [A: rat,B2: rat,F: rat > int,C: int] :
% 12.45/6.99 ( ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8193_order__le__less__subst2,axiom,
% 12.45/6.99 ! [A: num,B2: num,F: num > real,C: real] :
% 12.45/6.99 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8194_order__le__less__subst2,axiom,
% 12.45/6.99 ! [A: num,B2: num,F: num > rat,C: rat] :
% 12.45/6.99 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8195_order__le__less__subst2,axiom,
% 12.45/6.99 ! [A: num,B2: num,F: num > num,C: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8196_order__le__less__subst2,axiom,
% 12.45/6.99 ! [A: num,B2: num,F: num > nat,C: nat] :
% 12.45/6.99 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8197_order__le__less__subst2,axiom,
% 12.45/6.99 ! [A: num,B2: num,F: num > int,C: int] :
% 12.45/6.99 ( ( ord_less_eq_num @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8198_order__less__le__subst1,axiom,
% 12.45/6.99 ! [A: real,F: rat > real,B2: rat,C: rat] :
% 12.45/6.99 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8199_order__less__le__subst1,axiom,
% 12.45/6.99 ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 12.45/6.99 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8200_order__less__le__subst1,axiom,
% 12.45/6.99 ! [A: num,F: rat > num,B2: rat,C: rat] :
% 12.45/6.99 ( ( ord_less_num @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8201_order__less__le__subst1,axiom,
% 12.45/6.99 ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 12.45/6.99 ( ( ord_less_nat @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8202_order__less__le__subst1,axiom,
% 12.45/6.99 ! [A: int,F: rat > int,B2: rat,C: rat] :
% 12.45/6.99 ( ( ord_less_int @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_eq_rat @ B2 @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8203_order__less__le__subst1,axiom,
% 12.45/6.99 ! [A: real,F: num > real,B2: num,C: num] :
% 12.45/6.99 ( ( ord_less_real @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8204_order__less__le__subst1,axiom,
% 12.45/6.99 ! [A: rat,F: num > rat,B2: num,C: num] :
% 12.45/6.99 ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8205_order__less__le__subst1,axiom,
% 12.45/6.99 ! [A: num,F: num > num,B2: num,C: num] :
% 12.45/6.99 ( ( ord_less_num @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8206_order__less__le__subst1,axiom,
% 12.45/6.99 ! [A: nat,F: num > nat,B2: num,C: num] :
% 12.45/6.99 ( ( ord_less_nat @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8207_order__less__le__subst1,axiom,
% 12.45/6.99 ! [A: int,F: num > int,B2: num,C: num] :
% 12.45/6.99 ( ( ord_less_int @ A @ ( F @ B2 ) )
% 12.45/6.99 => ( ( ord_less_eq_num @ B2 @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8208_order__less__le__subst2,axiom,
% 12.45/6.99 ! [A: real,B2: real,F: real > real,C: real] :
% 12.45/6.99 ( ( ord_less_real @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: real,Y4: real] :
% 12.45/6.99 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8209_order__less__le__subst2,axiom,
% 12.45/6.99 ! [A: rat,B2: rat,F: rat > real,C: real] :
% 12.45/6.99 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8210_order__less__le__subst2,axiom,
% 12.45/6.99 ! [A: num,B2: num,F: num > real,C: real] :
% 12.45/6.99 ( ( ord_less_num @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8211_order__less__le__subst2,axiom,
% 12.45/6.99 ! [A: nat,B2: nat,F: nat > real,C: real] :
% 12.45/6.99 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.99 ( ( ord_less_nat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8212_order__less__le__subst2,axiom,
% 12.45/6.99 ! [A: int,B2: int,F: int > real,C: real] :
% 12.45/6.99 ( ( ord_less_int @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: int,Y4: int] :
% 12.45/6.99 ( ( ord_less_int @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8213_order__less__le__subst2,axiom,
% 12.45/6.99 ! [A: real,B2: real,F: real > rat,C: rat] :
% 12.45/6.99 ( ( ord_less_real @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: real,Y4: real] :
% 12.45/6.99 ( ( ord_less_real @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8214_order__less__le__subst2,axiom,
% 12.45/6.99 ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 12.45/6.99 ( ( ord_less_rat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: rat,Y4: rat] :
% 12.45/6.99 ( ( ord_less_rat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8215_order__less__le__subst2,axiom,
% 12.45/6.99 ! [A: num,B2: num,F: num > rat,C: rat] :
% 12.45/6.99 ( ( ord_less_num @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: num,Y4: num] :
% 12.45/6.99 ( ( ord_less_num @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8216_order__less__le__subst2,axiom,
% 12.45/6.99 ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 12.45/6.99 ( ( ord_less_nat @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: nat,Y4: nat] :
% 12.45/6.99 ( ( ord_less_nat @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8217_order__less__le__subst2,axiom,
% 12.45/6.99 ! [A: int,B2: int,F: int > rat,C: rat] :
% 12.45/6.99 ( ( ord_less_int @ A @ B2 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 12.45/6.99 => ( ! [X3: int,Y4: int] :
% 12.45/6.99 ( ( ord_less_int @ X3 @ Y4 )
% 12.45/6.99 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.45/6.99 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8218_linorder__le__less__linear,axiom,
% 12.45/6.99 ! [X4: real,Y3: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.99 | ( ord_less_real @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8219_linorder__le__less__linear,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.99 | ( ord_less_rat @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8220_linorder__le__less__linear,axiom,
% 12.45/6.99 ! [X4: num,Y3: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X4 @ Y3 )
% 12.45/6.99 | ( ord_less_num @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8221_linorder__le__less__linear,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 12.45/6.99 | ( ord_less_nat @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8222_linorder__le__less__linear,axiom,
% 12.45/6.99 ! [X4: int,Y3: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.99 | ( ord_less_int @ Y3 @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8223_order__le__imp__less__or__eq,axiom,
% 12.45/6.99 ! [X4: real,Y3: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_real @ X4 @ Y3 )
% 12.45/6.99 | ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8224_order__le__imp__less__or__eq,axiom,
% 12.45/6.99 ! [X4: set_nat,Y3: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_set_nat @ X4 @ Y3 )
% 12.45/6.99 | ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8225_order__le__imp__less__or__eq,axiom,
% 12.45/6.99 ! [X4: rat,Y3: rat] :
% 12.45/6.99 ( ( ord_less_eq_rat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_rat @ X4 @ Y3 )
% 12.45/6.99 | ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8226_order__le__imp__less__or__eq,axiom,
% 12.45/6.99 ! [X4: num,Y3: num] :
% 12.45/6.99 ( ( ord_less_eq_num @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_num @ X4 @ Y3 )
% 12.45/6.99 | ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8227_order__le__imp__less__or__eq,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.99 | ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8228_order__le__imp__less__or__eq,axiom,
% 12.45/6.99 ! [X4: int,Y3: int] :
% 12.45/6.99 ( ( ord_less_eq_int @ X4 @ Y3 )
% 12.45/6.99 => ( ( ord_less_int @ X4 @ Y3 )
% 12.45/6.99 | ( X4 = Y3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8229_and__not__numerals_I8_J,axiom,
% 12.45/6.99 ! [M: num,N: num] :
% 12.45/6.99 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.45/6.99 = ( 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 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8230_not__int__rec,axiom,
% 12.45/6.99 ( bit_ri7919022796975470100ot_int
% 12.45/6.99 = ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8231__C5_Ohyps_C_I11_J,axiom,
% 12.45/6.99 ( ( mi != ma )
% 12.45/6.99 => ! [I4: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 12.45/6.99 => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 12.45/6.99 = I4 )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 12.45/6.99 & ! [X5: nat] :
% 12.45/6.99 ( ( ( ( vEBT_VEBT_high @ X5 @ na )
% 12.45/6.99 = I4 )
% 12.45/6.99 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
% 12.45/6.99 => ( ( ord_less_nat @ mi @ X5 )
% 12.45/6.99 & ( ord_less_eq_nat @ X5 @ ma ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8232__C20_C,axiom,
% 12.45/6.99 ( ( za = mi )
% 12.45/6.99 | ( za = ma )
% 12.45/6.99 | ( ( ord_less_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
% 12.45/6.99 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8233_abd,axiom,
% 12.45/6.99 ( ( za = ma )
% 12.45/6.99 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8234_aaa,axiom,
% 12.45/6.99 ( ( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/6.99 = ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8235__C33_C,axiom,
% 12.45/6.99 ~ ? [U2: nat] :
% 12.45/6.99 ( ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U2 )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ U2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8236__092_060open_062_092_060exists_062miny_O_Aboth__member__options_A_ItreeList_A_B_Asc_J_Aminy_092_060close_062,axiom,
% 12.45/6.99 ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ sc ) @ X_1 ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8237_abf,axiom,
% 12.45/6.99 vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8238_abe,axiom,
% 12.45/6.99 vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8239_aa,axiom,
% 12.45/6.99 ( ( za = ma )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8240_nth__equalityI,axiom,
% 12.45/6.99 ! [Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 12.45/6.99 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 12.45/6.99 = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 12.45/6.99 => ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 12.45/6.99 => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
% 12.45/6.99 = ( nth_VEBT_VEBT @ Ys2 @ I3 ) ) )
% 12.45/6.99 => ( Xs2 = Ys2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8241_nth__equalityI,axiom,
% 12.45/6.99 ! [Xs2: list_o,Ys2: list_o] :
% 12.45/6.99 ( ( ( size_size_list_o @ Xs2 )
% 12.45/6.99 = ( size_size_list_o @ Ys2 ) )
% 12.45/6.99 => ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 12.45/6.99 => ( ( nth_o @ Xs2 @ I3 )
% 12.45/6.99 = ( nth_o @ Ys2 @ I3 ) ) )
% 12.45/6.99 => ( Xs2 = Ys2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8242_nth__equalityI,axiom,
% 12.45/6.99 ! [Xs2: list_nat,Ys2: list_nat] :
% 12.45/6.99 ( ( ( size_size_list_nat @ Xs2 )
% 12.45/6.99 = ( size_size_list_nat @ Ys2 ) )
% 12.45/6.99 => ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 12.45/6.99 => ( ( nth_nat @ Xs2 @ I3 )
% 12.45/6.99 = ( nth_nat @ Ys2 @ I3 ) ) )
% 12.45/6.99 => ( Xs2 = Ys2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8243_nth__equalityI,axiom,
% 12.45/6.99 ! [Xs2: list_int,Ys2: list_int] :
% 12.45/6.99 ( ( ( size_size_list_int @ Xs2 )
% 12.45/6.99 = ( size_size_list_int @ Ys2 ) )
% 12.45/6.99 => ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 12.45/6.99 => ( ( nth_int @ Xs2 @ I3 )
% 12.45/6.99 = ( nth_int @ Ys2 @ I3 ) ) )
% 12.45/6.99 => ( Xs2 = Ys2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8244_Skolem__list__nth,axiom,
% 12.45/6.99 ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 12.45/6.99 ( ( ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ K )
% 12.45/6.99 => ? [X6: vEBT_VEBT] : ( P @ I2 @ X6 ) ) )
% 12.45/6.99 = ( ? [Xs: list_VEBT_VEBT] :
% 12.45/6.99 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 12.45/6.99 = K )
% 12.45/6.99 & ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ K )
% 12.45/6.99 => ( P @ I2 @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8245_Skolem__list__nth,axiom,
% 12.45/6.99 ! [K: nat,P: nat > $o > $o] :
% 12.45/6.99 ( ( ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ K )
% 12.45/6.99 => ? [X6: $o] : ( P @ I2 @ X6 ) ) )
% 12.45/6.99 = ( ? [Xs: list_o] :
% 12.45/6.99 ( ( ( size_size_list_o @ Xs )
% 12.45/6.99 = K )
% 12.45/6.99 & ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ K )
% 12.45/6.99 => ( P @ I2 @ ( nth_o @ Xs @ I2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8246_Skolem__list__nth,axiom,
% 12.45/6.99 ! [K: nat,P: nat > nat > $o] :
% 12.45/6.99 ( ( ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ K )
% 12.45/6.99 => ? [X6: nat] : ( P @ I2 @ X6 ) ) )
% 12.45/6.99 = ( ? [Xs: list_nat] :
% 12.45/6.99 ( ( ( size_size_list_nat @ Xs )
% 12.45/6.99 = K )
% 12.45/6.99 & ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ K )
% 12.45/6.99 => ( P @ I2 @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8247_Skolem__list__nth,axiom,
% 12.45/6.99 ! [K: nat,P: nat > int > $o] :
% 12.45/6.99 ( ( ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ K )
% 12.45/6.99 => ? [X6: int] : ( P @ I2 @ X6 ) ) )
% 12.45/6.99 = ( ? [Xs: list_int] :
% 12.45/6.99 ( ( ( size_size_list_int @ Xs )
% 12.45/6.99 = K )
% 12.45/6.99 & ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ K )
% 12.45/6.99 => ( P @ I2 @ ( nth_int @ Xs @ I2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8248_list__eq__iff__nth__eq,axiom,
% 12.45/6.99 ( ( ^ [Y6: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : ( Y6 = Z4 ) )
% 12.45/6.99 = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 12.45/6.99 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 12.45/6.99 = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 12.45/6.99 & ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 12.45/6.99 => ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 12.45/6.99 = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8249_list__eq__iff__nth__eq,axiom,
% 12.45/6.99 ( ( ^ [Y6: list_o,Z4: list_o] : ( Y6 = Z4 ) )
% 12.45/6.99 = ( ^ [Xs: list_o,Ys3: list_o] :
% 12.45/6.99 ( ( ( size_size_list_o @ Xs )
% 12.45/6.99 = ( size_size_list_o @ Ys3 ) )
% 12.45/6.99 & ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 12.45/6.99 => ( ( nth_o @ Xs @ I2 )
% 12.45/6.99 = ( nth_o @ Ys3 @ I2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8250_list__eq__iff__nth__eq,axiom,
% 12.45/6.99 ( ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) )
% 12.45/6.99 = ( ^ [Xs: list_nat,Ys3: list_nat] :
% 12.45/6.99 ( ( ( size_size_list_nat @ Xs )
% 12.45/6.99 = ( size_size_list_nat @ Ys3 ) )
% 12.45/6.99 & ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 12.45/6.99 => ( ( nth_nat @ Xs @ I2 )
% 12.45/6.99 = ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8251_list__eq__iff__nth__eq,axiom,
% 12.45/6.99 ( ( ^ [Y6: list_int,Z4: list_int] : ( Y6 = Z4 ) )
% 12.45/6.99 = ( ^ [Xs: list_int,Ys3: list_int] :
% 12.45/6.99 ( ( ( size_size_list_int @ Xs )
% 12.45/6.99 = ( size_size_list_int @ Ys3 ) )
% 12.45/6.99 & ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 12.45/6.99 => ( ( nth_int @ Xs @ I2 )
% 12.45/6.99 = ( nth_int @ Ys3 @ I2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8252_num_Osize_I4_J,axiom,
% 12.45/6.99 ( ( size_size_num @ one )
% 12.45/6.99 = zero_zero_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8253_num_Osize_I5_J,axiom,
% 12.45/6.99 ! [X2: num] :
% 12.45/6.99 ( ( size_size_num @ ( bit0 @ X2 ) )
% 12.45/6.99 = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8254_num_Osize_I6_J,axiom,
% 12.45/6.99 ! [X32: num] :
% 12.45/6.99 ( ( size_size_num @ ( bit1 @ X32 ) )
% 12.45/6.99 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8255_in__children__def,axiom,
% 12.45/6.99 ( vEBT_V5917875025757280293ildren
% 12.45/6.99 = ( ^ [N2: nat,TreeList: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N2 ) ) @ ( vEBT_VEBT_low @ X @ N2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8256_abc,axiom,
% 12.45/6.99 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8257__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
% 12.45/6.99 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8258_abh,axiom,
% 12.45/6.99 vEBT_vebt_member @ summary @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8259_valid__0__not,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT] :
% 12.45/6.99 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8260_valid__tree__deg__neq__0,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT] :
% 12.45/6.99 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8261__C5_Ohyps_C_I2_J,axiom,
% 12.45/6.99 vEBT_invar_vebt @ summary @ m ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8262_deg__not__0,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8263_both__member__options__equiv__member,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( vEBT_V8194947554948674370ptions @ T @ X4 )
% 12.45/6.99 = ( vEBT_vebt_member @ T @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8264_valid__member__both__member__options,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( vEBT_V8194947554948674370ptions @ T @ X4 )
% 12.45/6.99 => ( vEBT_vebt_member @ T @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8265__092_060open_062both__member__options_Asummary_Asc_092_060close_062,axiom,
% 12.45/6.99 vEBT_V8194947554948674370ptions @ summary @ sc ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8266__092_060open_062vebt__member_Asummary_Asc_092_060close_062,axiom,
% 12.45/6.99 vEBT_vebt_member @ summary @ sc ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8267_member__correct,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( vEBT_vebt_member @ T @ X4 )
% 12.45/6.99 = ( member_nat @ X4 @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8268_member__bound,axiom,
% 12.45/6.99 ! [Tree: vEBT_VEBT,X4: nat,N: nat] :
% 12.45/6.99 ( ( vEBT_vebt_member @ Tree @ X4 )
% 12.45/6.99 => ( ( vEBT_invar_vebt @ Tree @ N )
% 12.45/6.99 => ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8269_abg,axiom,
% 12.45/6.99 vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8270__092_060open_062length_AtreeList_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_Am_092_060close_062,axiom,
% 12.45/6.99 ( ( ( size_s6755466524823107622T_VEBT @ treeList )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 12.45/6.99 & ( vEBT_invar_vebt @ summary @ m ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8271__C5_Ohyps_C_I7_J,axiom,
% 12.45/6.99 ! [I4: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 12.45/6.99 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ X6 ) )
% 12.45/6.99 = ( vEBT_V8194947554948674370ptions @ summary @ I4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8272_post__member__pre__member,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,X4: nat,Y3: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.99 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.99 => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X4 ) @ Y3 )
% 12.45/6.99 => ( ( vEBT_vebt_member @ T @ Y3 )
% 12.45/6.99 | ( X4 = Y3 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8273_valid__insert__both__member__options__pres,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,X4: nat,Y3: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.99 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.99 => ( ( vEBT_V8194947554948674370ptions @ T @ X4 )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X4 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8274_valid__insert__both__member__options__add,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X4 ) @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8275_both__member__options__ding,axiom,
% 12.45/6.99 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 12.45/6.99 => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 12.45/6.99 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8276_deg__deg__n,axiom,
% 12.45/6.99 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 12.45/6.99 => ( Deg = N ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8277_deg__SUcn__Node,axiom,
% 12.45/6.99 ! [Tree: vEBT_VEBT,N: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 12.45/6.99 => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 12.45/6.99 ( Tree
% 12.45/6.99 = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8278__C5_Ohyps_C_I8_J,axiom,
% 12.45/6.99 ( ( mi = ma )
% 12.45/6.99 => ! [X5: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 12.45/6.99 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8279_set__vebt__set__vebt_H__valid,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( vEBT_set_vebt @ T )
% 12.45/6.99 = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8280_buildup__gives__valid,axiom,
% 12.45/6.99 ! [N: nat] :
% 12.45/6.99 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/6.99 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8281__092_060open_062is__succ__in__set_A_Iset__vebt_H_Asummary_J_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_Asc_092_060close_062,axiom,
% 12.45/6.99 vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8282_set__n__deg__not__0,axiom,
% 12.45/6.99 ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
% 12.45/6.99 ( ! [X3: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X3 @ N ) )
% 12.45/6.99 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.99 => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8283_inthall,axiom,
% 12.45/6.99 ! [Xs2: list_Extended_enat,P: extended_enat > $o,N: nat] :
% 12.45/6.99 ( ! [X3: extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ X3 @ ( set_Extended_enat2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X3 ) )
% 12.45/6.99 => ( ( ord_less_nat @ N @ ( size_s3941691890525107288d_enat @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_Extended_enat @ Xs2 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8284_inthall,axiom,
% 12.45/6.99 ! [Xs2: list_complex,P: complex > $o,N: nat] :
% 12.45/6.99 ( ! [X3: complex] :
% 12.45/6.99 ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X3 ) )
% 12.45/6.99 => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8285_inthall,axiom,
% 12.45/6.99 ! [Xs2: list_real,P: real > $o,N: nat] :
% 12.45/6.99 ( ! [X3: real] :
% 12.45/6.99 ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X3 ) )
% 12.45/6.99 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8286_inthall,axiom,
% 12.45/6.99 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 12.45/6.99 ( ! [X3: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X3 ) )
% 12.45/6.99 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8287_inthall,axiom,
% 12.45/6.99 ! [Xs2: list_o,P: $o > $o,N: nat] :
% 12.45/6.99 ( ! [X3: $o] :
% 12.45/6.99 ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X3 ) )
% 12.45/6.99 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8288_inthall,axiom,
% 12.45/6.99 ! [Xs2: list_nat,P: nat > $o,N: nat] :
% 12.45/6.99 ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X3 ) )
% 12.45/6.99 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8289_inthall,axiom,
% 12.45/6.99 ! [Xs2: list_int,P: int > $o,N: nat] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X3 ) )
% 12.45/6.99 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8290_succ__member,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,X4: nat,Y3: nat] :
% 12.45/6.99 ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 @ Y3 )
% 12.45/6.99 = ( ( vEBT_vebt_member @ T @ Y3 )
% 12.45/6.99 & ( ord_less_nat @ X4 @ Y3 )
% 12.45/6.99 & ! [Z5: nat] :
% 12.45/6.99 ( ( ( vEBT_vebt_member @ T @ Z5 )
% 12.45/6.99 & ( ord_less_nat @ X4 @ Z5 ) )
% 12.45/6.99 => ( ord_less_eq_nat @ Y3 @ Z5 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8291_length__pos__if__in__set,axiom,
% 12.45/6.99 ! [X4: extended_enat,Xs2: list_Extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs2 ) )
% 12.45/6.99 => ( ord_less_nat @ zero_zero_nat @ ( size_s3941691890525107288d_enat @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8292_length__pos__if__in__set,axiom,
% 12.45/6.99 ! [X4: complex,Xs2: list_complex] :
% 12.45/6.99 ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
% 12.45/6.99 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8293_length__pos__if__in__set,axiom,
% 12.45/6.99 ! [X4: real,Xs2: list_real] :
% 12.45/6.99 ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
% 12.45/6.99 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8294_length__pos__if__in__set,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 12.45/6.99 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8295_length__pos__if__in__set,axiom,
% 12.45/6.99 ! [X4: $o,Xs2: list_o] :
% 12.45/6.99 ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
% 12.45/6.99 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8296_length__pos__if__in__set,axiom,
% 12.45/6.99 ! [X4: nat,Xs2: list_nat] :
% 12.45/6.99 ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
% 12.45/6.99 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8297_length__pos__if__in__set,axiom,
% 12.45/6.99 ! [X4: int,Xs2: list_int] :
% 12.45/6.99 ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
% 12.45/6.99 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8298_nth__mem,axiom,
% 12.45/6.99 ! [N: nat,Xs2: list_Extended_enat] :
% 12.45/6.99 ( ( ord_less_nat @ N @ ( size_s3941691890525107288d_enat @ Xs2 ) )
% 12.45/6.99 => ( member_Extended_enat @ ( nth_Extended_enat @ Xs2 @ N ) @ ( set_Extended_enat2 @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8299_nth__mem,axiom,
% 12.45/6.99 ! [N: nat,Xs2: list_complex] :
% 12.45/6.99 ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 12.45/6.99 => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8300_nth__mem,axiom,
% 12.45/6.99 ! [N: nat,Xs2: list_real] :
% 12.45/6.99 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 12.45/6.99 => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8301_nth__mem,axiom,
% 12.45/6.99 ! [N: nat,Xs2: list_VEBT_VEBT] :
% 12.45/6.99 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 12.45/6.99 => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8302_nth__mem,axiom,
% 12.45/6.99 ! [N: nat,Xs2: list_o] :
% 12.45/6.99 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 12.45/6.99 => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8303_nth__mem,axiom,
% 12.45/6.99 ! [N: nat,Xs2: list_nat] :
% 12.45/6.99 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 12.45/6.99 => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8304_nth__mem,axiom,
% 12.45/6.99 ! [N: nat,Xs2: list_int] :
% 12.45/6.99 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 12.45/6.99 => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8305_list__ball__nth,axiom,
% 12.45/6.99 ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 12.45/6.99 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 12.45/6.99 => ( ! [X3: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X3 ) )
% 12.45/6.99 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8306_list__ball__nth,axiom,
% 12.45/6.99 ! [N: nat,Xs2: list_o,P: $o > $o] :
% 12.45/6.99 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 12.45/6.99 => ( ! [X3: $o] :
% 12.45/6.99 ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X3 ) )
% 12.45/6.99 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8307_list__ball__nth,axiom,
% 12.45/6.99 ! [N: nat,Xs2: list_nat,P: nat > $o] :
% 12.45/6.99 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 12.45/6.99 => ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X3 ) )
% 12.45/6.99 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8308_list__ball__nth,axiom,
% 12.45/6.99 ! [N: nat,Xs2: list_int,P: int > $o] :
% 12.45/6.99 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 12.45/6.99 => ( ! [X3: int] :
% 12.45/6.99 ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X3 ) )
% 12.45/6.99 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8309_in__set__conv__nth,axiom,
% 12.45/6.99 ! [X4: extended_enat,Xs2: list_Extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs2 ) )
% 12.45/6.99 = ( ? [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_s3941691890525107288d_enat @ Xs2 ) )
% 12.45/6.99 & ( ( nth_Extended_enat @ Xs2 @ I2 )
% 12.45/6.99 = X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8310_in__set__conv__nth,axiom,
% 12.45/6.99 ! [X4: complex,Xs2: list_complex] :
% 12.45/6.99 ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
% 12.45/6.99 = ( ? [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 12.45/6.99 & ( ( nth_complex @ Xs2 @ I2 )
% 12.45/6.99 = X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8311_in__set__conv__nth,axiom,
% 12.45/6.99 ! [X4: real,Xs2: list_real] :
% 12.45/6.99 ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
% 12.45/6.99 = ( ? [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
% 12.45/6.99 & ( ( nth_real @ Xs2 @ I2 )
% 12.45/6.99 = X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8312_in__set__conv__nth,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 12.45/6.99 = ( ? [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 12.45/6.99 & ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
% 12.45/6.99 = X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8313_in__set__conv__nth,axiom,
% 12.45/6.99 ! [X4: $o,Xs2: list_o] :
% 12.45/6.99 ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
% 12.45/6.99 = ( ? [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 12.45/6.99 & ( ( nth_o @ Xs2 @ I2 )
% 12.45/6.99 = X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8314_in__set__conv__nth,axiom,
% 12.45/6.99 ! [X4: nat,Xs2: list_nat] :
% 12.45/6.99 ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
% 12.45/6.99 = ( ? [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 12.45/6.99 & ( ( nth_nat @ Xs2 @ I2 )
% 12.45/6.99 = X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8315_in__set__conv__nth,axiom,
% 12.45/6.99 ! [X4: int,Xs2: list_int] :
% 12.45/6.99 ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
% 12.45/6.99 = ( ? [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 12.45/6.99 & ( ( nth_int @ Xs2 @ I2 )
% 12.45/6.99 = X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8316_all__nth__imp__all__set,axiom,
% 12.45/6.99 ! [Xs2: list_Extended_enat,P: extended_enat > $o,X4: extended_enat] :
% 12.45/6.99 ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( size_s3941691890525107288d_enat @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_Extended_enat @ Xs2 @ I3 ) ) )
% 12.45/6.99 => ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8317_all__nth__imp__all__set,axiom,
% 12.45/6.99 ! [Xs2: list_complex,P: complex > $o,X4: complex] :
% 12.45/6.99 ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_complex @ Xs2 @ I3 ) ) )
% 12.45/6.99 => ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8318_all__nth__imp__all__set,axiom,
% 12.45/6.99 ! [Xs2: list_real,P: real > $o,X4: real] :
% 12.45/6.99 ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_real @ Xs2 @ I3 ) ) )
% 12.45/6.99 => ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8319_all__nth__imp__all__set,axiom,
% 12.45/6.99 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X4: vEBT_VEBT] :
% 12.45/6.99 ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
% 12.45/6.99 => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8320_all__nth__imp__all__set,axiom,
% 12.45/6.99 ! [Xs2: list_o,P: $o > $o,X4: $o] :
% 12.45/6.99 ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_o @ Xs2 @ I3 ) ) )
% 12.45/6.99 => ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8321_all__nth__imp__all__set,axiom,
% 12.45/6.99 ! [Xs2: list_nat,P: nat > $o,X4: nat] :
% 12.45/6.99 ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
% 12.45/6.99 => ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8322_all__nth__imp__all__set,axiom,
% 12.45/6.99 ! [Xs2: list_int,P: int > $o,X4: int] :
% 12.45/6.99 ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_int @ Xs2 @ I3 ) ) )
% 12.45/6.99 => ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8323_all__set__conv__all__nth,axiom,
% 12.45/6.99 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 12.45/6.99 ( ( ! [X: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X ) ) )
% 12.45/6.99 = ( ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8324_all__set__conv__all__nth,axiom,
% 12.45/6.99 ! [Xs2: list_o,P: $o > $o] :
% 12.45/6.99 ( ( ! [X: $o] :
% 12.45/6.99 ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X ) ) )
% 12.45/6.99 = ( ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_o @ Xs2 @ I2 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8325_all__set__conv__all__nth,axiom,
% 12.45/6.99 ! [Xs2: list_nat,P: nat > $o] :
% 12.45/6.99 ( ( ! [X: nat] :
% 12.45/6.99 ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X ) ) )
% 12.45/6.99 = ( ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8326_all__set__conv__all__nth,axiom,
% 12.45/6.99 ! [Xs2: list_int,P: int > $o] :
% 12.45/6.99 ( ( ! [X: int] :
% 12.45/6.99 ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 12.45/6.99 => ( P @ X ) ) )
% 12.45/6.99 = ( ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 12.45/6.99 => ( P @ ( nth_int @ Xs2 @ I2 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8327_is__succ__in__set__def,axiom,
% 12.45/6.99 ( vEBT_is_succ_in_set
% 12.45/6.99 = ( ^ [Xs: set_nat,X: nat,Y: nat] :
% 12.45/6.99 ( ( member_nat @ Y @ Xs )
% 12.45/6.99 & ( ord_less_nat @ X @ Y )
% 12.45/6.99 & ! [Z5: nat] :
% 12.45/6.99 ( ( member_nat @ Z5 @ Xs )
% 12.45/6.99 => ( ( ord_less_nat @ X @ Z5 )
% 12.45/6.99 => ( ord_less_eq_nat @ Y @ Z5 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8328_inrange,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8329_fgh,axiom,
% 12.45/6.99 ( ( vEBT_VEBT_set_vebt @ ( nth_VEBT_VEBT @ treeList @ sc ) )
% 12.45/6.99 != bot_bot_set_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8330__092_060open_062vebt__member_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Az_A_092_060and_062_Ax_A_060_Az_092_060close_062,axiom,
% 12.45/6.99 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ za )
% 12.45/6.99 & ( ord_less_nat @ xa @ za ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8331__092_060open_062vebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_092_060close_062,axiom,
% 12.45/6.99 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 = ( some_nat @ sc ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8332_buildup__gives__empty,axiom,
% 12.45/6.99 ! [N: nat] :
% 12.45/6.99 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 12.45/6.99 = bot_bot_set_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8333_power__shift,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat,Z: nat] :
% 12.45/6.99 ( ( ( power_power_nat @ X4 @ Y3 )
% 12.45/6.99 = Z )
% 12.45/6.99 = ( ( vEBT_VEBT_power @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) )
% 12.45/6.99 = ( some_nat @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8334__C5_Ohyps_C_I1_J,axiom,
% 12.45/6.99 ! [X5: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 12.45/6.99 => ( ( vEBT_invar_vebt @ X5 @ na )
% 12.45/6.99 & ! [Xa2: nat,Xb2: nat] :
% 12.45/6.99 ( ( ( vEBT_vebt_succ @ X5 @ Xa2 )
% 12.45/6.99 = ( some_nat @ Xb2 ) )
% 12.45/6.99 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ X5 ) @ Xa2 @ Xb2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8335_mi__eq__ma__no__ch,axiom,
% 12.45/6.99 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 12.45/6.99 => ( ( Mi = Ma )
% 12.45/6.99 => ( ! [X5: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/6.99 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
% 12.45/6.99 & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8336__C5_Ohyps_C_I3_J,axiom,
% 12.45/6.99 ! [X4: nat,Sx: nat] :
% 12.45/6.99 ( ( ( vEBT_vebt_succ @ summary @ X4 )
% 12.45/6.99 = ( some_nat @ Sx ) )
% 12.45/6.99 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X4 @ Sx ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8337_atLeastAtMost__iff,axiom,
% 12.45/6.99 ! [I: extended_enat,L2: extended_enat,U: extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ I @ ( set_or5403411693681687835d_enat @ L2 @ U ) )
% 12.45/6.99 = ( ( ord_le2932123472753598470d_enat @ L2 @ I )
% 12.45/6.99 & ( ord_le2932123472753598470d_enat @ I @ U ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8338_atLeastAtMost__iff,axiom,
% 12.45/6.99 ! [I: set_nat,L2: set_nat,U: set_nat] :
% 12.45/6.99 ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L2 @ U ) )
% 12.45/6.99 = ( ( ord_less_eq_set_nat @ L2 @ I )
% 12.45/6.99 & ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8339_atLeastAtMost__iff,axiom,
% 12.45/6.99 ! [I: rat,L2: rat,U: rat] :
% 12.45/6.99 ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L2 @ U ) )
% 12.45/6.99 = ( ( ord_less_eq_rat @ L2 @ I )
% 12.45/6.99 & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8340_atLeastAtMost__iff,axiom,
% 12.45/6.99 ! [I: num,L2: num,U: num] :
% 12.45/6.99 ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L2 @ U ) )
% 12.45/6.99 = ( ( ord_less_eq_num @ L2 @ I )
% 12.45/6.99 & ( ord_less_eq_num @ I @ U ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8341_atLeastAtMost__iff,axiom,
% 12.45/6.99 ! [I: nat,L2: nat,U: nat] :
% 12.45/6.99 ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 12.45/6.99 = ( ( ord_less_eq_nat @ L2 @ I )
% 12.45/6.99 & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8342_atLeastAtMost__iff,axiom,
% 12.45/6.99 ! [I: int,L2: int,U: int] :
% 12.45/6.99 ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 12.45/6.99 = ( ( ord_less_eq_int @ L2 @ I )
% 12.45/6.99 & ( ord_less_eq_int @ I @ U ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8343_atLeastAtMost__iff,axiom,
% 12.45/6.99 ! [I: real,L2: real,U: real] :
% 12.45/6.99 ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L2 @ U ) )
% 12.45/6.99 = ( ( ord_less_eq_real @ L2 @ I )
% 12.45/6.99 & ( ord_less_eq_real @ I @ U ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8344_Icc__eq__Icc,axiom,
% 12.45/6.99 ! [L2: set_nat,H2: set_nat,L4: set_nat,H3: set_nat] :
% 12.45/6.99 ( ( ( set_or4548717258645045905et_nat @ L2 @ H2 )
% 12.45/6.99 = ( set_or4548717258645045905et_nat @ L4 @ H3 ) )
% 12.45/6.99 = ( ( ( L2 = L4 )
% 12.45/6.99 & ( H2 = H3 ) )
% 12.45/6.99 | ( ~ ( ord_less_eq_set_nat @ L2 @ H2 )
% 12.45/6.99 & ~ ( ord_less_eq_set_nat @ L4 @ H3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8345_Icc__eq__Icc,axiom,
% 12.45/6.99 ! [L2: rat,H2: rat,L4: rat,H3: rat] :
% 12.45/6.99 ( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
% 12.45/6.99 = ( set_or633870826150836451st_rat @ L4 @ H3 ) )
% 12.45/6.99 = ( ( ( L2 = L4 )
% 12.45/6.99 & ( H2 = H3 ) )
% 12.45/6.99 | ( ~ ( ord_less_eq_rat @ L2 @ H2 )
% 12.45/6.99 & ~ ( ord_less_eq_rat @ L4 @ H3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8346_Icc__eq__Icc,axiom,
% 12.45/6.99 ! [L2: num,H2: num,L4: num,H3: num] :
% 12.45/6.99 ( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
% 12.45/6.99 = ( set_or7049704709247886629st_num @ L4 @ H3 ) )
% 12.45/6.99 = ( ( ( L2 = L4 )
% 12.45/6.99 & ( H2 = H3 ) )
% 12.45/6.99 | ( ~ ( ord_less_eq_num @ L2 @ H2 )
% 12.45/6.99 & ~ ( ord_less_eq_num @ L4 @ H3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8347_Icc__eq__Icc,axiom,
% 12.45/6.99 ! [L2: nat,H2: nat,L4: nat,H3: nat] :
% 12.45/6.99 ( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
% 12.45/6.99 = ( set_or1269000886237332187st_nat @ L4 @ H3 ) )
% 12.45/6.99 = ( ( ( L2 = L4 )
% 12.45/6.99 & ( H2 = H3 ) )
% 12.45/6.99 | ( ~ ( ord_less_eq_nat @ L2 @ H2 )
% 12.45/6.99 & ~ ( ord_less_eq_nat @ L4 @ H3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8348_Icc__eq__Icc,axiom,
% 12.45/6.99 ! [L2: int,H2: int,L4: int,H3: int] :
% 12.45/6.99 ( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
% 12.45/6.99 = ( set_or1266510415728281911st_int @ L4 @ H3 ) )
% 12.45/6.99 = ( ( ( L2 = L4 )
% 12.45/6.99 & ( H2 = H3 ) )
% 12.45/6.99 | ( ~ ( ord_less_eq_int @ L2 @ H2 )
% 12.45/6.99 & ~ ( ord_less_eq_int @ L4 @ H3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8349_Icc__eq__Icc,axiom,
% 12.45/6.99 ! [L2: real,H2: real,L4: real,H3: real] :
% 12.45/6.99 ( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
% 12.45/6.99 = ( set_or1222579329274155063t_real @ L4 @ H3 ) )
% 12.45/6.99 = ( ( ( L2 = L4 )
% 12.45/6.99 & ( H2 = H3 ) )
% 12.45/6.99 | ( ~ ( ord_less_eq_real @ L2 @ H2 )
% 12.45/6.99 & ~ ( ord_less_eq_real @ L4 @ H3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8350_insert__simp__mima,axiom,
% 12.45/6.99 ! [X4: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 12.45/6.99 ( ( ( X4 = Mi )
% 12.45/6.99 | ( X4 = Ma ) )
% 12.45/6.99 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 12.45/6.99 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8351_Diff__cancel,axiom,
% 12.45/6.99 ! [A3: set_Extended_enat] :
% 12.45/6.99 ( ( minus_925952699566721837d_enat @ A3 @ A3 )
% 12.45/6.99 = bot_bo7653980558646680370d_enat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8352_Diff__cancel,axiom,
% 12.45/6.99 ! [A3: set_real] :
% 12.45/6.99 ( ( minus_minus_set_real @ A3 @ A3 )
% 12.45/6.99 = bot_bot_set_real ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8353_Diff__cancel,axiom,
% 12.45/6.99 ! [A3: set_int] :
% 12.45/6.99 ( ( minus_minus_set_int @ A3 @ A3 )
% 12.45/6.99 = bot_bot_set_int ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8354_Diff__cancel,axiom,
% 12.45/6.99 ! [A3: set_nat] :
% 12.45/6.99 ( ( minus_minus_set_nat @ A3 @ A3 )
% 12.45/6.99 = bot_bot_set_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8355_empty__Diff,axiom,
% 12.45/6.99 ! [A3: set_Extended_enat] :
% 12.45/6.99 ( ( minus_925952699566721837d_enat @ bot_bo7653980558646680370d_enat @ A3 )
% 12.45/6.99 = bot_bo7653980558646680370d_enat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8356_empty__Diff,axiom,
% 12.45/6.99 ! [A3: set_real] :
% 12.45/6.99 ( ( minus_minus_set_real @ bot_bot_set_real @ A3 )
% 12.45/6.99 = bot_bot_set_real ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8357_empty__Diff,axiom,
% 12.45/6.99 ! [A3: set_int] :
% 12.45/6.99 ( ( minus_minus_set_int @ bot_bot_set_int @ A3 )
% 12.45/6.99 = bot_bot_set_int ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8358_empty__Diff,axiom,
% 12.45/6.99 ! [A3: set_nat] :
% 12.45/6.99 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
% 12.45/6.99 = bot_bot_set_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8359_Diff__empty,axiom,
% 12.45/6.99 ! [A3: set_Extended_enat] :
% 12.45/6.99 ( ( minus_925952699566721837d_enat @ A3 @ bot_bo7653980558646680370d_enat )
% 12.45/6.99 = A3 ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8360_Diff__empty,axiom,
% 12.45/6.99 ! [A3: set_real] :
% 12.45/6.99 ( ( minus_minus_set_real @ A3 @ bot_bot_set_real )
% 12.45/6.99 = A3 ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8361_Diff__empty,axiom,
% 12.45/6.99 ! [A3: set_int] :
% 12.45/6.99 ( ( minus_minus_set_int @ A3 @ bot_bot_set_int )
% 12.45/6.99 = A3 ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8362_Diff__empty,axiom,
% 12.45/6.99 ! [A3: set_nat] :
% 12.45/6.99 ( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
% 12.45/6.99 = A3 ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8363_mi__ma__2__deg,axiom,
% 12.45/6.99 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 12.45/6.99 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 12.45/6.99 & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8364_succ__min,axiom,
% 12.45/6.99 ! [Deg: nat,X4: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 12.45/6.99 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 12.45/6.99 => ( ( ord_less_nat @ X4 @ Mi )
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 12.45/6.99 = ( some_nat @ Mi ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8365_atLeastatMost__empty__iff,axiom,
% 12.45/6.99 ! [A: extended_enat,B2: extended_enat] :
% 12.45/6.99 ( ( ( set_or5403411693681687835d_enat @ A @ B2 )
% 12.45/6.99 = bot_bo7653980558646680370d_enat )
% 12.45/6.99 = ( ~ ( ord_le2932123472753598470d_enat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8366_atLeastatMost__empty__iff,axiom,
% 12.45/6.99 ! [A: set_nat,B2: set_nat] :
% 12.45/6.99 ( ( ( set_or4548717258645045905et_nat @ A @ B2 )
% 12.45/6.99 = bot_bot_set_set_nat )
% 12.45/6.99 = ( ~ ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8367_atLeastatMost__empty__iff,axiom,
% 12.45/6.99 ! [A: rat,B2: rat] :
% 12.45/6.99 ( ( ( set_or633870826150836451st_rat @ A @ B2 )
% 12.45/6.99 = bot_bot_set_rat )
% 12.45/6.99 = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8368_atLeastatMost__empty__iff,axiom,
% 12.45/6.99 ! [A: num,B2: num] :
% 12.45/6.99 ( ( ( set_or7049704709247886629st_num @ A @ B2 )
% 12.45/6.99 = bot_bot_set_num )
% 12.45/6.99 = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8369_atLeastatMost__empty__iff,axiom,
% 12.45/6.99 ! [A: nat,B2: nat] :
% 12.45/6.99 ( ( ( set_or1269000886237332187st_nat @ A @ B2 )
% 12.45/6.99 = bot_bot_set_nat )
% 12.45/6.99 = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8370_atLeastatMost__empty__iff,axiom,
% 12.45/6.99 ! [A: int,B2: int] :
% 12.45/6.99 ( ( ( set_or1266510415728281911st_int @ A @ B2 )
% 12.45/6.99 = bot_bot_set_int )
% 12.45/6.99 = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8371_atLeastatMost__empty__iff,axiom,
% 12.45/6.99 ! [A: real,B2: real] :
% 12.45/6.99 ( ( ( set_or1222579329274155063t_real @ A @ B2 )
% 12.45/6.99 = bot_bot_set_real )
% 12.45/6.99 = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8372_atLeastatMost__empty__iff2,axiom,
% 12.45/6.99 ! [A: extended_enat,B2: extended_enat] :
% 12.45/6.99 ( ( bot_bo7653980558646680370d_enat
% 12.45/6.99 = ( set_or5403411693681687835d_enat @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_le2932123472753598470d_enat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8373_atLeastatMost__empty__iff2,axiom,
% 12.45/6.99 ! [A: set_nat,B2: set_nat] :
% 12.45/6.99 ( ( bot_bot_set_set_nat
% 12.45/6.99 = ( set_or4548717258645045905et_nat @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8374_atLeastatMost__empty__iff2,axiom,
% 12.45/6.99 ! [A: rat,B2: rat] :
% 12.45/6.99 ( ( bot_bot_set_rat
% 12.45/6.99 = ( set_or633870826150836451st_rat @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8375_atLeastatMost__empty__iff2,axiom,
% 12.45/6.99 ! [A: num,B2: num] :
% 12.45/6.99 ( ( bot_bot_set_num
% 12.45/6.99 = ( set_or7049704709247886629st_num @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8376_atLeastatMost__empty__iff2,axiom,
% 12.45/6.99 ! [A: nat,B2: nat] :
% 12.45/6.99 ( ( bot_bot_set_nat
% 12.45/6.99 = ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8377_atLeastatMost__empty__iff2,axiom,
% 12.45/6.99 ! [A: int,B2: int] :
% 12.45/6.99 ( ( bot_bot_set_int
% 12.45/6.99 = ( set_or1266510415728281911st_int @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8378_atLeastatMost__empty__iff2,axiom,
% 12.45/6.99 ! [A: real,B2: real] :
% 12.45/6.99 ( ( bot_bot_set_real
% 12.45/6.99 = ( set_or1222579329274155063t_real @ A @ B2 ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8379_atLeastatMost__subset__iff,axiom,
% 12.45/6.99 ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
% 12.45/6.99 ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B2 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_set_nat @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_set_nat @ C @ A )
% 12.45/6.99 & ( ord_less_eq_set_nat @ B2 @ D ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8380_atLeastatMost__subset__iff,axiom,
% 12.45/6.99 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.45/6.99 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_rat @ C @ A )
% 12.45/6.99 & ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8381_atLeastatMost__subset__iff,axiom,
% 12.45/6.99 ! [A: num,B2: num,C: num,D: num] :
% 12.45/6.99 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_num @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_num @ C @ A )
% 12.45/6.99 & ( ord_less_eq_num @ B2 @ D ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8382_atLeastatMost__subset__iff,axiom,
% 12.45/6.99 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_nat @ C @ A )
% 12.45/6.99 & ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8383_atLeastatMost__subset__iff,axiom,
% 12.45/6.99 ! [A: int,B2: int,C: int,D: int] :
% 12.45/6.99 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_int @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_int @ C @ A )
% 12.45/6.99 & ( ord_less_eq_int @ B2 @ D ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8384_atLeastatMost__subset__iff,axiom,
% 12.45/6.99 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.99 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 12.45/6.99 = ( ~ ( ord_less_eq_real @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_real @ C @ A )
% 12.45/6.99 & ( ord_less_eq_real @ B2 @ D ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8385_atLeastatMost__empty,axiom,
% 12.45/6.99 ! [B2: extended_enat,A: extended_enat] :
% 12.45/6.99 ( ( ord_le72135733267957522d_enat @ B2 @ A )
% 12.45/6.99 => ( ( set_or5403411693681687835d_enat @ A @ B2 )
% 12.45/6.99 = bot_bo7653980558646680370d_enat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8386_atLeastatMost__empty,axiom,
% 12.45/6.99 ! [B2: rat,A: rat] :
% 12.45/6.99 ( ( ord_less_rat @ B2 @ A )
% 12.45/6.99 => ( ( set_or633870826150836451st_rat @ A @ B2 )
% 12.45/6.99 = bot_bot_set_rat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8387_atLeastatMost__empty,axiom,
% 12.45/6.99 ! [B2: num,A: num] :
% 12.45/6.99 ( ( ord_less_num @ B2 @ A )
% 12.45/6.99 => ( ( set_or7049704709247886629st_num @ A @ B2 )
% 12.45/6.99 = bot_bot_set_num ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8388_atLeastatMost__empty,axiom,
% 12.45/6.99 ! [B2: nat,A: nat] :
% 12.45/6.99 ( ( ord_less_nat @ B2 @ A )
% 12.45/6.99 => ( ( set_or1269000886237332187st_nat @ A @ B2 )
% 12.45/6.99 = bot_bot_set_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8389_atLeastatMost__empty,axiom,
% 12.45/6.99 ! [B2: int,A: int] :
% 12.45/6.99 ( ( ord_less_int @ B2 @ A )
% 12.45/6.99 => ( ( set_or1266510415728281911st_int @ A @ B2 )
% 12.45/6.99 = bot_bot_set_int ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8390_atLeastatMost__empty,axiom,
% 12.45/6.99 ! [B2: real,A: real] :
% 12.45/6.99 ( ( ord_less_real @ B2 @ A )
% 12.45/6.99 => ( ( set_or1222579329274155063t_real @ A @ B2 )
% 12.45/6.99 = bot_bot_set_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8391_Diff__eq__empty__iff,axiom,
% 12.45/6.99 ! [A3: set_Extended_enat,B4: set_Extended_enat] :
% 12.45/6.99 ( ( ( minus_925952699566721837d_enat @ A3 @ B4 )
% 12.45/6.99 = bot_bo7653980558646680370d_enat )
% 12.45/6.99 = ( ord_le7203529160286727270d_enat @ A3 @ B4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8392_Diff__eq__empty__iff,axiom,
% 12.45/6.99 ! [A3: set_real,B4: set_real] :
% 12.45/6.99 ( ( ( minus_minus_set_real @ A3 @ B4 )
% 12.45/6.99 = bot_bot_set_real )
% 12.45/6.99 = ( ord_less_eq_set_real @ A3 @ B4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8393_Diff__eq__empty__iff,axiom,
% 12.45/6.99 ! [A3: set_int,B4: set_int] :
% 12.45/6.99 ( ( ( minus_minus_set_int @ A3 @ B4 )
% 12.45/6.99 = bot_bot_set_int )
% 12.45/6.99 = ( ord_less_eq_set_int @ A3 @ B4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8394_Diff__eq__empty__iff,axiom,
% 12.45/6.99 ! [A3: set_nat,B4: set_nat] :
% 12.45/6.99 ( ( ( minus_minus_set_nat @ A3 @ B4 )
% 12.45/6.99 = bot_bot_set_nat )
% 12.45/6.99 = ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8395__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062sc_O_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 12.45/6.99 ~ ! [Sc: nat] :
% 12.45/6.99 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 != ( some_nat @ Sc ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8396_both__member__options__from__complete__tree__to__child,axiom,
% 12.45/6.99 ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 12.45/6.99 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 12.45/6.99 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 | ( X4 = Mi )
% 12.45/6.99 | ( X4 = Ma ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8397_member__inv,axiom,
% 12.45/6.99 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
% 12.45/6.99 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 12.45/6.99 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 12.45/6.99 & ( ( X4 = Mi )
% 12.45/6.99 | ( X4 = Ma )
% 12.45/6.99 | ( ( ord_less_nat @ X4 @ Ma )
% 12.45/6.99 & ( ord_less_nat @ Mi @ X4 )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/6.99 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8398_both__member__options__from__chilf__to__complete__tree,axiom,
% 12.45/6.99 ! [X4: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 12.45/6.99 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/6.99 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 12.45/6.99 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8399__092_060open_062Some_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Asc_J_092_060close_062,axiom,
% 12.45/6.99 ( ( some_nat @ miny )
% 12.45/6.99 = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ sc ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8400_greater__shift,axiom,
% 12.45/6.99 ( ord_less_nat
% 12.45/6.99 = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8401_lesseq__shift,axiom,
% 12.45/6.99 ( ord_less_eq_nat
% 12.45/6.99 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8402_False,axiom,
% 12.45/6.99 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 != none_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8403_not__psubset__empty,axiom,
% 12.45/6.99 ! [A3: set_Extended_enat] :
% 12.45/6.99 ~ ( ord_le2529575680413868914d_enat @ A3 @ bot_bo7653980558646680370d_enat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8404_not__psubset__empty,axiom,
% 12.45/6.99 ! [A3: set_real] :
% 12.45/6.99 ~ ( ord_less_set_real @ A3 @ bot_bot_set_real ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8405_not__psubset__empty,axiom,
% 12.45/6.99 ! [A3: set_nat] :
% 12.45/6.99 ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8406_not__psubset__empty,axiom,
% 12.45/6.99 ! [A3: set_int] :
% 12.45/6.99 ~ ( ord_less_set_int @ A3 @ bot_bot_set_int ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8407_bot_Onot__eq__extremum,axiom,
% 12.45/6.99 ! [A: set_Extended_enat] :
% 12.45/6.99 ( ( A != bot_bo7653980558646680370d_enat )
% 12.45/6.99 = ( ord_le2529575680413868914d_enat @ bot_bo7653980558646680370d_enat @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8408_bot_Onot__eq__extremum,axiom,
% 12.45/6.99 ! [A: set_real] :
% 12.45/6.99 ( ( A != bot_bot_set_real )
% 12.45/6.99 = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8409_bot_Onot__eq__extremum,axiom,
% 12.45/6.99 ! [A: set_nat] :
% 12.45/6.99 ( ( A != bot_bot_set_nat )
% 12.45/6.99 = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8410_bot_Onot__eq__extremum,axiom,
% 12.45/6.99 ! [A: set_int] :
% 12.45/6.99 ( ( A != bot_bot_set_int )
% 12.45/6.99 = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8411_bot_Onot__eq__extremum,axiom,
% 12.45/6.99 ! [A: nat] :
% 12.45/6.99 ( ( A != bot_bot_nat )
% 12.45/6.99 = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8412_bot_Oextremum__strict,axiom,
% 12.45/6.99 ! [A: set_Extended_enat] :
% 12.45/6.99 ~ ( ord_le2529575680413868914d_enat @ A @ bot_bo7653980558646680370d_enat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8413_bot_Oextremum__strict,axiom,
% 12.45/6.99 ! [A: set_real] :
% 12.45/6.99 ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8414_bot_Oextremum__strict,axiom,
% 12.45/6.99 ! [A: set_nat] :
% 12.45/6.99 ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8415_bot_Oextremum__strict,axiom,
% 12.45/6.99 ! [A: set_int] :
% 12.45/6.99 ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8416_bot_Oextremum__strict,axiom,
% 12.45/6.99 ! [A: nat] :
% 12.45/6.99 ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8417_bot_Oextremum,axiom,
% 12.45/6.99 ! [A: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ bot_bo7653980558646680370d_enat @ A ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8418_bot_Oextremum,axiom,
% 12.45/6.99 ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8419_bot_Oextremum,axiom,
% 12.45/6.99 ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8420_bot_Oextremum,axiom,
% 12.45/6.99 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8421_bot_Oextremum,axiom,
% 12.45/6.99 ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8422_bot_Oextremum__unique,axiom,
% 12.45/6.99 ! [A: set_Extended_enat] :
% 12.45/6.99 ( ( ord_le7203529160286727270d_enat @ A @ bot_bo7653980558646680370d_enat )
% 12.45/6.99 = ( A = bot_bo7653980558646680370d_enat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8423_bot_Oextremum__unique,axiom,
% 12.45/6.99 ! [A: set_real] :
% 12.45/6.99 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 12.45/6.99 = ( A = bot_bot_set_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8424_bot_Oextremum__unique,axiom,
% 12.45/6.99 ! [A: set_int] :
% 12.45/6.99 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 12.45/6.99 = ( A = bot_bot_set_int ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8425_bot_Oextremum__unique,axiom,
% 12.45/6.99 ! [A: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 12.45/6.99 = ( A = bot_bot_set_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8426_bot_Oextremum__unique,axiom,
% 12.45/6.99 ! [A: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 12.45/6.99 = ( A = bot_bot_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8427_bot_Oextremum__uniqueI,axiom,
% 12.45/6.99 ! [A: set_Extended_enat] :
% 12.45/6.99 ( ( ord_le7203529160286727270d_enat @ A @ bot_bo7653980558646680370d_enat )
% 12.45/6.99 => ( A = bot_bo7653980558646680370d_enat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8428_bot_Oextremum__uniqueI,axiom,
% 12.45/6.99 ! [A: set_real] :
% 12.45/6.99 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 12.45/6.99 => ( A = bot_bot_set_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8429_bot_Oextremum__uniqueI,axiom,
% 12.45/6.99 ! [A: set_int] :
% 12.45/6.99 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 12.45/6.99 => ( A = bot_bot_set_int ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8430_bot_Oextremum__uniqueI,axiom,
% 12.45/6.99 ! [A: set_nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 12.45/6.99 => ( A = bot_bot_set_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8431_bot_Oextremum__uniqueI,axiom,
% 12.45/6.99 ! [A: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 12.45/6.99 => ( A = bot_bot_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8432_diff__shunt__var,axiom,
% 12.45/6.99 ! [X4: set_Extended_enat,Y3: set_Extended_enat] :
% 12.45/6.99 ( ( ( minus_925952699566721837d_enat @ X4 @ Y3 )
% 12.45/6.99 = bot_bo7653980558646680370d_enat )
% 12.45/6.99 = ( ord_le7203529160286727270d_enat @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8433_diff__shunt__var,axiom,
% 12.45/6.99 ! [X4: set_real,Y3: set_real] :
% 12.45/6.99 ( ( ( minus_minus_set_real @ X4 @ Y3 )
% 12.45/6.99 = bot_bot_set_real )
% 12.45/6.99 = ( ord_less_eq_set_real @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8434_diff__shunt__var,axiom,
% 12.45/6.99 ! [X4: set_int,Y3: set_int] :
% 12.45/6.99 ( ( ( minus_minus_set_int @ X4 @ Y3 )
% 12.45/6.99 = bot_bot_set_int )
% 12.45/6.99 = ( ord_less_eq_set_int @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8435_diff__shunt__var,axiom,
% 12.45/6.99 ! [X4: set_nat,Y3: set_nat] :
% 12.45/6.99 ( ( ( minus_minus_set_nat @ X4 @ Y3 )
% 12.45/6.99 = bot_bot_set_nat )
% 12.45/6.99 = ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8436_ex__nat__less,axiom,
% 12.45/6.99 ! [N: nat,P: nat > $o] :
% 12.45/6.99 ( ( ? [M6: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M6 @ N )
% 12.45/6.99 & ( P @ M6 ) ) )
% 12.45/6.99 = ( ? [X: nat] :
% 12.45/6.99 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 12.45/6.99 & ( P @ X ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8437_all__nat__less,axiom,
% 12.45/6.99 ! [N: nat,P: nat > $o] :
% 12.45/6.99 ( ( ! [M6: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M6 @ N )
% 12.45/6.99 => ( P @ M6 ) ) )
% 12.45/6.99 = ( ! [X: nat] :
% 12.45/6.99 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 12.45/6.99 => ( P @ X ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8438_atLeastatMost__psubset__iff,axiom,
% 12.45/6.99 ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
% 12.45/6.99 ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B2 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
% 12.45/6.99 = ( ( ~ ( ord_less_eq_set_nat @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_set_nat @ C @ A )
% 12.45/6.99 & ( ord_less_eq_set_nat @ B2 @ D )
% 12.45/6.99 & ( ( ord_less_set_nat @ C @ A )
% 12.45/6.99 | ( ord_less_set_nat @ B2 @ D ) ) ) )
% 12.45/6.99 & ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8439_atLeastatMost__psubset__iff,axiom,
% 12.45/6.99 ! [A: rat,B2: rat,C: rat,D: rat] :
% 12.45/6.99 ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 12.45/6.99 = ( ( ~ ( ord_less_eq_rat @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_rat @ C @ A )
% 12.45/6.99 & ( ord_less_eq_rat @ B2 @ D )
% 12.45/6.99 & ( ( ord_less_rat @ C @ A )
% 12.45/6.99 | ( ord_less_rat @ B2 @ D ) ) ) )
% 12.45/6.99 & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8440_atLeastatMost__psubset__iff,axiom,
% 12.45/6.99 ! [A: num,B2: num,C: num,D: num] :
% 12.45/6.99 ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 12.45/6.99 = ( ( ~ ( ord_less_eq_num @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_num @ C @ A )
% 12.45/6.99 & ( ord_less_eq_num @ B2 @ D )
% 12.45/6.99 & ( ( ord_less_num @ C @ A )
% 12.45/6.99 | ( ord_less_num @ B2 @ D ) ) ) )
% 12.45/6.99 & ( ord_less_eq_num @ C @ D ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8441_atLeastatMost__psubset__iff,axiom,
% 12.45/6.99 ! [A: nat,B2: nat,C: nat,D: nat] :
% 12.45/6.99 ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 12.45/6.99 = ( ( ~ ( ord_less_eq_nat @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_nat @ C @ A )
% 12.45/6.99 & ( ord_less_eq_nat @ B2 @ D )
% 12.45/6.99 & ( ( ord_less_nat @ C @ A )
% 12.45/6.99 | ( ord_less_nat @ B2 @ D ) ) ) )
% 12.45/6.99 & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8442_atLeastatMost__psubset__iff,axiom,
% 12.45/6.99 ! [A: int,B2: int,C: int,D: int] :
% 12.45/6.99 ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 12.45/6.99 = ( ( ~ ( ord_less_eq_int @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_int @ C @ A )
% 12.45/6.99 & ( ord_less_eq_int @ B2 @ D )
% 12.45/6.99 & ( ( ord_less_int @ C @ A )
% 12.45/6.99 | ( ord_less_int @ B2 @ D ) ) ) )
% 12.45/6.99 & ( ord_less_eq_int @ C @ D ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8443_atLeastatMost__psubset__iff,axiom,
% 12.45/6.99 ! [A: real,B2: real,C: real,D: real] :
% 12.45/6.99 ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 12.45/6.99 = ( ( ~ ( ord_less_eq_real @ A @ B2 )
% 12.45/6.99 | ( ( ord_less_eq_real @ C @ A )
% 12.45/6.99 & ( ord_less_eq_real @ B2 @ D )
% 12.45/6.99 & ( ( ord_less_real @ C @ A )
% 12.45/6.99 | ( ord_less_real @ B2 @ D ) ) ) )
% 12.45/6.99 & ( ord_less_eq_real @ C @ D ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8444_invar__vebt_Ointros_I4_J,axiom,
% 12.45/6.99 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 12.45/6.99 ( ! [X3: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X3 @ N ) )
% 12.45/6.99 => ( ( vEBT_invar_vebt @ Summary @ M )
% 12.45/6.99 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.99 => ( ( M = N )
% 12.45/6.99 => ( ( Deg
% 12.45/6.99 = ( plus_plus_nat @ N @ M ) )
% 12.45/6.99 => ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.99 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
% 12.45/6.99 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 12.45/6.99 => ( ( ( Mi = Ma )
% 12.45/6.99 => ! [X3: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/6.99 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 12.45/6.99 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 12.45/6.99 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 12.45/6.99 => ( ( ( Mi != Ma )
% 12.45/6.99 => ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.99 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 12.45/6.99 = I3 )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 12.45/6.99 & ! [X3: nat] :
% 12.45/6.99 ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 12.45/6.99 = I3 )
% 12.45/6.99 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 12.45/6.99 => ( ( ord_less_nat @ Mi @ X3 )
% 12.45/6.99 & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8445_invar__vebt_Ointros_I5_J,axiom,
% 12.45/6.99 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 12.45/6.99 ( ! [X3: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X3 @ N ) )
% 12.45/6.99 => ( ( vEBT_invar_vebt @ Summary @ M )
% 12.45/6.99 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.99 => ( ( M
% 12.45/6.99 = ( suc @ N ) )
% 12.45/6.99 => ( ( Deg
% 12.45/6.99 = ( plus_plus_nat @ N @ M ) )
% 12.45/6.99 => ( ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.99 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
% 12.45/6.99 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 12.45/6.99 => ( ( ( Mi = Ma )
% 12.45/6.99 => ! [X3: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/6.99 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 12.45/6.99 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 12.45/6.99 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 12.45/6.99 => ( ( ( Mi != Ma )
% 12.45/6.99 => ! [I3: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.99 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 12.45/6.99 = I3 )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 12.45/6.99 & ! [X3: nat] :
% 12.45/6.99 ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 12.45/6.99 = I3 )
% 12.45/6.99 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 12.45/6.99 => ( ( ord_less_nat @ Mi @ X3 )
% 12.45/6.99 & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8446_mintlistlength,axiom,
% 12.45/6.99 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 12.45/6.99 => ( ( Mi != Ma )
% 12.45/6.99 => ( ( ord_less_nat @ Mi @ Ma )
% 12.45/6.99 & ? [M4: nat] :
% 12.45/6.99 ( ( ( some_nat @ M4 )
% 12.45/6.99 = ( vEBT_vebt_mint @ Summary ) )
% 12.45/6.99 & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8447__092_060open_062min__in__set_A_Iset__vebt_H_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_Aminy_092_060close_062,axiom,
% 12.45/6.99 vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ miny ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8448_scmem,axiom,
% 12.45/6.99 vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ miny ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8449__092_060open_062invar__vebt_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_An_092_060close_062,axiom,
% 12.45/6.99 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ na ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8450_mint__member,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ( vEBT_vebt_mint @ T )
% 12.45/6.99 = ( some_nat @ Maxi ) )
% 12.45/6.99 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8451_mint__corr__help,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,Mini: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ( vEBT_vebt_mint @ T )
% 12.45/6.99 = ( some_nat @ Mini ) )
% 12.45/6.99 => ( ( vEBT_vebt_member @ T @ X4 )
% 12.45/6.99 => ( ord_less_eq_nat @ Mini @ X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8452_mint__corr__help__empty,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ( vEBT_vebt_mint @ T )
% 12.45/6.99 = none_nat )
% 12.45/6.99 => ( ( vEBT_VEBT_set_vebt @ T )
% 12.45/6.99 = bot_bot_set_nat ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8453_mint__sound,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 )
% 12.45/6.99 => ( ( vEBT_vebt_mint @ T )
% 12.45/6.99 = ( some_nat @ X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8454_mint__corr,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ( vEBT_vebt_mint @ T )
% 12.45/6.99 = ( some_nat @ X4 ) )
% 12.45/6.99 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8455__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062miny_O_ASome_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Asc_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 12.45/6.99 ~ ! [Miny: nat] :
% 12.45/6.99 ( ( some_nat @ Miny )
% 12.45/6.99 != ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ sc ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8456_misiz,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,M: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ( some_nat @ M )
% 12.45/6.99 = ( vEBT_vebt_mint @ T ) )
% 12.45/6.99 => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8457_bit__0__eq,axiom,
% 12.45/6.99 ( ( bit_se1146084159140164899it_int @ zero_zero_int )
% 12.45/6.99 = bot_bot_nat_o ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8458_bit__0__eq,axiom,
% 12.45/6.99 ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
% 12.45/6.99 = bot_bot_nat_o ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8459_succ__list__to__short,axiom,
% 12.45/6.99 ! [Deg: nat,Mi: nat,X4: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 12.45/6.99 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 12.45/6.99 => ( ( ord_less_eq_nat @ Mi @ X4 )
% 12.45/6.99 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 12.45/6.99 = none_nat ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8460_nested__mint,axiom,
% 12.45/6.99 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 12.45/6.99 => ( ( N
% 12.45/6.99 = ( suc @ ( suc @ Va2 ) ) )
% 12.45/6.99 => ( ~ ( ord_less_nat @ Ma @ Mi )
% 12.45/6.99 => ( ( Ma != Mi )
% 12.45/6.99 => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8461__092_060open_062Some_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_092_060close_062,axiom,
% 12.45/6.99 ( ( some_nat @ miny )
% 12.45/6.99 = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8462_local_Opower__def,axiom,
% 12.45/6.99 ( vEBT_VEBT_power
% 12.45/6.99 = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8463_bot__enat__def,axiom,
% 12.45/6.99 bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8464_bot__nat__def,axiom,
% 12.45/6.99 bot_bot_nat = zero_zero_nat ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8465_aset_I2_J,axiom,
% 12.45/6.99 ! [D4: int,A3: set_int,P: int > $o,Q: int > $o] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ! [Xa2: int] :
% 12.45/6.99 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb2: int] :
% 12.45/6.99 ( ( member_int @ Xb2 @ A3 )
% 12.45/6.99 => ( X3
% 12.45/6.99 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 12.45/6.99 => ( ( P @ X3 )
% 12.45/6.99 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 12.45/6.99 => ( ! [X3: int] :
% 12.45/6.99 ( ! [Xa2: int] :
% 12.45/6.99 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb2: int] :
% 12.45/6.99 ( ( member_int @ Xb2 @ A3 )
% 12.45/6.99 => ( X3
% 12.45/6.99 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 12.45/6.99 => ( ( Q @ X3 )
% 12.45/6.99 => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ A3 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ( P @ X5 )
% 12.45/6.99 | ( Q @ X5 ) )
% 12.45/6.99 => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
% 12.45/6.99 | ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8466_aset_I1_J,axiom,
% 12.45/6.99 ! [D4: int,A3: set_int,P: int > $o,Q: int > $o] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ! [Xa2: int] :
% 12.45/6.99 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb2: int] :
% 12.45/6.99 ( ( member_int @ Xb2 @ A3 )
% 12.45/6.99 => ( X3
% 12.45/6.99 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 12.45/6.99 => ( ( P @ X3 )
% 12.45/6.99 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 12.45/6.99 => ( ! [X3: int] :
% 12.45/6.99 ( ! [Xa2: int] :
% 12.45/6.99 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb2: int] :
% 12.45/6.99 ( ( member_int @ Xb2 @ A3 )
% 12.45/6.99 => ( X3
% 12.45/6.99 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 12.45/6.99 => ( ( Q @ X3 )
% 12.45/6.99 => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ A3 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ( P @ X5 )
% 12.45/6.99 & ( Q @ X5 ) )
% 12.45/6.99 => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
% 12.45/6.99 & ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8467_bset_I2_J,axiom,
% 12.45/6.99 ! [D4: int,B4: set_int,P: int > $o,Q: int > $o] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ! [Xa2: int] :
% 12.45/6.99 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb2: int] :
% 12.45/6.99 ( ( member_int @ Xb2 @ B4 )
% 12.45/6.99 => ( X3
% 12.45/6.99 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 12.45/6.99 => ( ( P @ X3 )
% 12.45/6.99 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 12.45/6.99 => ( ! [X3: int] :
% 12.45/6.99 ( ! [Xa2: int] :
% 12.45/6.99 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb2: int] :
% 12.45/6.99 ( ( member_int @ Xb2 @ B4 )
% 12.45/6.99 => ( X3
% 12.45/6.99 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 12.45/6.99 => ( ( Q @ X3 )
% 12.45/6.99 => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ B4 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ( P @ X5 )
% 12.45/6.99 | ( Q @ X5 ) )
% 12.45/6.99 => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
% 12.45/6.99 | ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8468_bset_I1_J,axiom,
% 12.45/6.99 ! [D4: int,B4: set_int,P: int > $o,Q: int > $o] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ! [Xa2: int] :
% 12.45/6.99 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb2: int] :
% 12.45/6.99 ( ( member_int @ Xb2 @ B4 )
% 12.45/6.99 => ( X3
% 12.45/6.99 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 12.45/6.99 => ( ( P @ X3 )
% 12.45/6.99 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 12.45/6.99 => ( ! [X3: int] :
% 12.45/6.99 ( ! [Xa2: int] :
% 12.45/6.99 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb2: int] :
% 12.45/6.99 ( ( member_int @ Xb2 @ B4 )
% 12.45/6.99 => ( X3
% 12.45/6.99 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 12.45/6.99 => ( ( Q @ X3 )
% 12.45/6.99 => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ B4 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ( P @ X5 )
% 12.45/6.99 & ( Q @ X5 ) )
% 12.45/6.99 => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
% 12.45/6.99 & ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8469_bset_I9_J,axiom,
% 12.45/6.99 ! [D: int,D4: int,B4: set_int,T: int] :
% 12.45/6.99 ( ( dvd_dvd_int @ D @ D4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ B4 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 12.45/6.99 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8470_bset_I10_J,axiom,
% 12.45/6.99 ! [D: int,D4: int,B4: set_int,T: int] :
% 12.45/6.99 ( ( dvd_dvd_int @ D @ D4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ B4 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 12.45/6.99 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8471_aset_I9_J,axiom,
% 12.45/6.99 ! [D: int,D4: int,A3: set_int,T: int] :
% 12.45/6.99 ( ( dvd_dvd_int @ D @ D4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ A3 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 12.45/6.99 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8472_aset_I10_J,axiom,
% 12.45/6.99 ! [D: int,D4: int,A3: set_int,T: int] :
% 12.45/6.99 ( ( dvd_dvd_int @ D @ D4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ A3 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 12.45/6.99 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8473_vebt__succ_Osimps_I4_J,axiom,
% 12.45/6.99 ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
% 12.45/6.99 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
% 12.45/6.99 = none_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8474_periodic__finite__ex,axiom,
% 12.45/6.99 ! [D: int,P: int > $o] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D )
% 12.45/6.99 => ( ! [X3: int,K2: int] :
% 12.45/6.99 ( ( P @ X3 )
% 12.45/6.99 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
% 12.45/6.99 => ( ( ? [X6: int] : ( P @ X6 ) )
% 12.45/6.99 = ( ? [X: int] :
% 12.45/6.99 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 12.45/6.99 & ( P @ X ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8475_aset_I7_J,axiom,
% 12.45/6.99 ! [D4: int,A3: set_int,T: int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ A3 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ord_less_int @ T @ X5 )
% 12.45/6.99 => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8476_aset_I5_J,axiom,
% 12.45/6.99 ! [D4: int,T: int,A3: set_int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ( ( member_int @ T @ A3 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ A3 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ord_less_int @ X5 @ T )
% 12.45/6.99 => ( ord_less_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8477_aset_I4_J,axiom,
% 12.45/6.99 ! [D4: int,T: int,A3: set_int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ( ( member_int @ T @ A3 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ A3 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( X5 != T )
% 12.45/6.99 => ( ( plus_plus_int @ X5 @ D4 )
% 12.45/6.99 != T ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8478_aset_I3_J,axiom,
% 12.45/6.99 ! [D4: int,T: int,A3: set_int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ A3 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( X5 = T )
% 12.45/6.99 => ( ( plus_plus_int @ X5 @ D4 )
% 12.45/6.99 = T ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8479_bset_I7_J,axiom,
% 12.45/6.99 ! [D4: int,T: int,B4: set_int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ( ( member_int @ T @ B4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ B4 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ord_less_int @ T @ X5 )
% 12.45/6.99 => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8480_bset_I5_J,axiom,
% 12.45/6.99 ! [D4: int,B4: set_int,T: int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ B4 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ord_less_int @ X5 @ T )
% 12.45/6.99 => ( ord_less_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8481_bset_I4_J,axiom,
% 12.45/6.99 ! [D4: int,T: int,B4: set_int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ( ( member_int @ T @ B4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ B4 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( X5 != T )
% 12.45/6.99 => ( ( minus_minus_int @ X5 @ D4 )
% 12.45/6.99 != T ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8482_bset_I3_J,axiom,
% 12.45/6.99 ! [D4: int,T: int,B4: set_int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ B4 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( X5 = T )
% 12.45/6.99 => ( ( minus_minus_int @ X5 @ D4 )
% 12.45/6.99 = T ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8483_vebt__succ_Osimps_I5_J,axiom,
% 12.45/6.99 ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 12.45/6.99 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
% 12.45/6.99 = none_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8484_aset_I8_J,axiom,
% 12.45/6.99 ! [D4: int,A3: set_int,T: int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ A3 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ord_less_eq_int @ T @ X5 )
% 12.45/6.99 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8485_aset_I6_J,axiom,
% 12.45/6.99 ! [D4: int,T: int,A3: set_int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ A3 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ord_less_eq_int @ X5 @ T )
% 12.45/6.99 => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8486_bset_I8_J,axiom,
% 12.45/6.99 ! [D4: int,T: int,B4: set_int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ B4 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ord_less_eq_int @ T @ X5 )
% 12.45/6.99 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8487_bset_I6_J,axiom,
% 12.45/6.99 ! [D4: int,B4: set_int,T: int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ! [X5: int] :
% 12.45/6.99 ( ! [Xa3: int] :
% 12.45/6.99 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb3: int] :
% 12.45/6.99 ( ( member_int @ Xb3 @ B4 )
% 12.45/6.99 => ( X5
% 12.45/6.99 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 12.45/6.99 => ( ( ord_less_eq_int @ X5 @ T )
% 12.45/6.99 => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8488_cpmi,axiom,
% 12.45/6.99 ! [D4: int,P: int > $o,P4: int > $o,B4: set_int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ( ? [Z3: int] :
% 12.45/6.99 ! [X3: int] :
% 12.45/6.99 ( ( ord_less_int @ X3 @ Z3 )
% 12.45/6.99 => ( ( P @ X3 )
% 12.45/6.99 = ( P4 @ X3 ) ) )
% 12.45/6.99 => ( ! [X3: int] :
% 12.45/6.99 ( ! [Xa2: int] :
% 12.45/6.99 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb2: int] :
% 12.45/6.99 ( ( member_int @ Xb2 @ B4 )
% 12.45/6.99 => ( X3
% 12.45/6.99 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 12.45/6.99 => ( ( P @ X3 )
% 12.45/6.99 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 12.45/6.99 => ( ! [X3: int,K2: int] :
% 12.45/6.99 ( ( P4 @ X3 )
% 12.45/6.99 = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 12.45/6.99 => ( ( ? [X6: int] : ( P @ X6 ) )
% 12.45/6.99 = ( ? [X: int] :
% 12.45/6.99 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 & ( P4 @ X ) )
% 12.45/6.99 | ? [X: int] :
% 12.45/6.99 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 & ? [Y: int] :
% 12.45/6.99 ( ( member_int @ Y @ B4 )
% 12.45/6.99 & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8489_cppi,axiom,
% 12.45/6.99 ! [D4: int,P: int > $o,P4: int > $o,A3: set_int] :
% 12.45/6.99 ( ( ord_less_int @ zero_zero_int @ D4 )
% 12.45/6.99 => ( ? [Z3: int] :
% 12.45/6.99 ! [X3: int] :
% 12.45/6.99 ( ( ord_less_int @ Z3 @ X3 )
% 12.45/6.99 => ( ( P @ X3 )
% 12.45/6.99 = ( P4 @ X3 ) ) )
% 12.45/6.99 => ( ! [X3: int] :
% 12.45/6.99 ( ! [Xa2: int] :
% 12.45/6.99 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 => ! [Xb2: int] :
% 12.45/6.99 ( ( member_int @ Xb2 @ A3 )
% 12.45/6.99 => ( X3
% 12.45/6.99 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 12.45/6.99 => ( ( P @ X3 )
% 12.45/6.99 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 12.45/6.99 => ( ! [X3: int,K2: int] :
% 12.45/6.99 ( ( P4 @ X3 )
% 12.45/6.99 = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 12.45/6.99 => ( ( ? [X6: int] : ( P @ X6 ) )
% 12.45/6.99 = ( ? [X: int] :
% 12.45/6.99 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 & ( P4 @ X ) )
% 12.45/6.99 | ? [X: int] :
% 12.45/6.99 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 12.45/6.99 & ? [Y: int] :
% 12.45/6.99 ( ( member_int @ Y @ A3 )
% 12.45/6.99 & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8490__092_060open_062res_A_061_Athe_A_ISome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_092_060close_062,axiom,
% 12.45/6.99 ( res
% 12.45/6.99 = ( the_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8491_summaxma,axiom,
% 12.45/6.99 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 12.45/6.99 => ( ( Mi != Ma )
% 12.45/6.99 => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 12.45/6.99 = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8492_maxt__sound,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 )
% 12.45/6.99 => ( ( vEBT_vebt_maxt @ T )
% 12.45/6.99 = ( some_nat @ X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8493_maxt__corr,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ( vEBT_vebt_maxt @ T )
% 12.45/6.99 = ( some_nat @ X4 ) )
% 12.45/6.99 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8494_add__shift,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat,Z: nat] :
% 12.45/6.99 ( ( ( plus_plus_nat @ X4 @ Y3 )
% 12.45/6.99 = Z )
% 12.45/6.99 = ( ( vEBT_VEBT_add @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) )
% 12.45/6.99 = ( some_nat @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8495_mul__shift,axiom,
% 12.45/6.99 ! [X4: nat,Y3: nat,Z: nat] :
% 12.45/6.99 ( ( ( times_times_nat @ X4 @ Y3 )
% 12.45/6.99 = Z )
% 12.45/6.99 = ( ( vEBT_VEBT_mul @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) )
% 12.45/6.99 = ( some_nat @ Z ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8496_maxbmo,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,X4: nat] :
% 12.45/6.99 ( ( ( vEBT_vebt_maxt @ T )
% 12.45/6.99 = ( some_nat @ X4 ) )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ T @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8497_add__def,axiom,
% 12.45/6.99 ( vEBT_VEBT_add
% 12.45/6.99 = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8498_mul__def,axiom,
% 12.45/6.99 ( vEBT_VEBT_mul
% 12.45/6.99 = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8499_maxt__member,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ( vEBT_vebt_maxt @ T )
% 12.45/6.99 = ( some_nat @ Maxi ) )
% 12.45/6.99 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8500_maxt__corr__help,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat,Maxi: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ( vEBT_vebt_maxt @ T )
% 12.45/6.99 = ( some_nat @ Maxi ) )
% 12.45/6.99 => ( ( vEBT_vebt_member @ T @ X4 )
% 12.45/6.99 => ( ord_less_eq_nat @ X4 @ Maxi ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8501_maxt__corr__help__empty,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( ( vEBT_vebt_maxt @ T )
% 12.45/6.99 = none_nat )
% 12.45/6.99 => ( ( vEBT_VEBT_set_vebt @ T )
% 12.45/6.99 = bot_bot_set_nat ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8502__092_060open_062vebt__succ_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ASome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_092_060close_062,axiom,
% 12.45/6.99 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 12.45/6.99 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8503__C2_C,axiom,
% 12.45/6.99 ( ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 = none_nat )
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 12.45/6.99 = none_nat ) )
% 12.45/6.99 & ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 != none_nat )
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 12.45/6.99 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8504__092_060open_062vebt__member_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_A_Ithe_A_ISome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_J_092_060close_062,axiom,
% 12.45/6.99 vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ( the_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8505__C1_C,axiom,
% 12.45/6.99 ( ( ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 != none_nat )
% 12.45/6.99 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 12.45/6.99 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 12.45/6.99 & ( ~ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 != none_nat )
% 12.45/6.99 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 12.45/6.99 => ( ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 = none_nat )
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 12.45/6.99 = none_nat ) )
% 12.45/6.99 & ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 != none_nat )
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 12.45/6.99 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8506_i1,axiom,
% 12.45/6.99 ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 = none_nat )
% 12.45/6.99 | ~ ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8507_divmod__step__eq,axiom,
% 12.45/6.99 ! [L2: num,R2: nat,Q2: nat] :
% 12.45/6.99 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
% 12.45/6.99 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
% 12.45/6.99 = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L2 ) ) ) ) )
% 12.45/6.99 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
% 12.45/6.99 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
% 12.45/6.99 = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8508_divmod__step__eq,axiom,
% 12.45/6.99 ! [L2: num,R2: int,Q2: int] :
% 12.45/6.99 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
% 12.45/6.99 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/6.99 = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L2 ) ) ) ) )
% 12.45/6.99 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
% 12.45/6.99 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/6.99 = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8509_divmod__step__eq,axiom,
% 12.45/6.99 ! [L2: num,R2: code_integer,Q2: code_integer] :
% 12.45/6.99 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
% 12.45/6.99 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
% 12.45/6.99 = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
% 12.45/6.99 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
% 12.45/6.99 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
% 12.45/6.99 = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8510_less__shift,axiom,
% 12.45/6.99 ( ord_less_nat
% 12.45/6.99 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8511_fold__atLeastAtMost__nat_Ocases,axiom,
% 12.45/6.99 ! [X4: produc3368934014287244435at_num] :
% 12.45/6.99 ~ ! [F3: nat > num > num,A5: nat,B5: nat,Acc2: num] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc851828971589881931at_num @ F3 @ ( produc1195630363706982562at_num @ A5 @ ( product_Pair_nat_num @ B5 @ Acc2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8512_fold__atLeastAtMost__nat_Ocases,axiom,
% 12.45/6.99 ! [X4: produc4471711990508489141at_nat] :
% 12.45/6.99 ~ ! [F3: nat > nat > nat,A5: nat,B5: nat,Acc2: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ Acc2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8513_xor__num_Ocases,axiom,
% 12.45/6.99 ! [X4: product_prod_num_num] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 != ( product_Pair_num_num @ one @ one ) )
% 12.45/6.99 => ( ! [N4: num] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) )
% 12.45/6.99 => ( ! [N4: num] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) )
% 12.45/6.99 => ( ! [M4: num] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
% 12.45/6.99 => ( ! [M4: num,N4: num] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) )
% 12.45/6.99 => ( ! [M4: num,N4: num] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) )
% 12.45/6.99 => ( ! [M4: num] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
% 12.45/6.99 => ( ! [M4: num,N4: num] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) )
% 12.45/6.99 => ~ ! [M4: num,N4: num] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8514_minNullmin,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT] :
% 12.45/6.99 ( ( vEBT_VEBT_minNull @ T )
% 12.45/6.99 => ( ( vEBT_vebt_mint @ T )
% 12.45/6.99 = none_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8515_minminNull,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT] :
% 12.45/6.99 ( ( ( vEBT_vebt_mint @ T )
% 12.45/6.99 = none_nat )
% 12.45/6.99 => ( vEBT_VEBT_minNull @ T ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8516_vebt__member_Osimps_I4_J,axiom,
% 12.45/6.99 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
% 12.45/6.99 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X4 ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8517_succ__less__length__list,axiom,
% 12.45/6.99 ! [Deg: nat,Mi: nat,X4: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 12.45/6.99 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 12.45/6.99 => ( ( ord_less_eq_nat @ Mi @ X4 )
% 12.45/6.99 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 12.45/6.99 = ( if_option_nat
% 12.45/6.99 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 != none_nat )
% 12.45/6.99 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 12.45/6.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 @ ( if_option_nat
% 12.45/6.99 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 = none_nat )
% 12.45/6.99 @ none_nat
% 12.45/6.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8518_not__min__Null__member,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT] :
% 12.45/6.99 ( ~ ( vEBT_VEBT_minNull @ T )
% 12.45/6.99 => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8519_min__Null__member,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,X4: nat] :
% 12.45/6.99 ( ( vEBT_VEBT_minNull @ T )
% 12.45/6.99 => ~ ( vEBT_vebt_member @ T @ X4 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8520_set__vebt_H__def,axiom,
% 12.45/6.99 ( vEBT_VEBT_set_vebt
% 12.45/6.99 = ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8521_succ__greatereq__min,axiom,
% 12.45/6.99 ! [Deg: nat,Mi: nat,X4: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 12.45/6.99 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 12.45/6.99 => ( ( ord_less_eq_nat @ Mi @ X4 )
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 12.45/6.99 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/6.99 @ ( if_option_nat
% 12.45/6.99 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 != none_nat )
% 12.45/6.99 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 12.45/6.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 @ ( if_option_nat
% 12.45/6.99 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 = none_nat )
% 12.45/6.99 @ none_nat
% 12.45/6.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 12.45/6.99 @ none_nat ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8522_nat__less__as__int,axiom,
% 12.45/6.99 ( ord_less_nat
% 12.45/6.99 = ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8523_less__set__def,axiom,
% 12.45/6.99 ( ord_le2529575680413868914d_enat
% 12.45/6.99 = ( ^ [A6: set_Extended_enat,B7: set_Extended_enat] :
% 12.45/6.99 ( ord_le8499522857272258027enat_o
% 12.45/6.99 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 )
% 12.45/6.99 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ B7 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8524_less__set__def,axiom,
% 12.45/6.99 ( ord_less_set_complex
% 12.45/6.99 = ( ^ [A6: set_complex,B7: set_complex] :
% 12.45/6.99 ( ord_less_complex_o
% 12.45/6.99 @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 12.45/6.99 @ ^ [X: complex] : ( member_complex @ X @ B7 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8525_less__set__def,axiom,
% 12.45/6.99 ( ord_less_set_real
% 12.45/6.99 = ( ^ [A6: set_real,B7: set_real] :
% 12.45/6.99 ( ord_less_real_o
% 12.45/6.99 @ ^ [X: real] : ( member_real @ X @ A6 )
% 12.45/6.99 @ ^ [X: real] : ( member_real @ X @ B7 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8526_less__set__def,axiom,
% 12.45/6.99 ( ord_less_set_nat
% 12.45/6.99 = ( ^ [A6: set_nat,B7: set_nat] :
% 12.45/6.99 ( ord_less_nat_o
% 12.45/6.99 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 12.45/6.99 @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8527_less__set__def,axiom,
% 12.45/6.99 ( ord_less_set_int
% 12.45/6.99 = ( ^ [A6: set_int,B7: set_int] :
% 12.45/6.99 ( ord_less_int_o
% 12.45/6.99 @ ^ [X: int] : ( member_int @ X @ A6 )
% 12.45/6.99 @ ^ [X: int] : ( member_int @ X @ B7 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8528_strict__subset__divisors__dvd,axiom,
% 12.45/6.99 ! [A: complex,B2: complex] :
% 12.45/6.99 ( ( ord_less_set_complex
% 12.45/6.99 @ ( collect_complex
% 12.45/6.99 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
% 12.45/6.99 @ ( collect_complex
% 12.45/6.99 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B2 ) ) )
% 12.45/6.99 = ( ( dvd_dvd_complex @ A @ B2 )
% 12.45/6.99 & ~ ( dvd_dvd_complex @ B2 @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8529_strict__subset__divisors__dvd,axiom,
% 12.45/6.99 ! [A: real,B2: real] :
% 12.45/6.99 ( ( ord_less_set_real
% 12.45/6.99 @ ( collect_real
% 12.45/6.99 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
% 12.45/6.99 @ ( collect_real
% 12.45/6.99 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B2 ) ) )
% 12.45/6.99 = ( ( dvd_dvd_real @ A @ B2 )
% 12.45/6.99 & ~ ( dvd_dvd_real @ B2 @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8530_strict__subset__divisors__dvd,axiom,
% 12.45/6.99 ! [A: nat,B2: nat] :
% 12.45/6.99 ( ( ord_less_set_nat
% 12.45/6.99 @ ( collect_nat
% 12.45/6.99 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
% 12.45/6.99 @ ( collect_nat
% 12.45/6.99 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B2 ) ) )
% 12.45/6.99 = ( ( dvd_dvd_nat @ A @ B2 )
% 12.45/6.99 & ~ ( dvd_dvd_nat @ B2 @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8531_strict__subset__divisors__dvd,axiom,
% 12.45/6.99 ! [A: int,B2: int] :
% 12.45/6.99 ( ( ord_less_set_int
% 12.45/6.99 @ ( collect_int
% 12.45/6.99 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
% 12.45/6.99 @ ( collect_int
% 12.45/6.99 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B2 ) ) )
% 12.45/6.99 = ( ( dvd_dvd_int @ A @ B2 )
% 12.45/6.99 & ~ ( dvd_dvd_int @ B2 @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8532_strict__subset__divisors__dvd,axiom,
% 12.45/6.99 ! [A: code_integer,B2: code_integer] :
% 12.45/6.99 ( ( ord_le1307284697595431911nteger
% 12.45/6.99 @ ( collect_Code_integer
% 12.45/6.99 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
% 12.45/6.99 @ ( collect_Code_integer
% 12.45/6.99 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B2 ) ) )
% 12.45/6.99 = ( ( dvd_dvd_Code_integer @ A @ B2 )
% 12.45/6.99 & ~ ( dvd_dvd_Code_integer @ B2 @ A ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8533_set__diff__eq,axiom,
% 12.45/6.99 ( minus_925952699566721837d_enat
% 12.45/6.99 = ( ^ [A6: set_Extended_enat,B7: set_Extended_enat] :
% 12.45/6.99 ( collec4429806609662206161d_enat
% 12.45/6.99 @ ^ [X: extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ X @ A6 )
% 12.45/6.99 & ~ ( member_Extended_enat @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8534_set__diff__eq,axiom,
% 12.45/6.99 ( minus_811609699411566653omplex
% 12.45/6.99 = ( ^ [A6: set_complex,B7: set_complex] :
% 12.45/6.99 ( collect_complex
% 12.45/6.99 @ ^ [X: complex] :
% 12.45/6.99 ( ( member_complex @ X @ A6 )
% 12.45/6.99 & ~ ( member_complex @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8535_set__diff__eq,axiom,
% 12.45/6.99 ( minus_minus_set_real
% 12.45/6.99 = ( ^ [A6: set_real,B7: set_real] :
% 12.45/6.99 ( collect_real
% 12.45/6.99 @ ^ [X: real] :
% 12.45/6.99 ( ( member_real @ X @ A6 )
% 12.45/6.99 & ~ ( member_real @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8536_set__diff__eq,axiom,
% 12.45/6.99 ( minus_7954133019191499631st_nat
% 12.45/6.99 = ( ^ [A6: set_list_nat,B7: set_list_nat] :
% 12.45/6.99 ( collect_list_nat
% 12.45/6.99 @ ^ [X: list_nat] :
% 12.45/6.99 ( ( member_list_nat @ X @ A6 )
% 12.45/6.99 & ~ ( member_list_nat @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8537_set__diff__eq,axiom,
% 12.45/6.99 ( minus_minus_set_int
% 12.45/6.99 = ( ^ [A6: set_int,B7: set_int] :
% 12.45/6.99 ( collect_int
% 12.45/6.99 @ ^ [X: int] :
% 12.45/6.99 ( ( member_int @ X @ A6 )
% 12.45/6.99 & ~ ( member_int @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8538_set__diff__eq,axiom,
% 12.45/6.99 ( minus_minus_set_nat
% 12.45/6.99 = ( ^ [A6: set_nat,B7: set_nat] :
% 12.45/6.99 ( collect_nat
% 12.45/6.99 @ ^ [X: nat] :
% 12.45/6.99 ( ( member_nat @ X @ A6 )
% 12.45/6.99 & ~ ( member_nat @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8539_minus__set__def,axiom,
% 12.45/6.99 ( minus_925952699566721837d_enat
% 12.45/6.99 = ( ^ [A6: set_Extended_enat,B7: set_Extended_enat] :
% 12.45/6.99 ( collec4429806609662206161d_enat
% 12.45/6.99 @ ( minus_2020553357622893040enat_o
% 12.45/6.99 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 )
% 12.45/6.99 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8540_minus__set__def,axiom,
% 12.45/6.99 ( minus_811609699411566653omplex
% 12.45/6.99 = ( ^ [A6: set_complex,B7: set_complex] :
% 12.45/6.99 ( collect_complex
% 12.45/6.99 @ ( minus_8727706125548526216plex_o
% 12.45/6.99 @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 12.45/6.99 @ ^ [X: complex] : ( member_complex @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8541_minus__set__def,axiom,
% 12.45/6.99 ( minus_minus_set_real
% 12.45/6.99 = ( ^ [A6: set_real,B7: set_real] :
% 12.45/6.99 ( collect_real
% 12.45/6.99 @ ( minus_minus_real_o
% 12.45/6.99 @ ^ [X: real] : ( member_real @ X @ A6 )
% 12.45/6.99 @ ^ [X: real] : ( member_real @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8542_minus__set__def,axiom,
% 12.45/6.99 ( minus_7954133019191499631st_nat
% 12.45/6.99 = ( ^ [A6: set_list_nat,B7: set_list_nat] :
% 12.45/6.99 ( collect_list_nat
% 12.45/6.99 @ ( minus_1139252259498527702_nat_o
% 12.45/6.99 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
% 12.45/6.99 @ ^ [X: list_nat] : ( member_list_nat @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8543_minus__set__def,axiom,
% 12.45/6.99 ( minus_minus_set_int
% 12.45/6.99 = ( ^ [A6: set_int,B7: set_int] :
% 12.45/6.99 ( collect_int
% 12.45/6.99 @ ( minus_minus_int_o
% 12.45/6.99 @ ^ [X: int] : ( member_int @ X @ A6 )
% 12.45/6.99 @ ^ [X: int] : ( member_int @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8544_minus__set__def,axiom,
% 12.45/6.99 ( minus_minus_set_nat
% 12.45/6.99 = ( ^ [A6: set_nat,B7: set_nat] :
% 12.45/6.99 ( collect_nat
% 12.45/6.99 @ ( minus_minus_nat_o
% 12.45/6.99 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 12.45/6.99 @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8545_numeral__code_I2_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 12.45/6.99 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8546_numeral__code_I2_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 12.45/6.99 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8547_numeral__code_I2_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 12.45/6.99 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8548_numeral__code_I2_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 12.45/6.99 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8549_numeral__code_I2_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 12.45/6.99 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8550_lambda__zero,axiom,
% 12.45/6.99 ( ( ^ [H: real] : zero_zero_real )
% 12.45/6.99 = ( times_times_real @ zero_zero_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8551_lambda__zero,axiom,
% 12.45/6.99 ( ( ^ [H: rat] : zero_zero_rat )
% 12.45/6.99 = ( times_times_rat @ zero_zero_rat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8552_lambda__zero,axiom,
% 12.45/6.99 ( ( ^ [H: nat] : zero_zero_nat )
% 12.45/6.99 = ( times_times_nat @ zero_zero_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8553_lambda__zero,axiom,
% 12.45/6.99 ( ( ^ [H: int] : zero_zero_int )
% 12.45/6.99 = ( times_times_int @ zero_zero_int ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8554_mult__commute__abs,axiom,
% 12.45/6.99 ! [C: real] :
% 12.45/6.99 ( ( ^ [X: real] : ( times_times_real @ X @ C ) )
% 12.45/6.99 = ( times_times_real @ C ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8555_mult__commute__abs,axiom,
% 12.45/6.99 ! [C: rat] :
% 12.45/6.99 ( ( ^ [X: rat] : ( times_times_rat @ X @ C ) )
% 12.45/6.99 = ( times_times_rat @ C ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8556_mult__commute__abs,axiom,
% 12.45/6.99 ! [C: nat] :
% 12.45/6.99 ( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
% 12.45/6.99 = ( times_times_nat @ C ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8557_mult__commute__abs,axiom,
% 12.45/6.99 ! [C: int] :
% 12.45/6.99 ( ( ^ [X: int] : ( times_times_int @ X @ C ) )
% 12.45/6.99 = ( times_times_int @ C ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8558_lambda__one,axiom,
% 12.45/6.99 ( ( ^ [X: complex] : X )
% 12.45/6.99 = ( times_times_complex @ one_one_complex ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8559_lambda__one,axiom,
% 12.45/6.99 ( ( ^ [X: real] : X )
% 12.45/6.99 = ( times_times_real @ one_one_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8560_lambda__one,axiom,
% 12.45/6.99 ( ( ^ [X: rat] : X )
% 12.45/6.99 = ( times_times_rat @ one_one_rat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8561_lambda__one,axiom,
% 12.45/6.99 ( ( ^ [X: nat] : X )
% 12.45/6.99 = ( times_times_nat @ one_one_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8562_lambda__one,axiom,
% 12.45/6.99 ( ( ^ [X: int] : X )
% 12.45/6.99 = ( times_times_int @ one_one_int ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8563_subset__divisors__dvd,axiom,
% 12.45/6.99 ! [A: complex,B2: complex] :
% 12.45/6.99 ( ( ord_le211207098394363844omplex
% 12.45/6.99 @ ( collect_complex
% 12.45/6.99 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
% 12.45/6.99 @ ( collect_complex
% 12.45/6.99 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B2 ) ) )
% 12.45/6.99 = ( dvd_dvd_complex @ A @ B2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8564_subset__divisors__dvd,axiom,
% 12.45/6.99 ! [A: real,B2: real] :
% 12.45/6.99 ( ( ord_less_eq_set_real
% 12.45/6.99 @ ( collect_real
% 12.45/6.99 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
% 12.45/6.99 @ ( collect_real
% 12.45/6.99 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B2 ) ) )
% 12.45/6.99 = ( dvd_dvd_real @ A @ B2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8565_subset__divisors__dvd,axiom,
% 12.45/6.99 ! [A: int,B2: int] :
% 12.45/6.99 ( ( ord_less_eq_set_int
% 12.45/6.99 @ ( collect_int
% 12.45/6.99 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
% 12.45/6.99 @ ( collect_int
% 12.45/6.99 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B2 ) ) )
% 12.45/6.99 = ( dvd_dvd_int @ A @ B2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8566_subset__divisors__dvd,axiom,
% 12.45/6.99 ! [A: code_integer,B2: code_integer] :
% 12.45/6.99 ( ( ord_le7084787975880047091nteger
% 12.45/6.99 @ ( collect_Code_integer
% 12.45/6.99 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
% 12.45/6.99 @ ( collect_Code_integer
% 12.45/6.99 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B2 ) ) )
% 12.45/6.99 = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8567_subset__divisors__dvd,axiom,
% 12.45/6.99 ! [A: nat,B2: nat] :
% 12.45/6.99 ( ( ord_less_eq_set_nat
% 12.45/6.99 @ ( collect_nat
% 12.45/6.99 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
% 12.45/6.99 @ ( collect_nat
% 12.45/6.99 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B2 ) ) )
% 12.45/6.99 = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8568_nat__leq__as__int,axiom,
% 12.45/6.99 ( ord_less_eq_nat
% 12.45/6.99 = ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8569_power__numeral__even,axiom,
% 12.45/6.99 ! [Z: complex,W: num] :
% 12.45/6.99 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 12.45/6.99 = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8570_power__numeral__even,axiom,
% 12.45/6.99 ! [Z: real,W: num] :
% 12.45/6.99 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 12.45/6.99 = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8571_power__numeral__even,axiom,
% 12.45/6.99 ! [Z: rat,W: num] :
% 12.45/6.99 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 12.45/6.99 = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8572_power__numeral__even,axiom,
% 12.45/6.99 ! [Z: nat,W: num] :
% 12.45/6.99 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 12.45/6.99 = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8573_power__numeral__even,axiom,
% 12.45/6.99 ! [Z: int,W: num] :
% 12.45/6.99 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 12.45/6.99 = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8574_numeral__code_I3_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 12.45/6.99 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8575_numeral__code_I3_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 12.45/6.99 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8576_numeral__code_I3_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 12.45/6.99 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8577_numeral__code_I3_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 12.45/6.99 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8578_numeral__code_I3_J,axiom,
% 12.45/6.99 ! [N: num] :
% 12.45/6.99 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 12.45/6.99 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8579_power__numeral__odd,axiom,
% 12.45/6.99 ! [Z: complex,W: num] :
% 12.45/6.99 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 12.45/6.99 = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8580_power__numeral__odd,axiom,
% 12.45/6.99 ! [Z: real,W: num] :
% 12.45/6.99 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 12.45/6.99 = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8581_power__numeral__odd,axiom,
% 12.45/6.99 ! [Z: rat,W: num] :
% 12.45/6.99 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 12.45/6.99 = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8582_power__numeral__odd,axiom,
% 12.45/6.99 ! [Z: nat,W: num] :
% 12.45/6.99 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 12.45/6.99 = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8583_power__numeral__odd,axiom,
% 12.45/6.99 ! [Z: int,W: num] :
% 12.45/6.99 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 12.45/6.99 = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8584_nat__plus__as__int,axiom,
% 12.45/6.99 ( plus_plus_nat
% 12.45/6.99 = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8585_nat__times__as__int,axiom,
% 12.45/6.99 ( times_times_nat
% 12.45/6.99 = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8586_nat__minus__as__int,axiom,
% 12.45/6.99 ( minus_minus_nat
% 12.45/6.99 = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8587_nat__div__as__int,axiom,
% 12.45/6.99 ( divide_divide_nat
% 12.45/6.99 = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8588_nat__mod__as__int,axiom,
% 12.45/6.99 ( modulo_modulo_nat
% 12.45/6.99 = ( ^ [A4: nat,B3: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8589_set__conv__nth,axiom,
% 12.45/6.99 ( set_complex2
% 12.45/6.99 = ( ^ [Xs: list_complex] :
% 12.45/6.99 ( collect_complex
% 12.45/6.99 @ ^ [Uu: complex] :
% 12.45/6.99 ? [I2: nat] :
% 12.45/6.99 ( ( Uu
% 12.45/6.99 = ( nth_complex @ Xs @ I2 ) )
% 12.45/6.99 & ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8590_set__conv__nth,axiom,
% 12.45/6.99 ( set_real2
% 12.45/6.99 = ( ^ [Xs: list_real] :
% 12.45/6.99 ( collect_real
% 12.45/6.99 @ ^ [Uu: real] :
% 12.45/6.99 ? [I2: nat] :
% 12.45/6.99 ( ( Uu
% 12.45/6.99 = ( nth_real @ Xs @ I2 ) )
% 12.45/6.99 & ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8591_set__conv__nth,axiom,
% 12.45/6.99 ( set_list_nat2
% 12.45/6.99 = ( ^ [Xs: list_list_nat] :
% 12.45/6.99 ( collect_list_nat
% 12.45/6.99 @ ^ [Uu: list_nat] :
% 12.45/6.99 ? [I2: nat] :
% 12.45/6.99 ( ( Uu
% 12.45/6.99 = ( nth_list_nat @ Xs @ I2 ) )
% 12.45/6.99 & ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8592_set__conv__nth,axiom,
% 12.45/6.99 ( set_VEBT_VEBT2
% 12.45/6.99 = ( ^ [Xs: list_VEBT_VEBT] :
% 12.45/6.99 ( collect_VEBT_VEBT
% 12.45/6.99 @ ^ [Uu: vEBT_VEBT] :
% 12.45/6.99 ? [I2: nat] :
% 12.45/6.99 ( ( Uu
% 12.45/6.99 = ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 12.45/6.99 & ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8593_set__conv__nth,axiom,
% 12.45/6.99 ( set_o2
% 12.45/6.99 = ( ^ [Xs: list_o] :
% 12.45/6.99 ( collect_o
% 12.45/6.99 @ ^ [Uu: $o] :
% 12.45/6.99 ? [I2: nat] :
% 12.45/6.99 ( ( Uu
% 12.45/6.99 = ( nth_o @ Xs @ I2 ) )
% 12.45/6.99 & ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8594_set__conv__nth,axiom,
% 12.45/6.99 ( set_nat2
% 12.45/6.99 = ( ^ [Xs: list_nat] :
% 12.45/6.99 ( collect_nat
% 12.45/6.99 @ ^ [Uu: nat] :
% 12.45/6.99 ? [I2: nat] :
% 12.45/6.99 ( ( Uu
% 12.45/6.99 = ( nth_nat @ Xs @ I2 ) )
% 12.45/6.99 & ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8595_set__conv__nth,axiom,
% 12.45/6.99 ( set_int2
% 12.45/6.99 = ( ^ [Xs: list_int] :
% 12.45/6.99 ( collect_int
% 12.45/6.99 @ ^ [Uu: int] :
% 12.45/6.99 ? [I2: nat] :
% 12.45/6.99 ( ( Uu
% 12.45/6.99 = ( nth_int @ Xs @ I2 ) )
% 12.45/6.99 & ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8596_diff__nat__eq__if,axiom,
% 12.45/6.99 ! [Z6: int,Z: int] :
% 12.45/6.99 ( ( ( ord_less_int @ Z6 @ zero_zero_int )
% 12.45/6.99 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 12.45/6.99 = ( nat2 @ Z ) ) )
% 12.45/6.99 & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
% 12.45/6.99 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 12.45/6.99 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8597_set__decode__def,axiom,
% 12.45/6.99 ( nat_set_decode
% 12.45/6.99 = ( ^ [X: nat] :
% 12.45/6.99 ( collect_nat
% 12.45/6.99 @ ^ [N2: nat] :
% 12.45/6.99 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8598_signed__take__bit__code,axiom,
% 12.45/6.99 ( bit_ri6519982836138164636nteger
% 12.45/6.99 = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( bit_se9216721137139052372nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A4 ) @ N2 ) @ ( plus_p5714425477246183910nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A4 ) @ ( bit_se7788150548672797655nteger @ ( suc @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) @ ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8599_signed__take__bit__code,axiom,
% 12.45/6.99 ( bit_ri631733984087533419it_int
% 12.45/6.99 = ( ^ [N2: nat,A4: int] : ( if_int @ ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A4 ) @ N2 ) @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A4 ) @ ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ one_one_int ) ) ) @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8600_pochhammer__code,axiom,
% 12.45/6.99 ( comm_s2602460028002588243omplex
% 12.45/6.99 = ( ^ [A4: complex,N2: nat] :
% 12.45/6.99 ( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
% 12.45/6.99 @ ( set_fo1517530859248394432omplex
% 12.45/6.99 @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ O ) ) )
% 12.45/6.99 @ zero_zero_nat
% 12.45/6.99 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 12.45/6.99 @ one_one_complex ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8601_pochhammer__code,axiom,
% 12.45/6.99 ( comm_s7457072308508201937r_real
% 12.45/6.99 = ( ^ [A4: real,N2: nat] :
% 12.45/6.99 ( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
% 12.45/6.99 @ ( set_fo3111899725591712190t_real
% 12.45/6.99 @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ O ) ) )
% 12.45/6.99 @ zero_zero_nat
% 12.45/6.99 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 12.45/6.99 @ one_one_real ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8602_pochhammer__code,axiom,
% 12.45/6.99 ( comm_s4028243227959126397er_rat
% 12.45/6.99 = ( ^ [A4: rat,N2: nat] :
% 12.45/6.99 ( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
% 12.45/6.99 @ ( set_fo1949268297981939178at_rat
% 12.45/6.99 @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ O ) ) )
% 12.45/6.99 @ zero_zero_nat
% 12.45/6.99 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 12.45/6.99 @ one_one_rat ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8603_pochhammer__code,axiom,
% 12.45/6.99 ( comm_s4660882817536571857er_int
% 12.45/6.99 = ( ^ [A4: int,N2: nat] :
% 12.45/6.99 ( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
% 12.45/6.99 @ ( set_fo2581907887559384638at_int
% 12.45/6.99 @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ O ) ) )
% 12.45/6.99 @ zero_zero_nat
% 12.45/6.99 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 12.45/6.99 @ one_one_int ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8604_pochhammer__code,axiom,
% 12.45/6.99 ( comm_s4663373288045622133er_nat
% 12.45/6.99 = ( ^ [A4: nat,N2: nat] :
% 12.45/6.99 ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
% 12.45/6.99 @ ( set_fo2584398358068434914at_nat
% 12.45/6.99 @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 12.45/6.99 @ zero_zero_nat
% 12.45/6.99 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 12.45/6.99 @ one_one_nat ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8605_gbinomial__code,axiom,
% 12.45/6.99 ( gbinomial_complex
% 12.45/6.99 = ( ^ [A4: complex,K3: nat] :
% 12.45/6.99 ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
% 12.45/6.99 @ ( divide1717551699836669952omplex
% 12.45/6.99 @ ( set_fo1517530859248394432omplex
% 12.45/6.99 @ ^ [L: nat] : ( times_times_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ L ) ) )
% 12.45/6.99 @ zero_zero_nat
% 12.45/6.99 @ ( minus_minus_nat @ K3 @ one_one_nat )
% 12.45/6.99 @ one_one_complex )
% 12.45/6.99 @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8606_gbinomial__code,axiom,
% 12.45/6.99 ( gbinomial_rat
% 12.45/6.99 = ( ^ [A4: rat,K3: nat] :
% 12.45/6.99 ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
% 12.45/6.99 @ ( divide_divide_rat
% 12.45/6.99 @ ( set_fo1949268297981939178at_rat
% 12.45/6.99 @ ^ [L: nat] : ( times_times_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ L ) ) )
% 12.45/6.99 @ zero_zero_nat
% 12.45/6.99 @ ( minus_minus_nat @ K3 @ one_one_nat )
% 12.45/6.99 @ one_one_rat )
% 12.45/6.99 @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8607_gbinomial__code,axiom,
% 12.45/6.99 ( gbinomial_real
% 12.45/6.99 = ( ^ [A4: real,K3: nat] :
% 12.45/6.99 ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
% 12.45/6.99 @ ( divide_divide_real
% 12.45/6.99 @ ( set_fo3111899725591712190t_real
% 12.45/6.99 @ ^ [L: nat] : ( times_times_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ L ) ) )
% 12.45/6.99 @ zero_zero_nat
% 12.45/6.99 @ ( minus_minus_nat @ K3 @ one_one_nat )
% 12.45/6.99 @ one_one_real )
% 12.45/6.99 @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8608_vebt__member_Osimps_I5_J,axiom,
% 12.45/6.99 ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
% 12.45/6.99 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
% 12.45/6.99 = ( ( X4 != Mi )
% 12.45/6.99 => ( ( X4 != Ma )
% 12.45/6.99 => ( ~ ( ord_less_nat @ X4 @ Mi )
% 12.45/6.99 & ( ~ ( ord_less_nat @ X4 @ Mi )
% 12.45/6.99 => ( ~ ( ord_less_nat @ Ma @ X4 )
% 12.45/6.99 & ( ~ ( ord_less_nat @ Ma @ X4 )
% 12.45/6.99 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/6.99 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8609_vebt__succ_Osimps_I6_J,axiom,
% 12.45/6.99 ! [X4: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 12.45/6.99 ( ( ( ord_less_nat @ X4 @ Mi )
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
% 12.45/6.99 = ( some_nat @ Mi ) ) )
% 12.45/6.99 & ( ~ ( ord_less_nat @ X4 @ Mi )
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
% 12.45/6.99 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/6.99 @ ( if_option_nat
% 12.45/6.99 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 != none_nat )
% 12.45/6.99 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 12.45/6.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 @ ( if_option_nat
% 12.45/6.99 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 = none_nat )
% 12.45/6.99 @ none_nat
% 12.45/6.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 12.45/6.99 @ none_nat ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8610_of__int__code__if,axiom,
% 12.45/6.99 ( ring_1_of_int_real
% 12.45/6.99 = ( ^ [K3: int] :
% 12.45/6.99 ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
% 12.45/6.99 @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
% 12.45/6.99 @ ( if_real
% 12.45/6.99 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.99 = zero_zero_int )
% 12.45/6.99 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8611_of__int__code__if,axiom,
% 12.45/6.99 ( ring_1_of_int_int
% 12.45/6.99 = ( ^ [K3: int] :
% 12.45/6.99 ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
% 12.45/6.99 @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
% 12.45/6.99 @ ( if_int
% 12.45/6.99 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.99 = zero_zero_int )
% 12.45/6.99 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8612_of__int__code__if,axiom,
% 12.45/6.99 ( ring_17405671764205052669omplex
% 12.45/6.99 = ( ^ [K3: int] :
% 12.45/6.99 ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
% 12.45/6.99 @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
% 12.45/6.99 @ ( if_complex
% 12.45/6.99 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.99 = zero_zero_int )
% 12.45/6.99 @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8613_of__int__code__if,axiom,
% 12.45/6.99 ( ring_18347121197199848620nteger
% 12.45/6.99 = ( ^ [K3: int] :
% 12.45/6.99 ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 12.45/6.99 @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
% 12.45/6.99 @ ( if_Code_integer
% 12.45/6.99 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.99 = zero_zero_int )
% 12.45/6.99 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8614_of__int__code__if,axiom,
% 12.45/6.99 ( ring_1_of_int_rat
% 12.45/6.99 = ( ^ [K3: int] :
% 12.45/6.99 ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
% 12.45/6.99 @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
% 12.45/6.99 @ ( if_rat
% 12.45/6.99 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/6.99 = zero_zero_int )
% 12.45/6.99 @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8615_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 12.45/6.99 ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
% 12.45/6.99 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X4 )
% 12.45/6.99 = ( ( X4 = Mi )
% 12.45/6.99 | ( X4 = Ma )
% 12.45/6.99 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/6.99 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8616_monoseq__arctan__series,axiom,
% 12.45/6.99 ! [X4: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.99 => ( topolo6980174941875973593q_real
% 12.45/6.99 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8617_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 12.45/6.99 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
% 12.45/6.99 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X4 )
% 12.45/6.99 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/6.99 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8618_buildup__nothing__in__min__max,axiom,
% 12.45/6.99 ! [N: nat,X4: nat] :
% 12.45/6.99 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X4 ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8619_buildup__nothing__in__leaf,axiom,
% 12.45/6.99 ! [N: nat,X4: nat] :
% 12.45/6.99 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X4 ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8620_both__member__options__def,axiom,
% 12.45/6.99 ( vEBT_V8194947554948674370ptions
% 12.45/6.99 = ( ^ [T3: vEBT_VEBT,X: nat] :
% 12.45/6.99 ( ( vEBT_V5719532721284313246member @ T3 @ X )
% 12.45/6.99 | ( vEBT_VEBT_membermima @ T3 @ X ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8621_member__valid__both__member__options,axiom,
% 12.45/6.99 ! [Tree: vEBT_VEBT,N: nat,X4: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ Tree @ N )
% 12.45/6.99 => ( ( vEBT_vebt_member @ Tree @ X4 )
% 12.45/6.99 => ( ( vEBT_V5719532721284313246member @ Tree @ X4 )
% 12.45/6.99 | ( vEBT_VEBT_membermima @ Tree @ X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8622_monoseq__realpow,axiom,
% 12.45/6.99 ! [X4: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/6.99 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/6.99 => ( topolo6980174941875973593q_real @ ( power_power_real @ X4 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8623_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/6.99 ( ( vEBT_VEBT_membermima @ X4 @ Xa )
% 12.45/6.99 => ( ! [Mi2: nat,Ma2: nat] :
% 12.45/6.99 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.45/6.99 => ~ ( ( Xa = Mi2 )
% 12.45/6.99 | ( Xa = Ma2 ) ) )
% 12.45/6.99 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [Vc2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 12.45/6.99 => ~ ( ( Xa = Mi2 )
% 12.45/6.99 | ( Xa = Ma2 )
% 12.45/6.99 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( 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 ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 12.45/6.99 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [Vd2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 12.45/6.99 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( 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 ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8624_vebt__member_Oelims_I2_J,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/6.99 ( ( vEBT_vebt_member @ X4 @ Xa )
% 12.45/6.99 => ( ! [A5: $o,B5: $o] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/6.99 => ~ ( ( ( Xa = zero_zero_nat )
% 12.45/6.99 => A5 )
% 12.45/6.99 & ( ( Xa != zero_zero_nat )
% 12.45/6.99 => ( ( ( Xa = one_one_nat )
% 12.45/6.99 => B5 )
% 12.45/6.99 & ( Xa = one_one_nat ) ) ) ) )
% 12.45/6.99 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [Summary2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 12.45/6.99 => ~ ( ( Xa != Mi2 )
% 12.45/6.99 => ( ( Xa != Ma2 )
% 12.45/6.99 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/6.99 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/6.99 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/6.99 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/6.99 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8625_vebt__insert_Osimps_I3_J,axiom,
% 12.45/6.99 ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
% 12.45/6.99 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X4 )
% 12.45/6.99 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8626_pochhammer__times__pochhammer__half,axiom,
% 12.45/6.99 ! [Z: complex,N: nat] :
% 12.45/6.99 ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 12.45/6.99 = ( groups6464643781859351333omplex
% 12.45/6.99 @ ^ [K3: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8627_pochhammer__times__pochhammer__half,axiom,
% 12.45/6.99 ! [Z: real,N: nat] :
% 12.45/6.99 ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 12.45/6.99 = ( groups129246275422532515t_real
% 12.45/6.99 @ ^ [K3: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8628_pochhammer__times__pochhammer__half,axiom,
% 12.45/6.99 ! [Z: rat,N: nat] :
% 12.45/6.99 ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 12.45/6.99 = ( groups73079841787564623at_rat
% 12.45/6.99 @ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8629_Leaf__0__not,axiom,
% 12.45/6.99 ! [A: $o,B2: $o] :
% 12.45/6.99 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ zero_zero_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8630_deg__1__Leafy,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT,N: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/6.99 => ( ( N = one_one_nat )
% 12.45/6.99 => ? [A5: $o,B5: $o] :
% 12.45/6.99 ( T
% 12.45/6.99 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8631_deg__1__Leaf,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 12.45/6.99 => ? [A5: $o,B5: $o] :
% 12.45/6.99 ( T
% 12.45/6.99 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8632_deg1Leaf,axiom,
% 12.45/6.99 ! [T: vEBT_VEBT] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 12.45/6.99 = ( ? [A4: $o,B3: $o] :
% 12.45/6.99 ( T
% 12.45/6.99 = ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8633_prod_Ocl__ivl__Suc,axiom,
% 12.45/6.99 ! [N: nat,M: nat,G: nat > complex] :
% 12.45/6.99 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 12.45/6.99 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = one_one_complex ) )
% 12.45/6.99 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 12.45/6.99 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8634_prod_Ocl__ivl__Suc,axiom,
% 12.45/6.99 ! [N: nat,M: nat,G: nat > real] :
% 12.45/6.99 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 12.45/6.99 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = one_one_real ) )
% 12.45/6.99 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 12.45/6.99 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8635_prod_Ocl__ivl__Suc,axiom,
% 12.45/6.99 ! [N: nat,M: nat,G: nat > rat] :
% 12.45/6.99 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 12.45/6.99 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = one_one_rat ) )
% 12.45/6.99 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 12.45/6.99 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8636_prod_Ocl__ivl__Suc,axiom,
% 12.45/6.99 ! [N: nat,M: nat,G: nat > nat] :
% 12.45/6.99 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 12.45/6.99 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = one_one_nat ) )
% 12.45/6.99 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 12.45/6.99 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8637_prod_Ocl__ivl__Suc,axiom,
% 12.45/6.99 ! [N: nat,M: nat,G: nat > int] :
% 12.45/6.99 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 12.45/6.99 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = one_one_int ) )
% 12.45/6.99 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 12.45/6.99 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8638_mod__prod__eq,axiom,
% 12.45/6.99 ! [F: nat > nat,A: nat,A3: set_nat] :
% 12.45/6.99 ( ( modulo_modulo_nat
% 12.45/6.99 @ ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [I2: nat] : ( modulo_modulo_nat @ ( F @ I2 ) @ A )
% 12.45/6.99 @ A3 )
% 12.45/6.99 @ A )
% 12.45/6.99 = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8639_mod__prod__eq,axiom,
% 12.45/6.99 ! [F: nat > int,A: int,A3: set_nat] :
% 12.45/6.99 ( ( modulo_modulo_int
% 12.45/6.99 @ ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [I2: nat] : ( modulo_modulo_int @ ( F @ I2 ) @ A )
% 12.45/6.99 @ A3 )
% 12.45/6.99 @ A )
% 12.45/6.99 = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A3 ) @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8640_mod__prod__eq,axiom,
% 12.45/6.99 ! [F: int > int,A: int,A3: set_int] :
% 12.45/6.99 ( ( modulo_modulo_int
% 12.45/6.99 @ ( groups1705073143266064639nt_int
% 12.45/6.99 @ ^ [I2: int] : ( modulo_modulo_int @ ( F @ I2 ) @ A )
% 12.45/6.99 @ A3 )
% 12.45/6.99 @ A )
% 12.45/6.99 = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A3 ) @ A ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8641_vebt__insert_Ocases,axiom,
% 12.45/6.99 ! [X4: produc9072475918466114483BT_nat] :
% 12.45/6.99 ( ! [A5: $o,B5: $o,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X3 ) )
% 12.45/6.99 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ X3 ) )
% 12.45/6.99 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X3 ) )
% 12.45/6.99 => ( ! [V3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) )
% 12.45/6.99 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8642_vebt__insert_Osimps_I1_J,axiom,
% 12.45/6.99 ! [X4: nat,A: $o,B2: $o] :
% 12.45/6.99 ( ( ( X4 = zero_zero_nat )
% 12.45/6.99 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
% 12.45/6.99 = ( vEBT_Leaf @ $true @ B2 ) ) )
% 12.45/6.99 & ( ( X4 != zero_zero_nat )
% 12.45/6.99 => ( ( ( X4 = one_one_nat )
% 12.45/6.99 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
% 12.45/6.99 = ( vEBT_Leaf @ A @ $true ) ) )
% 12.45/6.99 & ( ( X4 != one_one_nat )
% 12.45/6.99 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
% 12.45/6.99 = ( vEBT_Leaf @ A @ B2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8643_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 12.45/6.99 ! [G: nat > nat,M: nat,N: nat] :
% 12.45/6.99 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.45/6.99 = ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8644_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 12.45/6.99 ! [G: nat > int,M: nat,N: nat] :
% 12.45/6.99 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.45/6.99 = ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8645_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 12.45/6.99 ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 12.45/6.99 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 12.45/6.99 = ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8646_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 12.45/6.99 ! [G: nat > int,M: nat,K: nat,N: nat] :
% 12.45/6.99 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 12.45/6.99 = ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8647_vebt__member_Ocases,axiom,
% 12.45/6.99 ! [X4: produc9072475918466114483BT_nat] :
% 12.45/6.99 ( ! [A5: $o,B5: $o,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X3 ) )
% 12.45/6.99 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) @ X3 ) )
% 12.45/6.99 => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) @ X3 ) )
% 12.45/6.99 => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
% 12.45/6.99 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8648_vebt__succ_Ocases,axiom,
% 12.45/6.99 ! [X4: produc9072475918466114483BT_nat] :
% 12.45/6.99 ( ! [Uu2: $o,B5: $o] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) )
% 12.45/6.99 => ( ! [Uv: $o,Uw: $o,N4: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N4 ) ) )
% 12.45/6.99 => ( ! [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Va3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) @ Va3 ) )
% 12.45/6.99 => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
% 12.45/6.99 => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
% 12.45/6.99 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8649_VEBT__internal_Omembermima_Ocases,axiom,
% 12.45/6.99 ! [X4: produc9072475918466114483BT_nat] :
% 12.45/6.99 ( ! [Uu2: $o,Uv: $o,Uw: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Uw ) )
% 12.45/6.99 => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) @ Uz ) )
% 12.45/6.99 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X3 ) )
% 12.45/6.99 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ X3 ) )
% 12.45/6.99 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ X3 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8650_VEBT__internal_Onaive__member_Ocases,axiom,
% 12.45/6.99 ! [X4: produc9072475918466114483BT_nat] :
% 12.45/6.99 ( ! [A5: $o,B5: $o,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X3 ) )
% 12.45/6.99 => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) @ Ux ) )
% 12.45/6.99 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ X3 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8651_invar__vebt_Ointros_I1_J,axiom,
% 12.45/6.99 ! [A: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ zero_zero_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8652_vebt__member_Osimps_I1_J,axiom,
% 12.45/6.99 ! [A: $o,B2: $o,X4: nat] :
% 12.45/6.99 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
% 12.45/6.99 = ( ( ( X4 = zero_zero_nat )
% 12.45/6.99 => A )
% 12.45/6.99 & ( ( X4 != zero_zero_nat )
% 12.45/6.99 => ( ( ( X4 = one_one_nat )
% 12.45/6.99 => B2 )
% 12.45/6.99 & ( X4 = one_one_nat ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8653_vebt__buildup_Osimps_I2_J,axiom,
% 12.45/6.99 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 12.45/6.99 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8654_vebt__succ_Osimps_I2_J,axiom,
% 12.45/6.99 ! [Uv2: $o,Uw2: $o,N: nat] :
% 12.45/6.99 ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
% 12.45/6.99 = none_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8655_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 12.45/6.99 ! [A: $o,B2: $o,X4: nat] :
% 12.45/6.99 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B2 ) @ X4 )
% 12.45/6.99 = ( ( ( X4 = zero_zero_nat )
% 12.45/6.99 => A )
% 12.45/6.99 & ( ( X4 != zero_zero_nat )
% 12.45/6.99 => ( ( ( X4 = one_one_nat )
% 12.45/6.99 => B2 )
% 12.45/6.99 & ( X4 = one_one_nat ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8656_prod_OatLeastAtMost__rev,axiom,
% 12.45/6.99 ! [G: nat > nat,N: nat,M: nat] :
% 12.45/6.99 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 12.45/6.99 = ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8657_prod_OatLeastAtMost__rev,axiom,
% 12.45/6.99 ! [G: nat > int,N: nat,M: nat] :
% 12.45/6.99 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 12.45/6.99 = ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8658_vebt__succ_Osimps_I3_J,axiom,
% 12.45/6.99 ! [Ux2: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
% 12.45/6.99 ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy @ Uz2 ) @ Va2 )
% 12.45/6.99 = none_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8659_prod_OatLeast0__atMost__Suc,axiom,
% 12.45/6.99 ! [G: nat > real,N: nat] :
% 12.45/6.99 ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8660_prod_OatLeast0__atMost__Suc,axiom,
% 12.45/6.99 ! [G: nat > rat,N: nat] :
% 12.45/6.99 ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8661_prod_OatLeast0__atMost__Suc,axiom,
% 12.45/6.99 ! [G: nat > nat,N: nat] :
% 12.45/6.99 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8662_prod_OatLeast0__atMost__Suc,axiom,
% 12.45/6.99 ! [G: nat > int,N: nat] :
% 12.45/6.99 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8663_prod_OatLeast__Suc__atMost,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > real] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.99 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 12.45/6.99 = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8664_prod_OatLeast__Suc__atMost,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > rat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.99 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 12.45/6.99 = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8665_prod_OatLeast__Suc__atMost,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.99 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 12.45/6.99 = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8666_prod_OatLeast__Suc__atMost,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > int] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.99 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 12.45/6.99 = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8667_prod_Onat__ivl__Suc_H,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > real] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.45/6.99 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8668_prod_Onat__ivl__Suc_H,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > rat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.45/6.99 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8669_prod_Onat__ivl__Suc_H,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.45/6.99 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8670_prod_Onat__ivl__Suc_H,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > int] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.45/6.99 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8671_vebt__insert_Osimps_I4_J,axiom,
% 12.45/6.99 ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
% 12.45/6.99 ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X4 )
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8672_prod_OSuc__reindex__ivl,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > real] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.99 => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_real @ ( G @ M )
% 12.45/6.99 @ ( groups129246275422532515t_real
% 12.45/6.99 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8673_prod_OSuc__reindex__ivl,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > rat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.99 => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_rat @ ( G @ M )
% 12.45/6.99 @ ( groups73079841787564623at_rat
% 12.45/6.99 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8674_prod_OSuc__reindex__ivl,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.99 => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_nat @ ( G @ M )
% 12.45/6.99 @ ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8675_prod_OSuc__reindex__ivl,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > int] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ N )
% 12.45/6.99 => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 12.45/6.99 = ( times_times_int @ ( G @ M )
% 12.45/6.99 @ ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8676_fact__prod,axiom,
% 12.45/6.99 ( semiri5044797733671781792omplex
% 12.45/6.99 = ( ^ [N2: nat] :
% 12.45/6.99 ( semiri8010041392384452111omplex
% 12.45/6.99 @ ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [X: nat] : X
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8677_fact__prod,axiom,
% 12.45/6.99 ( semiri773545260158071498ct_rat
% 12.45/6.99 = ( ^ [N2: nat] :
% 12.45/6.99 ( semiri681578069525770553at_rat
% 12.45/6.99 @ ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [X: nat] : X
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8678_fact__prod,axiom,
% 12.45/6.99 ( semiri1406184849735516958ct_int
% 12.45/6.99 = ( ^ [N2: nat] :
% 12.45/6.99 ( semiri1314217659103216013at_int
% 12.45/6.99 @ ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [X: nat] : X
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8679_fact__prod,axiom,
% 12.45/6.99 ( semiri1408675320244567234ct_nat
% 12.45/6.99 = ( ^ [N2: nat] :
% 12.45/6.99 ( semiri1316708129612266289at_nat
% 12.45/6.99 @ ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [X: nat] : X
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8680_fact__prod,axiom,
% 12.45/6.99 ( semiri2265585572941072030t_real
% 12.45/6.99 = ( ^ [N2: nat] :
% 12.45/6.99 ( semiri5074537144036343181t_real
% 12.45/6.99 @ ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [X: nat] : X
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8681_prod__atLeastAtMost__code,axiom,
% 12.45/6.99 ! [F: nat > complex,A: nat,B2: nat] :
% 12.45/6.99 ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 12.45/6.99 = ( set_fo1517530859248394432omplex
% 12.45/6.99 @ ^ [A4: nat] : ( times_times_complex @ ( F @ A4 ) )
% 12.45/6.99 @ A
% 12.45/6.99 @ B2
% 12.45/6.99 @ one_one_complex ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8682_prod__atLeastAtMost__code,axiom,
% 12.45/6.99 ! [F: nat > real,A: nat,B2: nat] :
% 12.45/6.99 ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 12.45/6.99 = ( set_fo3111899725591712190t_real
% 12.45/6.99 @ ^ [A4: nat] : ( times_times_real @ ( F @ A4 ) )
% 12.45/6.99 @ A
% 12.45/6.99 @ B2
% 12.45/6.99 @ one_one_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8683_prod__atLeastAtMost__code,axiom,
% 12.45/6.99 ! [F: nat > rat,A: nat,B2: nat] :
% 12.45/6.99 ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 12.45/6.99 = ( set_fo1949268297981939178at_rat
% 12.45/6.99 @ ^ [A4: nat] : ( times_times_rat @ ( F @ A4 ) )
% 12.45/6.99 @ A
% 12.45/6.99 @ B2
% 12.45/6.99 @ one_one_rat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8684_prod__atLeastAtMost__code,axiom,
% 12.45/6.99 ! [F: nat > nat,A: nat,B2: nat] :
% 12.45/6.99 ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 12.45/6.99 = ( set_fo2584398358068434914at_nat
% 12.45/6.99 @ ^ [A4: nat] : ( times_times_nat @ ( F @ A4 ) )
% 12.45/6.99 @ A
% 12.45/6.99 @ B2
% 12.45/6.99 @ one_one_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8685_prod__atLeastAtMost__code,axiom,
% 12.45/6.99 ! [F: nat > int,A: nat,B2: nat] :
% 12.45/6.99 ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 12.45/6.99 = ( set_fo2581907887559384638at_int
% 12.45/6.99 @ ^ [A4: nat] : ( times_times_int @ ( F @ A4 ) )
% 12.45/6.99 @ A
% 12.45/6.99 @ B2
% 12.45/6.99 @ one_one_int ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8686_prod_Oub__add__nat,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > real,P2: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 12.45/6.99 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
% 12.45/6.99 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8687_prod_Oub__add__nat,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > rat,P2: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 12.45/6.99 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
% 12.45/6.99 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8688_prod_Oub__add__nat,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > nat,P2: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 12.45/6.99 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
% 12.45/6.99 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8689_prod_Oub__add__nat,axiom,
% 12.45/6.99 ! [M: nat,N: nat,G: nat > int,P2: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 12.45/6.99 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
% 12.45/6.99 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8690_vebt__succ_Osimps_I1_J,axiom,
% 12.45/6.99 ! [B2: $o,Uu3: $o] :
% 12.45/6.99 ( ( B2
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu3 @ B2 ) @ zero_zero_nat )
% 12.45/6.99 = ( some_nat @ one_one_nat ) ) )
% 12.45/6.99 & ( ~ B2
% 12.45/6.99 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu3 @ B2 ) @ zero_zero_nat )
% 12.45/6.99 = none_nat ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8691_fact__eq__fact__times,axiom,
% 12.45/6.99 ! [N: nat,M: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.99 => ( ( semiri1408675320244567234ct_nat @ M )
% 12.45/6.99 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 12.45/6.99 @ ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [X: nat] : X
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8692_pochhammer__Suc__prod,axiom,
% 12.45/6.99 ! [A: complex,N: nat] :
% 12.45/6.99 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 12.45/6.99 = ( groups6464643781859351333omplex
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8693_pochhammer__Suc__prod,axiom,
% 12.45/6.99 ! [A: real,N: nat] :
% 12.45/6.99 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 12.45/6.99 = ( groups129246275422532515t_real
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8694_pochhammer__Suc__prod,axiom,
% 12.45/6.99 ! [A: rat,N: nat] :
% 12.45/6.99 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 12.45/6.99 = ( groups73079841787564623at_rat
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8695_pochhammer__Suc__prod,axiom,
% 12.45/6.99 ! [A: nat,N: nat] :
% 12.45/6.99 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 12.45/6.99 = ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8696_pochhammer__Suc__prod,axiom,
% 12.45/6.99 ! [A: int,N: nat] :
% 12.45/6.99 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 12.45/6.99 = ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8697_pochhammer__prod__rev,axiom,
% 12.45/6.99 ( comm_s2602460028002588243omplex
% 12.45/6.99 = ( ^ [A4: complex,N2: nat] :
% 12.45/6.99 ( groups6464643781859351333omplex
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N2 @ I2 ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8698_pochhammer__prod__rev,axiom,
% 12.45/6.99 ( comm_s7457072308508201937r_real
% 12.45/6.99 = ( ^ [A4: real,N2: nat] :
% 12.45/6.99 ( groups129246275422532515t_real
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I2 ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8699_pochhammer__prod__rev,axiom,
% 12.45/6.99 ( comm_s4028243227959126397er_rat
% 12.45/6.99 = ( ^ [A4: rat,N2: nat] :
% 12.45/6.99 ( groups73079841787564623at_rat
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I2 ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8700_pochhammer__prod__rev,axiom,
% 12.45/6.99 ( comm_s4663373288045622133er_nat
% 12.45/6.99 = ( ^ [A4: nat,N2: nat] :
% 12.45/6.99 ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I2 ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8701_pochhammer__prod__rev,axiom,
% 12.45/6.99 ( comm_s4660882817536571857er_int
% 12.45/6.99 = ( ^ [A4: int,N2: nat] :
% 12.45/6.99 ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I2 ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8702_fact__div__fact,axiom,
% 12.45/6.99 ! [N: nat,M: nat] :
% 12.45/6.99 ( ( ord_less_eq_nat @ N @ M )
% 12.45/6.99 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 12.45/6.99 = ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [X: nat] : X
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8703_prod_Oin__pairs,axiom,
% 12.45/6.99 ! [G: nat > real,M: nat,N: nat] :
% 12.45/6.99 ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 12.45/6.99 = ( groups129246275422532515t_real
% 12.45/6.99 @ ^ [I2: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8704_prod_Oin__pairs,axiom,
% 12.45/6.99 ! [G: nat > rat,M: nat,N: nat] :
% 12.45/6.99 ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 12.45/6.99 = ( groups73079841787564623at_rat
% 12.45/6.99 @ ^ [I2: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8705_prod_Oin__pairs,axiom,
% 12.45/6.99 ! [G: nat > nat,M: nat,N: nat] :
% 12.45/6.99 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 12.45/6.99 = ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [I2: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8706_prod_Oin__pairs,axiom,
% 12.45/6.99 ! [G: nat > int,M: nat,N: nat] :
% 12.45/6.99 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 12.45/6.99 = ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [I2: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8707_pochhammer__Suc__prod__rev,axiom,
% 12.45/6.99 ! [A: complex,N: nat] :
% 12.45/6.99 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 12.45/6.99 = ( groups6464643781859351333omplex
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N @ I2 ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8708_pochhammer__Suc__prod__rev,axiom,
% 12.45/6.99 ! [A: real,N: nat] :
% 12.45/6.99 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 12.45/6.99 = ( groups129246275422532515t_real
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I2 ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8709_pochhammer__Suc__prod__rev,axiom,
% 12.45/6.99 ! [A: rat,N: nat] :
% 12.45/6.99 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 12.45/6.99 = ( groups73079841787564623at_rat
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I2 ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8710_pochhammer__Suc__prod__rev,axiom,
% 12.45/6.99 ! [A: nat,N: nat] :
% 12.45/6.99 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 12.45/6.99 = ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I2 ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8711_pochhammer__Suc__prod__rev,axiom,
% 12.45/6.99 ! [A: int,N: nat] :
% 12.45/6.99 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 12.45/6.99 = ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [I2: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I2 ) ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8712_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 12.45/6.99 ( ( ( vEBT_VEBT_membermima @ X4 @ Xa )
% 12.45/6.99 = Y3 )
% 12.45/6.99 => ( ( ? [Uu2: $o,Uv: $o] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 12.45/6.99 => Y3 )
% 12.45/6.99 => ( ( ? [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
% 12.45/6.99 => Y3 )
% 12.45/6.99 => ( ! [Mi2: nat,Ma2: nat] :
% 12.45/6.99 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( ~ ( ( Xa = Mi2 )
% 12.45/6.99 | ( Xa = Ma2 ) ) ) ) )
% 12.45/6.99 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [Vc2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( ~ ( ( Xa = Mi2 )
% 12.45/6.99 | ( Xa = Ma2 )
% 12.45/6.99 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( 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 ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 12.45/6.99 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [Vd2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( 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 ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8713_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/6.99 ( ~ ( vEBT_VEBT_membermima @ X4 @ Xa )
% 12.45/6.99 => ( ! [Uu2: $o,Uv: $o] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( vEBT_Leaf @ Uu2 @ Uv ) )
% 12.45/6.99 => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
% 12.45/6.99 => ( ! [Mi2: nat,Ma2: nat] :
% 12.45/6.99 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.45/6.99 => ( ( Xa = Mi2 )
% 12.45/6.99 | ( Xa = Ma2 ) ) )
% 12.45/6.99 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [Vc2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 12.45/6.99 => ( ( Xa = Mi2 )
% 12.45/6.99 | ( Xa = Ma2 )
% 12.45/6.99 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( 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 ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 12.45/6.99 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [Vd2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 12.45/6.99 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( 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 ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8714_gbinomial__Suc,axiom,
% 12.45/6.99 ! [A: complex,K: nat] :
% 12.45/6.99 ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 12.45/6.99 = ( divide1717551699836669952omplex
% 12.45/6.99 @ ( groups6464643781859351333omplex
% 12.45/6.99 @ ^ [I2: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 12.45/6.99 @ ( semiri5044797733671781792omplex @ ( suc @ K ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8715_gbinomial__Suc,axiom,
% 12.45/6.99 ! [A: rat,K: nat] :
% 12.45/6.99 ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 12.45/6.99 = ( divide_divide_rat
% 12.45/6.99 @ ( groups73079841787564623at_rat
% 12.45/6.99 @ ^ [I2: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 12.45/6.99 @ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8716_gbinomial__Suc,axiom,
% 12.45/6.99 ! [A: real,K: nat] :
% 12.45/6.99 ( ( gbinomial_real @ A @ ( suc @ K ) )
% 12.45/6.99 = ( divide_divide_real
% 12.45/6.99 @ ( groups129246275422532515t_real
% 12.45/6.99 @ ^ [I2: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 12.45/6.99 @ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8717_gbinomial__Suc,axiom,
% 12.45/6.99 ! [A: nat,K: nat] :
% 12.45/6.99 ( ( gbinomial_nat @ A @ ( suc @ K ) )
% 12.45/6.99 = ( divide_divide_nat
% 12.45/6.99 @ ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [I2: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 12.45/6.99 @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8718_gbinomial__Suc,axiom,
% 12.45/6.99 ! [A: int,K: nat] :
% 12.45/6.99 ( ( gbinomial_int @ A @ ( suc @ K ) )
% 12.45/6.99 = ( divide_divide_int
% 12.45/6.99 @ ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [I2: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I2 ) )
% 12.45/6.99 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 12.45/6.99 @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8719_vebt__member_Oelims_I1_J,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 12.45/6.99 ( ( ( vEBT_vebt_member @ X4 @ Xa )
% 12.45/6.99 = Y3 )
% 12.45/6.99 => ( ! [A5: $o,B5: $o] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( ~ ( ( ( Xa = zero_zero_nat )
% 12.45/6.99 => A5 )
% 12.45/6.99 & ( ( Xa != zero_zero_nat )
% 12.45/6.99 => ( ( ( Xa = one_one_nat )
% 12.45/6.99 => B5 )
% 12.45/6.99 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 12.45/6.99 => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 12.45/6.99 => Y3 )
% 12.45/6.99 => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
% 12.45/6.99 => Y3 )
% 12.45/6.99 => ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 12.45/6.99 => Y3 )
% 12.45/6.99 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [Summary2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( ~ ( ( Xa != Mi2 )
% 12.45/6.99 => ( ( Xa != Ma2 )
% 12.45/6.99 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/6.99 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/6.99 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/6.99 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/6.99 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8720_vebt__member_Oelims_I3_J,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/6.99 ( ~ ( vEBT_vebt_member @ X4 @ Xa )
% 12.45/6.99 => ( ! [A5: $o,B5: $o] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/6.99 => ( ( ( Xa = zero_zero_nat )
% 12.45/6.99 => A5 )
% 12.45/6.99 & ( ( Xa != zero_zero_nat )
% 12.45/6.99 => ( ( ( Xa = one_one_nat )
% 12.45/6.99 => B5 )
% 12.45/6.99 & ( Xa = one_one_nat ) ) ) ) )
% 12.45/6.99 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 12.45/6.99 => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
% 12.45/6.99 => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 12.45/6.99 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [Summary2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 12.45/6.99 => ( ( Xa != Mi2 )
% 12.45/6.99 => ( ( Xa != Ma2 )
% 12.45/6.99 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/6.99 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/6.99 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/6.99 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/6.99 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8721_invar__vebt_Osimps,axiom,
% 12.45/6.99 ( vEBT_invar_vebt
% 12.45/6.99 = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 12.45/6.99 ( ( ? [A4: $o,B3: $o] :
% 12.45/6.99 ( A1
% 12.45/6.99 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.45/6.99 & ( A22
% 12.45/6.99 = ( suc @ zero_zero_nat ) ) )
% 12.45/6.99 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 12.45/6.99 ( ( A1
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 12.45/6.99 & ! [X: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X @ N2 ) )
% 12.45/6.99 & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 12.45/6.99 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 12.45/6.99 & ( A22
% 12.45/6.99 = ( plus_plus_nat @ N2 @ N2 ) )
% 12.45/6.99 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 12.45/6.99 & ! [X: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.45/6.99 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/6.99 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 12.45/6.99 ( ( A1
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 12.45/6.99 & ! [X: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X @ N2 ) )
% 12.45/6.99 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 12.45/6.99 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 12.45/6.99 & ( A22
% 12.45/6.99 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 12.45/6.99 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 12.45/6.99 & ! [X: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.45/6.99 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/6.99 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 12.45/6.99 ( ( A1
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 12.45/6.99 & ! [X: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X @ N2 ) )
% 12.45/6.99 & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 12.45/6.99 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 12.45/6.99 & ( A22
% 12.45/6.99 = ( plus_plus_nat @ N2 @ N2 ) )
% 12.45/6.99 & ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 12.45/6.99 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 12.45/6.99 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 12.45/6.99 & ( ( Mi3 = Ma3 )
% 12.45/6.99 => ! [X: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.45/6.99 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/6.99 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.45/6.99 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 12.45/6.99 & ( ( Mi3 != Ma3 )
% 12.45/6.99 => ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 12.45/6.99 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 12.45/6.99 = I2 )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 12.45/6.99 & ! [X: nat] :
% 12.45/6.99 ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 12.45/6.99 = I2 )
% 12.45/6.99 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 12.45/6.99 => ( ( ord_less_nat @ Mi3 @ X )
% 12.45/6.99 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
% 12.45/6.99 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 12.45/6.99 ( ( A1
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 12.45/6.99 & ! [X: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X @ N2 ) )
% 12.45/6.99 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 12.45/6.99 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 12.45/6.99 & ( A22
% 12.45/6.99 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 12.45/6.99 & ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 12.45/6.99 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 12.45/6.99 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 12.45/6.99 & ( ( Mi3 = Ma3 )
% 12.45/6.99 => ! [X: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.45/6.99 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/6.99 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.45/6.99 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 12.45/6.99 & ( ( Mi3 != Ma3 )
% 12.45/6.99 => ! [I2: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 12.45/6.99 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 12.45/6.99 = I2 )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 12.45/6.99 & ! [X: nat] :
% 12.45/6.99 ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 12.45/6.99 = I2 )
% 12.45/6.99 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 12.45/6.99 => ( ( ord_less_nat @ Mi3 @ X )
% 12.45/6.99 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8722_invar__vebt_Ocases,axiom,
% 12.45/6.99 ! [A12: vEBT_VEBT,A23: nat] :
% 12.45/6.99 ( ( vEBT_invar_vebt @ A12 @ A23 )
% 12.45/6.99 => ( ( ? [A5: $o,B5: $o] :
% 12.45/6.99 ( A12
% 12.45/6.99 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/6.99 => ( A23
% 12.45/6.99 != ( suc @ zero_zero_nat ) ) )
% 12.45/6.99 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 12.45/6.99 ( ( A12
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/6.99 => ( ( A23 = Deg2 )
% 12.45/6.99 => ( ! [X5: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 12.45/6.99 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 12.45/6.99 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.45/6.99 => ( ( M4 = N4 )
% 12.45/6.99 => ( ( Deg2
% 12.45/6.99 = ( plus_plus_nat @ N4 @ M4 ) )
% 12.45/6.99 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 12.45/6.99 => ~ ! [X5: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/6.99 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 12.45/6.99 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 12.45/6.99 ( ( A12
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/6.99 => ( ( A23 = Deg2 )
% 12.45/6.99 => ( ! [X5: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 12.45/6.99 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 12.45/6.99 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.45/6.99 => ( ( M4
% 12.45/6.99 = ( suc @ N4 ) )
% 12.45/6.99 => ( ( Deg2
% 12.45/6.99 = ( plus_plus_nat @ N4 @ M4 ) )
% 12.45/6.99 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 12.45/6.99 => ~ ! [X5: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/6.99 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 12.45/6.99 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 12.45/6.99 ( ( A12
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/6.99 => ( ( A23 = Deg2 )
% 12.45/6.99 => ( ! [X5: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 12.45/6.99 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 12.45/6.99 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.45/6.99 => ( ( M4 = N4 )
% 12.45/6.99 => ( ( Deg2
% 12.45/6.99 = ( plus_plus_nat @ N4 @ M4 ) )
% 12.45/6.99 => ( ! [I4: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.45/6.99 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 12.45/6.99 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 12.45/6.99 => ( ( ( Mi2 = Ma2 )
% 12.45/6.99 => ! [X5: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/6.99 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 12.45/6.99 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 12.45/6.99 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/6.99 => ~ ( ( Mi2 != Ma2 )
% 12.45/6.99 => ! [I4: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.45/6.99 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 12.45/6.99 = I4 )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 12.45/6.99 & ! [X5: nat] :
% 12.45/6.99 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 12.45/6.99 = I4 )
% 12.45/6.99 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 12.45/6.99 => ( ( ord_less_nat @ Mi2 @ X5 )
% 12.45/6.99 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 12.45/6.99 => ~ ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 12.45/6.99 ( ( A12
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/6.99 => ( ( A23 = Deg2 )
% 12.45/6.99 => ( ! [X5: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 12.45/6.99 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 12.45/6.99 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.45/6.99 => ( ( M4
% 12.45/6.99 = ( suc @ N4 ) )
% 12.45/6.99 => ( ( Deg2
% 12.45/6.99 = ( plus_plus_nat @ N4 @ M4 ) )
% 12.45/6.99 => ( ! [I4: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.45/6.99 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 12.45/6.99 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 12.45/6.99 => ( ( ( Mi2 = Ma2 )
% 12.45/6.99 => ! [X5: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/6.99 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 12.45/6.99 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 12.45/6.99 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/6.99 => ~ ( ( Mi2 != Ma2 )
% 12.45/6.99 => ! [I4: nat] :
% 12.45/6.99 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.45/6.99 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 12.45/6.99 = I4 )
% 12.45/6.99 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 12.45/6.99 & ! [X5: nat] :
% 12.45/6.99 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 12.45/6.99 = I4 )
% 12.45/6.99 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 12.45/6.99 => ( ( ord_less_nat @ Mi2 @ X5 )
% 12.45/6.99 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8723_invar__vebt_Ointros_I2_J,axiom,
% 12.45/6.99 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 12.45/6.99 ( ! [X3: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X3 @ N ) )
% 12.45/6.99 => ( ( vEBT_invar_vebt @ Summary @ M )
% 12.45/6.99 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.99 => ( ( M = N )
% 12.45/6.99 => ( ( Deg
% 12.45/6.99 = ( plus_plus_nat @ N @ M ) )
% 12.45/6.99 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 12.45/6.99 => ( ! [X3: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/6.99 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8724_invar__vebt_Ointros_I3_J,axiom,
% 12.45/6.99 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 12.45/6.99 ( ! [X3: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ X3 @ N ) )
% 12.45/6.99 => ( ( vEBT_invar_vebt @ Summary @ M )
% 12.45/6.99 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.45/6.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.45/6.99 => ( ( M
% 12.45/6.99 = ( suc @ N ) )
% 12.45/6.99 => ( ( Deg
% 12.45/6.99 = ( plus_plus_nat @ N @ M ) )
% 12.45/6.99 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 12.45/6.99 => ( ! [X3: vEBT_VEBT] :
% 12.45/6.99 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/6.99 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
% 12.45/6.99 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8725_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 12.45/6.99 ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
% 12.45/6.99 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X4 )
% 12.45/6.99 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/6.99 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8726_vebt__succ_Oelims,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xa: nat,Y3: option_nat] :
% 12.45/6.99 ( ( ( vEBT_vebt_succ @ X4 @ Xa )
% 12.45/6.99 = Y3 )
% 12.45/6.99 => ( ! [Uu2: $o,B5: $o] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Leaf @ Uu2 @ B5 ) )
% 12.45/6.99 => ( ( Xa = zero_zero_nat )
% 12.45/6.99 => ~ ( ( B5
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( some_nat @ one_one_nat ) ) )
% 12.45/6.99 & ( ~ B5
% 12.45/6.99 => ( Y3 = none_nat ) ) ) ) )
% 12.45/6.99 => ( ( ? [Uv: $o,Uw: $o] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Leaf @ Uv @ Uw ) )
% 12.45/6.99 => ( ? [N4: nat] :
% 12.45/6.99 ( Xa
% 12.45/6.99 = ( suc @ N4 ) )
% 12.45/6.99 => ( Y3 != none_nat ) ) )
% 12.45/6.99 => ( ( ? [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) )
% 12.45/6.99 => ( Y3 != none_nat ) )
% 12.45/6.99 => ( ( ? [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 12.45/6.99 => ( Y3 != none_nat ) )
% 12.45/6.99 => ( ( ? [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 12.45/6.99 => ( Y3 != none_nat ) )
% 12.45/6.99 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 12.45/6.99 => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( some_nat @ Mi2 ) ) )
% 12.45/6.99 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 @ ( if_option_nat
% 12.45/6.99 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 != none_nat )
% 12.45/6.99 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 12.45/6.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 @ ( if_option_nat
% 12.45/6.99 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 = none_nat )
% 12.45/6.99 @ none_nat
% 12.45/6.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 12.45/6.99 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8727_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/6.99 ( ~ ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 12.45/6.99 => ( ! [A5: $o,B5: $o] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/6.99 => ( ( ( Xa = zero_zero_nat )
% 12.45/6.99 => A5 )
% 12.45/6.99 & ( ( Xa != zero_zero_nat )
% 12.45/6.99 => ( ( ( Xa = one_one_nat )
% 12.45/6.99 => B5 )
% 12.45/6.99 & ( Xa = one_one_nat ) ) ) ) )
% 12.45/6.99 => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
% 12.45/6.99 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [S3: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 12.45/6.99 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( 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 ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8728_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/6.99 ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 12.45/6.99 => ( ! [A5: $o,B5: $o] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/6.99 => ~ ( ( ( Xa = zero_zero_nat )
% 12.45/6.99 => A5 )
% 12.45/6.99 & ( ( Xa != zero_zero_nat )
% 12.45/6.99 => ( ( ( Xa = one_one_nat )
% 12.45/6.99 => B5 )
% 12.45/6.99 & ( Xa = one_one_nat ) ) ) ) )
% 12.45/6.99 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [S3: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 12.45/6.99 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( 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 ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8729_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 12.45/6.99 ( ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 12.45/6.99 = Y3 )
% 12.45/6.99 => ( ! [A5: $o,B5: $o] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( ~ ( ( ( Xa = zero_zero_nat )
% 12.45/6.99 => A5 )
% 12.45/6.99 & ( ( Xa != zero_zero_nat )
% 12.45/6.99 => ( ( ( Xa = one_one_nat )
% 12.45/6.99 => B5 )
% 12.45/6.99 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 12.45/6.99 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
% 12.45/6.99 => Y3 )
% 12.45/6.99 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 12.45/6.99 ( ? [S3: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 => ( 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 ) ) ) ) ) )
% 12.45/6.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8730_vebt__maxt_Oelims,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Y3: option_nat] :
% 12.45/6.99 ( ( ( vEBT_vebt_maxt @ X4 )
% 12.45/6.99 = Y3 )
% 12.45/6.99 => ( ! [A5: $o,B5: $o] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/6.99 => ~ ( ( B5
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( some_nat @ one_one_nat ) ) )
% 12.45/6.99 & ( ~ B5
% 12.45/6.99 => ( ( A5
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( some_nat @ zero_zero_nat ) ) )
% 12.45/6.99 & ( ~ A5
% 12.45/6.99 => ( Y3 = none_nat ) ) ) ) ) )
% 12.45/6.99 => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 12.45/6.99 => ( Y3 != none_nat ) )
% 12.45/6.99 => ~ ! [Mi2: nat,Ma2: nat] :
% 12.45/6.99 ( ? [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
% 12.45/6.99 => ( Y3
% 12.45/6.99 != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8731_vebt__mint_Oelims,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Y3: option_nat] :
% 12.45/6.99 ( ( ( vEBT_vebt_mint @ X4 )
% 12.45/6.99 = Y3 )
% 12.45/6.99 => ( ! [A5: $o,B5: $o] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/6.99 => ~ ( ( A5
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( some_nat @ zero_zero_nat ) ) )
% 12.45/6.99 & ( ~ A5
% 12.45/6.99 => ( ( B5
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( some_nat @ one_one_nat ) ) )
% 12.45/6.99 & ( ~ B5
% 12.45/6.99 => ( Y3 = none_nat ) ) ) ) ) )
% 12.45/6.99 => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 12.45/6.99 => ( Y3 != none_nat ) )
% 12.45/6.99 => ~ ! [Mi2: nat] :
% 12.45/6.99 ( ? [Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 12.45/6.99 ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
% 12.45/6.99 => ( Y3
% 12.45/6.99 != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8732_prod_Oempty,axiom,
% 12.45/6.99 ! [G: extended_enat > complex] :
% 12.45/6.99 ( ( groups4622424608036095791omplex @ G @ bot_bo7653980558646680370d_enat )
% 12.45/6.99 = one_one_complex ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8733_prod_Oempty,axiom,
% 12.45/6.99 ! [G: extended_enat > real] :
% 12.45/6.99 ( ( groups97031904164794029t_real @ G @ bot_bo7653980558646680370d_enat )
% 12.45/6.99 = one_one_real ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8734_prod_Oempty,axiom,
% 12.45/6.99 ! [G: extended_enat > rat] :
% 12.45/6.99 ( ( groups2245840878043517529at_rat @ G @ bot_bo7653980558646680370d_enat )
% 12.45/6.99 = one_one_rat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8735_prod_Oempty,axiom,
% 12.45/6.99 ! [G: extended_enat > nat] :
% 12.45/6.99 ( ( groups2880970938130013265at_nat @ G @ bot_bo7653980558646680370d_enat )
% 12.45/6.99 = one_one_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8736_prod_Oempty,axiom,
% 12.45/6.99 ! [G: extended_enat > int] :
% 12.45/6.99 ( ( groups2878480467620962989at_int @ G @ bot_bo7653980558646680370d_enat )
% 12.45/6.99 = one_one_int ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8737_prod_Oempty,axiom,
% 12.45/6.99 ! [G: real > complex] :
% 12.45/6.99 ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
% 12.45/6.99 = one_one_complex ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8738_prod_Oempty,axiom,
% 12.45/6.99 ! [G: real > real] :
% 12.45/6.99 ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
% 12.45/6.99 = one_one_real ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8739_prod_Oempty,axiom,
% 12.45/6.99 ! [G: real > rat] :
% 12.45/6.99 ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
% 12.45/6.99 = one_one_rat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8740_prod_Oempty,axiom,
% 12.45/6.99 ! [G: real > nat] :
% 12.45/6.99 ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
% 12.45/6.99 = one_one_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8741_prod_Oempty,axiom,
% 12.45/6.99 ! [G: real > int] :
% 12.45/6.99 ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
% 12.45/6.99 = one_one_int ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8742_vebt__maxt_Osimps_I1_J,axiom,
% 12.45/6.99 ! [B2: $o,A: $o] :
% 12.45/6.99 ( ( B2
% 12.45/6.99 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 12.45/6.99 = ( some_nat @ one_one_nat ) ) )
% 12.45/6.99 & ( ~ B2
% 12.45/6.99 => ( ( A
% 12.45/6.99 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 12.45/6.99 = ( some_nat @ zero_zero_nat ) ) )
% 12.45/6.99 & ( ~ A
% 12.45/6.99 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 12.45/6.99 = none_nat ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8743_prod_Oneutral__const,axiom,
% 12.45/6.99 ! [A3: set_nat] :
% 12.45/6.99 ( ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [Uu: nat] : one_one_nat
% 12.45/6.99 @ A3 )
% 12.45/6.99 = one_one_nat ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8744_prod_Oneutral__const,axiom,
% 12.45/6.99 ! [A3: set_nat] :
% 12.45/6.99 ( ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [Uu: nat] : one_one_int
% 12.45/6.99 @ A3 )
% 12.45/6.99 = one_one_int ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8745_prod_Oneutral__const,axiom,
% 12.45/6.99 ! [A3: set_int] :
% 12.45/6.99 ( ( groups1705073143266064639nt_int
% 12.45/6.99 @ ^ [Uu: int] : one_one_int
% 12.45/6.99 @ A3 )
% 12.45/6.99 = one_one_int ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8746_prod__int__eq,axiom,
% 12.45/6.99 ! [I: nat,J: nat] :
% 12.45/6.99 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 12.45/6.99 = ( groups1705073143266064639nt_int
% 12.45/6.99 @ ^ [X: int] : X
% 12.45/6.99 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8747_prod__int__plus__eq,axiom,
% 12.45/6.99 ! [I: nat,J: nat] :
% 12.45/6.99 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 12.45/6.99 = ( groups1705073143266064639nt_int
% 12.45/6.99 @ ^ [X: int] : X
% 12.45/6.99 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8748_prod_Onot__neutral__contains__not__neutral,axiom,
% 12.45/6.99 ! [G: extended_enat > complex,A3: set_Extended_enat] :
% 12.45/6.99 ( ( ( groups4622424608036095791omplex @ G @ A3 )
% 12.45/6.99 != one_one_complex )
% 12.45/6.99 => ~ ! [A5: extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ A5 @ A3 )
% 12.45/6.99 => ( ( G @ A5 )
% 12.45/6.99 = one_one_complex ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8749_prod_Onot__neutral__contains__not__neutral,axiom,
% 12.45/6.99 ! [G: complex > complex,A3: set_complex] :
% 12.45/6.99 ( ( ( groups3708469109370488835omplex @ G @ A3 )
% 12.45/6.99 != one_one_complex )
% 12.45/6.99 => ~ ! [A5: complex] :
% 12.45/6.99 ( ( member_complex @ A5 @ A3 )
% 12.45/6.99 => ( ( G @ A5 )
% 12.45/6.99 = one_one_complex ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8750_prod_Onot__neutral__contains__not__neutral,axiom,
% 12.45/6.99 ! [G: real > complex,A3: set_real] :
% 12.45/6.99 ( ( ( groups713298508707869441omplex @ G @ A3 )
% 12.45/6.99 != one_one_complex )
% 12.45/6.99 => ~ ! [A5: real] :
% 12.45/6.99 ( ( member_real @ A5 @ A3 )
% 12.45/6.99 => ( ( G @ A5 )
% 12.45/6.99 = one_one_complex ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8751_prod_Onot__neutral__contains__not__neutral,axiom,
% 12.45/6.99 ! [G: nat > complex,A3: set_nat] :
% 12.45/6.99 ( ( ( groups6464643781859351333omplex @ G @ A3 )
% 12.45/6.99 != one_one_complex )
% 12.45/6.99 => ~ ! [A5: nat] :
% 12.45/6.99 ( ( member_nat @ A5 @ A3 )
% 12.45/6.99 => ( ( G @ A5 )
% 12.45/6.99 = one_one_complex ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8752_prod_Onot__neutral__contains__not__neutral,axiom,
% 12.45/6.99 ! [G: int > complex,A3: set_int] :
% 12.45/6.99 ( ( ( groups7440179247065528705omplex @ G @ A3 )
% 12.45/6.99 != one_one_complex )
% 12.45/6.99 => ~ ! [A5: int] :
% 12.45/6.99 ( ( member_int @ A5 @ A3 )
% 12.45/6.99 => ( ( G @ A5 )
% 12.45/6.99 = one_one_complex ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8753_prod_Onot__neutral__contains__not__neutral,axiom,
% 12.45/6.99 ! [G: extended_enat > real,A3: set_Extended_enat] :
% 12.45/6.99 ( ( ( groups97031904164794029t_real @ G @ A3 )
% 12.45/6.99 != one_one_real )
% 12.45/6.99 => ~ ! [A5: extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ A5 @ A3 )
% 12.45/6.99 => ( ( G @ A5 )
% 12.45/6.99 = one_one_real ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8754_prod_Onot__neutral__contains__not__neutral,axiom,
% 12.45/6.99 ! [G: complex > real,A3: set_complex] :
% 12.45/6.99 ( ( ( groups766887009212190081x_real @ G @ A3 )
% 12.45/6.99 != one_one_real )
% 12.45/6.99 => ~ ! [A5: complex] :
% 12.45/6.99 ( ( member_complex @ A5 @ A3 )
% 12.45/6.99 => ( ( G @ A5 )
% 12.45/6.99 = one_one_real ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8755_prod_Onot__neutral__contains__not__neutral,axiom,
% 12.45/6.99 ! [G: real > real,A3: set_real] :
% 12.45/6.99 ( ( ( groups1681761925125756287l_real @ G @ A3 )
% 12.45/6.99 != one_one_real )
% 12.45/6.99 => ~ ! [A5: real] :
% 12.45/6.99 ( ( member_real @ A5 @ A3 )
% 12.45/6.99 => ( ( G @ A5 )
% 12.45/6.99 = one_one_real ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8756_prod_Onot__neutral__contains__not__neutral,axiom,
% 12.45/6.99 ! [G: nat > real,A3: set_nat] :
% 12.45/6.99 ( ( ( groups129246275422532515t_real @ G @ A3 )
% 12.45/6.99 != one_one_real )
% 12.45/6.99 => ~ ! [A5: nat] :
% 12.45/6.99 ( ( member_nat @ A5 @ A3 )
% 12.45/6.99 => ( ( G @ A5 )
% 12.45/6.99 = one_one_real ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8757_prod_Onot__neutral__contains__not__neutral,axiom,
% 12.45/6.99 ! [G: int > real,A3: set_int] :
% 12.45/6.99 ( ( ( groups2316167850115554303t_real @ G @ A3 )
% 12.45/6.99 != one_one_real )
% 12.45/6.99 => ~ ! [A5: int] :
% 12.45/6.99 ( ( member_int @ A5 @ A3 )
% 12.45/6.99 => ( ( G @ A5 )
% 12.45/6.99 = one_one_real ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8758_prod_Oneutral,axiom,
% 12.45/6.99 ! [A3: set_nat,G: nat > nat] :
% 12.45/6.99 ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ A3 )
% 12.45/6.99 => ( ( G @ X3 )
% 12.45/6.99 = one_one_nat ) )
% 12.45/6.99 => ( ( groups708209901874060359at_nat @ G @ A3 )
% 12.45/6.99 = one_one_nat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8759_prod_Oneutral,axiom,
% 12.45/6.99 ! [A3: set_nat,G: nat > int] :
% 12.45/6.99 ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ A3 )
% 12.45/6.99 => ( ( G @ X3 )
% 12.45/6.99 = one_one_int ) )
% 12.45/6.99 => ( ( groups705719431365010083at_int @ G @ A3 )
% 12.45/6.99 = one_one_int ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8760_prod_Oneutral,axiom,
% 12.45/6.99 ! [A3: set_int,G: int > int] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ( member_int @ X3 @ A3 )
% 12.45/6.99 => ( ( G @ X3 )
% 12.45/6.99 = one_one_int ) )
% 12.45/6.99 => ( ( groups1705073143266064639nt_int @ G @ A3 )
% 12.45/6.99 = one_one_int ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8761_prod_Odistrib,axiom,
% 12.45/6.99 ! [G: nat > nat,H2: nat > nat,A3: set_nat] :
% 12.45/6.99 ( ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [X: nat] : ( times_times_nat @ ( G @ X ) @ ( H2 @ X ) )
% 12.45/6.99 @ A3 )
% 12.45/6.99 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A3 ) @ ( groups708209901874060359at_nat @ H2 @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8762_prod_Odistrib,axiom,
% 12.45/6.99 ! [G: nat > int,H2: nat > int,A3: set_nat] :
% 12.45/6.99 ( ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [X: nat] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
% 12.45/6.99 @ A3 )
% 12.45/6.99 = ( times_times_int @ ( groups705719431365010083at_int @ G @ A3 ) @ ( groups705719431365010083at_int @ H2 @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8763_prod_Odistrib,axiom,
% 12.45/6.99 ! [G: int > int,H2: int > int,A3: set_int] :
% 12.45/6.99 ( ( groups1705073143266064639nt_int
% 12.45/6.99 @ ^ [X: int] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
% 12.45/6.99 @ A3 )
% 12.45/6.99 = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A3 ) @ ( groups1705073143266064639nt_int @ H2 @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8764_prod__power__distrib,axiom,
% 12.45/6.99 ! [F: nat > nat,A3: set_nat,N: nat] :
% 12.45/6.99 ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ N )
% 12.45/6.99 = ( groups708209901874060359at_nat
% 12.45/6.99 @ ^ [X: nat] : ( power_power_nat @ ( F @ X ) @ N )
% 12.45/6.99 @ A3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8765_prod__power__distrib,axiom,
% 12.45/6.99 ! [F: nat > int,A3: set_nat,N: nat] :
% 12.45/6.99 ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A3 ) @ N )
% 12.45/6.99 = ( groups705719431365010083at_int
% 12.45/6.99 @ ^ [X: nat] : ( power_power_int @ ( F @ X ) @ N )
% 12.45/6.99 @ A3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8766_prod__power__distrib,axiom,
% 12.45/6.99 ! [F: int > int,A3: set_int,N: nat] :
% 12.45/6.99 ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A3 ) @ N )
% 12.45/6.99 = ( groups1705073143266064639nt_int
% 12.45/6.99 @ ^ [X: int] : ( power_power_int @ ( F @ X ) @ N )
% 12.45/6.99 @ A3 ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8767_prod__nonneg,axiom,
% 12.45/6.99 ! [A3: set_nat,F: nat > nat] :
% 12.45/6.99 ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8768_prod__nonneg,axiom,
% 12.45/6.99 ! [A3: set_nat,F: nat > int] :
% 12.45/6.99 ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8769_prod__nonneg,axiom,
% 12.45/6.99 ! [A3: set_int,F: int > int] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ( member_int @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8770_prod__mono,axiom,
% 12.45/6.99 ! [A3: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
% 12.45/6.99 ( ! [I3: extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ I3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 12.45/6.99 & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A3 ) @ ( groups97031904164794029t_real @ G @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8771_prod__mono,axiom,
% 12.45/6.99 ! [A3: set_complex,F: complex > real,G: complex > real] :
% 12.45/6.99 ( ! [I3: complex] :
% 12.45/6.99 ( ( member_complex @ I3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 12.45/6.99 & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A3 ) @ ( groups766887009212190081x_real @ G @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8772_prod__mono,axiom,
% 12.45/6.99 ! [A3: set_real,F: real > real,G: real > real] :
% 12.45/6.99 ( ! [I3: real] :
% 12.45/6.99 ( ( member_real @ I3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 12.45/6.99 & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A3 ) @ ( groups1681761925125756287l_real @ G @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8773_prod__mono,axiom,
% 12.45/6.99 ! [A3: set_nat,F: nat > real,G: nat > real] :
% 12.45/6.99 ( ! [I3: nat] :
% 12.45/6.99 ( ( member_nat @ I3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 12.45/6.99 & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A3 ) @ ( groups129246275422532515t_real @ G @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8774_prod__mono,axiom,
% 12.45/6.99 ! [A3: set_int,F: int > real,G: int > real] :
% 12.45/6.99 ( ! [I3: int] :
% 12.45/6.99 ( ( member_int @ I3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 12.45/6.99 & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A3 ) @ ( groups2316167850115554303t_real @ G @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8775_prod__mono,axiom,
% 12.45/6.99 ! [A3: set_Extended_enat,F: extended_enat > rat,G: extended_enat > rat] :
% 12.45/6.99 ( ! [I3: extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ I3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 12.45/6.99 & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ ( groups2245840878043517529at_rat @ F @ A3 ) @ ( groups2245840878043517529at_rat @ G @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8776_prod__mono,axiom,
% 12.45/6.99 ! [A3: set_complex,F: complex > rat,G: complex > rat] :
% 12.45/6.99 ( ! [I3: complex] :
% 12.45/6.99 ( ( member_complex @ I3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 12.45/6.99 & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) @ ( groups225925009352817453ex_rat @ G @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8777_prod__mono,axiom,
% 12.45/6.99 ! [A3: set_real,F: real > rat,G: real > rat] :
% 12.45/6.99 ( ! [I3: real] :
% 12.45/6.99 ( ( member_real @ I3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 12.45/6.99 & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) @ ( groups4061424788464935467al_rat @ G @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8778_prod__mono,axiom,
% 12.45/6.99 ! [A3: set_nat,F: nat > rat,G: nat > rat] :
% 12.45/6.99 ( ! [I3: nat] :
% 12.45/6.99 ( ( member_nat @ I3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 12.45/6.99 & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ ( groups73079841787564623at_rat @ G @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8779_prod__mono,axiom,
% 12.45/6.99 ! [A3: set_int,F: int > rat,G: int > rat] :
% 12.45/6.99 ( ! [I3: int] :
% 12.45/6.99 ( ( member_int @ I3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 12.45/6.99 & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ ( groups1072433553688619179nt_rat @ G @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8780_prod__pos,axiom,
% 12.45/6.99 ! [A3: set_nat,F: nat > nat] :
% 12.45/6.99 ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8781_prod__pos,axiom,
% 12.45/6.99 ! [A3: set_nat,F: nat > int] :
% 12.45/6.99 ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8782_prod__pos,axiom,
% 12.45/6.99 ! [A3: set_int,F: int > int] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ( member_int @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8783_prod__ge__1,axiom,
% 12.45/6.99 ! [A3: set_Extended_enat,F: extended_enat > real] :
% 12.45/6.99 ( ! [X3: extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ one_one_real @ ( groups97031904164794029t_real @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8784_prod__ge__1,axiom,
% 12.45/6.99 ! [A3: set_complex,F: complex > real] :
% 12.45/6.99 ( ! [X3: complex] :
% 12.45/6.99 ( ( member_complex @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8785_prod__ge__1,axiom,
% 12.45/6.99 ! [A3: set_real,F: real > real] :
% 12.45/6.99 ( ! [X3: real] :
% 12.45/6.99 ( ( member_real @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8786_prod__ge__1,axiom,
% 12.45/6.99 ! [A3: set_nat,F: nat > real] :
% 12.45/6.99 ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8787_prod__ge__1,axiom,
% 12.45/6.99 ! [A3: set_int,F: int > real] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ( member_int @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8788_prod__ge__1,axiom,
% 12.45/6.99 ! [A3: set_Extended_enat,F: extended_enat > rat] :
% 12.45/6.99 ( ! [X3: extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ one_one_rat @ ( groups2245840878043517529at_rat @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8789_prod__ge__1,axiom,
% 12.45/6.99 ! [A3: set_complex,F: complex > rat] :
% 12.45/6.99 ( ! [X3: complex] :
% 12.45/6.99 ( ( member_complex @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8790_prod__ge__1,axiom,
% 12.45/6.99 ! [A3: set_real,F: real > rat] :
% 12.45/6.99 ( ! [X3: real] :
% 12.45/6.99 ( ( member_real @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8791_prod__ge__1,axiom,
% 12.45/6.99 ! [A3: set_nat,F: nat > rat] :
% 12.45/6.99 ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8792_prod__ge__1,axiom,
% 12.45/6.99 ! [A3: set_int,F: int > rat] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ( member_int @ X3 @ A3 )
% 12.45/6.99 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8793_prod__le__1,axiom,
% 12.45/6.99 ! [A3: set_Extended_enat,F: extended_enat > real] :
% 12.45/6.99 ( ! [X3: extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ X3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 12.45/6.99 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A3 ) @ one_one_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8794_prod__le__1,axiom,
% 12.45/6.99 ! [A3: set_complex,F: complex > real] :
% 12.45/6.99 ( ! [X3: complex] :
% 12.45/6.99 ( ( member_complex @ X3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 12.45/6.99 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A3 ) @ one_one_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8795_prod__le__1,axiom,
% 12.45/6.99 ! [A3: set_real,F: real > real] :
% 12.45/6.99 ( ! [X3: real] :
% 12.45/6.99 ( ( member_real @ X3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 12.45/6.99 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A3 ) @ one_one_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8796_prod__le__1,axiom,
% 12.45/6.99 ! [A3: set_nat,F: nat > real] :
% 12.45/6.99 ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 12.45/6.99 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A3 ) @ one_one_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8797_prod__le__1,axiom,
% 12.45/6.99 ! [A3: set_int,F: int > real] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ( member_int @ X3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 12.45/6.99 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 12.45/6.99 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A3 ) @ one_one_real ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8798_prod__le__1,axiom,
% 12.45/6.99 ! [A3: set_Extended_enat,F: extended_enat > rat] :
% 12.45/6.99 ( ! [X3: extended_enat] :
% 12.45/6.99 ( ( member_Extended_enat @ X3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 12.45/6.99 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ ( groups2245840878043517529at_rat @ F @ A3 ) @ one_one_rat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8799_prod__le__1,axiom,
% 12.45/6.99 ! [A3: set_complex,F: complex > rat] :
% 12.45/6.99 ( ! [X3: complex] :
% 12.45/6.99 ( ( member_complex @ X3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 12.45/6.99 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) @ one_one_rat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8800_prod__le__1,axiom,
% 12.45/6.99 ! [A3: set_real,F: real > rat] :
% 12.45/6.99 ( ! [X3: real] :
% 12.45/6.99 ( ( member_real @ X3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 12.45/6.99 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) @ one_one_rat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8801_prod__le__1,axiom,
% 12.45/6.99 ! [A3: set_nat,F: nat > rat] :
% 12.45/6.99 ( ! [X3: nat] :
% 12.45/6.99 ( ( member_nat @ X3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 12.45/6.99 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ one_one_rat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8802_prod__le__1,axiom,
% 12.45/6.99 ! [A3: set_int,F: int > rat] :
% 12.45/6.99 ( ! [X3: int] :
% 12.45/6.99 ( ( member_int @ X3 @ A3 )
% 12.45/6.99 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 12.45/6.99 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 12.45/6.99 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ one_one_rat ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8803_vebt__mint_Osimps_I1_J,axiom,
% 12.45/6.99 ! [A: $o,B2: $o] :
% 12.45/6.99 ( ( A
% 12.45/6.99 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 12.45/6.99 = ( some_nat @ zero_zero_nat ) ) )
% 12.45/6.99 & ( ~ A
% 12.45/6.99 => ( ( B2
% 12.45/6.99 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 12.45/6.99 = ( some_nat @ one_one_nat ) ) )
% 12.45/6.99 & ( ~ B2
% 12.45/6.99 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 12.45/6.99 = none_nat ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8804_vebt__succ_Opelims,axiom,
% 12.45/6.99 ! [X4: vEBT_VEBT,Xa: nat,Y3: option_nat] :
% 12.45/6.99 ( ( ( vEBT_vebt_succ @ X4 @ Xa )
% 12.45/6.99 = Y3 )
% 12.45/6.99 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/6.99 => ( ! [Uu2: $o,B5: $o] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Leaf @ Uu2 @ B5 ) )
% 12.45/6.99 => ( ( Xa = zero_zero_nat )
% 12.45/6.99 => ( ( ( B5
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( some_nat @ one_one_nat ) ) )
% 12.45/6.99 & ( ~ B5
% 12.45/6.99 => ( Y3 = none_nat ) ) )
% 12.45/6.99 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) ) ) ) )
% 12.45/6.99 => ( ! [Uv: $o,Uw: $o] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Leaf @ Uv @ Uw ) )
% 12.45/6.99 => ! [N4: nat] :
% 12.45/6.99 ( ( Xa
% 12.45/6.99 = ( suc @ N4 ) )
% 12.45/6.99 => ( ( Y3 = none_nat )
% 12.45/6.99 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N4 ) ) ) ) ) )
% 12.45/6.99 => ( ! [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) )
% 12.45/6.99 => ( ( Y3 = none_nat )
% 12.45/6.99 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy2 @ Uz ) @ Xa ) ) ) )
% 12.45/6.99 => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 12.45/6.99 => ( ( Y3 = none_nat )
% 12.45/6.99 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
% 12.45/6.99 => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 12.45/6.99 => ( ( Y3 = none_nat )
% 12.45/6.99 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
% 12.45/6.99 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/6.99 ( ( X4
% 12.45/6.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 12.45/6.99 => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( some_nat @ Mi2 ) ) )
% 12.45/6.99 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/6.99 => ( Y3
% 12.45/6.99 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/6.99 @ ( if_option_nat
% 12.45/6.99 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 != none_nat )
% 12.45/6.99 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 12.45/6.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/6.99 @ ( if_option_nat
% 12.45/6.99 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/6.99 = none_nat )
% 12.45/6.99 @ none_nat
% 12.45/6.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 12.45/6.99 @ none_nat ) ) ) )
% 12.45/6.99 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8805_ln__series,axiom,
% 12.45/6.99 ! [X4: real] :
% 12.45/6.99 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/6.99 => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.45/6.99 => ( ( ln_ln_real @ X4 )
% 12.45/6.99 = ( suminf_real
% 12.45/6.99 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8806_arctan__series,axiom,
% 12.45/6.99 ! [X4: real] :
% 12.45/6.99 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/6.99 => ( ( arctan @ X4 )
% 12.45/6.99 = ( suminf_real
% 12.45/6.99 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8807_divmod__algorithm__code_I7_J,axiom,
% 12.45/6.99 ! [M: num,N: num] :
% 12.45/6.99 ( ( ( ord_less_eq_num @ M @ N )
% 12.45/6.99 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/6.99 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 12.45/6.99 & ( ~ ( ord_less_eq_num @ M @ N )
% 12.45/6.99 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/6.99 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8808_divmod__algorithm__code_I7_J,axiom,
% 12.45/6.99 ! [M: num,N: num] :
% 12.45/6.99 ( ( ( ord_less_eq_num @ M @ N )
% 12.45/6.99 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/6.99 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 12.45/6.99 & ( ~ ( ord_less_eq_num @ M @ N )
% 12.45/6.99 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/6.99 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 12.45/6.99
% 12.45/6.99 thf(fact_8809_divmod__algorithm__code_I7_J,axiom,
% 12.45/6.99 ! [M: num,N: num] :
% 12.45/6.99 ( ( ( ord_less_eq_num @ M @ N )
% 12.45/6.99 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_num @ M @ N )
% 12.45/7.00 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8810_divmod__algorithm__code_I2_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( unique5052692396658037445od_int @ M @ one )
% 12.45/7.00 = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8811_divmod__algorithm__code_I2_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( unique5055182867167087721od_nat @ M @ one )
% 12.45/7.00 = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8812_divmod__algorithm__code_I2_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( unique3479559517661332726nteger @ M @ one )
% 12.45/7.00 = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8813_powser__zero,axiom,
% 12.45/7.00 ! [F: nat > complex] :
% 12.45/7.00 ( ( suminf_complex
% 12.45/7.00 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
% 12.45/7.00 = ( F @ zero_zero_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8814_powser__zero,axiom,
% 12.45/7.00 ! [F: nat > real] :
% 12.45/7.00 ( ( suminf_real
% 12.45/7.00 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
% 12.45/7.00 = ( F @ zero_zero_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8815_divmod__algorithm__code_I3_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 12.45/7.00 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8816_divmod__algorithm__code_I3_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 12.45/7.00 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8817_divmod__algorithm__code_I3_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 12.45/7.00 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8818_divmod__algorithm__code_I4_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 12.45/7.00 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8819_divmod__algorithm__code_I4_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 12.45/7.00 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8820_divmod__algorithm__code_I4_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 12.45/7.00 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8821_divmod__algorithm__code_I8_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( ( ord_less_num @ M @ N )
% 12.45/7.00 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_num @ M @ N )
% 12.45/7.00 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8822_divmod__algorithm__code_I8_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( ( ord_less_num @ M @ N )
% 12.45/7.00 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_num @ M @ N )
% 12.45/7.00 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8823_divmod__algorithm__code_I8_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( ( ord_less_num @ M @ N )
% 12.45/7.00 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_num @ M @ N )
% 12.45/7.00 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8824_divmod__int__def,axiom,
% 12.45/7.00 ( unique5052692396658037445od_int
% 12.45/7.00 = ( ^ [M6: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8825_divmod__def,axiom,
% 12.45/7.00 ( unique5052692396658037445od_int
% 12.45/7.00 = ( ^ [M6: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8826_divmod__def,axiom,
% 12.45/7.00 ( unique5055182867167087721od_nat
% 12.45/7.00 = ( ^ [M6: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8827_divmod__def,axiom,
% 12.45/7.00 ( unique3479559517661332726nteger
% 12.45/7.00 = ( ^ [M6: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8828_divmod_H__nat__def,axiom,
% 12.45/7.00 ( unique5055182867167087721od_nat
% 12.45/7.00 = ( ^ [M6: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8829_divmod__divmod__step,axiom,
% 12.45/7.00 ( unique5055182867167087721od_nat
% 12.45/7.00 = ( ^ [M6: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8830_divmod__divmod__step,axiom,
% 12.45/7.00 ( unique5052692396658037445od_int
% 12.45/7.00 = ( ^ [M6: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8831_divmod__divmod__step,axiom,
% 12.45/7.00 ( unique3479559517661332726nteger
% 12.45/7.00 = ( ^ [M6: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M6 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M6 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M6 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8832_pi__series,axiom,
% 12.45/7.00 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.45/7.00 = ( suminf_real
% 12.45/7.00 @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8833_vebt__member_Opelims_I3_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/7.00 ( ~ ( vEBT_vebt_member @ X4 @ Xa )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [A5: $o,B5: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
% 12.45/7.00 => ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => A5 )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( ( ( Xa = one_one_nat )
% 12.45/7.00 => B5 )
% 12.45/7.00 & ( Xa = one_one_nat ) ) ) ) ) )
% 12.45/7.00 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) @ Xa ) ) )
% 12.45/7.00 => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) @ Xa ) ) )
% 12.45/7.00 => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) )
% 12.45/7.00 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
% 12.45/7.00 => ( ( Xa != Mi2 )
% 12.45/7.00 => ( ( Xa != Ma2 )
% 12.45/7.00 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/7.00 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/7.00 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/7.00 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/7.00 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8834_vebt__member_Opelims_I1_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 12.45/7.00 ( ( ( vEBT_vebt_member @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [A5: $o,B5: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => A5 )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( ( ( Xa = one_one_nat )
% 12.45/7.00 => B5 )
% 12.45/7.00 & ( Xa = one_one_nat ) ) ) ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
% 12.45/7.00 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 12.45/7.00 => ( ~ Y3
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) @ Xa ) ) ) )
% 12.45/7.00 => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) )
% 12.45/7.00 => ( ~ Y3
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz ) @ Xa ) ) ) )
% 12.45/7.00 => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 12.45/7.00 => ( ~ Y3
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
% 12.45/7.00 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( ( Xa != Mi2 )
% 12.45/7.00 => ( ( Xa != Ma2 )
% 12.45/7.00 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/7.00 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/7.00 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/7.00 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/7.00 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8835_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 12.45/7.00 ( ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [A5: $o,B5: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => A5 )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( ( ( Xa = one_one_nat )
% 12.45/7.00 => B5 )
% 12.45/7.00 & ( Xa = one_one_nat ) ) ) ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
% 12.45/7.00 => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
% 12.45/7.00 => ( ~ Y3
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) @ Xa ) ) ) )
% 12.45/7.00 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( 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 ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8836_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/7.00 ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [A5: $o,B5: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
% 12.45/7.00 => ~ ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => A5 )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( ( ( Xa = one_one_nat )
% 12.45/7.00 => B5 )
% 12.45/7.00 & ( Xa = one_one_nat ) ) ) ) ) )
% 12.45/7.00 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 12.45/7.00 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( 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 ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8837_vebt__member_Opelims_I2_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/7.00 ( ( vEBT_vebt_member @ X4 @ Xa )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [A5: $o,B5: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
% 12.45/7.00 => ~ ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => A5 )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( ( ( Xa = one_one_nat )
% 12.45/7.00 => B5 )
% 12.45/7.00 & ( Xa = one_one_nat ) ) ) ) ) )
% 12.45/7.00 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
% 12.45/7.00 => ~ ( ( Xa != Mi2 )
% 12.45/7.00 => ( ( Xa != Ma2 )
% 12.45/7.00 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/7.00 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 12.45/7.00 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/7.00 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 12.45/7.00 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8838_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/7.00 ( ~ ( vEBT_V5719532721284313246member @ X4 @ Xa )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [A5: $o,B5: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
% 12.45/7.00 => ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => A5 )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( ( ( Xa = one_one_nat )
% 12.45/7.00 => B5 )
% 12.45/7.00 & ( Xa = one_one_nat ) ) ) ) ) )
% 12.45/7.00 => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw ) @ Xa ) ) )
% 12.45/7.00 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 12.45/7.00 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( 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 ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8839_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 12.45/7.00 ( ( ( vEBT_VEBT_membermima @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [Uu2: $o,Uv: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 12.45/7.00 => ( ~ Y3
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) ) ) )
% 12.45/7.00 => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
% 12.45/7.00 => ( ~ Y3
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) @ Xa ) ) ) )
% 12.45/7.00 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( ( Xa = Mi2 )
% 12.45/7.00 | ( Xa = Ma2 ) ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) ) ) )
% 12.45/7.00 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( ( Xa = Mi2 )
% 12.45/7.00 | ( Xa = Ma2 )
% 12.45/7.00 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( 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 ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) ) ) )
% 12.45/7.00 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( 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 ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8840_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/7.00 ( ~ ( vEBT_VEBT_membermima @ X4 @ Xa )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [Uu2: $o,Uv: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) ) )
% 12.45/7.00 => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy2 ) @ Xa ) ) )
% 12.45/7.00 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
% 12.45/7.00 => ( ( Xa = Mi2 )
% 12.45/7.00 | ( Xa = Ma2 ) ) ) )
% 12.45/7.00 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 12.45/7.00 => ( ( Xa = Mi2 )
% 12.45/7.00 | ( Xa = Ma2 )
% 12.45/7.00 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( 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 ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 12.45/7.00 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 12.45/7.00 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( 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 ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8841_suminf__geometric,axiom,
% 12.45/7.00 ! [C: real] :
% 12.45/7.00 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 12.45/7.00 => ( ( suminf_real @ ( power_power_real @ C ) )
% 12.45/7.00 = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8842_suminf__geometric,axiom,
% 12.45/7.00 ! [C: complex] :
% 12.45/7.00 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 12.45/7.00 => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 12.45/7.00 = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8843_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/7.00 ( ( vEBT_VEBT_membermima @ X4 @ Xa )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
% 12.45/7.00 => ~ ( ( Xa = Mi2 )
% 12.45/7.00 | ( Xa = Ma2 ) ) ) )
% 12.45/7.00 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 12.45/7.00 => ~ ( ( Xa = Mi2 )
% 12.45/7.00 | ( Xa = Ma2 )
% 12.45/7.00 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( 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 ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 12.45/7.00 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 12.45/7.00 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 => ( 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 ) ) ) ) ) )
% 12.45/7.00 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8844_minus__one__div__numeral,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8845_one__div__minus__numeral,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8846_dvd__numeral__simp,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8847_dvd__numeral__simp,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.45/7.00 = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8848_dvd__numeral__simp,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 12.45/7.00 = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8849_minus__numeral__div__numeral,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8850_divides__aux__eq,axiom,
% 12.45/7.00 ! [Q2: nat,R2: nat] :
% 12.45/7.00 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
% 12.45/7.00 = ( R2 = zero_zero_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8851_divides__aux__eq,axiom,
% 12.45/7.00 ! [Q2: int,R2: int] :
% 12.45/7.00 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/7.00 = ( R2 = zero_zero_int ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8852_Divides_Oadjust__div__eq,axiom,
% 12.45/7.00 ! [Q2: int,R2: int] :
% 12.45/7.00 ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/7.00 = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8853_numeral__div__minus__numeral,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8854_neg__eucl__rel__int__mult__2,axiom,
% 12.45/7.00 ! [B2: int,A: int,Q2: int,R2: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 12.45/7.00 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/7.00 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8855_summable__arctan__series,axiom,
% 12.45/7.00 ! [X4: real] :
% 12.45/7.00 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/7.00 => ( summable_real
% 12.45/7.00 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8856_pos__eucl__rel__int__mult__2,axiom,
% 12.45/7.00 ! [B2: int,A: int,Q2: int,R2: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 12.45/7.00 => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/7.00 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8857_product__nth,axiom,
% 12.45/7.00 ! [N: nat,Xs2: list_num,Ys2: list_num] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys2 ) ) )
% 12.45/7.00 => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys2 ) @ N )
% 12.45/7.00 = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) @ ( nth_num @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8858_product__nth,axiom,
% 12.45/7.00 ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 12.45/7.00 => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys2 ) @ N )
% 12.45/7.00 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8859_product__nth,axiom,
% 12.45/7.00 ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_o] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) )
% 12.45/7.00 => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys2 ) @ N )
% 12.45/7.00 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8860_product__nth,axiom,
% 12.45/7.00 ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_nat] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) )
% 12.45/7.00 => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys2 ) @ N )
% 12.45/7.00 = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8861_product__nth,axiom,
% 12.45/7.00 ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_int] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) )
% 12.45/7.00 => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys2 ) @ N )
% 12.45/7.00 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8862_product__nth,axiom,
% 12.45/7.00 ! [N: nat,Xs2: list_o,Ys2: list_VEBT_VEBT] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 12.45/7.00 => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys2 ) @ N )
% 12.45/7.00 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8863_product__nth,axiom,
% 12.45/7.00 ! [N: nat,Xs2: list_o,Ys2: list_o] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) )
% 12.45/7.00 => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys2 ) @ N )
% 12.45/7.00 = ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8864_product__nth,axiom,
% 12.45/7.00 ! [N: nat,Xs2: list_o,Ys2: list_nat] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) )
% 12.45/7.00 => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys2 ) @ N )
% 12.45/7.00 = ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8865_product__nth,axiom,
% 12.45/7.00 ! [N: nat,Xs2: list_o,Ys2: list_int] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) )
% 12.45/7.00 => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys2 ) @ N )
% 12.45/7.00 = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8866_product__nth,axiom,
% 12.45/7.00 ! [N: nat,Xs2: list_nat,Ys2: list_num] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_num @ Ys2 ) ) )
% 12.45/7.00 => ( ( nth_Pr8326237132889035090at_num @ ( product_nat_num @ Xs2 @ Ys2 ) @ N )
% 12.45/7.00 = ( product_Pair_nat_num @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) @ ( nth_num @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8867_summable__iff__shift,axiom,
% 12.45/7.00 ! [F: nat > real,K: nat] :
% 12.45/7.00 ( ( summable_real
% 12.45/7.00 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 12.45/7.00 = ( summable_real @ F ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8868_summable__cmult__iff,axiom,
% 12.45/7.00 ! [C: real,F: nat > real] :
% 12.45/7.00 ( ( summable_real
% 12.45/7.00 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 12.45/7.00 = ( ( C = zero_zero_real )
% 12.45/7.00 | ( summable_real @ F ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8869_summable__divide__iff,axiom,
% 12.45/7.00 ! [F: nat > complex,C: complex] :
% 12.45/7.00 ( ( summable_complex
% 12.45/7.00 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 12.45/7.00 = ( ( C = zero_zero_complex )
% 12.45/7.00 | ( summable_complex @ F ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8870_summable__divide__iff,axiom,
% 12.45/7.00 ! [F: nat > real,C: real] :
% 12.45/7.00 ( ( summable_real
% 12.45/7.00 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 12.45/7.00 = ( ( C = zero_zero_real )
% 12.45/7.00 | ( summable_real @ F ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8871_length__product,axiom,
% 12.45/7.00 ! [Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 12.45/7.00 ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys2 ) )
% 12.45/7.00 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8872_length__product,axiom,
% 12.45/7.00 ! [Xs2: list_VEBT_VEBT,Ys2: list_o] :
% 12.45/7.00 ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys2 ) )
% 12.45/7.00 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8873_length__product,axiom,
% 12.45/7.00 ! [Xs2: list_VEBT_VEBT,Ys2: list_nat] :
% 12.45/7.00 ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys2 ) )
% 12.45/7.00 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8874_length__product,axiom,
% 12.45/7.00 ! [Xs2: list_VEBT_VEBT,Ys2: list_int] :
% 12.45/7.00 ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys2 ) )
% 12.45/7.00 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8875_length__product,axiom,
% 12.45/7.00 ! [Xs2: list_o,Ys2: list_VEBT_VEBT] :
% 12.45/7.00 ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys2 ) )
% 12.45/7.00 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8876_length__product,axiom,
% 12.45/7.00 ! [Xs2: list_o,Ys2: list_o] :
% 12.45/7.00 ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys2 ) )
% 12.45/7.00 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8877_length__product,axiom,
% 12.45/7.00 ! [Xs2: list_o,Ys2: list_nat] :
% 12.45/7.00 ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys2 ) )
% 12.45/7.00 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8878_length__product,axiom,
% 12.45/7.00 ! [Xs2: list_o,Ys2: list_int] :
% 12.45/7.00 ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys2 ) )
% 12.45/7.00 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8879_length__product,axiom,
% 12.45/7.00 ! [Xs2: list_nat,Ys2: list_VEBT_VEBT] :
% 12.45/7.00 ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys2 ) )
% 12.45/7.00 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8880_length__product,axiom,
% 12.45/7.00 ! [Xs2: list_nat,Ys2: list_o] :
% 12.45/7.00 ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys2 ) )
% 12.45/7.00 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8881_summable__geometric__iff,axiom,
% 12.45/7.00 ! [C: complex] :
% 12.45/7.00 ( ( summable_complex @ ( power_power_complex @ C ) )
% 12.45/7.00 = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8882_unique__quotient,axiom,
% 12.45/7.00 ! [A: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
% 12.45/7.00 ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/7.00 => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 12.45/7.00 => ( Q2 = Q5 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8883_unique__remainder,axiom,
% 12.45/7.00 ! [A: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
% 12.45/7.00 ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/7.00 => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 12.45/7.00 => ( R2 = R4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8884_eucl__rel__int__by0,axiom,
% 12.45/7.00 ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8885_summable__rabs__comparison__test,axiom,
% 12.45/7.00 ! [F: nat > real,G: nat > real] :
% 12.45/7.00 ( ? [N7: nat] :
% 12.45/7.00 ! [N4: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ N7 @ N4 )
% 12.45/7.00 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 12.45/7.00 => ( ( summable_real @ G )
% 12.45/7.00 => ( summable_real
% 12.45/7.00 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8886_div__int__unique,axiom,
% 12.45/7.00 ! [K: int,L2: int,Q2: int,R2: int] :
% 12.45/7.00 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/7.00 => ( ( divide_divide_int @ K @ L2 )
% 12.45/7.00 = Q2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8887_mod__int__unique,axiom,
% 12.45/7.00 ! [K: int,L2: int,Q2: int,R2: int] :
% 12.45/7.00 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/7.00 => ( ( modulo_modulo_int @ K @ L2 )
% 12.45/7.00 = R2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8888_summable__rabs,axiom,
% 12.45/7.00 ! [F: nat > real] :
% 12.45/7.00 ( ( summable_real
% 12.45/7.00 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
% 12.45/7.00 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 12.45/7.00 @ ( suminf_real
% 12.45/7.00 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8889_eucl__rel__int__dividesI,axiom,
% 12.45/7.00 ! [L2: int,K: int,Q2: int] :
% 12.45/7.00 ( ( L2 != zero_zero_int )
% 12.45/7.00 => ( ( K
% 12.45/7.00 = ( times_times_int @ Q2 @ L2 ) )
% 12.45/7.00 => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8890_eucl__rel__int,axiom,
% 12.45/7.00 ! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8891_summable__power__series,axiom,
% 12.45/7.00 ! [F: nat > real,Z: real] :
% 12.45/7.00 ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
% 12.45/7.00 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 12.45/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 12.45/7.00 => ( ( ord_less_real @ Z @ one_one_real )
% 12.45/7.00 => ( summable_real
% 12.45/7.00 @ ^ [I2: nat] : ( times_times_real @ ( F @ I2 ) @ ( power_power_real @ Z @ I2 ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8892_zminus1__lemma,axiom,
% 12.45/7.00 ! [A: int,B2: int,Q2: int,R2: int] :
% 12.45/7.00 ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/7.00 => ( ( B2 != zero_zero_int )
% 12.45/7.00 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B2 @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B2 @ R2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8893_eucl__rel__int__iff,axiom,
% 12.45/7.00 ! [K: int,L2: int,Q2: int,R2: int] :
% 12.45/7.00 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.45/7.00 = ( ( K
% 12.45/7.00 = ( plus_plus_int @ ( times_times_int @ L2 @ Q2 ) @ R2 ) )
% 12.45/7.00 & ( ( ord_less_int @ zero_zero_int @ L2 )
% 12.45/7.00 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 12.45/7.00 & ( ord_less_int @ R2 @ L2 ) ) )
% 12.45/7.00 & ( ~ ( ord_less_int @ zero_zero_int @ L2 )
% 12.45/7.00 => ( ( ( ord_less_int @ L2 @ zero_zero_int )
% 12.45/7.00 => ( ( ord_less_int @ L2 @ R2 )
% 12.45/7.00 & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 12.45/7.00 & ( ~ ( ord_less_int @ L2 @ zero_zero_int )
% 12.45/7.00 => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8894_eucl__rel__int__remainderI,axiom,
% 12.45/7.00 ! [R2: int,L2: int,K: int,Q2: int] :
% 12.45/7.00 ( ( ( sgn_sgn_int @ R2 )
% 12.45/7.00 = ( sgn_sgn_int @ L2 ) )
% 12.45/7.00 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L2 ) )
% 12.45/7.00 => ( ( K
% 12.45/7.00 = ( plus_plus_int @ ( times_times_int @ Q2 @ L2 ) @ R2 ) )
% 12.45/7.00 => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8895_eucl__rel__int_Osimps,axiom,
% 12.45/7.00 ( eucl_rel_int
% 12.45/7.00 = ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
% 12.45/7.00 ( ? [K3: int] :
% 12.45/7.00 ( ( A1 = K3 )
% 12.45/7.00 & ( A22 = zero_zero_int )
% 12.45/7.00 & ( A32
% 12.45/7.00 = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 12.45/7.00 | ? [L: int,K3: int,Q4: int] :
% 12.45/7.00 ( ( A1 = K3 )
% 12.45/7.00 & ( A22 = L )
% 12.45/7.00 & ( A32
% 12.45/7.00 = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 12.45/7.00 & ( L != zero_zero_int )
% 12.45/7.00 & ( K3
% 12.45/7.00 = ( times_times_int @ Q4 @ L ) ) )
% 12.45/7.00 | ? [R5: int,L: int,K3: int,Q4: int] :
% 12.45/7.00 ( ( A1 = K3 )
% 12.45/7.00 & ( A22 = L )
% 12.45/7.00 & ( A32
% 12.45/7.00 = ( product_Pair_int_int @ Q4 @ R5 ) )
% 12.45/7.00 & ( ( sgn_sgn_int @ R5 )
% 12.45/7.00 = ( sgn_sgn_int @ L ) )
% 12.45/7.00 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
% 12.45/7.00 & ( K3
% 12.45/7.00 = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R5 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8896_eucl__rel__int_Ocases,axiom,
% 12.45/7.00 ! [A12: int,A23: int,A33: product_prod_int_int] :
% 12.45/7.00 ( ( eucl_rel_int @ A12 @ A23 @ A33 )
% 12.45/7.00 => ( ( ( A23 = zero_zero_int )
% 12.45/7.00 => ( A33
% 12.45/7.00 != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 12.45/7.00 => ( ! [Q3: int] :
% 12.45/7.00 ( ( A33
% 12.45/7.00 = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 12.45/7.00 => ( ( A23 != zero_zero_int )
% 12.45/7.00 => ( A12
% 12.45/7.00 != ( times_times_int @ Q3 @ A23 ) ) ) )
% 12.45/7.00 => ~ ! [R3: int,Q3: int] :
% 12.45/7.00 ( ( A33
% 12.45/7.00 = ( product_Pair_int_int @ Q3 @ R3 ) )
% 12.45/7.00 => ( ( ( sgn_sgn_int @ R3 )
% 12.45/7.00 = ( sgn_sgn_int @ A23 ) )
% 12.45/7.00 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
% 12.45/7.00 => ( A12
% 12.45/7.00 != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8897_and__int_Oelims,axiom,
% 12.45/7.00 ! [X4: int,Xa: int,Y3: int] :
% 12.45/7.00 ( ( ( bit_se725231765392027082nd_int @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.45/7.00 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( uminus_uminus_int
% 12.45/7.00 @ ( zero_n2684676970156552555ol_int
% 12.45/7.00 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
% 12.45/7.00 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 12.45/7.00 & ( ~ ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.45/7.00 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( plus_plus_int
% 12.45/7.00 @ ( zero_n2684676970156552555ol_int
% 12.45/7.00 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
% 12.45/7.00 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 12.45/7.00 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8898_and__int_Osimps,axiom,
% 12.45/7.00 ( bit_se725231765392027082nd_int
% 12.45/7.00 = ( ^ [K3: int,L: int] :
% 12.45/7.00 ( if_int
% 12.45/7.00 @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.45/7.00 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.45/7.00 @ ( uminus_uminus_int
% 12.45/7.00 @ ( zero_n2684676970156552555ol_int
% 12.45/7.00 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 12.45/7.00 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 12.45/7.00 @ ( plus_plus_int
% 12.45/7.00 @ ( zero_n2684676970156552555ol_int
% 12.45/7.00 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 12.45/7.00 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 12.45/7.00 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8899_vebt__buildup_Oelims,axiom,
% 12.45/7.00 ! [X4: nat,Y3: vEBT_VEBT] :
% 12.45/7.00 ( ( ( vEBT_vebt_buildup @ X4 )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ( X4 = zero_zero_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( vEBT_Leaf @ $false @ $false ) ) )
% 12.45/7.00 => ( ( ( X4
% 12.45/7.00 = ( suc @ zero_zero_nat ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( vEBT_Leaf @ $false @ $false ) ) )
% 12.45/7.00 => ~ ! [Va: nat] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( suc @ ( suc @ Va ) ) )
% 12.45/7.00 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( 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 ) ) ) ) ) ) )
% 12.45/7.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8900_sin__paired,axiom,
% 12.45/7.00 ! [X4: real] :
% 12.45/7.00 ( sums_real
% 12.45/7.00 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 12.45/7.00 @ ( sin_real @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8901_atLeastAtMostPlus1__int__conv,axiom,
% 12.45/7.00 ! [M: int,N: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 12.45/7.00 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 12.45/7.00 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8902_simp__from__to,axiom,
% 12.45/7.00 ( set_or1266510415728281911st_int
% 12.45/7.00 = ( ^ [I2: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I2 ) @ bot_bot_set_int @ ( insert_int @ I2 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8903_power__half__series,axiom,
% 12.45/7.00 ( sums_real
% 12.45/7.00 @ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
% 12.45/7.00 @ one_one_real ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8904_sums__if_H,axiom,
% 12.45/7.00 ! [G: nat > real,X4: real] :
% 12.45/7.00 ( ( sums_real @ G @ X4 )
% 12.45/7.00 => ( sums_real
% 12.45/7.00 @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8905_sums__if,axiom,
% 12.45/7.00 ! [G: nat > real,X4: real,F: nat > real,Y3: real] :
% 12.45/7.00 ( ( sums_real @ G @ X4 )
% 12.45/7.00 => ( ( sums_real @ F @ Y3 )
% 12.45/7.00 => ( sums_real
% 12.45/7.00 @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 @ ( plus_plus_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8906_cos__paired,axiom,
% 12.45/7.00 ! [X4: real] :
% 12.45/7.00 ( sums_real
% 12.45/7.00 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 12.45/7.00 @ ( cos_real @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8907_vebt__buildup_Osimps_I3_J,axiom,
% 12.45/7.00 ! [Va2: nat] :
% 12.45/7.00 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 12.45/7.00 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 12.45/7.00 = ( 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 ) ) ) ) ) ) )
% 12.45/7.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 12.45/7.00 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 12.45/7.00 = ( 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 ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8908_and__int_Opelims,axiom,
% 12.45/7.00 ! [X4: int,Xa: int,Y3: int] :
% 12.45/7.00 ( ( ( bit_se725231765392027082nd_int @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X4 @ Xa ) )
% 12.45/7.00 => ~ ( ( ( ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.45/7.00 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( uminus_uminus_int
% 12.45/7.00 @ ( zero_n2684676970156552555ol_int
% 12.45/7.00 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
% 12.45/7.00 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 12.45/7.00 & ( ~ ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.45/7.00 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( plus_plus_int
% 12.45/7.00 @ ( zero_n2684676970156552555ol_int
% 12.45/7.00 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
% 12.45/7.00 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 12.45/7.00 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 12.45/7.00 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X4 @ Xa ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8909_and__int_Opsimps,axiom,
% 12.45/7.00 ! [K: int,L2: int] :
% 12.45/7.00 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
% 12.45/7.00 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.45/7.00 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.45/7.00 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 12.45/7.00 = ( uminus_uminus_int
% 12.45/7.00 @ ( zero_n2684676970156552555ol_int
% 12.45/7.00 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 12.45/7.00 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
% 12.45/7.00 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.45/7.00 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.45/7.00 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 12.45/7.00 = ( plus_plus_int
% 12.45/7.00 @ ( zero_n2684676970156552555ol_int
% 12.45/7.00 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 12.45/7.00 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 12.45/7.00 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8910_obtain__set__succ,axiom,
% 12.45/7.00 ! [X4: nat,Z: nat,A3: set_nat,B4: set_nat] :
% 12.45/7.00 ( ( ord_less_nat @ X4 @ Z )
% 12.45/7.00 => ( ( vEBT_VEBT_max_in_set @ A3 @ Z )
% 12.45/7.00 => ( ( finite_finite_nat @ B4 )
% 12.45/7.00 => ( ( A3 = B4 )
% 12.45/7.00 => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A3 @ X4 @ X_1 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8911_set__vebt__finite,axiom,
% 12.45/7.00 ! [T: vEBT_VEBT,N: nat] :
% 12.45/7.00 ( ( vEBT_invar_vebt @ T @ N )
% 12.45/7.00 => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8912_succ__none__empty,axiom,
% 12.45/7.00 ! [Xs2: set_nat,A: nat] :
% 12.45/7.00 ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_1 )
% 12.45/7.00 => ( ( finite_finite_nat @ Xs2 )
% 12.45/7.00 => ~ ? [X5: nat] :
% 12.45/7.00 ( ( member_nat @ X5 @ Xs2 )
% 12.45/7.00 & ( ord_less_nat @ A @ X5 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8913_finite__atLeastAtMost,axiom,
% 12.45/7.00 ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8914_finite__nat__set__iff__bounded,axiom,
% 12.45/7.00 ( finite_finite_nat
% 12.45/7.00 = ( ^ [N8: set_nat] :
% 12.45/7.00 ? [M6: nat] :
% 12.45/7.00 ! [X: nat] :
% 12.45/7.00 ( ( member_nat @ X @ N8 )
% 12.45/7.00 => ( ord_less_nat @ X @ M6 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8915_bounded__nat__set__is__finite,axiom,
% 12.45/7.00 ! [N3: set_nat,N: nat] :
% 12.45/7.00 ( ! [X3: nat] :
% 12.45/7.00 ( ( member_nat @ X3 @ N3 )
% 12.45/7.00 => ( ord_less_nat @ X3 @ N ) )
% 12.45/7.00 => ( finite_finite_nat @ N3 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8916_finite__nat__set__iff__bounded__le,axiom,
% 12.45/7.00 ( finite_finite_nat
% 12.45/7.00 = ( ^ [N8: set_nat] :
% 12.45/7.00 ? [M6: nat] :
% 12.45/7.00 ! [X: nat] :
% 12.45/7.00 ( ( member_nat @ X @ N8 )
% 12.45/7.00 => ( ord_less_eq_nat @ X @ M6 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8917_finite__M__bounded__by__nat,axiom,
% 12.45/7.00 ! [P: nat > $o,I: nat] :
% 12.45/7.00 ( finite_finite_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [K3: nat] :
% 12.45/7.00 ( ( P @ K3 )
% 12.45/7.00 & ( ord_less_nat @ K3 @ I ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8918_finite__less__ub,axiom,
% 12.45/7.00 ! [F: nat > nat,U: nat] :
% 12.45/7.00 ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
% 12.45/7.00 => ( finite_finite_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8919_atLeast0__atMost__Suc,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 12.45/7.00 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8920_Icc__eq__insert__lb__nat,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ M @ N )
% 12.45/7.00 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 12.45/7.00 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8921_atLeastAtMostSuc__conv,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.45/7.00 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 12.45/7.00 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8922_atLeastAtMost__insertL,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ M @ N )
% 12.45/7.00 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 12.45/7.00 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8923_finite__divisors__nat,axiom,
% 12.45/7.00 ! [M: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/7.00 => ( finite_finite_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8924_subset__eq__atLeast0__atMost__finite,axiom,
% 12.45/7.00 ! [N3: set_nat,N: nat] :
% 12.45/7.00 ( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 12.45/7.00 => ( finite_finite_nat @ N3 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8925_set__decode__plus__power__2,axiom,
% 12.45/7.00 ! [N: nat,Z: nat] :
% 12.45/7.00 ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 12.45/7.00 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 12.45/7.00 = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8926_and__int_Opinduct,axiom,
% 12.45/7.00 ! [A0: int,A12: int,P: int > int > $o] :
% 12.45/7.00 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 12.45/7.00 => ( ! [K2: int,L3: int] :
% 12.45/7.00 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L3 ) )
% 12.45/7.00 => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.45/7.00 & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.45/7.00 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 => ( P @ K2 @ L3 ) ) )
% 12.45/7.00 => ( P @ A0 @ A12 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8927_finite__Collect__le__nat,axiom,
% 12.45/7.00 ! [K: nat] :
% 12.45/7.00 ( finite_finite_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8928_finite__Collect__less__nat,axiom,
% 12.45/7.00 ! [K: nat] :
% 12.45/7.00 ( finite_finite_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8929_finite__atLeastAtMost__int,axiom,
% 12.45/7.00 ! [L2: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8930_finite__interval__int4,axiom,
% 12.45/7.00 ! [A: int,B2: int] :
% 12.45/7.00 ( finite_finite_int
% 12.45/7.00 @ ( collect_int
% 12.45/7.00 @ ^ [I2: int] :
% 12.45/7.00 ( ( ord_less_int @ A @ I2 )
% 12.45/7.00 & ( ord_less_int @ I2 @ B2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8931_finite__interval__int3,axiom,
% 12.45/7.00 ! [A: int,B2: int] :
% 12.45/7.00 ( finite_finite_int
% 12.45/7.00 @ ( collect_int
% 12.45/7.00 @ ^ [I2: int] :
% 12.45/7.00 ( ( ord_less_int @ A @ I2 )
% 12.45/7.00 & ( ord_less_eq_int @ I2 @ B2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8932_finite__interval__int2,axiom,
% 12.45/7.00 ! [A: int,B2: int] :
% 12.45/7.00 ( finite_finite_int
% 12.45/7.00 @ ( collect_int
% 12.45/7.00 @ ^ [I2: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ A @ I2 )
% 12.45/7.00 & ( ord_less_int @ I2 @ B2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8933_finite__nth__roots,axiom,
% 12.45/7.00 ! [N: nat,C: complex] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( finite3207457112153483333omplex
% 12.45/7.00 @ ( collect_complex
% 12.45/7.00 @ ^ [Z5: complex] :
% 12.45/7.00 ( ( power_power_complex @ Z5 @ N )
% 12.45/7.00 = C ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8934_set__encode__insert,axiom,
% 12.45/7.00 ! [A3: set_nat,N: nat] :
% 12.45/7.00 ( ( finite_finite_nat @ A3 )
% 12.45/7.00 => ( ~ ( member_nat @ N @ A3 )
% 12.45/7.00 => ( ( nat_set_encode @ ( insert_nat @ N @ A3 ) )
% 12.45/7.00 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8935_upto_Opinduct,axiom,
% 12.45/7.00 ! [A0: int,A12: int,P: int > int > $o] :
% 12.45/7.00 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 12.45/7.00 => ( ! [I3: int,J2: int] :
% 12.45/7.00 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
% 12.45/7.00 => ( ( ( ord_less_eq_int @ I3 @ J2 )
% 12.45/7.00 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
% 12.45/7.00 => ( P @ I3 @ J2 ) ) )
% 12.45/7.00 => ( P @ A0 @ A12 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8936_set__encode__def,axiom,
% 12.45/7.00 ( nat_set_encode
% 12.45/7.00 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8937_mask__eq__sum__exp__nat,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 12.45/7.00 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8938_gauss__sum__nat,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [X: nat] : X
% 12.45/7.00 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 12.45/7.00 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8939_arith__series__nat,axiom,
% 12.45/7.00 ! [A: nat,D: nat,N: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I2 @ D ) )
% 12.45/7.00 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 12.45/7.00 = ( 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 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8940_Sum__Icc__nat,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [X: nat] : X
% 12.45/7.00 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 12.45/7.00 = ( 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 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8941_even__set__encode__iff,axiom,
% 12.45/7.00 ! [A3: set_nat] :
% 12.45/7.00 ( ( finite_finite_nat @ A3 )
% 12.45/7.00 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A3 ) )
% 12.45/7.00 = ( ~ ( member_nat @ zero_zero_nat @ A3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8942_Maclaurin__minus__cos__expansion,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 12.45/7.00 => ? [T2: real] :
% 12.45/7.00 ( ( ord_less_real @ X4 @ T2 )
% 12.45/7.00 & ( ord_less_real @ T2 @ zero_zero_real )
% 12.45/7.00 & ( ( cos_real @ X4 )
% 12.45/7.00 = ( plus_plus_real
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8943_Maclaurin__cos__expansion2,axiom,
% 12.45/7.00 ! [X4: real,N: nat] :
% 12.45/7.00 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ? [T2: real] :
% 12.45/7.00 ( ( ord_less_real @ zero_zero_real @ T2 )
% 12.45/7.00 & ( ord_less_real @ T2 @ X4 )
% 12.45/7.00 & ( ( cos_real @ X4 )
% 12.45/7.00 = ( plus_plus_real
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8944_Maclaurin__sin__expansion3,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ? [T2: real] :
% 12.45/7.00 ( ( ord_less_real @ zero_zero_real @ T2 )
% 12.45/7.00 & ( ord_less_real @ T2 @ X4 )
% 12.45/7.00 & ( ( sin_real @ X4 )
% 12.45/7.00 = ( plus_plus_real
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8945_Maclaurin__sin__expansion4,axiom,
% 12.45/7.00 ! [X4: real,N: nat] :
% 12.45/7.00 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ? [T2: real] :
% 12.45/7.00 ( ( ord_less_real @ zero_zero_real @ T2 )
% 12.45/7.00 & ( ord_less_eq_real @ T2 @ X4 )
% 12.45/7.00 & ( ( sin_real @ X4 )
% 12.45/7.00 = ( plus_plus_real
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8946_finite__lessThan,axiom,
% 12.45/7.00 ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8947_lessThan__0,axiom,
% 12.45/7.00 ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 12.45/7.00 = bot_bot_set_nat ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8948_sumr__cos__zero__one,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 12.45/7.00 = one_one_real ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8949_lessThan__Suc,axiom,
% 12.45/7.00 ! [K: nat] :
% 12.45/7.00 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 12.45/7.00 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8950_lessThan__empty__iff,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ( set_ord_lessThan_nat @ N )
% 12.45/7.00 = bot_bot_set_nat )
% 12.45/7.00 = ( N = zero_zero_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8951_lessThan__nat__numeral,axiom,
% 12.45/7.00 ! [K: num] :
% 12.45/7.00 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 12.45/7.00 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8952_sum__nth__roots,axiom,
% 12.45/7.00 ! [N: nat,C: complex] :
% 12.45/7.00 ( ( ord_less_nat @ one_one_nat @ N )
% 12.45/7.00 => ( ( groups7754918857620584856omplex
% 12.45/7.00 @ ^ [X: complex] : X
% 12.45/7.00 @ ( collect_complex
% 12.45/7.00 @ ^ [Z5: complex] :
% 12.45/7.00 ( ( power_power_complex @ Z5 @ N )
% 12.45/7.00 = C ) ) )
% 12.45/7.00 = zero_zero_complex ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8953_sum__roots__unity,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_less_nat @ one_one_nat @ N )
% 12.45/7.00 => ( ( groups7754918857620584856omplex
% 12.45/7.00 @ ^ [X: complex] : X
% 12.45/7.00 @ ( collect_complex
% 12.45/7.00 @ ^ [Z5: complex] :
% 12.45/7.00 ( ( power_power_complex @ Z5 @ N )
% 12.45/7.00 = one_one_complex ) ) )
% 12.45/7.00 = zero_zero_complex ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8954_Maclaurin__lemma,axiom,
% 12.45/7.00 ! [H2: real,F: real > real,J: nat > real,N: nat] :
% 12.45/7.00 ( ( ord_less_real @ zero_zero_real @ H2 )
% 12.45/7.00 => ? [B8: real] :
% 12.45/7.00 ( ( F @ H2 )
% 12.45/7.00 = ( plus_plus_real
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8955_sum__split__even__odd,axiom,
% 12.45/7.00 ! [F: nat > real,G: nat > real,N: nat] :
% 12.45/7.00 ( ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [I2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ ( F @ I2 ) @ ( G @ I2 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/7.00 = ( plus_plus_real
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [I2: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ one_one_nat ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8956_Maclaurin__exp__le,axiom,
% 12.45/7.00 ! [X4: real,N: nat] :
% 12.45/7.00 ? [T2: real] :
% 12.45/7.00 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 12.45/7.00 & ( ( exp_real @ X4 )
% 12.45/7.00 = ( plus_plus_real
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X4 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8957_Maclaurin__sin__bound,axiom,
% 12.45/7.00 ! [X4: real,N: nat] :
% 12.45/7.00 ( ord_less_eq_real
% 12.45/7.00 @ ( abs_abs_real
% 12.45/7.00 @ ( minus_minus_real @ ( sin_real @ X4 )
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 12.45/7.00 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X4 ) @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8958_Sum__Icc__int,axiom,
% 12.45/7.00 ! [M: int,N: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ M @ N )
% 12.45/7.00 => ( ( groups4538972089207619220nt_int
% 12.45/7.00 @ ^ [X: int] : X
% 12.45/7.00 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 12.45/7.00 = ( 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 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8959_sum__pos__lt__pair,axiom,
% 12.45/7.00 ! [F: nat > real,K: nat] :
% 12.45/7.00 ( ( summable_real @ F )
% 12.45/7.00 => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
% 12.45/7.00 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8960_Maclaurin__exp__lt,axiom,
% 12.45/7.00 ! [X4: real,N: nat] :
% 12.45/7.00 ( ( X4 != zero_zero_real )
% 12.45/7.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ? [T2: real] :
% 12.45/7.00 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T2 ) )
% 12.45/7.00 & ( ord_less_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 12.45/7.00 & ( ( exp_real @ X4 )
% 12.45/7.00 = ( plus_plus_real
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X4 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8961_Maclaurin__sin__expansion,axiom,
% 12.45/7.00 ! [X4: real,N: nat] :
% 12.45/7.00 ? [T2: real] :
% 12.45/7.00 ( ( sin_real @ X4 )
% 12.45/7.00 = ( plus_plus_real
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8962_Maclaurin__sin__expansion2,axiom,
% 12.45/7.00 ! [X4: real,N: nat] :
% 12.45/7.00 ? [T2: real] :
% 12.45/7.00 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 12.45/7.00 & ( ( sin_real @ X4 )
% 12.45/7.00 = ( plus_plus_real
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8963_Maclaurin__cos__expansion,axiom,
% 12.45/7.00 ! [X4: real,N: nat] :
% 12.45/7.00 ? [T2: real] :
% 12.45/7.00 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 12.45/7.00 & ( ( cos_real @ X4 )
% 12.45/7.00 = ( plus_plus_real
% 12.45/7.00 @ ( groups6591440286371151544t_real
% 12.45/7.00 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T2 @ ( 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 @ X4 @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8964_bij__betw__roots__unity,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( bij_betw_nat_complex
% 12.45/7.00 @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.45/7.00 @ ( set_ord_lessThan_nat @ N )
% 12.45/7.00 @ ( collect_complex
% 12.45/7.00 @ ^ [Z5: complex] :
% 12.45/7.00 ( ( power_power_complex @ Z5 @ N )
% 12.45/7.00 = one_one_complex ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8965_finite__atMost,axiom,
% 12.45/7.00 ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8966_atMost__0,axiom,
% 12.45/7.00 ( ( set_ord_atMost_nat @ zero_zero_nat )
% 12.45/7.00 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8967_atMost__atLeast0,axiom,
% 12.45/7.00 ( set_ord_atMost_nat
% 12.45/7.00 = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8968_lessThan__Suc__atMost,axiom,
% 12.45/7.00 ! [K: nat] :
% 12.45/7.00 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 12.45/7.00 = ( set_ord_atMost_nat @ K ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8969_atMost__Suc,axiom,
% 12.45/7.00 ! [K: nat] :
% 12.45/7.00 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 12.45/7.00 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8970_atMost__nat__numeral,axiom,
% 12.45/7.00 ! [K: num] :
% 12.45/7.00 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 12.45/7.00 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8971_sum__choose__upper,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [K3: nat] : ( binomial @ K3 @ M )
% 12.45/7.00 @ ( set_ord_atMost_nat @ N ) )
% 12.45/7.00 = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8972_sum__choose__lower,axiom,
% 12.45/7.00 ! [R2: nat,N: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
% 12.45/7.00 @ ( set_ord_atMost_nat @ N ) )
% 12.45/7.00 = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8973_choose__rising__sum_I2_J,axiom,
% 12.45/7.00 ! [N: nat,M: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 12.45/7.00 @ ( set_ord_atMost_nat @ M ) )
% 12.45/7.00 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8974_choose__rising__sum_I1_J,axiom,
% 12.45/7.00 ! [N: nat,M: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 12.45/7.00 @ ( set_ord_atMost_nat @ M ) )
% 12.45/7.00 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8975_atLeast1__atMost__eq__remove0,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.45/7.00 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8976_sum__choose__diagonal,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ M @ N )
% 12.45/7.00 => ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 12.45/7.00 @ ( set_ord_atMost_nat @ M ) )
% 12.45/7.00 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8977_vandermonde,axiom,
% 12.45/7.00 ! [M: nat,N: nat,R2: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
% 12.45/7.00 @ ( set_ord_atMost_nat @ R2 ) )
% 12.45/7.00 = ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8978_choose__row__sum,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 12.45/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8979_binomial,axiom,
% 12.45/7.00 ! [A: nat,B2: nat,N: nat] :
% 12.45/7.00 ( ( power_power_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
% 12.45/7.00 = ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N @ K3 ) ) )
% 12.45/7.00 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8980_polynomial__product__nat,axiom,
% 12.45/7.00 ! [M: nat,A: nat > nat,N: nat,B2: nat > nat,X4: nat] :
% 12.45/7.00 ( ! [I3: nat] :
% 12.45/7.00 ( ( ord_less_nat @ M @ I3 )
% 12.45/7.00 => ( ( A @ I3 )
% 12.45/7.00 = zero_zero_nat ) )
% 12.45/7.00 => ( ! [J2: nat] :
% 12.45/7.00 ( ( ord_less_nat @ N @ J2 )
% 12.45/7.00 => ( ( B2 @ J2 )
% 12.45/7.00 = zero_zero_nat ) )
% 12.45/7.00 => ( ( times_times_nat
% 12.45/7.00 @ ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( power_power_nat @ X4 @ I2 ) )
% 12.45/7.00 @ ( set_ord_atMost_nat @ M ) )
% 12.45/7.00 @ ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [J3: nat] : ( times_times_nat @ ( B2 @ J3 ) @ ( power_power_nat @ X4 @ J3 ) )
% 12.45/7.00 @ ( set_ord_atMost_nat @ N ) ) )
% 12.45/7.00 = ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [R5: nat] :
% 12.45/7.00 ( times_times_nat
% 12.45/7.00 @ ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 12.45/7.00 @ ( set_ord_atMost_nat @ R5 ) )
% 12.45/7.00 @ ( power_power_nat @ X4 @ R5 ) )
% 12.45/7.00 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8981_choose__square__sum,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/7.00 @ ( set_ord_atMost_nat @ N ) )
% 12.45/7.00 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8982_binomial__r__part__sum,axiom,
% 12.45/7.00 ! [M: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 12.45/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8983_choose__linear__sum,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [I2: nat] : ( times_times_nat @ I2 @ ( binomial @ N @ I2 ) )
% 12.45/7.00 @ ( set_ord_atMost_nat @ N ) )
% 12.45/7.00 = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8984_of__nat__id,axiom,
% 12.45/7.00 ( semiri1316708129612266289at_nat
% 12.45/7.00 = ( ^ [N2: nat] : N2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8985_real__scaleR__def,axiom,
% 12.45/7.00 real_V1485227260804924795R_real = times_times_real ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8986_complex__scaleR,axiom,
% 12.45/7.00 ! [R2: real,A: real,B2: real] :
% 12.45/7.00 ( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B2 ) )
% 12.45/7.00 = ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8987_bij__betw__nth__root__unity,axiom,
% 12.45/7.00 ! [C: complex,N: nat] :
% 12.45/7.00 ( ( C != zero_zero_complex )
% 12.45/7.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 12.45/7.00 @ ( collect_complex
% 12.45/7.00 @ ^ [Z5: complex] :
% 12.45/7.00 ( ( power_power_complex @ Z5 @ N )
% 12.45/7.00 = one_one_complex ) )
% 12.45/7.00 @ ( collect_complex
% 12.45/7.00 @ ^ [Z5: complex] :
% 12.45/7.00 ( ( power_power_complex @ Z5 @ N )
% 12.45/7.00 = C ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8988_infinite__int__iff__unbounded,axiom,
% 12.45/7.00 ! [S2: set_int] :
% 12.45/7.00 ( ( ~ ( finite_finite_int @ S2 ) )
% 12.45/7.00 = ( ! [M6: int] :
% 12.45/7.00 ? [N2: int] :
% 12.45/7.00 ( ( ord_less_int @ M6 @ ( abs_abs_int @ N2 ) )
% 12.45/7.00 & ( member_int @ N2 @ S2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8989_Arg__def,axiom,
% 12.45/7.00 ( arg
% 12.45/7.00 = ( ^ [Z5: complex] :
% 12.45/7.00 ( if_real @ ( Z5 = zero_zero_complex ) @ zero_zero_real
% 12.45/7.00 @ ( fChoice_real
% 12.45/7.00 @ ^ [A4: real] :
% 12.45/7.00 ( ( ( sgn_sgn_complex @ Z5 )
% 12.45/7.00 = ( cis @ A4 ) )
% 12.45/7.00 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
% 12.45/7.00 & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8990_real__root__zero,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( root @ N @ zero_zero_real )
% 12.45/7.00 = zero_zero_real ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8991_real__root__Suc__0,axiom,
% 12.45/7.00 ! [X4: real] :
% 12.45/7.00 ( ( root @ ( suc @ zero_zero_nat ) @ X4 )
% 12.45/7.00 = X4 ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8992_real__root__eq__iff,axiom,
% 12.45/7.00 ! [N: nat,X4: real,Y3: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ( root @ N @ X4 )
% 12.45/7.00 = ( root @ N @ Y3 ) )
% 12.45/7.00 = ( X4 = Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8993_root__0,axiom,
% 12.45/7.00 ! [X4: real] :
% 12.45/7.00 ( ( root @ zero_zero_nat @ X4 )
% 12.45/7.00 = zero_zero_real ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8994_real__root__eq__0__iff,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ( root @ N @ X4 )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 = ( X4 = zero_zero_real ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8995_real__root__less__iff,axiom,
% 12.45/7.00 ! [N: nat,X4: real,Y3: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) )
% 12.45/7.00 = ( ord_less_real @ X4 @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8996_real__root__le__iff,axiom,
% 12.45/7.00 ! [N: nat,X4: real,Y3: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_eq_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) )
% 12.45/7.00 = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8997_real__root__one,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( root @ N @ one_one_real )
% 12.45/7.00 = one_one_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8998_real__root__eq__1__iff,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ( root @ N @ X4 )
% 12.45/7.00 = one_one_real )
% 12.45/7.00 = ( X4 = one_one_real ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_8999_real__root__gt__0__iff,axiom,
% 12.45/7.00 ! [N: nat,Y3: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 12.45/7.00 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9000_real__root__lt__0__iff,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ ( root @ N @ X4 ) @ zero_zero_real )
% 12.45/7.00 = ( ord_less_real @ X4 @ zero_zero_real ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9001_real__root__le__0__iff,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_eq_real @ ( root @ N @ X4 ) @ zero_zero_real )
% 12.45/7.00 = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9002_real__root__ge__0__iff,axiom,
% 12.45/7.00 ! [N: nat,Y3: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 12.45/7.00 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9003_real__root__gt__1__iff,axiom,
% 12.45/7.00 ! [N: nat,Y3: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
% 12.45/7.00 = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9004_real__root__lt__1__iff,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ ( root @ N @ X4 ) @ one_one_real )
% 12.45/7.00 = ( ord_less_real @ X4 @ one_one_real ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9005_real__root__ge__1__iff,axiom,
% 12.45/7.00 ! [N: nat,Y3: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
% 12.45/7.00 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9006_real__root__le__1__iff,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_eq_real @ ( root @ N @ X4 ) @ one_one_real )
% 12.45/7.00 = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9007_real__root__pow__pos2,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ( ( power_power_real @ ( root @ N @ X4 ) @ N )
% 12.45/7.00 = X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9008_real__root__minus,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( root @ N @ ( uminus_uminus_real @ X4 ) )
% 12.45/7.00 = ( uminus_uminus_real @ ( root @ N @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9009_real__root__inverse,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( root @ N @ ( inverse_inverse_real @ X4 ) )
% 12.45/7.00 = ( inverse_inverse_real @ ( root @ N @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9010_real__root__commute,axiom,
% 12.45/7.00 ! [M: nat,N: nat,X4: real] :
% 12.45/7.00 ( ( root @ M @ ( root @ N @ X4 ) )
% 12.45/7.00 = ( root @ N @ ( root @ M @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9011_real__root__mult,axiom,
% 12.45/7.00 ! [N: nat,X4: real,Y3: real] :
% 12.45/7.00 ( ( root @ N @ ( times_times_real @ X4 @ Y3 ) )
% 12.45/7.00 = ( times_times_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9012_real__root__mult__exp,axiom,
% 12.45/7.00 ! [M: nat,N: nat,X4: real] :
% 12.45/7.00 ( ( root @ ( times_times_nat @ M @ N ) @ X4 )
% 12.45/7.00 = ( root @ M @ ( root @ N @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9013_real__root__divide,axiom,
% 12.45/7.00 ! [N: nat,X4: real,Y3: real] :
% 12.45/7.00 ( ( root @ N @ ( divide_divide_real @ X4 @ Y3 ) )
% 12.45/7.00 = ( divide_divide_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9014_real__root__pos__pos__le,axiom,
% 12.45/7.00 ! [X4: real,N: nat] :
% 12.45/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9015_real__root__less__mono,axiom,
% 12.45/7.00 ! [N: nat,X4: real,Y3: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ X4 @ Y3 )
% 12.45/7.00 => ( ord_less_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9016_real__root__le__mono,axiom,
% 12.45/7.00 ! [N: nat,X4: real,Y3: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_eq_real @ X4 @ Y3 )
% 12.45/7.00 => ( ord_less_eq_real @ ( root @ N @ X4 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9017_real__root__power,axiom,
% 12.45/7.00 ! [N: nat,X4: real,K: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( root @ N @ ( power_power_real @ X4 @ K ) )
% 12.45/7.00 = ( power_power_real @ ( root @ N @ X4 ) @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9018_real__root__abs,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( root @ N @ ( abs_abs_real @ X4 ) )
% 12.45/7.00 = ( abs_abs_real @ ( root @ N @ X4 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9019_sgn__root,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( sgn_sgn_real @ ( root @ N @ X4 ) )
% 12.45/7.00 = ( sgn_sgn_real @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9020_real__root__gt__zero,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X4 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9021_real__root__strict__decreasing,axiom,
% 12.45/7.00 ! [N: nat,N3: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_nat @ N @ N3 )
% 12.45/7.00 => ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/7.00 => ( ord_less_real @ ( root @ N3 @ X4 ) @ ( root @ N @ X4 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9022_sqrt__def,axiom,
% 12.45/7.00 ( sqrt
% 12.45/7.00 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9023_root__abs__power,axiom,
% 12.45/7.00 ! [N: nat,Y3: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
% 12.45/7.00 = ( abs_abs_real @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9024_real__root__pos__pos,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X4 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9025_real__root__strict__increasing,axiom,
% 12.45/7.00 ! [N: nat,N3: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_nat @ N @ N3 )
% 12.45/7.00 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/7.00 => ( ord_less_real @ ( root @ N @ X4 ) @ ( root @ N3 @ X4 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9026_real__root__decreasing,axiom,
% 12.45/7.00 ! [N: nat,N3: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.45/7.00 => ( ( ord_less_eq_real @ one_one_real @ X4 )
% 12.45/7.00 => ( ord_less_eq_real @ ( root @ N3 @ X4 ) @ ( root @ N @ X4 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9027_real__root__pow__pos,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ( ( power_power_real @ ( root @ N @ X4 ) @ N )
% 12.45/7.00 = X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9028_real__root__pos__unique,axiom,
% 12.45/7.00 ! [N: nat,Y3: real,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.45/7.00 => ( ( ( power_power_real @ Y3 @ N )
% 12.45/7.00 = X4 )
% 12.45/7.00 => ( ( root @ N @ X4 )
% 12.45/7.00 = Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9029_real__root__power__cancel,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ( ( root @ N @ ( power_power_real @ X4 @ N ) )
% 12.45/7.00 = X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9030_odd__real__root__pow,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/7.00 => ( ( power_power_real @ ( root @ N @ X4 ) @ N )
% 12.45/7.00 = X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9031_odd__real__root__unique,axiom,
% 12.45/7.00 ! [N: nat,Y3: real,X4: real] :
% 12.45/7.00 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/7.00 => ( ( ( power_power_real @ Y3 @ N )
% 12.45/7.00 = X4 )
% 12.45/7.00 => ( ( root @ N @ X4 )
% 12.45/7.00 = Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9032_odd__real__root__power__cancel,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/7.00 => ( ( root @ N @ ( power_power_real @ X4 @ N ) )
% 12.45/7.00 = X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9033_real__root__increasing,axiom,
% 12.45/7.00 ! [N: nat,N3: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.45/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/7.00 => ( ord_less_eq_real @ ( root @ N @ X4 ) @ ( root @ N3 @ X4 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9034_sgn__power__root,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X4 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X4 ) ) @ N ) )
% 12.45/7.00 = X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9035_root__sgn__power,axiom,
% 12.45/7.00 ! [N: nat,Y3: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
% 12.45/7.00 = Y3 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9036_ln__root,axiom,
% 12.45/7.00 ! [N: nat,B2: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/7.00 => ( ( ln_ln_real @ ( root @ N @ B2 ) )
% 12.45/7.00 = ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9037_log__root,axiom,
% 12.45/7.00 ! [N: nat,A: real,B2: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.45/7.00 => ( ( log @ B2 @ ( root @ N @ A ) )
% 12.45/7.00 = ( divide_divide_real @ ( log @ B2 @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9038_log__base__root,axiom,
% 12.45/7.00 ! [N: nat,B2: real,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ zero_zero_real @ B2 )
% 12.45/7.00 => ( ( log @ ( root @ N @ B2 ) @ X4 )
% 12.45/7.00 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X4 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9039_split__root,axiom,
% 12.45/7.00 ! [P: real > $o,N: nat,X4: real] :
% 12.45/7.00 ( ( P @ ( root @ N @ X4 ) )
% 12.45/7.00 = ( ( ( N = zero_zero_nat )
% 12.45/7.00 => ( P @ zero_zero_real ) )
% 12.45/7.00 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ! [Y: real] :
% 12.45/7.00 ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 12.45/7.00 = X4 )
% 12.45/7.00 => ( P @ Y ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9040_infinite__nat__iff__unbounded,axiom,
% 12.45/7.00 ! [S2: set_nat] :
% 12.45/7.00 ( ( ~ ( finite_finite_nat @ S2 ) )
% 12.45/7.00 = ( ! [M6: nat] :
% 12.45/7.00 ? [N2: nat] :
% 12.45/7.00 ( ( ord_less_nat @ M6 @ N2 )
% 12.45/7.00 & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9041_unbounded__k__infinite,axiom,
% 12.45/7.00 ! [K: nat,S2: set_nat] :
% 12.45/7.00 ( ! [M4: nat] :
% 12.45/7.00 ( ( ord_less_nat @ K @ M4 )
% 12.45/7.00 => ? [N6: nat] :
% 12.45/7.00 ( ( ord_less_nat @ M4 @ N6 )
% 12.45/7.00 & ( member_nat @ N6 @ S2 ) ) )
% 12.45/7.00 => ~ ( finite_finite_nat @ S2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9042_infinite__nat__iff__unbounded__le,axiom,
% 12.45/7.00 ! [S2: set_nat] :
% 12.45/7.00 ( ( ~ ( finite_finite_nat @ S2 ) )
% 12.45/7.00 = ( ! [M6: nat] :
% 12.45/7.00 ? [N2: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ M6 @ N2 )
% 12.45/7.00 & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9043_root__powr__inverse,axiom,
% 12.45/7.00 ! [N: nat,X4: real] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.00 => ( ( root @ N @ X4 )
% 12.45/7.00 = ( powr_real @ X4 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9044_divmod__BitM__2__eq,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 12.45/7.00 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9045_insert__simp__excp,axiom,
% 12.45/7.00 ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X4: nat,Ma: nat,Summary: vEBT_VEBT] :
% 12.45/7.00 ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/7.00 => ( ( ord_less_nat @ X4 @ Mi )
% 12.45/7.00 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 12.45/7.00 => ( ( X4 != Ma )
% 12.45/7.00 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X4 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9046_insert__simp__norm,axiom,
% 12.45/7.00 ! [X4: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 12.45/7.00 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/7.00 => ( ( ord_less_nat @ Mi @ X4 )
% 12.45/7.00 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 12.45/7.00 => ( ( X4 != Ma )
% 12.45/7.00 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X4 )
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X4 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9047_max__Suc__Suc,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.45/7.00 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9048_max__0R,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_max_nat @ N @ zero_zero_nat )
% 12.45/7.00 = N ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9049_max__0L,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_max_nat @ zero_zero_nat @ N )
% 12.45/7.00 = N ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9050_max__nat_Oright__neutral,axiom,
% 12.45/7.00 ! [A: nat] :
% 12.45/7.00 ( ( ord_max_nat @ A @ zero_zero_nat )
% 12.45/7.00 = A ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9051_max__nat_Oneutr__eq__iff,axiom,
% 12.45/7.00 ! [A: nat,B2: nat] :
% 12.45/7.00 ( ( zero_zero_nat
% 12.45/7.00 = ( ord_max_nat @ A @ B2 ) )
% 12.45/7.00 = ( ( A = zero_zero_nat )
% 12.45/7.00 & ( B2 = zero_zero_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9052_max__nat_Oleft__neutral,axiom,
% 12.45/7.00 ! [A: nat] :
% 12.45/7.00 ( ( ord_max_nat @ zero_zero_nat @ A )
% 12.45/7.00 = A ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9053_max__nat_Oeq__neutr__iff,axiom,
% 12.45/7.00 ! [A: nat,B2: nat] :
% 12.45/7.00 ( ( ( ord_max_nat @ A @ B2 )
% 12.45/7.00 = zero_zero_nat )
% 12.45/7.00 = ( ( A = zero_zero_nat )
% 12.45/7.00 & ( B2 = zero_zero_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9054_max__numeral__Suc,axiom,
% 12.45/7.00 ! [K: num,N: nat] :
% 12.45/7.00 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 12.45/7.00 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9055_max__Suc__numeral,axiom,
% 12.45/7.00 ! [N: nat,K: num] :
% 12.45/7.00 ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.45/7.00 = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9056_pred__numeral__simps_I2_J,axiom,
% 12.45/7.00 ! [K: num] :
% 12.45/7.00 ( ( pred_numeral @ ( bit0 @ K ) )
% 12.45/7.00 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9057_nat__add__max__right,axiom,
% 12.45/7.00 ! [M: nat,N: nat,Q2: nat] :
% 12.45/7.00 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 12.45/7.00 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9058_nat__add__max__left,axiom,
% 12.45/7.00 ! [M: nat,N: nat,Q2: nat] :
% 12.45/7.00 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 12.45/7.00 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9059_nat__mult__max__right,axiom,
% 12.45/7.00 ! [M: nat,N: nat,Q2: nat] :
% 12.45/7.00 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 12.45/7.00 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9060_nat__mult__max__left,axiom,
% 12.45/7.00 ! [M: nat,N: nat,Q2: nat] :
% 12.45/7.00 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 12.45/7.00 = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9061_semiring__norm_I26_J,axiom,
% 12.45/7.00 ( ( bitM @ one )
% 12.45/7.00 = one ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9062_nat__minus__add__max,axiom,
% 12.45/7.00 ! [N: nat,M: nat] :
% 12.45/7.00 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 12.45/7.00 = ( ord_max_nat @ N @ M ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9063_semiring__norm_I27_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bitM @ ( bit0 @ N ) )
% 12.45/7.00 = ( bit1 @ ( bitM @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9064_semiring__norm_I28_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bitM @ ( bit1 @ N ) )
% 12.45/7.00 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9065_inc__BitM__eq,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( inc @ ( bitM @ N ) )
% 12.45/7.00 = ( bit0 @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9066_BitM__inc__eq,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bitM @ ( inc @ N ) )
% 12.45/7.00 = ( bit1 @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9067_eval__nat__numeral_I2_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 12.45/7.00 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9068_one__plus__BitM,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 12.45/7.00 = ( bit0 @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9069_BitM__plus__one,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 12.45/7.00 = ( bit0 @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9070_vebt__insert_Osimps_I5_J,axiom,
% 12.45/7.00 ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
% 12.45/7.00 ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X4 )
% 12.45/7.00 = ( if_VEBT_VEBT
% 12.45/7.00 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.45/7.00 & ~ ( ( X4 = Mi )
% 12.45/7.00 | ( X4 = Ma ) ) )
% 12.45/7.00 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ X4 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 12.45/7.00 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9071_vebt__insert_Oelims,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
% 12.45/7.00 ( ( ( vEBT_vebt_insert @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ! [A5: $o,B5: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.00 => ~ ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( vEBT_Leaf @ $true @ B5 ) ) )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( ( ( Xa = one_one_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ $true ) ) )
% 12.45/7.00 & ( ( Xa != one_one_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) ) )
% 12.45/7.00 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) ) )
% 12.45/7.00 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
% 12.45/7.00 => ( ! [V3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) ) )
% 12.45/7.00 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( if_VEBT_VEBT
% 12.45/7.00 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 & ~ ( ( Xa = Mi2 )
% 12.45/7.00 | ( Xa = Ma2 ) ) )
% 12.45/7.00 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 12.45/7.00 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9072_vebt__insert_Opelims,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
% 12.45/7.00 ( ( ( vEBT_vebt_insert @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [A5: $o,B5: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.00 => ( ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( vEBT_Leaf @ $true @ B5 ) ) )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( ( ( Xa = one_one_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ $true ) ) )
% 12.45/7.00 & ( ( Xa != one_one_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
% 12.45/7.00 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ Xa ) ) ) )
% 12.45/7.00 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
% 12.45/7.00 => ( ! [V3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
% 12.45/7.00 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( if_VEBT_VEBT
% 12.45/7.00 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 12.45/7.00 & ~ ( ( Xa = Mi2 )
% 12.45/7.00 | ( Xa = Ma2 ) ) )
% 12.45/7.00 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 12.45/7.00 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9073_vebt__buildup_Opelims,axiom,
% 12.45/7.00 ! [X4: nat,Y3: vEBT_VEBT] :
% 12.45/7.00 ( ( ( vEBT_vebt_buildup @ X4 )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X4 )
% 12.45/7.00 => ( ( ( X4 = zero_zero_nat )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( vEBT_Leaf @ $false @ $false ) )
% 12.45/7.00 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 12.45/7.00 => ( ( ( X4
% 12.45/7.00 = ( suc @ zero_zero_nat ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( vEBT_Leaf @ $false @ $false ) )
% 12.45/7.00 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 12.45/7.00 => ~ ! [Va: nat] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( suc @ ( suc @ Va ) ) )
% 12.45/7.00 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( 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 ) ) ) ) ) ) )
% 12.45/7.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( 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 ) ) ) ) ) ) ) ) )
% 12.45/7.00 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9074_max__enat__simps_I2_J,axiom,
% 12.45/7.00 ! [Q2: extended_enat] :
% 12.45/7.00 ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 12.45/7.00 = Q2 ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9075_max__enat__simps_I3_J,axiom,
% 12.45/7.00 ! [Q2: extended_enat] :
% 12.45/7.00 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 12.45/7.00 = Q2 ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9076_divmod__step__nat__def,axiom,
% 12.45/7.00 ( unique5026877609467782581ep_nat
% 12.45/7.00 = ( ^ [L: num] :
% 12.45/7.00 ( produc2626176000494625587at_nat
% 12.45/7.00 @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ 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 @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9077_divmod__step__int__def,axiom,
% 12.45/7.00 ( unique5024387138958732305ep_int
% 12.45/7.00 = ( ^ [L: num] :
% 12.45/7.00 ( produc4245557441103728435nt_int
% 12.45/7.00 @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ 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 @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9078_divmod__nat__if,axiom,
% 12.45/7.00 ( divmod_nat
% 12.45/7.00 = ( ^ [M6: nat,N2: nat] :
% 12.45/7.00 ( if_Pro6206227464963214023at_nat
% 12.45/7.00 @ ( ( N2 = zero_zero_nat )
% 12.45/7.00 | ( ord_less_nat @ M6 @ N2 ) )
% 12.45/7.00 @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
% 12.45/7.00 @ ( produc2626176000494625587at_nat
% 12.45/7.00 @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
% 12.45/7.00 @ ( divmod_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9079_arctan__def,axiom,
% 12.45/7.00 ( arctan
% 12.45/7.00 = ( ^ [Y: real] :
% 12.45/7.00 ( the_real
% 12.45/7.00 @ ^ [X: real] :
% 12.45/7.00 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.45/7.00 & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/7.00 & ( ( tan_real @ X )
% 12.45/7.00 = Y ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9080_ln__neg__is__const,axiom,
% 12.45/7.00 ! [X4: real] :
% 12.45/7.00 ( ( ord_less_eq_real @ X4 @ zero_zero_real )
% 12.45/7.00 => ( ( ln_ln_real @ X4 )
% 12.45/7.00 = ( the_real
% 12.45/7.00 @ ^ [X: real] : $false ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9081_arccos__def,axiom,
% 12.45/7.00 ( arccos
% 12.45/7.00 = ( ^ [Y: real] :
% 12.45/7.00 ( the_real
% 12.45/7.00 @ ^ [X: real] :
% 12.45/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.45/7.00 & ( ord_less_eq_real @ X @ pi )
% 12.45/7.00 & ( ( cos_real @ X )
% 12.45/7.00 = Y ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9082_divmod__nat__def,axiom,
% 12.45/7.00 ( divmod_nat
% 12.45/7.00 = ( ^ [M6: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N2 ) @ ( modulo_modulo_nat @ M6 @ N2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9083_pi__half,axiom,
% 12.45/7.00 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.45/7.00 = ( the_real
% 12.45/7.00 @ ^ [X: real] :
% 12.45/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.45/7.00 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.45/7.00 & ( ( cos_real @ X )
% 12.45/7.00 = zero_zero_real ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9084_pi__def,axiom,
% 12.45/7.00 ( pi
% 12.45/7.00 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 12.45/7.00 @ ( the_real
% 12.45/7.00 @ ^ [X: real] :
% 12.45/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.45/7.00 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.45/7.00 & ( ( cos_real @ X )
% 12.45/7.00 = zero_zero_real ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9085_arcsin__def,axiom,
% 12.45/7.00 ( arcsin
% 12.45/7.00 = ( ^ [Y: real] :
% 12.45/7.00 ( the_real
% 12.45/7.00 @ ^ [X: real] :
% 12.45/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.45/7.00 & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/7.00 & ( ( sin_real @ X )
% 12.45/7.00 = Y ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9086_or__int__unfold,axiom,
% 12.45/7.00 ( bit_se1409905431419307370or_int
% 12.45/7.00 = ( ^ [K3: int,L: int] :
% 12.45/7.00 ( if_int
% 12.45/7.00 @ ( ( K3
% 12.45/7.00 = ( uminus_uminus_int @ one_one_int ) )
% 12.45/7.00 | ( L
% 12.45/7.00 = ( uminus_uminus_int @ one_one_int ) ) )
% 12.45/7.00 @ ( uminus_uminus_int @ one_one_int )
% 12.45/7.00 @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9087_horner__sum__of__bool__2__less,axiom,
% 12.45/7.00 ! [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 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9088_or__nonnegative__int__iff,axiom,
% 12.45/7.00 ! [K: int,L2: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
% 12.45/7.00 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.45/7.00 & ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9089_or__negative__int__iff,axiom,
% 12.45/7.00 ! [K: int,L2: int] :
% 12.45/7.00 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ zero_zero_int )
% 12.45/7.00 = ( ( ord_less_int @ K @ zero_zero_int )
% 12.45/7.00 | ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9090_or__minus__numerals_I2_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9091_or__minus__numerals_I6_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 12.45/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9092_or__minus__minus__numerals,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( 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 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9093_and__minus__minus__numerals,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( 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 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9094_bit__or__int__iff,axiom,
% 12.45/7.00 ! [K: int,L2: int,N: nat] :
% 12.45/7.00 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N )
% 12.45/7.00 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 12.45/7.00 | ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9095_OR__lower,axiom,
% 12.45/7.00 ! [X4: int,Y3: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/7.00 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 12.45/7.00 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X4 @ Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9096_or__greater__eq,axiom,
% 12.45/7.00 ! [L2: int,K: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 12.45/7.00 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9097_plus__and__or,axiom,
% 12.45/7.00 ! [X4: int,Y3: int] :
% 12.45/7.00 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X4 @ Y3 ) @ ( bit_se1409905431419307370or_int @ X4 @ Y3 ) )
% 12.45/7.00 = ( plus_plus_int @ X4 @ Y3 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9098_or__int__def,axiom,
% 12.45/7.00 ( bit_se1409905431419307370or_int
% 12.45/7.00 = ( ^ [K3: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9099_or__not__numerals_I1_J,axiom,
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.45/7.00 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9100_xor__int__def,axiom,
% 12.45/7.00 ( bit_se6526347334894502574or_int
% 12.45/7.00 = ( ^ [K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9101_concat__bit__def,axiom,
% 12.45/7.00 ( bit_concat_bit
% 12.45/7.00 = ( ^ [N2: nat,K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9102_set__bit__int__def,axiom,
% 12.45/7.00 ( bit_se7879613467334960850it_int
% 12.45/7.00 = ( ^ [N2: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9103_or__not__numerals_I4_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.45/7.00 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9104_or__not__numerals_I2_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.45/7.00 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9105_or__not__numerals_I3_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.45/7.00 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9106_or__not__numerals_I7_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.45/7.00 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9107_or__not__numerals_I6_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.45/7.00 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9108_OR__upper,axiom,
% 12.45/7.00 ! [X4: int,N: nat,Y3: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.45/7.00 => ( ( ord_less_int @ X4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/7.00 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.45/7.00 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X4 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9109_or__not__numerals_I5_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.45/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9110_or__not__numerals_I9_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.45/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9111_or__not__numerals_I8_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.45/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9112_or__int__rec,axiom,
% 12.45/7.00 ( bit_se1409905431419307370or_int
% 12.45/7.00 = ( ^ [K3: int,L: int] :
% 12.45/7.00 ( plus_plus_int
% 12.45/7.00 @ ( zero_n2684676970156552555ol_int
% 12.45/7.00 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 12.45/7.00 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 12.45/7.00 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9113_or__minus__numerals_I5_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 12.45/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9114_or__minus__numerals_I1_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9115_or__nat__numerals_I2_J,axiom,
% 12.45/7.00 ! [Y3: num] :
% 12.45/7.00 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 12.45/7.00 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9116_or__nat__numerals_I4_J,axiom,
% 12.45/7.00 ! [X4: num] :
% 12.45/7.00 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 12.45/7.00 = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9117_or__nat__numerals_I1_J,axiom,
% 12.45/7.00 ! [Y3: num] :
% 12.45/7.00 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 12.45/7.00 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9118_or__nat__numerals_I3_J,axiom,
% 12.45/7.00 ! [X4: num] :
% 12.45/7.00 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
% 12.45/7.00 = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9119_or__minus__numerals_I4_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9120_or__minus__numerals_I8_J,axiom,
% 12.45/7.00 ! [N: num,M: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9121_or__minus__numerals_I7_J,axiom,
% 12.45/7.00 ! [N: num,M: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9122_or__minus__numerals_I3_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9123_or__not__num__neg_Osimps_I1_J,axiom,
% 12.45/7.00 ( ( bit_or_not_num_neg @ one @ one )
% 12.45/7.00 = one ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9124_set__bit__nat__def,axiom,
% 12.45/7.00 ( bit_se7882103937844011126it_nat
% 12.45/7.00 = ( ^ [M6: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9125_or__not__num__neg_Osimps_I4_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 12.45/7.00 = ( bit0 @ one ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9126_or__not__num__neg_Osimps_I6_J,axiom,
% 12.45/7.00 ! [N: num,M: num] :
% 12.45/7.00 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 12.45/7.00 = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9127_or__not__num__neg_Osimps_I3_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 12.45/7.00 = ( bit1 @ M ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9128_or__not__num__neg_Osimps_I7_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 12.45/7.00 = one ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9129_or__not__num__neg_Osimps_I5_J,axiom,
% 12.45/7.00 ! [N: num,M: num] :
% 12.45/7.00 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 12.45/7.00 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9130_or__not__num__neg_Osimps_I9_J,axiom,
% 12.45/7.00 ! [N: num,M: num] :
% 12.45/7.00 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 12.45/7.00 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9131_or__nat__def,axiom,
% 12.45/7.00 ( bit_se1412395901928357646or_nat
% 12.45/7.00 = ( ^ [M6: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9132_or__not__num__neg_Osimps_I2_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 12.45/7.00 = ( bit1 @ M ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9133_or__not__num__neg_Osimps_I8_J,axiom,
% 12.45/7.00 ! [N: num,M: num] :
% 12.45/7.00 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 12.45/7.00 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9134_or__not__num__neg_Oelims,axiom,
% 12.45/7.00 ! [X4: num,Xa: num,Y3: num] :
% 12.45/7.00 ( ( ( bit_or_not_num_neg @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3 != one ) ) )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ! [M4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( bit1 @ M4 ) ) ) )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ! [M4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( bit1 @ M4 ) ) ) )
% 12.45/7.00 => ( ( ? [N4: num] :
% 12.45/7.00 ( X4
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( bit0 @ one ) ) ) )
% 12.45/7.00 => ( ! [N4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ! [M4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 12.45/7.00 => ( ! [N4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ! [M4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( bit0 @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 12.45/7.00 => ( ( ? [N4: num] :
% 12.45/7.00 ( X4
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3 != one ) ) )
% 12.45/7.00 => ( ! [N4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ! [M4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 12.45/7.00 => ~ ! [N4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ! [M4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9135_numeral__or__not__num__eq,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9136_int__numeral__not__or__num__neg,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9137_int__numeral__or__not__num__neg,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9138_or__Suc__0__eq,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.45/7.00 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9139_Suc__0__or__eq,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.45/7.00 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9140_or__nat__rec,axiom,
% 12.45/7.00 ( bit_se1412395901928357646or_nat
% 12.45/7.00 = ( ^ [M6: nat,N2: nat] :
% 12.45/7.00 ( plus_plus_nat
% 12.45/7.00 @ ( zero_n2687167440665602831ol_nat
% 12.45/7.00 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 12.45/7.00 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 12.45/7.00 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9141_or__nat__unfold,axiom,
% 12.45/7.00 ( bit_se1412395901928357646or_nat
% 12.45/7.00 = ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9142_Sum__Ico__nat,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [X: nat] : X
% 12.45/7.00 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 12.45/7.00 = ( 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 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9143_sum__power2,axiom,
% 12.45/7.00 ! [K: nat] :
% 12.45/7.00 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 12.45/7.00 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9144_finite__atLeastLessThan,axiom,
% 12.45/7.00 ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9145_atLeastLessThan__singleton,axiom,
% 12.45/7.00 ! [M: nat] :
% 12.45/7.00 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 12.45/7.00 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9146_all__nat__less__eq,axiom,
% 12.45/7.00 ! [N: nat,P: nat > $o] :
% 12.45/7.00 ( ( ! [M6: nat] :
% 12.45/7.00 ( ( ord_less_nat @ M6 @ N )
% 12.45/7.00 => ( P @ M6 ) ) )
% 12.45/7.00 = ( ! [X: nat] :
% 12.45/7.00 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 12.45/7.00 => ( P @ X ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9147_ex__nat__less__eq,axiom,
% 12.45/7.00 ! [N: nat,P: nat > $o] :
% 12.45/7.00 ( ( ? [M6: nat] :
% 12.45/7.00 ( ( ord_less_nat @ M6 @ N )
% 12.45/7.00 & ( P @ M6 ) ) )
% 12.45/7.00 = ( ? [X: nat] :
% 12.45/7.00 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 12.45/7.00 & ( P @ X ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9148_atLeastLessThanSuc__atLeastAtMost,axiom,
% 12.45/7.00 ! [L2: nat,U: nat] :
% 12.45/7.00 ( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U ) )
% 12.45/7.00 = ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9149_lessThan__atLeast0,axiom,
% 12.45/7.00 ( set_ord_lessThan_nat
% 12.45/7.00 = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9150_atLeastLessThan0,axiom,
% 12.45/7.00 ! [M: nat] :
% 12.45/7.00 ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 12.45/7.00 = bot_bot_set_nat ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9151_atLeast0__lessThan__Suc,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 12.45/7.00 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9152_subset__eq__atLeast0__lessThan__finite,axiom,
% 12.45/7.00 ! [N3: set_nat,N: nat] :
% 12.45/7.00 ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 12.45/7.00 => ( finite_finite_nat @ N3 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9153_atLeastLessThanSuc,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ( ord_less_eq_nat @ M @ N )
% 12.45/7.00 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 12.45/7.00 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_nat @ M @ N )
% 12.45/7.00 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 12.45/7.00 = bot_bot_set_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9154_prod__Suc__fact,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 12.45/7.00 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9155_prod__Suc__Suc__fact,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 12.45/7.00 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9156_atLeastLessThan__nat__numeral,axiom,
% 12.45/7.00 ! [M: nat,K: num] :
% 12.45/7.00 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 12.45/7.00 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 12.45/7.00 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 12.45/7.00 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 12.45/7.00 = bot_bot_set_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9157_atLeast1__lessThan__eq__remove0,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.45/7.00 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9158_Chebyshev__sum__upper__nat,axiom,
% 12.45/7.00 ! [N: nat,A: nat > nat,B2: nat > nat] :
% 12.45/7.00 ( ! [I3: nat,J2: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ I3 @ J2 )
% 12.45/7.00 => ( ( ord_less_nat @ J2 @ N )
% 12.45/7.00 => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
% 12.45/7.00 => ( ! [I3: nat,J2: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ I3 @ J2 )
% 12.45/7.00 => ( ( ord_less_nat @ J2 @ N )
% 12.45/7.00 => ( ord_less_eq_nat @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
% 12.45/7.00 => ( ord_less_eq_nat
% 12.45/7.00 @ ( times_times_nat @ N
% 12.45/7.00 @ ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( B2 @ I2 ) )
% 12.45/7.00 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 12.45/7.00 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9159_finite__atLeastLessThan__int,axiom,
% 12.45/7.00 ! [L2: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9160_finite__atLeastZeroLessThan__int,axiom,
% 12.45/7.00 ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9161_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 12.45/7.00 ! [L2: int,U: int] :
% 12.45/7.00 ( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
% 12.45/7.00 = ( set_or1266510415728281911st_int @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9162_valid__eq,axiom,
% 12.45/7.00 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9163_valid__eq1,axiom,
% 12.45/7.00 ! [T: vEBT_VEBT,D: nat] :
% 12.45/7.00 ( ( vEBT_invar_vebt @ T @ D )
% 12.45/7.00 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9164_valid__eq2,axiom,
% 12.45/7.00 ! [T: vEBT_VEBT,D: nat] :
% 12.45/7.00 ( ( vEBT_VEBT_valid @ T @ D )
% 12.45/7.00 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9165_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 12.45/7.00 ! [Uu3: $o,Uv2: $o,D: nat] :
% 12.45/7.00 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ D )
% 12.45/7.00 = ( D = one_one_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9166_divmod__step__integer__def,axiom,
% 12.45/7.00 ( unique4921790084139445826nteger
% 12.45/7.00 = ( ^ [L: num] :
% 12.45/7.00 ( produc6916734918728496179nteger
% 12.45/7.00 @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9167_csqrt_Osimps_I1_J,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( re @ ( csqrt @ Z ) )
% 12.45/7.00 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9168_complex__Re__numeral,axiom,
% 12.45/7.00 ! [V: num] :
% 12.45/7.00 ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 12.45/7.00 = ( numeral_numeral_real @ V ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9169_Re__divide__of__nat,axiom,
% 12.45/7.00 ! [Z: complex,N: nat] :
% 12.45/7.00 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 12.45/7.00 = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9170_Re__divide__of__real,axiom,
% 12.45/7.00 ! [Z: complex,R2: real] :
% 12.45/7.00 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 12.45/7.00 = ( divide_divide_real @ ( re @ Z ) @ R2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9171_Re__sgn,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( re @ ( sgn_sgn_complex @ Z ) )
% 12.45/7.00 = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9172_Re__divide__numeral,axiom,
% 12.45/7.00 ! [Z: complex,W: num] :
% 12.45/7.00 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 12.45/7.00 = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9173_plus__integer__code_I1_J,axiom,
% 12.45/7.00 ! [K: code_integer] :
% 12.45/7.00 ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
% 12.45/7.00 = K ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9174_plus__integer__code_I2_J,axiom,
% 12.45/7.00 ! [L2: code_integer] :
% 12.45/7.00 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L2 )
% 12.45/7.00 = L2 ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9175_times__integer__code_I1_J,axiom,
% 12.45/7.00 ! [K: code_integer] :
% 12.45/7.00 ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 12.45/7.00 = zero_z3403309356797280102nteger ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9176_times__integer__code_I2_J,axiom,
% 12.45/7.00 ! [L2: code_integer] :
% 12.45/7.00 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L2 )
% 12.45/7.00 = zero_z3403309356797280102nteger ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9177_divmod__integer_H__def,axiom,
% 12.45/7.00 ( unique3479559517661332726nteger
% 12.45/7.00 = ( ^ [M6: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9178_sgn__integer__code,axiom,
% 12.45/7.00 ( sgn_sgn_Code_integer
% 12.45/7.00 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9179_minus__integer__code_I2_J,axiom,
% 12.45/7.00 ! [L2: code_integer] :
% 12.45/7.00 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L2 )
% 12.45/7.00 = ( uminus1351360451143612070nteger @ L2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9180_minus__integer__code_I1_J,axiom,
% 12.45/7.00 ! [K: code_integer] :
% 12.45/7.00 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 12.45/7.00 = K ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9181_complex__Re__le__cmod,axiom,
% 12.45/7.00 ! [X4: complex] : ( ord_less_eq_real @ ( re @ X4 ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9182_one__complex_Osimps_I1_J,axiom,
% 12.45/7.00 ( ( re @ one_one_complex )
% 12.45/7.00 = one_one_real ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9183_plus__complex_Osimps_I1_J,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( re @ ( plus_plus_complex @ X4 @ Y3 ) )
% 12.45/7.00 = ( plus_plus_real @ ( re @ X4 ) @ ( re @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9184_scaleR__complex_Osimps_I1_J,axiom,
% 12.45/7.00 ! [R2: real,X4: complex] :
% 12.45/7.00 ( ( re @ ( real_V2046097035970521341omplex @ R2 @ X4 ) )
% 12.45/7.00 = ( times_times_real @ R2 @ ( re @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9185_minus__complex_Osimps_I1_J,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( re @ ( minus_minus_complex @ X4 @ Y3 ) )
% 12.45/7.00 = ( minus_minus_real @ ( re @ X4 ) @ ( re @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9186_abs__Re__le__cmod,axiom,
% 12.45/7.00 ! [X4: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9187_Re__csqrt,axiom,
% 12.45/7.00 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9188_one__natural_Orsp,axiom,
% 12.45/7.00 one_one_nat = one_one_nat ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9189_cmod__plus__Re__le__0__iff,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 12.45/7.00 = ( ( re @ Z )
% 12.45/7.00 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9190_cos__n__Re__cis__pow__n,axiom,
% 12.45/7.00 ! [N: nat,A: real] :
% 12.45/7.00 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 12.45/7.00 = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9191_csqrt_Ocode,axiom,
% 12.45/7.00 ( csqrt
% 12.45/7.00 = ( ^ [Z5: complex] :
% 12.45/7.00 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/7.00 @ ( times_times_real
% 12.45/7.00 @ ( if_real
% 12.45/7.00 @ ( ( im @ Z5 )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 @ one_one_real
% 12.45/7.00 @ ( sgn_sgn_real @ ( im @ Z5 ) ) )
% 12.45/7.00 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9192_csqrt_Osimps_I2_J,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( im @ ( csqrt @ Z ) )
% 12.45/7.00 = ( times_times_real
% 12.45/7.00 @ ( if_real
% 12.45/7.00 @ ( ( im @ Z )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 @ one_one_real
% 12.45/7.00 @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 12.45/7.00 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9193_integer__of__int__code,axiom,
% 12.45/7.00 ( code_integer_of_int
% 12.45/7.00 = ( ^ [K3: int] :
% 12.45/7.00 ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 12.45/7.00 @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 12.45/7.00 @ ( if_Code_integer
% 12.45/7.00 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.45/7.00 = zero_zero_int )
% 12.45/7.00 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9194_Im__i__times,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
% 12.45/7.00 = ( re @ Z ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9195_Im__divide__of__real,axiom,
% 12.45/7.00 ! [Z: complex,R2: real] :
% 12.45/7.00 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 12.45/7.00 = ( divide_divide_real @ ( im @ Z ) @ R2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9196_Im__sgn,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( im @ ( sgn_sgn_complex @ Z ) )
% 12.45/7.00 = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9197_Re__power__real,axiom,
% 12.45/7.00 ! [X4: complex,N: nat] :
% 12.45/7.00 ( ( ( im @ X4 )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 => ( ( re @ ( power_power_complex @ X4 @ N ) )
% 12.45/7.00 = ( power_power_real @ ( re @ X4 ) @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9198_Re__i__times,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
% 12.45/7.00 = ( uminus_uminus_real @ ( im @ Z ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9199_Im__divide__numeral,axiom,
% 12.45/7.00 ! [Z: complex,W: num] :
% 12.45/7.00 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 12.45/7.00 = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9200_Im__divide__of__nat,axiom,
% 12.45/7.00 ! [Z: complex,N: nat] :
% 12.45/7.00 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 12.45/7.00 = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9201_csqrt__of__real__nonneg,axiom,
% 12.45/7.00 ! [X4: complex] :
% 12.45/7.00 ( ( ( im @ X4 )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X4 ) )
% 12.45/7.00 => ( ( csqrt @ X4 )
% 12.45/7.00 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X4 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9202_csqrt__minus,axiom,
% 12.45/7.00 ! [X4: complex] :
% 12.45/7.00 ( ( ( ord_less_real @ ( im @ X4 ) @ zero_zero_real )
% 12.45/7.00 | ( ( ( im @ X4 )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X4 ) ) ) )
% 12.45/7.00 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X4 ) )
% 12.45/7.00 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X4 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9203_csqrt__of__real__nonpos,axiom,
% 12.45/7.00 ! [X4: complex] :
% 12.45/7.00 ( ( ( im @ X4 )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 => ( ( ord_less_eq_real @ ( re @ X4 ) @ zero_zero_real )
% 12.45/7.00 => ( ( csqrt @ X4 )
% 12.45/7.00 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X4 ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9204_divide__integer_Oabs__eq,axiom,
% 12.45/7.00 ! [Xa: int,X4: int] :
% 12.45/7.00 ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 12.45/7.00 = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9205_modulo__integer_Oabs__eq,axiom,
% 12.45/7.00 ! [Xa: int,X4: int] :
% 12.45/7.00 ( ( modulo364778990260209775nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 12.45/7.00 = ( code_integer_of_int @ ( modulo_modulo_int @ Xa @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9206_less__integer__code_I1_J,axiom,
% 12.45/7.00 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9207_less__integer_Oabs__eq,axiom,
% 12.45/7.00 ! [Xa: int,X4: int] :
% 12.45/7.00 ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 12.45/7.00 = ( ord_less_int @ Xa @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9208_abs__integer__code,axiom,
% 12.45/7.00 ( abs_abs_Code_integer
% 12.45/7.00 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9209_imaginary__unit_Osimps_I2_J,axiom,
% 12.45/7.00 ( ( im @ imaginary_unit )
% 12.45/7.00 = one_one_real ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9210_plus__integer_Oabs__eq,axiom,
% 12.45/7.00 ! [Xa: int,X4: int] :
% 12.45/7.00 ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 12.45/7.00 = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9211_times__integer_Oabs__eq,axiom,
% 12.45/7.00 ! [Xa: int,X4: int] :
% 12.45/7.00 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 12.45/7.00 = ( code_integer_of_int @ ( times_times_int @ Xa @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9212_minus__integer_Oabs__eq,axiom,
% 12.45/7.00 ! [Xa: int,X4: int] :
% 12.45/7.00 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
% 12.45/7.00 = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9213_plus__complex_Osimps_I2_J,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( im @ ( plus_plus_complex @ X4 @ Y3 ) )
% 12.45/7.00 = ( plus_plus_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9214_scaleR__complex_Osimps_I2_J,axiom,
% 12.45/7.00 ! [R2: real,X4: complex] :
% 12.45/7.00 ( ( im @ ( real_V2046097035970521341omplex @ R2 @ X4 ) )
% 12.45/7.00 = ( times_times_real @ R2 @ ( im @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9215_minus__complex_Osimps_I2_J,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( im @ ( minus_minus_complex @ X4 @ Y3 ) )
% 12.45/7.00 = ( minus_minus_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9216_abs__Im__le__cmod,axiom,
% 12.45/7.00 ! [X4: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9217_times__complex_Osimps_I2_J,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( im @ ( times_times_complex @ X4 @ Y3 ) )
% 12.45/7.00 = ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9218_cmod__Im__le__iff,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( ( re @ X4 )
% 12.45/7.00 = ( re @ Y3 ) )
% 12.45/7.00 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 12.45/7.00 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X4 ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9219_cmod__Re__le__iff,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( ( im @ X4 )
% 12.45/7.00 = ( im @ Y3 ) )
% 12.45/7.00 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 12.45/7.00 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X4 ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9220_times__complex_Osimps_I1_J,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( re @ ( times_times_complex @ X4 @ Y3 ) )
% 12.45/7.00 = ( minus_minus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9221_plus__complex_Ocode,axiom,
% 12.45/7.00 ( plus_plus_complex
% 12.45/7.00 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) @ ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9222_scaleR__complex_Ocode,axiom,
% 12.45/7.00 ( real_V2046097035970521341omplex
% 12.45/7.00 = ( ^ [R5: real,X: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X ) ) @ ( times_times_real @ R5 @ ( im @ X ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9223_minus__complex_Ocode,axiom,
% 12.45/7.00 ( minus_minus_complex
% 12.45/7.00 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) @ ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9224_csqrt__principal,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 12.45/7.00 | ( ( ( re @ ( csqrt @ Z ) )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9225_cmod__le,axiom,
% 12.45/7.00 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9226_sin__n__Im__cis__pow__n,axiom,
% 12.45/7.00 ! [N: nat,A: real] :
% 12.45/7.00 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 12.45/7.00 = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9227_Re__exp,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( re @ ( exp_complex @ Z ) )
% 12.45/7.00 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9228_Im__exp,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( im @ ( exp_complex @ Z ) )
% 12.45/7.00 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9229_complex__eq,axiom,
% 12.45/7.00 ! [A: complex] :
% 12.45/7.00 ( A
% 12.45/7.00 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9230_times__complex_Ocode,axiom,
% 12.45/7.00 ( times_times_complex
% 12.45/7.00 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9231_exp__eq__polar,axiom,
% 12.45/7.00 ( exp_complex
% 12.45/7.00 = ( ^ [Z5: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z5 ) ) ) @ ( cis @ ( im @ Z5 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9232_cmod__power2,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/7.00 = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9233_Im__power2,axiom,
% 12.45/7.00 ! [X4: complex] :
% 12.45/7.00 ( ( im @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/7.00 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X4 ) ) @ ( im @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9234_Re__power2,axiom,
% 12.45/7.00 ! [X4: complex] :
% 12.45/7.00 ( ( re @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/7.00 = ( minus_minus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9235_complex__eq__0,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( Z = zero_zero_complex )
% 12.45/7.00 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/7.00 = zero_zero_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9236_norm__complex__def,axiom,
% 12.45/7.00 ( real_V1022390504157884413omplex
% 12.45/7.00 = ( ^ [Z5: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9237_inverse__complex_Osimps_I1_J,axiom,
% 12.45/7.00 ! [X4: complex] :
% 12.45/7.00 ( ( re @ ( invers8013647133539491842omplex @ X4 ) )
% 12.45/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9238_complex__neq__0,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( Z != zero_zero_complex )
% 12.45/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9239_Re__divide,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( re @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
% 12.45/7.00 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9240_csqrt__unique,axiom,
% 12.45/7.00 ! [W: complex,Z: complex] :
% 12.45/7.00 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.45/7.00 = Z )
% 12.45/7.00 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 12.45/7.00 | ( ( ( re @ W )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 12.45/7.00 => ( ( csqrt @ Z )
% 12.45/7.00 = W ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9241_csqrt__square,axiom,
% 12.45/7.00 ! [B2: complex] :
% 12.45/7.00 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B2 ) )
% 12.45/7.00 | ( ( ( re @ B2 )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B2 ) ) ) )
% 12.45/7.00 => ( ( csqrt @ ( power_power_complex @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/7.00 = B2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9242_inverse__complex_Osimps_I2_J,axiom,
% 12.45/7.00 ! [X4: complex] :
% 12.45/7.00 ( ( im @ ( invers8013647133539491842omplex @ X4 ) )
% 12.45/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9243_Im__divide,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( im @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
% 12.45/7.00 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9244_complex__abs__le__norm,axiom,
% 12.45/7.00 ! [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 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9245_complex__unit__circle,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( Z != zero_zero_complex )
% 12.45/7.00 => ( ( 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 ) ) ) )
% 12.45/7.00 = one_one_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9246_inverse__complex_Ocode,axiom,
% 12.45/7.00 ( invers8013647133539491842omplex
% 12.45/7.00 = ( ^ [X: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9247_Complex__divide,axiom,
% 12.45/7.00 ( divide1717551699836669952omplex
% 12.45/7.00 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9248_Im__Reals__divide,axiom,
% 12.45/7.00 ! [R2: complex,Z: complex] :
% 12.45/7.00 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 12.45/7.00 => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 12.45/7.00 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9249_Re__Reals__divide,axiom,
% 12.45/7.00 ! [R2: complex,Z: complex] :
% 12.45/7.00 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 12.45/7.00 => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 12.45/7.00 = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9250_Re__divide__Reals,axiom,
% 12.45/7.00 ! [R2: complex,Z: complex] :
% 12.45/7.00 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 12.45/7.00 => ( ( re @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 12.45/7.00 = ( divide_divide_real @ ( re @ Z ) @ ( re @ R2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9251_real__eq__imaginary__iff,axiom,
% 12.45/7.00 ! [Y3: complex,X4: complex] :
% 12.45/7.00 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 12.45/7.00 => ( ( member_complex @ X4 @ real_V2521375963428798218omplex )
% 12.45/7.00 => ( ( X4
% 12.45/7.00 = ( times_times_complex @ imaginary_unit @ Y3 ) )
% 12.45/7.00 = ( ( X4 = zero_zero_complex )
% 12.45/7.00 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9252_imaginary__eq__real__iff,axiom,
% 12.45/7.00 ! [Y3: complex,X4: complex] :
% 12.45/7.00 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 12.45/7.00 => ( ( member_complex @ X4 @ real_V2521375963428798218omplex )
% 12.45/7.00 => ( ( ( times_times_complex @ imaginary_unit @ Y3 )
% 12.45/7.00 = X4 )
% 12.45/7.00 = ( ( X4 = zero_zero_complex )
% 12.45/7.00 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9253_Im__divide__Reals,axiom,
% 12.45/7.00 ! [R2: complex,Z: complex] :
% 12.45/7.00 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 12.45/7.00 => ( ( im @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 12.45/7.00 = ( divide_divide_real @ ( im @ Z ) @ ( re @ R2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9254_complex__diff__cnj,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 12.45/7.00 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9255_complex__mult__cnj,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 12.45/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9256_integer__of__num_I3_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 12.45/7.00 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9257_complex__cnj__mult,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( cnj @ ( times_times_complex @ X4 @ Y3 ) )
% 12.45/7.00 = ( times_times_complex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9258_complex__cnj__divide,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( cnj @ ( divide1717551699836669952omplex @ X4 @ Y3 ) )
% 12.45/7.00 = ( divide1717551699836669952omplex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9259_complex__cnj__add,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( cnj @ ( plus_plus_complex @ X4 @ Y3 ) )
% 12.45/7.00 = ( plus_plus_complex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9260_complex__cnj__diff,axiom,
% 12.45/7.00 ! [X4: complex,Y3: complex] :
% 12.45/7.00 ( ( cnj @ ( minus_minus_complex @ X4 @ Y3 ) )
% 12.45/7.00 = ( minus_minus_complex @ ( cnj @ X4 ) @ ( cnj @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9261_complex__In__mult__cnj__zero,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 12.45/7.00 = zero_zero_real ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9262_Re__complex__div__eq__0,axiom,
% 12.45/7.00 ! [A: complex,B2: complex] :
% 12.45/7.00 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
% 12.45/7.00 = zero_zero_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9263_Im__complex__div__eq__0,axiom,
% 12.45/7.00 ! [A: complex,B2: complex] :
% 12.45/7.00 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 12.45/7.00 = zero_zero_real )
% 12.45/7.00 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
% 12.45/7.00 = zero_zero_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9264_complex__mod__sqrt__Re__mult__cnj,axiom,
% 12.45/7.00 ( real_V1022390504157884413omplex
% 12.45/7.00 = ( ^ [Z5: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z5 @ ( cnj @ Z5 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9265_integer__of__num__triv_I1_J,axiom,
% 12.45/7.00 ( ( code_integer_of_num @ one )
% 12.45/7.00 = one_one_Code_integer ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9266_Re__complex__div__lt__0,axiom,
% 12.45/7.00 ! [A: complex,B2: complex] :
% 12.45/7.00 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 12.45/7.00 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9267_Re__complex__div__gt__0,axiom,
% 12.45/7.00 ! [A: complex,B2: complex] :
% 12.45/7.00 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 12.45/7.00 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9268_Re__complex__div__ge__0,axiom,
% 12.45/7.00 ! [A: complex,B2: complex] :
% 12.45/7.00 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 12.45/7.00 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9269_Re__complex__div__le__0,axiom,
% 12.45/7.00 ! [A: complex,B2: complex] :
% 12.45/7.00 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 12.45/7.00 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9270_Im__complex__div__lt__0,axiom,
% 12.45/7.00 ! [A: complex,B2: complex] :
% 12.45/7.00 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 12.45/7.00 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9271_Im__complex__div__gt__0,axiom,
% 12.45/7.00 ! [A: complex,B2: complex] :
% 12.45/7.00 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 12.45/7.00 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9272_Im__complex__div__ge__0,axiom,
% 12.45/7.00 ! [A: complex,B2: complex] :
% 12.45/7.00 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 12.45/7.00 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9273_Im__complex__div__le__0,axiom,
% 12.45/7.00 ! [A: complex,B2: complex] :
% 12.45/7.00 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 12.45/7.00 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9274_integer__of__num_I2_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 12.45/7.00 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9275_complex__mod__mult__cnj,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 12.45/7.00 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9276_complex__div__gt__0,axiom,
% 12.45/7.00 ! [A: complex,B2: complex] :
% 12.45/7.00 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 12.45/7.00 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) )
% 12.45/7.00 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 12.45/7.00 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9277_integer__of__num__triv_I2_J,axiom,
% 12.45/7.00 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 12.45/7.00 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9278_complex__norm__square,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/7.00 = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9279_complex__add__cnj,axiom,
% 12.45/7.00 ! [Z: complex] :
% 12.45/7.00 ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 12.45/7.00 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9280_complex__div__cnj,axiom,
% 12.45/7.00 ( divide1717551699836669952omplex
% 12.45/7.00 = ( ^ [A4: complex,B3: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A4 @ ( cnj @ B3 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9281_cnj__add__mult__eq__Re,axiom,
% 12.45/7.00 ! [Z: complex,W: complex] :
% 12.45/7.00 ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 12.45/7.00 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9282_bit__cut__integer__def,axiom,
% 12.45/7.00 ( code_bit_cut_integer
% 12.45/7.00 = ( ^ [K3: code_integer] :
% 12.45/7.00 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.45/7.00 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9283_divmod__integer__def,axiom,
% 12.45/7.00 ( code_divmod_integer
% 12.45/7.00 = ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L ) @ ( modulo364778990260209775nteger @ K3 @ L ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9284_card__lessThan,axiom,
% 12.45/7.00 ! [U: nat] :
% 12.45/7.00 ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
% 12.45/7.00 = U ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9285_card__Collect__less__nat,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( finite_card_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
% 12.45/7.00 = N ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9286_card__atMost,axiom,
% 12.45/7.00 ! [U: nat] :
% 12.45/7.00 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 12.45/7.00 = ( suc @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9287_card__atLeastLessThan,axiom,
% 12.45/7.00 ! [L2: nat,U: nat] :
% 12.45/7.00 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
% 12.45/7.00 = ( minus_minus_nat @ U @ L2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9288_card__Collect__le__nat,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( finite_card_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
% 12.45/7.00 = ( suc @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9289_card__atLeastAtMost,axiom,
% 12.45/7.00 ! [L2: nat,U: nat] :
% 12.45/7.00 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 12.45/7.00 = ( minus_minus_nat @ ( suc @ U ) @ L2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9290_card__atLeastLessThan__int,axiom,
% 12.45/7.00 ! [L2: int,U: int] :
% 12.45/7.00 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L2 @ U ) )
% 12.45/7.00 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9291_card__atLeastAtMost__int,axiom,
% 12.45/7.00 ! [L2: int,U: int] :
% 12.45/7.00 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 12.45/7.00 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L2 ) @ one_one_int ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9292_card__less__Suc2,axiom,
% 12.45/7.00 ! [M7: set_nat,I: nat] :
% 12.45/7.00 ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 12.45/7.00 => ( ( finite_card_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [K3: nat] :
% 12.45/7.00 ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 12.45/7.00 & ( ord_less_nat @ K3 @ I ) ) ) )
% 12.45/7.00 = ( finite_card_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [K3: nat] :
% 12.45/7.00 ( ( member_nat @ K3 @ M7 )
% 12.45/7.00 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9293_card__less__Suc,axiom,
% 12.45/7.00 ! [M7: set_nat,I: nat] :
% 12.45/7.00 ( ( member_nat @ zero_zero_nat @ M7 )
% 12.45/7.00 => ( ( suc
% 12.45/7.00 @ ( finite_card_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [K3: nat] :
% 12.45/7.00 ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 12.45/7.00 & ( ord_less_nat @ K3 @ I ) ) ) ) )
% 12.45/7.00 = ( finite_card_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [K3: nat] :
% 12.45/7.00 ( ( member_nat @ K3 @ M7 )
% 12.45/7.00 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9294_card__less,axiom,
% 12.45/7.00 ! [M7: set_nat,I: nat] :
% 12.45/7.00 ( ( member_nat @ zero_zero_nat @ M7 )
% 12.45/7.00 => ( ( finite_card_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [K3: nat] :
% 12.45/7.00 ( ( member_nat @ K3 @ M7 )
% 12.45/7.00 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
% 12.45/7.00 != zero_zero_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9295_card__atLeastZeroLessThan__int,axiom,
% 12.45/7.00 ! [U: int] :
% 12.45/7.00 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
% 12.45/7.00 = ( nat2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9296_subset__card__intvl__is__intvl,axiom,
% 12.45/7.00 ! [A3: set_nat,K: nat] :
% 12.45/7.00 ( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) )
% 12.45/7.00 => ( A3
% 12.45/7.00 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9297_subset__eq__atLeast0__lessThan__card,axiom,
% 12.45/7.00 ! [N3: set_nat,N: nat] :
% 12.45/7.00 ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 12.45/7.00 => ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9298_card__sum__le__nat__sum,axiom,
% 12.45/7.00 ! [S2: set_nat] :
% 12.45/7.00 ( ord_less_eq_nat
% 12.45/7.00 @ ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [X: nat] : X
% 12.45/7.00 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 12.45/7.00 @ ( groups3542108847815614940at_nat
% 12.45/7.00 @ ^ [X: nat] : X
% 12.45/7.00 @ S2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9299_card__nth__roots,axiom,
% 12.45/7.00 ! [C: complex,N: nat] :
% 12.45/7.00 ( ( C != zero_zero_complex )
% 12.45/7.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( finite_card_complex
% 12.45/7.00 @ ( collect_complex
% 12.45/7.00 @ ^ [Z5: complex] :
% 12.45/7.00 ( ( power_power_complex @ Z5 @ N )
% 12.45/7.00 = C ) ) )
% 12.45/7.00 = N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9300_card__roots__unity__eq,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( finite_card_complex
% 12.45/7.00 @ ( collect_complex
% 12.45/7.00 @ ^ [Z5: complex] :
% 12.45/7.00 ( ( power_power_complex @ Z5 @ N )
% 12.45/7.00 = one_one_complex ) ) )
% 12.45/7.00 = N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9301_bit__cut__integer__code,axiom,
% 12.45/7.00 ( code_bit_cut_integer
% 12.45/7.00 = ( ^ [K3: code_integer] :
% 12.45/7.00 ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 12.45/7.00 @ ( produc9125791028180074456eger_o
% 12.45/7.00 @ ^ [R5: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
% 12.45/7.00 @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9302_divmod__abs__def,axiom,
% 12.45/7.00 ( code_divmod_abs
% 12.45/7.00 = ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9303_divmod__integer__code,axiom,
% 12.45/7.00 ( code_divmod_integer
% 12.45/7.00 = ( ^ [K3: code_integer,L: code_integer] :
% 12.45/7.00 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 12.45/7.00 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
% 12.45/7.00 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L )
% 12.45/7.00 @ ( produc6916734918728496179nteger
% 12.45/7.00 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S4 ) ) )
% 12.45/7.00 @ ( code_divmod_abs @ K3 @ L ) ) )
% 12.45/7.00 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 12.45/7.00 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 12.45/7.00 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L )
% 12.45/7.00 @ ( produc6916734918728496179nteger
% 12.45/7.00 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S4 ) ) )
% 12.45/7.00 @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9304_vebt__maxt_Opelims,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Y3: option_nat] :
% 12.45/7.00 ( ( ( vEBT_vebt_maxt @ X4 )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X4 )
% 12.45/7.00 => ( ! [A5: $o,B5: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.00 => ( ( ( B5
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( some_nat @ one_one_nat ) ) )
% 12.45/7.00 & ( ~ B5
% 12.45/7.00 => ( ( A5
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( some_nat @ zero_zero_nat ) ) )
% 12.45/7.00 & ( ~ A5
% 12.45/7.00 => ( Y3 = none_nat ) ) ) ) )
% 12.45/7.00 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
% 12.45/7.00 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 12.45/7.00 => ( ( Y3 = none_nat )
% 12.45/7.00 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) ) ) )
% 12.45/7.00 => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( some_nat @ Ma2 ) )
% 12.45/7.00 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9305_vebt__mint_Opelims,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Y3: option_nat] :
% 12.45/7.00 ( ( ( vEBT_vebt_mint @ X4 )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X4 )
% 12.45/7.00 => ( ! [A5: $o,B5: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.00 => ( ( ( A5
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( some_nat @ zero_zero_nat ) ) )
% 12.45/7.00 & ( ~ A5
% 12.45/7.00 => ( ( B5
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( some_nat @ one_one_nat ) ) )
% 12.45/7.00 & ( ~ B5
% 12.45/7.00 => ( Y3 = none_nat ) ) ) ) )
% 12.45/7.00 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
% 12.45/7.00 => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) )
% 12.45/7.00 => ( ( Y3 = none_nat )
% 12.45/7.00 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw ) ) ) )
% 12.45/7.00 => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( some_nat @ Mi2 ) )
% 12.45/7.00 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy2 @ Uz ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9306_prod__decode__aux_Osimps,axiom,
% 12.45/7.00 ( nat_prod_decode_aux
% 12.45/7.00 = ( ^ [K3: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K3 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K3 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M6 @ ( suc @ K3 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9307_nat_Odisc__eq__case_I1_J,axiom,
% 12.45/7.00 ! [Nat: nat] :
% 12.45/7.00 ( ( Nat = zero_zero_nat )
% 12.45/7.00 = ( case_nat_o @ $true
% 12.45/7.00 @ ^ [Uu: nat] : $false
% 12.45/7.00 @ Nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9308_nat_Odisc__eq__case_I2_J,axiom,
% 12.45/7.00 ! [Nat: nat] :
% 12.45/7.00 ( ( Nat != zero_zero_nat )
% 12.45/7.00 = ( case_nat_o @ $false
% 12.45/7.00 @ ^ [Uu: nat] : $true
% 12.45/7.00 @ Nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9309_less__eq__nat_Osimps_I2_J,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 12.45/7.00 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9310_max__Suc2,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 12.45/7.00 = ( case_nat_nat @ ( suc @ N )
% 12.45/7.00 @ ^ [M2: nat] : ( suc @ ( ord_max_nat @ M2 @ N ) )
% 12.45/7.00 @ M ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9311_max__Suc1,axiom,
% 12.45/7.00 ! [N: nat,M: nat] :
% 12.45/7.00 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 12.45/7.00 = ( case_nat_nat @ ( suc @ N )
% 12.45/7.00 @ ^ [M2: nat] : ( suc @ ( ord_max_nat @ N @ M2 ) )
% 12.45/7.00 @ M ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9312_diff__Suc,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 12.45/7.00 = ( case_nat_nat @ zero_zero_nat
% 12.45/7.00 @ ^ [K3: nat] : K3
% 12.45/7.00 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9313_prod__decode__aux_Oelims,axiom,
% 12.45/7.00 ! [X4: nat,Xa: nat,Y3: product_prod_nat_nat] :
% 12.45/7.00 ( ( ( nat_prod_decode_aux @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ( ord_less_eq_nat @ Xa @ X4 )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X4 @ Xa ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_nat @ Xa @ X4 )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( nat_prod_decode_aux @ ( suc @ X4 ) @ ( minus_minus_nat @ Xa @ ( suc @ X4 ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9314_pred__def,axiom,
% 12.45/7.00 ( pred
% 12.45/7.00 = ( case_nat_nat @ zero_zero_nat
% 12.45/7.00 @ ^ [X23: nat] : X23 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9315_floor__real__def,axiom,
% 12.45/7.00 ( archim6058952711729229775r_real
% 12.45/7.00 = ( ^ [X: real] :
% 12.45/7.00 ( the_int
% 12.45/7.00 @ ^ [Z5: int] :
% 12.45/7.00 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X )
% 12.45/7.00 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9316_floor__rat__def,axiom,
% 12.45/7.00 ( archim3151403230148437115or_rat
% 12.45/7.00 = ( ^ [X: rat] :
% 12.45/7.00 ( the_int
% 12.45/7.00 @ ^ [Z5: int] :
% 12.45/7.00 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X )
% 12.45/7.00 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9317_drop__bit__numeral__minus__bit1,axiom,
% 12.45/7.00 ! [L2: num,K: num] :
% 12.45/7.00 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.45/7.00 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9318_Suc__0__mod__numeral,axiom,
% 12.45/7.00 ! [K: num] :
% 12.45/7.00 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 12.45/7.00 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9319_drop__bit__nonnegative__int__iff,axiom,
% 12.45/7.00 ! [N: nat,K: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 12.45/7.00 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9320_drop__bit__negative__int__iff,axiom,
% 12.45/7.00 ! [N: nat,K: int] :
% 12.45/7.00 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 12.45/7.00 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9321_drop__bit__minus__one,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 12.45/7.00 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9322_snd__divmod__nat,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
% 12.45/7.00 = ( modulo_modulo_nat @ M @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9323_drop__bit__Suc__minus__bit0,axiom,
% 12.45/7.00 ! [N: nat,K: num] :
% 12.45/7.00 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.45/7.00 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9324_drop__bit__numeral__minus__bit0,axiom,
% 12.45/7.00 ! [L2: num,K: num] :
% 12.45/7.00 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.45/7.00 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9325_drop__bit__Suc__minus__bit1,axiom,
% 12.45/7.00 ! [N: nat,K: num] :
% 12.45/7.00 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.45/7.00 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9326_abs__rat__def,axiom,
% 12.45/7.00 ( abs_abs_rat
% 12.45/7.00 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9327_sgn__rat__def,axiom,
% 12.45/7.00 ( sgn_sgn_rat
% 12.45/7.00 = ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9328_obtain__pos__sum,axiom,
% 12.45/7.00 ! [R2: rat] :
% 12.45/7.00 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 12.45/7.00 => ~ ! [S3: rat] :
% 12.45/7.00 ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 12.45/7.00 => ! [T2: rat] :
% 12.45/7.00 ( ( ord_less_rat @ zero_zero_rat @ T2 )
% 12.45/7.00 => ( R2
% 12.45/7.00 != ( plus_plus_rat @ S3 @ T2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9329_less__eq__rat__def,axiom,
% 12.45/7.00 ( ord_less_eq_rat
% 12.45/7.00 = ( ^ [X: rat,Y: rat] :
% 12.45/7.00 ( ( ord_less_rat @ X @ Y )
% 12.45/7.00 | ( X = Y ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9330_drop__bit__push__bit__int,axiom,
% 12.45/7.00 ! [M: nat,N: nat,K: int] :
% 12.45/7.00 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 12.45/7.00 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9331_drop__bit__int__def,axiom,
% 12.45/7.00 ( bit_se8568078237143864401it_int
% 12.45/7.00 = ( ^ [N2: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9332_rat__inverse__code,axiom,
% 12.45/7.00 ! [P2: rat] :
% 12.45/7.00 ( ( quotient_of @ ( inverse_inverse_rat @ P2 ) )
% 12.45/7.00 = ( produc4245557441103728435nt_int
% 12.45/7.00 @ ^ [A4: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A4 ) @ B3 ) @ ( abs_abs_int @ A4 ) ) )
% 12.45/7.00 @ ( quotient_of @ P2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9333_normalize__negative,axiom,
% 12.45/7.00 ! [Q2: int,P2: int] :
% 12.45/7.00 ( ( ord_less_int @ Q2 @ zero_zero_int )
% 12.45/7.00 => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 12.45/7.00 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P2 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9334_Suc__0__div__numeral,axiom,
% 12.45/7.00 ! [K: num] :
% 12.45/7.00 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 12.45/7.00 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9335_snd__divmod__integer,axiom,
% 12.45/7.00 ! [K: code_integer,L2: code_integer] :
% 12.45/7.00 ( ( produc6174133586879617921nteger @ ( code_divmod_integer @ K @ L2 ) )
% 12.45/7.00 = ( modulo364778990260209775nteger @ K @ L2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9336_snd__divmod__abs,axiom,
% 12.45/7.00 ! [K: code_integer,L2: code_integer] :
% 12.45/7.00 ( ( produc6174133586879617921nteger @ ( code_divmod_abs @ K @ L2 ) )
% 12.45/7.00 = ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9337_drop__bit__of__Suc__0,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.45/7.00 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9338_fst__divmod__nat,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 12.45/7.00 = ( divide_divide_nat @ M @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9339_quotient__of__denom__pos_H,axiom,
% 12.45/7.00 ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9340_diff__rat__def,axiom,
% 12.45/7.00 ( minus_minus_rat
% 12.45/7.00 = ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9341_divide__rat__def,axiom,
% 12.45/7.00 ( divide_divide_rat
% 12.45/7.00 = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9342_rat__divide__code,axiom,
% 12.45/7.00 ! [P2: rat,Q2: rat] :
% 12.45/7.00 ( ( quotient_of @ ( divide_divide_rat @ P2 @ Q2 ) )
% 12.45/7.00 = ( produc4245557441103728435nt_int
% 12.45/7.00 @ ^ [A4: int,C2: int] :
% 12.45/7.00 ( produc4245557441103728435nt_int
% 12.45/7.00 @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) ) )
% 12.45/7.00 @ ( quotient_of @ Q2 ) )
% 12.45/7.00 @ ( quotient_of @ P2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9343_rat__times__code,axiom,
% 12.45/7.00 ! [P2: rat,Q2: rat] :
% 12.45/7.00 ( ( quotient_of @ ( times_times_rat @ P2 @ Q2 ) )
% 12.45/7.00 = ( produc4245557441103728435nt_int
% 12.45/7.00 @ ^ [A4: int,C2: int] :
% 12.45/7.00 ( produc4245557441103728435nt_int
% 12.45/7.00 @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ B3 ) @ ( times_times_int @ C2 @ D2 ) ) )
% 12.45/7.00 @ ( quotient_of @ Q2 ) )
% 12.45/7.00 @ ( quotient_of @ P2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9344_drop__bit__nat__eq,axiom,
% 12.45/7.00 ! [N: nat,K: int] :
% 12.45/7.00 ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 12.45/7.00 = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9345_quotient__of__div,axiom,
% 12.45/7.00 ! [R2: rat,N: int,D: int] :
% 12.45/7.00 ( ( ( quotient_of @ R2 )
% 12.45/7.00 = ( product_Pair_int_int @ N @ D ) )
% 12.45/7.00 => ( R2
% 12.45/7.00 = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9346_rat__plus__code,axiom,
% 12.45/7.00 ! [P2: rat,Q2: rat] :
% 12.45/7.00 ( ( quotient_of @ ( plus_plus_rat @ P2 @ Q2 ) )
% 12.45/7.00 = ( produc4245557441103728435nt_int
% 12.45/7.00 @ ^ [A4: int,C2: int] :
% 12.45/7.00 ( produc4245557441103728435nt_int
% 12.45/7.00 @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B3 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
% 12.45/7.00 @ ( quotient_of @ Q2 ) )
% 12.45/7.00 @ ( quotient_of @ P2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9347_rat__minus__code,axiom,
% 12.45/7.00 ! [P2: rat,Q2: rat] :
% 12.45/7.00 ( ( quotient_of @ ( minus_minus_rat @ P2 @ Q2 ) )
% 12.45/7.00 = ( produc4245557441103728435nt_int
% 12.45/7.00 @ ^ [A4: int,C2: int] :
% 12.45/7.00 ( produc4245557441103728435nt_int
% 12.45/7.00 @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B3 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
% 12.45/7.00 @ ( quotient_of @ Q2 ) )
% 12.45/7.00 @ ( quotient_of @ P2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9348_quotient__of__denom__pos,axiom,
% 12.45/7.00 ! [R2: rat,P2: int,Q2: int] :
% 12.45/7.00 ( ( ( quotient_of @ R2 )
% 12.45/7.00 = ( product_Pair_int_int @ P2 @ Q2 ) )
% 12.45/7.00 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9349_normalize__denom__pos,axiom,
% 12.45/7.00 ! [R2: product_prod_int_int,P2: int,Q2: int] :
% 12.45/7.00 ( ( ( normalize @ R2 )
% 12.45/7.00 = ( product_Pair_int_int @ P2 @ Q2 ) )
% 12.45/7.00 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9350_normalize__crossproduct,axiom,
% 12.45/7.00 ! [Q2: int,S: int,P2: int,R2: int] :
% 12.45/7.00 ( ( Q2 != zero_zero_int )
% 12.45/7.00 => ( ( S != zero_zero_int )
% 12.45/7.00 => ( ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 12.45/7.00 = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
% 12.45/7.00 => ( ( times_times_int @ P2 @ S )
% 12.45/7.00 = ( times_times_int @ R2 @ Q2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9351_drop__bit__nat__def,axiom,
% 12.45/7.00 ( bit_se8570568707652914677it_nat
% 12.45/7.00 = ( ^ [N2: nat,M6: nat] : ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9352_one__mod__minus__numeral,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9353_minus__one__mod__numeral,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9354_numeral__mod__minus__numeral,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9355_fst__divmod__integer,axiom,
% 12.45/7.00 ! [K: code_integer,L2: code_integer] :
% 12.45/7.00 ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L2 ) )
% 12.45/7.00 = ( divide6298287555418463151nteger @ K @ L2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9356_fst__divmod__abs,axiom,
% 12.45/7.00 ! [K: code_integer,L2: code_integer] :
% 12.45/7.00 ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L2 ) )
% 12.45/7.00 = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9357_minus__numeral__mod__numeral,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9358_Divides_Oadjust__mod__def,axiom,
% 12.45/7.00 ( adjust_mod
% 12.45/7.00 = ( ^ [L: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L @ R5 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9359_bezw_Osimps,axiom,
% 12.45/7.00 ( bezw
% 12.45/7.00 = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9360_bezw_Oelims,axiom,
% 12.45/7.00 ! [X4: nat,Xa: nat,Y3: product_prod_int_int] :
% 12.45/7.00 ( ( ( bezw @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Xa ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9361_bezw__non__0,axiom,
% 12.45/7.00 ! [Y3: nat,X4: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 12.45/7.00 => ( ( bezw @ X4 @ Y3 )
% 12.45/7.00 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9362_bezw_Opelims,axiom,
% 12.45/7.00 ! [X4: nat,Xa: nat,Y3: product_prod_int_int] :
% 12.45/7.00 ( ( ( bezw @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
% 12.45/7.00 => ~ ( ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Xa ) ) ) ) ) ) ) )
% 12.45/7.00 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9363_normalize__def,axiom,
% 12.45/7.00 ( normalize
% 12.45/7.00 = ( ^ [P3: product_prod_int_int] :
% 12.45/7.00 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P3 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) )
% 12.45/7.00 @ ( if_Pro3027730157355071871nt_int
% 12.45/7.00 @ ( ( product_snd_int_int @ P3 )
% 12.45/7.00 = zero_zero_int )
% 12.45/7.00 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 12.45/7.00 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9364_Frct__code__post_I5_J,axiom,
% 12.45/7.00 ! [K: num] :
% 12.45/7.00 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 12.45/7.00 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9365_gcd__pos__int,axiom,
% 12.45/7.00 ! [M: int,N: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 12.45/7.00 = ( ( M != zero_zero_int )
% 12.45/7.00 | ( N != zero_zero_int ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9366_gcd__red__int,axiom,
% 12.45/7.00 ( gcd_gcd_int
% 12.45/7.00 = ( ^ [X: int,Y: int] : ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9367_bezout__int,axiom,
% 12.45/7.00 ! [X4: int,Y3: int] :
% 12.45/7.00 ? [U3: int,V3: int] :
% 12.45/7.00 ( ( plus_plus_int @ ( times_times_int @ U3 @ X4 ) @ ( times_times_int @ V3 @ Y3 ) )
% 12.45/7.00 = ( gcd_gcd_int @ X4 @ Y3 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9368_gcd__mult__distrib__int,axiom,
% 12.45/7.00 ! [K: int,M: int,N: int] :
% 12.45/7.00 ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 12.45/7.00 = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9369_gcd__le1__int,axiom,
% 12.45/7.00 ! [A: int,B2: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ A )
% 12.45/7.00 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ A ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9370_gcd__le2__int,axiom,
% 12.45/7.00 ! [B2: int,A: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/7.00 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ B2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9371_gcd__non__0__int,axiom,
% 12.45/7.00 ! [Y3: int,X4: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ Y3 )
% 12.45/7.00 => ( ( gcd_gcd_int @ X4 @ Y3 )
% 12.45/7.00 = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X4 @ Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9372_gcd__code__int,axiom,
% 12.45/7.00 ( gcd_gcd_int
% 12.45/7.00 = ( ^ [K3: int,L: int] : ( abs_abs_int @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9373_Frct__code__post_I6_J,axiom,
% 12.45/7.00 ! [K: num,L2: num] :
% 12.45/7.00 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
% 12.45/7.00 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9374_prod__decode__aux_Opelims,axiom,
% 12.45/7.00 ! [X4: nat,Xa: nat,Y3: product_prod_nat_nat] :
% 12.45/7.00 ( ( ( nat_prod_decode_aux @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
% 12.45/7.00 => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X4 )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X4 @ Xa ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_nat @ Xa @ X4 )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( nat_prod_decode_aux @ ( suc @ X4 ) @ ( minus_minus_nat @ Xa @ ( suc @ X4 ) ) ) ) ) )
% 12.45/7.00 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9375_finite__enumerate,axiom,
% 12.45/7.00 ! [S2: set_nat] :
% 12.45/7.00 ( ( finite_finite_nat @ S2 )
% 12.45/7.00 => ? [R3: nat > nat] :
% 12.45/7.00 ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
% 12.45/7.00 & ! [N6: nat] :
% 12.45/7.00 ( ( ord_less_nat @ N6 @ ( finite_card_nat @ S2 ) )
% 12.45/7.00 => ( member_nat @ ( R3 @ N6 ) @ S2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9376_gcd__1__nat,axiom,
% 12.45/7.00 ! [M: nat] :
% 12.45/7.00 ( ( gcd_gcd_nat @ M @ one_one_nat )
% 12.45/7.00 = one_one_nat ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9377_gcd__Suc__0,axiom,
% 12.45/7.00 ! [M: nat] :
% 12.45/7.00 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 12.45/7.00 = ( suc @ zero_zero_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9378_gcd__pos__nat,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 12.45/7.00 = ( ( M != zero_zero_nat )
% 12.45/7.00 | ( N != zero_zero_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9379_gcd__red__nat,axiom,
% 12.45/7.00 ( gcd_gcd_nat
% 12.45/7.00 = ( ^ [X: nat,Y: nat] : ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9380_gcd__mult__distrib__nat,axiom,
% 12.45/7.00 ! [K: nat,M: nat,N: nat] :
% 12.45/7.00 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 12.45/7.00 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9381_gcd__le2__nat,axiom,
% 12.45/7.00 ! [B2: nat,A: nat] :
% 12.45/7.00 ( ( B2 != zero_zero_nat )
% 12.45/7.00 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ B2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9382_gcd__le1__nat,axiom,
% 12.45/7.00 ! [A: nat,B2: nat] :
% 12.45/7.00 ( ( A != zero_zero_nat )
% 12.45/7.00 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ A ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9383_gcd__diff2__nat,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ M @ N )
% 12.45/7.00 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 12.45/7.00 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9384_gcd__diff1__nat,axiom,
% 12.45/7.00 ! [N: nat,M: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ N @ M )
% 12.45/7.00 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 12.45/7.00 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9385_gcd__non__0__nat,axiom,
% 12.45/7.00 ! [Y3: nat,X4: nat] :
% 12.45/7.00 ( ( Y3 != zero_zero_nat )
% 12.45/7.00 => ( ( gcd_gcd_nat @ X4 @ Y3 )
% 12.45/7.00 = ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9386_gcd__nat_Osimps,axiom,
% 12.45/7.00 ( gcd_gcd_nat
% 12.45/7.00 = ( ^ [X: nat,Y: nat] : ( if_nat @ ( Y = zero_zero_nat ) @ X @ ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9387_gcd__nat_Oelims,axiom,
% 12.45/7.00 ! [X4: nat,Xa: nat,Y3: nat] :
% 12.45/7.00 ( ( ( gcd_gcd_nat @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => ( Y3 = X4 ) )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9388_bezout__nat,axiom,
% 12.45/7.00 ! [A: nat,B2: nat] :
% 12.45/7.00 ( ( A != zero_zero_nat )
% 12.45/7.00 => ? [X3: nat,Y4: nat] :
% 12.45/7.00 ( ( times_times_nat @ A @ X3 )
% 12.45/7.00 = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9389_bezout__gcd__nat_H,axiom,
% 12.45/7.00 ! [B2: nat,A: nat] :
% 12.45/7.00 ? [X3: nat,Y4: nat] :
% 12.45/7.00 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( times_times_nat @ A @ X3 ) )
% 12.45/7.00 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B2 @ Y4 ) )
% 12.45/7.00 = ( gcd_gcd_nat @ A @ B2 ) ) )
% 12.45/7.00 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B2 @ X3 ) )
% 12.45/7.00 & ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
% 12.45/7.00 = ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9390_gcd__code__integer,axiom,
% 12.45/7.00 ( gcd_gcd_Code_integer
% 12.45/7.00 = ( ^ [K3: code_integer,L: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9391_bezw__aux,axiom,
% 12.45/7.00 ! [X4: nat,Y3: nat] :
% 12.45/7.00 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X4 @ Y3 ) )
% 12.45/7.00 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X4 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X4 ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X4 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9392_nat__descend__induct,axiom,
% 12.45/7.00 ! [N: nat,P: nat > $o,M: nat] :
% 12.45/7.00 ( ! [K2: nat] :
% 12.45/7.00 ( ( ord_less_nat @ N @ K2 )
% 12.45/7.00 => ( P @ K2 ) )
% 12.45/7.00 => ( ! [K2: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ K2 @ N )
% 12.45/7.00 => ( ! [I4: nat] :
% 12.45/7.00 ( ( ord_less_nat @ K2 @ I4 )
% 12.45/7.00 => ( P @ I4 ) )
% 12.45/7.00 => ( P @ K2 ) ) )
% 12.45/7.00 => ( P @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9393_gcd__nat_Opelims,axiom,
% 12.45/7.00 ! [X4: nat,Xa: nat,Y3: nat] :
% 12.45/7.00 ( ( ( gcd_gcd_nat @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
% 12.45/7.00 => ~ ( ( ( ( Xa = zero_zero_nat )
% 12.45/7.00 => ( Y3 = X4 ) )
% 12.45/7.00 & ( ( Xa != zero_zero_nat )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) ) )
% 12.45/7.00 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9394_divmod__integer__eq__cases,axiom,
% 12.45/7.00 ( code_divmod_integer
% 12.45/7.00 = ( ^ [K3: code_integer,L: code_integer] :
% 12.45/7.00 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 12.45/7.00 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 12.45/7.00 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
% 12.45/7.00 @ ( if_Pro6119634080678213985nteger
% 12.45/7.00 @ ( ( sgn_sgn_Code_integer @ K3 )
% 12.45/7.00 = ( sgn_sgn_Code_integer @ L ) )
% 12.45/7.00 @ ( code_divmod_abs @ K3 @ L )
% 12.45/7.00 @ ( produc6916734918728496179nteger
% 12.45/7.00 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S4 ) ) )
% 12.45/7.00 @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9395_card__greaterThanLessThan__int,axiom,
% 12.45/7.00 ! [L2: int,U: int] :
% 12.45/7.00 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U ) )
% 12.45/7.00 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9396_xor__minus__numerals_I2_J,axiom,
% 12.45/7.00 ! [K: int,N: num] :
% 12.45/7.00 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9397_xor__minus__numerals_I1_J,axiom,
% 12.45/7.00 ! [N: num,K: int] :
% 12.45/7.00 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 12.45/7.00 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9398_finite__greaterThanLessThan__int,axiom,
% 12.45/7.00 ! [L2: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9399_card_Ocomp__fun__commute__on,axiom,
% 12.45/7.00 ( ( comp_nat_nat_nat @ suc @ suc )
% 12.45/7.00 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9400_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 12.45/7.00 ! [L2: int,U: int] :
% 12.45/7.00 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 12.45/7.00 = ( set_or5832277885323065728an_int @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9401_sub__BitM__One__eq,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 12.45/7.00 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9402_finite__greaterThanLessThan,axiom,
% 12.45/7.00 ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9403_card__greaterThanLessThan,axiom,
% 12.45/7.00 ! [L2: nat,U: nat] :
% 12.45/7.00 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
% 12.45/7.00 = ( minus_minus_nat @ U @ ( suc @ L2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9404_atLeastSucLessThan__greaterThanLessThan,axiom,
% 12.45/7.00 ! [L2: nat,U: nat] :
% 12.45/7.00 ( ( set_or4665077453230672383an_nat @ ( suc @ L2 ) @ U )
% 12.45/7.00 = ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9405_tanh__real__bounds,axiom,
% 12.45/7.00 ! [X4: real] : ( member_real @ ( tanh_real @ X4 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9406_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 12.45/7.00 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 12.45/7.00 @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 12.45/7.00 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9407_Suc__funpow,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( compow_nat_nat @ N @ suc )
% 12.45/7.00 = ( plus_plus_nat @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9408_int__of__integer__code,axiom,
% 12.45/7.00 ( code_int_of_integer
% 12.45/7.00 = ( ^ [K3: code_integer] :
% 12.45/7.00 ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 12.45/7.00 @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 12.45/7.00 @ ( produc1553301316500091796er_int
% 12.45/7.00 @ ^ [L: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
% 12.45/7.00 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9409_plus__integer_Orep__eq,axiom,
% 12.45/7.00 ! [X4: code_integer,Xa: code_integer] :
% 12.45/7.00 ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X4 @ Xa ) )
% 12.45/7.00 = ( plus_plus_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9410_times__integer_Orep__eq,axiom,
% 12.45/7.00 ! [X4: code_integer,Xa: code_integer] :
% 12.45/7.00 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X4 @ Xa ) )
% 12.45/7.00 = ( times_times_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9411_minus__integer_Orep__eq,axiom,
% 12.45/7.00 ! [X4: code_integer,Xa: code_integer] :
% 12.45/7.00 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X4 @ Xa ) )
% 12.45/7.00 = ( minus_minus_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9412_divide__integer_Orep__eq,axiom,
% 12.45/7.00 ! [X4: code_integer,Xa: code_integer] :
% 12.45/7.00 ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X4 @ Xa ) )
% 12.45/7.00 = ( divide_divide_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9413_modulo__integer_Orep__eq,axiom,
% 12.45/7.00 ! [X4: code_integer,Xa: code_integer] :
% 12.45/7.00 ( ( code_int_of_integer @ ( modulo364778990260209775nteger @ X4 @ Xa ) )
% 12.45/7.00 = ( modulo_modulo_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9414_less__integer_Orep__eq,axiom,
% 12.45/7.00 ( ord_le6747313008572928689nteger
% 12.45/7.00 = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9415_integer__less__iff,axiom,
% 12.45/7.00 ( ord_le6747313008572928689nteger
% 12.45/7.00 = ( ^ [K3: code_integer,L: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9416_times__int_Oabs__eq,axiom,
% 12.45/7.00 ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
% 12.45/7.00 ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
% 12.45/7.00 = ( abs_Integ
% 12.45/7.00 @ ( produc27273713700761075at_nat
% 12.45/7.00 @ ^ [X: nat,Y: nat] :
% 12.45/7.00 ( produc2626176000494625587at_nat
% 12.45/7.00 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) )
% 12.45/7.00 @ Xa
% 12.45/7.00 @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9417_Gcd__remove0__nat,axiom,
% 12.45/7.00 ! [M7: set_nat] :
% 12.45/7.00 ( ( finite_finite_nat @ M7 )
% 12.45/7.00 => ( ( gcd_Gcd_nat @ M7 )
% 12.45/7.00 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9418_Gcd__nat__eq__one,axiom,
% 12.45/7.00 ! [N3: set_nat] :
% 12.45/7.00 ( ( member_nat @ one_one_nat @ N3 )
% 12.45/7.00 => ( ( gcd_Gcd_nat @ N3 )
% 12.45/7.00 = one_one_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9419_nat_Oabs__eq,axiom,
% 12.45/7.00 ! [X4: product_prod_nat_nat] :
% 12.45/7.00 ( ( nat2 @ ( abs_Integ @ X4 ) )
% 12.45/7.00 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9420_one__int__def,axiom,
% 12.45/7.00 ( one_one_int
% 12.45/7.00 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9421_less__int_Oabs__eq,axiom,
% 12.45/7.00 ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
% 12.45/7.00 ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
% 12.45/7.00 = ( produc8739625826339149834_nat_o
% 12.45/7.00 @ ^ [X: nat,Y: nat] :
% 12.45/7.00 ( produc6081775807080527818_nat_o
% 12.45/7.00 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) )
% 12.45/7.00 @ Xa
% 12.45/7.00 @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9422_less__eq__int_Oabs__eq,axiom,
% 12.45/7.00 ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
% 12.45/7.00 ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
% 12.45/7.00 = ( produc8739625826339149834_nat_o
% 12.45/7.00 @ ^ [X: nat,Y: nat] :
% 12.45/7.00 ( produc6081775807080527818_nat_o
% 12.45/7.00 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) )
% 12.45/7.00 @ Xa
% 12.45/7.00 @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9423_plus__int_Oabs__eq,axiom,
% 12.45/7.00 ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
% 12.45/7.00 ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
% 12.45/7.00 = ( abs_Integ
% 12.45/7.00 @ ( produc27273713700761075at_nat
% 12.45/7.00 @ ^ [X: nat,Y: nat] :
% 12.45/7.00 ( produc2626176000494625587at_nat
% 12.45/7.00 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
% 12.45/7.00 @ Xa
% 12.45/7.00 @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9424_minus__int_Oabs__eq,axiom,
% 12.45/7.00 ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
% 12.45/7.00 ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
% 12.45/7.00 = ( abs_Integ
% 12.45/7.00 @ ( produc27273713700761075at_nat
% 12.45/7.00 @ ^ [X: nat,Y: nat] :
% 12.45/7.00 ( produc2626176000494625587at_nat
% 12.45/7.00 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) )
% 12.45/7.00 @ Xa
% 12.45/7.00 @ X4 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9425_num__of__nat_Osimps_I2_J,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( num_of_nat @ ( suc @ N ) )
% 12.45/7.00 = ( inc @ ( num_of_nat @ N ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( num_of_nat @ ( suc @ N ) )
% 12.45/7.00 = one ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9426_num__of__nat__numeral__eq,axiom,
% 12.45/7.00 ! [Q2: num] :
% 12.45/7.00 ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 12.45/7.00 = Q2 ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9427_num__of__nat_Osimps_I1_J,axiom,
% 12.45/7.00 ( ( num_of_nat @ zero_zero_nat )
% 12.45/7.00 = one ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9428_numeral__num__of__nat,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 12.45/7.00 = N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9429_num__of__nat__One,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 12.45/7.00 => ( ( num_of_nat @ N )
% 12.45/7.00 = one ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9430_num__of__nat__double,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 12.45/7.00 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9431_num__of__nat__plus__distrib,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/7.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 12.45/7.00 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9432_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 12.45/7.00 ! [N: nat,J: nat,I: nat] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 12.45/7.00 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 12.45/7.00 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9433_less__eq__int_Orep__eq,axiom,
% 12.45/7.00 ( ord_less_eq_int
% 12.45/7.00 = ( ^ [X: int,Xa4: int] :
% 12.45/7.00 ( produc8739625826339149834_nat_o
% 12.45/7.00 @ ^ [Y: nat,Z5: nat] :
% 12.45/7.00 ( produc6081775807080527818_nat_o
% 12.45/7.00 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U4 @ Z5 ) ) )
% 12.45/7.00 @ ( rep_Integ @ X )
% 12.45/7.00 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9434_less__int_Orep__eq,axiom,
% 12.45/7.00 ( ord_less_int
% 12.45/7.00 = ( ^ [X: int,Xa4: int] :
% 12.45/7.00 ( produc8739625826339149834_nat_o
% 12.45/7.00 @ ^ [Y: nat,Z5: nat] :
% 12.45/7.00 ( produc6081775807080527818_nat_o
% 12.45/7.00 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U4 @ Z5 ) ) )
% 12.45/7.00 @ ( rep_Integ @ X )
% 12.45/7.00 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9435_nat_Orep__eq,axiom,
% 12.45/7.00 ( nat2
% 12.45/7.00 = ( ^ [X: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9436_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 12.45/7.00 ! [N: nat,J: nat,I: nat] :
% 12.45/7.00 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 12.45/7.00 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 12.45/7.00 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9437_prod__encode__def,axiom,
% 12.45/7.00 ( nat_prod_encode
% 12.45/7.00 = ( produc6842872674320459806at_nat
% 12.45/7.00 @ ^ [M6: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M6 @ N2 ) ) @ M6 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9438_pow_Osimps_I3_J,axiom,
% 12.45/7.00 ! [X4: num,Y3: num] :
% 12.45/7.00 ( ( pow @ X4 @ ( bit1 @ Y3 ) )
% 12.45/7.00 = ( times_times_num @ ( sqr @ ( pow @ X4 @ Y3 ) ) @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9439_finite__greaterThanAtMost,axiom,
% 12.45/7.00 ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9440_card__greaterThanAtMost,axiom,
% 12.45/7.00 ! [L2: nat,U: nat] :
% 12.45/7.00 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
% 12.45/7.00 = ( minus_minus_nat @ U @ L2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9441_sqr_Osimps_I2_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( sqr @ ( bit0 @ N ) )
% 12.45/7.00 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9442_sqr_Osimps_I1_J,axiom,
% 12.45/7.00 ( ( sqr @ one )
% 12.45/7.00 = one ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9443_atLeastSucAtMost__greaterThanAtMost,axiom,
% 12.45/7.00 ! [L2: nat,U: nat] :
% 12.45/7.00 ( ( set_or1269000886237332187st_nat @ ( suc @ L2 ) @ U )
% 12.45/7.00 = ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9444_sqr__conv__mult,axiom,
% 12.45/7.00 ( sqr
% 12.45/7.00 = ( ^ [X: num] : ( times_times_num @ X @ X ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9445_le__prod__encode__1,axiom,
% 12.45/7.00 ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9446_le__prod__encode__2,axiom,
% 12.45/7.00 ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9447_pow_Osimps_I2_J,axiom,
% 12.45/7.00 ! [X4: num,Y3: num] :
% 12.45/7.00 ( ( pow @ X4 @ ( bit0 @ Y3 ) )
% 12.45/7.00 = ( sqr @ ( pow @ X4 @ Y3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9448_sqr_Osimps_I3_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( sqr @ ( bit1 @ N ) )
% 12.45/7.00 = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9449_prod__encode__prod__decode__aux,axiom,
% 12.45/7.00 ! [K: nat,M: nat] :
% 12.45/7.00 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 12.45/7.00 = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9450_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 12.45/7.00 ( ( ( vEBT_VEBT_valid @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ? [Uu2: $o,Uv: $o] :
% 12.45/7.00 ( X4
% 12.45/7.00 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( Xa != one_one_nat ) ) )
% 12.45/7.00 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( ~ ( ( Deg2 = Xa )
% 12.45/7.00 & ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.45/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 & ( case_o184042715313410164at_nat
% 12.45/7.00 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 12.45/7.00 & ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 @ ( produc6081775807080527818_nat_o
% 12.45/7.00 @ ^ [Mi3: nat,Ma3: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.45/7.00 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 & ! [I2: nat] :
% 12.45/7.00 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 12.45/7.00 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.45/7.00 & ( ( Mi3 = Ma3 )
% 12.45/7.00 => ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 & ( ( Mi3 != Ma3 )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 12.45/7.00 & ! [X: nat] :
% 12.45/7.00 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 12.45/7.00 => ( ( ord_less_nat @ Mi3 @ X )
% 12.45/7.00 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 12.45/7.00 @ Mima ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9451_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/7.00 ( ( vEBT_VEBT_valid @ X4 @ Xa )
% 12.45/7.00 => ( ( ? [Uu2: $o,Uv: $o] :
% 12.45/7.00 ( X4
% 12.45/7.00 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 12.45/7.00 => ( Xa != one_one_nat ) )
% 12.45/7.00 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ~ ( ( Deg2 = Xa )
% 12.45/7.00 & ! [X5: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.45/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 & ( case_o184042715313410164at_nat
% 12.45/7.00 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 12.45/7.00 & ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 @ ( produc6081775807080527818_nat_o
% 12.45/7.00 @ ^ [Mi3: nat,Ma3: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.45/7.00 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 & ! [I2: nat] :
% 12.45/7.00 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 12.45/7.00 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.45/7.00 & ( ( Mi3 = Ma3 )
% 12.45/7.00 => ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 & ( ( Mi3 != Ma3 )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 12.45/7.00 & ! [X: nat] :
% 12.45/7.00 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 12.45/7.00 => ( ( ord_less_nat @ Mi3 @ X )
% 12.45/7.00 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 12.45/7.00 @ Mima ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9452_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/7.00 ( ~ ( vEBT_VEBT_valid @ X4 @ Xa )
% 12.45/7.00 => ( ( ? [Uu2: $o,Uv: $o] :
% 12.45/7.00 ( X4
% 12.45/7.00 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 12.45/7.00 => ( Xa = one_one_nat ) )
% 12.45/7.00 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( ( Deg2 = Xa )
% 12.45/7.00 & ! [X3: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.45/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 & ( case_o184042715313410164at_nat
% 12.45/7.00 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 12.45/7.00 & ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 @ ( produc6081775807080527818_nat_o
% 12.45/7.00 @ ^ [Mi3: nat,Ma3: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.45/7.00 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 & ! [I2: nat] :
% 12.45/7.00 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 12.45/7.00 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.45/7.00 & ( ( Mi3 = Ma3 )
% 12.45/7.00 => ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 & ( ( Mi3 != Ma3 )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 12.45/7.00 & ! [X: nat] :
% 12.45/7.00 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 12.45/7.00 => ( ( ord_less_nat @ Mi3 @ X )
% 12.45/7.00 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 12.45/7.00 @ Mima ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9453_finite__greaterThanAtMost__int,axiom,
% 12.45/7.00 ! [L2: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9454_card__greaterThanAtMost__int,axiom,
% 12.45/7.00 ! [L2: int,U: int] :
% 12.45/7.00 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L2 @ U ) )
% 12.45/7.00 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9455_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 12.45/7.00 ! [L2: int,U: int] :
% 12.45/7.00 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 12.45/7.00 = ( set_or6656581121297822940st_int @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9456_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 12.45/7.00 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
% 12.45/7.00 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
% 12.45/7.00 = ( ( Deg = Deg3 )
% 12.45/7.00 & ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/7.00 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.45/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 & ( case_o184042715313410164at_nat
% 12.45/7.00 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
% 12.45/7.00 & ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 @ ( produc6081775807080527818_nat_o
% 12.45/7.00 @ ^ [Mi3: nat,Ma3: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.45/7.00 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 12.45/7.00 & ! [I2: nat] :
% 12.45/7.00 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X6 ) )
% 12.45/7.00 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 12.45/7.00 & ( ( Mi3 = Ma3 )
% 12.45/7.00 => ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 & ( ( Mi3 != Ma3 )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 12.45/7.00 & ! [X: nat] :
% 12.45/7.00 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 12.45/7.00 => ( ( ord_less_nat @ Mi3 @ X )
% 12.45/7.00 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 12.45/7.00 @ Mima2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9457_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat,Y3: $o] :
% 12.45/7.00 ( ( ( vEBT_VEBT_valid @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [Uu2: $o,Uv: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( Xa = one_one_nat ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) ) ) )
% 12.45/7.00 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( ( Deg2 = Xa )
% 12.45/7.00 & ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.45/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 & ( case_o184042715313410164at_nat
% 12.45/7.00 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 12.45/7.00 & ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 @ ( produc6081775807080527818_nat_o
% 12.45/7.00 @ ^ [Mi3: nat,Ma3: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.45/7.00 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 & ! [I2: nat] :
% 12.45/7.00 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 12.45/7.00 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.45/7.00 & ( ( Mi3 = Ma3 )
% 12.45/7.00 => ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 & ( ( Mi3 != Ma3 )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 12.45/7.00 & ! [X: nat] :
% 12.45/7.00 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 12.45/7.00 => ( ( ord_less_nat @ Mi3 @ X )
% 12.45/7.00 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 12.45/7.00 @ Mima ) ) )
% 12.45/7.00 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9458_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/7.00 ( ( vEBT_VEBT_valid @ X4 @ Xa )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [Uu2: $o,Uv: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) )
% 12.45/7.00 => ( Xa != one_one_nat ) ) )
% 12.45/7.00 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 12.45/7.00 => ~ ( ( Deg2 = Xa )
% 12.45/7.00 & ! [X5: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.45/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 & ( case_o184042715313410164at_nat
% 12.45/7.00 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 12.45/7.00 & ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 @ ( produc6081775807080527818_nat_o
% 12.45/7.00 @ ^ [Mi3: nat,Ma3: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.45/7.00 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 & ! [I2: nat] :
% 12.45/7.00 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 12.45/7.00 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.45/7.00 & ( ( Mi3 = Ma3 )
% 12.45/7.00 => ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 & ( ( Mi3 != Ma3 )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 12.45/7.00 & ! [X: nat] :
% 12.45/7.00 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 12.45/7.00 => ( ( ord_less_nat @ Mi3 @ X )
% 12.45/7.00 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 12.45/7.00 @ Mima ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9459_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 12.45/7.00 ! [X4: vEBT_VEBT,Xa: nat] :
% 12.45/7.00 ( ~ ( vEBT_VEBT_valid @ X4 @ Xa )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
% 12.45/7.00 => ( ! [Uu2: $o,Uv: $o] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Leaf @ Uu2 @ Uv ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) )
% 12.45/7.00 => ( Xa = one_one_nat ) ) )
% 12.45/7.00 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/7.00 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 12.45/7.00 => ( ( Deg2 = Xa )
% 12.45/7.00 & ! [X3: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.00 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.45/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 & ( case_o184042715313410164at_nat
% 12.45/7.00 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 12.45/7.00 & ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 @ ( produc6081775807080527818_nat_o
% 12.45/7.00 @ ^ [Mi3: nat,Ma3: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.45/7.00 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 & ! [I2: nat] :
% 12.45/7.00 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.00 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X6 ) )
% 12.45/7.00 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.45/7.00 & ( ( Mi3 = Ma3 )
% 12.45/7.00 => ! [X: vEBT_VEBT] :
% 12.45/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.45/7.00 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
% 12.45/7.00 & ( ( Mi3 != Ma3 )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 12.45/7.00 & ! [X: nat] :
% 12.45/7.00 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.45/7.00 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 12.45/7.00 => ( ( ord_less_nat @ Mi3 @ X )
% 12.45/7.00 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 12.45/7.00 @ Mima ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9460_GreatestI__nat,axiom,
% 12.45/7.00 ! [P: nat > $o,K: nat,B2: nat] :
% 12.45/7.00 ( ( P @ K )
% 12.45/7.00 => ( ! [Y4: nat] :
% 12.45/7.00 ( ( P @ Y4 )
% 12.45/7.00 => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 12.45/7.00 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9461_Greatest__le__nat,axiom,
% 12.45/7.00 ! [P: nat > $o,K: nat,B2: nat] :
% 12.45/7.00 ( ( P @ K )
% 12.45/7.00 => ( ! [Y4: nat] :
% 12.45/7.00 ( ( P @ Y4 )
% 12.45/7.00 => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 12.45/7.00 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9462_GreatestI__ex__nat,axiom,
% 12.45/7.00 ! [P: nat > $o,B2: nat] :
% 12.45/7.00 ( ? [X_12: nat] : ( P @ X_12 )
% 12.45/7.00 => ( ! [Y4: nat] :
% 12.45/7.00 ( ( P @ Y4 )
% 12.45/7.00 => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 12.45/7.00 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9463_rat__floor__lemma,axiom,
% 12.45/7.00 ! [A: int,B2: int] :
% 12.45/7.00 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B2 ) ) @ ( fract @ A @ B2 ) )
% 12.45/7.00 & ( ord_less_rat @ ( fract @ A @ B2 ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9464_image__minus__const__atLeastLessThan__nat,axiom,
% 12.45/7.00 ! [C: nat,Y3: nat,X4: nat] :
% 12.45/7.00 ( ( ( ord_less_nat @ C @ Y3 )
% 12.45/7.00 => ( ( image_nat_nat
% 12.45/7.00 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 12.45/7.00 @ ( set_or4665077453230672383an_nat @ X4 @ Y3 ) )
% 12.45/7.00 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X4 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_nat @ C @ Y3 )
% 12.45/7.00 => ( ( ( ord_less_nat @ X4 @ Y3 )
% 12.45/7.00 => ( ( image_nat_nat
% 12.45/7.00 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 12.45/7.00 @ ( set_or4665077453230672383an_nat @ X4 @ Y3 ) )
% 12.45/7.00 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 12.45/7.00 & ( ~ ( ord_less_nat @ X4 @ Y3 )
% 12.45/7.00 => ( ( image_nat_nat
% 12.45/7.00 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 12.45/7.00 @ ( set_or4665077453230672383an_nat @ X4 @ Y3 ) )
% 12.45/7.00 = bot_bot_set_nat ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9465_take__bit__numeral__minus__numeral__int,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( case_option_int_num @ zero_zero_int
% 12.45/7.00 @ ^ [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 ) ) )
% 12.45/7.00 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9466_take__bit__num__simps_I1_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 12.45/7.00 = none_num ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9467_bij__betw__Suc,axiom,
% 12.45/7.00 ! [M7: set_nat,N3: set_nat] :
% 12.45/7.00 ( ( bij_betw_nat_nat @ suc @ M7 @ N3 )
% 12.45/7.00 = ( ( image_nat_nat @ suc @ M7 )
% 12.45/7.00 = N3 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9468_take__bit__num__simps_I2_J,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 12.45/7.00 = ( some_num @ one ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9469_take__bit__num__simps_I5_J,axiom,
% 12.45/7.00 ! [R2: num] :
% 12.45/7.00 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
% 12.45/7.00 = ( some_num @ one ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9470_image__Suc__atLeastAtMost,axiom,
% 12.45/7.00 ! [I: nat,J: nat] :
% 12.45/7.00 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 12.45/7.00 = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9471_image__Suc__atLeastLessThan,axiom,
% 12.45/7.00 ! [I: nat,J: nat] :
% 12.45/7.00 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 12.45/7.00 = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9472_mult__rat,axiom,
% 12.45/7.00 ! [A: int,B2: int,C: int,D: int] :
% 12.45/7.00 ( ( times_times_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 12.45/7.00 = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9473_divide__rat,axiom,
% 12.45/7.00 ! [A: int,B2: int,C: int,D: int] :
% 12.45/7.00 ( ( divide_divide_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 12.45/7.00 = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9474_floor__Fract,axiom,
% 12.45/7.00 ! [A: int,B2: int] :
% 12.45/7.00 ( ( archim3151403230148437115or_rat @ ( fract @ A @ B2 ) )
% 12.45/7.00 = ( divide_divide_int @ A @ B2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9475_less__rat,axiom,
% 12.45/7.00 ! [B2: int,D: int,A: int,C: int] :
% 12.45/7.00 ( ( B2 != zero_zero_int )
% 12.45/7.00 => ( ( D != zero_zero_int )
% 12.45/7.00 => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 12.45/7.00 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9476_add__rat,axiom,
% 12.45/7.00 ! [B2: int,D: int,A: int,C: int] :
% 12.45/7.00 ( ( B2 != zero_zero_int )
% 12.45/7.00 => ( ( D != zero_zero_int )
% 12.45/7.00 => ( ( plus_plus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 12.45/7.00 = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9477_le__rat,axiom,
% 12.45/7.00 ! [B2: int,D: int,A: int,C: int] :
% 12.45/7.00 ( ( B2 != zero_zero_int )
% 12.45/7.00 => ( ( D != zero_zero_int )
% 12.45/7.00 => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 12.45/7.00 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9478_diff__rat,axiom,
% 12.45/7.00 ! [B2: int,D: int,A: int,C: int] :
% 12.45/7.00 ( ( B2 != zero_zero_int )
% 12.45/7.00 => ( ( D != zero_zero_int )
% 12.45/7.00 => ( ( minus_minus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 12.45/7.00 = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9479_sgn__rat,axiom,
% 12.45/7.00 ! [A: int,B2: int] :
% 12.45/7.00 ( ( sgn_sgn_rat @ ( fract @ A @ B2 ) )
% 12.45/7.00 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9480_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( bit_take_bit_num @ N @ one )
% 12.45/7.00 = ( case_nat_option_num @ none_num
% 12.45/7.00 @ ^ [N2: nat] : ( some_num @ one )
% 12.45/7.00 @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9481_zero__notin__Suc__image,axiom,
% 12.45/7.00 ! [A3: set_nat] :
% 12.45/7.00 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9482_Rat__induct__pos,axiom,
% 12.45/7.00 ! [P: rat > $o,Q2: rat] :
% 12.45/7.00 ( ! [A5: int,B5: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ B5 )
% 12.45/7.00 => ( P @ ( fract @ A5 @ B5 ) ) )
% 12.45/7.00 => ( P @ Q2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9483_eq__rat_I1_J,axiom,
% 12.45/7.00 ! [B2: int,D: int,A: int,C: int] :
% 12.45/7.00 ( ( B2 != zero_zero_int )
% 12.45/7.00 => ( ( D != zero_zero_int )
% 12.45/7.00 => ( ( ( fract @ A @ B2 )
% 12.45/7.00 = ( fract @ C @ D ) )
% 12.45/7.00 = ( ( times_times_int @ A @ D )
% 12.45/7.00 = ( times_times_int @ C @ B2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9484_mult__rat__cancel,axiom,
% 12.45/7.00 ! [C: int,A: int,B2: int] :
% 12.45/7.00 ( ( C != zero_zero_int )
% 12.45/7.00 => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 12.45/7.00 = ( fract @ A @ B2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9485_Fract__coprime,axiom,
% 12.45/7.00 ! [A: int,B2: int] :
% 12.45/7.00 ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B2 ) ) @ ( divide_divide_int @ B2 @ ( gcd_gcd_int @ A @ B2 ) ) )
% 12.45/7.00 = ( fract @ A @ B2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9486_Fract__of__int__quotient,axiom,
% 12.45/7.00 ( fract
% 12.45/7.00 = ( ^ [K3: int,L: int] : ( divide_divide_rat @ ( ring_1_of_int_rat @ K3 ) @ ( ring_1_of_int_rat @ L ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9487_image__Suc__lessThan,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.00 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9488_image__Suc__atMost,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 12.45/7.00 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9489_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 12.45/7.00 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9490_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 12.45/7.00 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9491_lessThan__Suc__eq__insert__0,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 12.45/7.00 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9492_atMost__Suc__eq__insert__0,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 12.45/7.00 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9493_Fract__less__zero__iff,axiom,
% 12.45/7.00 ! [B2: int,A: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/7.00 => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
% 12.45/7.00 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9494_zero__less__Fract__iff,axiom,
% 12.45/7.00 ! [B2: int,A: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/7.00 => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
% 12.45/7.00 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9495_Fract__less__one__iff,axiom,
% 12.45/7.00 ! [B2: int,A: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/7.00 => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ one_one_rat )
% 12.45/7.00 = ( ord_less_int @ A @ B2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9496_one__less__Fract__iff,axiom,
% 12.45/7.00 ! [B2: int,A: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/7.00 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B2 ) )
% 12.45/7.00 = ( ord_less_int @ B2 @ A ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9497_Fract__add__one,axiom,
% 12.45/7.00 ! [N: int,M: int] :
% 12.45/7.00 ( ( N != zero_zero_int )
% 12.45/7.00 => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 12.45/7.00 = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9498_Fract__le__zero__iff,axiom,
% 12.45/7.00 ! [B2: int,A: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/7.00 => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
% 12.45/7.00 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9499_zero__le__Fract__iff,axiom,
% 12.45/7.00 ! [B2: int,A: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/7.00 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
% 12.45/7.00 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9500_one__le__Fract__iff,axiom,
% 12.45/7.00 ! [B2: int,A: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/7.00 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B2 ) )
% 12.45/7.00 = ( ord_less_eq_int @ B2 @ A ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9501_Fract__le__one__iff,axiom,
% 12.45/7.00 ! [B2: int,A: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ B2 )
% 12.45/7.00 => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ one_one_rat )
% 12.45/7.00 = ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9502_take__bit__num__def,axiom,
% 12.45/7.00 ( bit_take_bit_num
% 12.45/7.00 = ( ^ [N2: nat,M6: num] :
% 12.45/7.00 ( if_option_num
% 12.45/7.00 @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M6 ) )
% 12.45/7.00 = zero_zero_nat )
% 12.45/7.00 @ none_num
% 12.45/7.00 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9503_and__minus__numerals_I3_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.45/7.00 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9504_and__minus__numerals_I7_J,axiom,
% 12.45/7.00 ! [N: num,M: num] :
% 12.45/7.00 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 12.45/7.00 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9505_and__minus__numerals_I4_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.45/7.00 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9506_take__bit__num__simps_I4_J,axiom,
% 12.45/7.00 ! [N: nat,M: num] :
% 12.45/7.00 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 12.45/7.00 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9507_take__bit__num__simps_I3_J,axiom,
% 12.45/7.00 ! [N: nat,M: num] :
% 12.45/7.00 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 12.45/7.00 = ( case_o6005452278849405969um_num @ none_num
% 12.45/7.00 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 12.45/7.00 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9508_take__bit__num__simps_I7_J,axiom,
% 12.45/7.00 ! [R2: num,M: num] :
% 12.45/7.00 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
% 12.45/7.00 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9509_take__bit__num__simps_I6_J,axiom,
% 12.45/7.00 ! [R2: num,M: num] :
% 12.45/7.00 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 12.45/7.00 = ( case_o6005452278849405969um_num @ none_num
% 12.45/7.00 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 12.45/7.00 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9510_and__minus__numerals_I8_J,axiom,
% 12.45/7.00 ! [N: num,M: num] :
% 12.45/7.00 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 12.45/7.00 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9511_and__not__num_Osimps_I8_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.45/7.00 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.45/7.00 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.45/7.00 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9512_and__not__num_Osimps_I1_J,axiom,
% 12.45/7.00 ( ( bit_and_not_num @ one @ one )
% 12.45/7.00 = none_num ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9513_and__not__num_Osimps_I2_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 12.45/7.00 = ( some_num @ one ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9514_and__not__num_Osimps_I4_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 12.45/7.00 = ( some_num @ ( bit0 @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9515_and__not__num_Osimps_I3_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 12.45/7.00 = none_num ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9516_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 12.45/7.00 ! [N: nat,M: num] :
% 12.45/7.00 ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 12.45/7.00 = ( case_nat_option_num @ none_num
% 12.45/7.00 @ ^ [N2: nat] :
% 12.45/7.00 ( case_o6005452278849405969um_num @ none_num
% 12.45/7.00 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 12.45/7.00 @ ( bit_take_bit_num @ N2 @ M ) )
% 12.45/7.00 @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9517_image__int__atLeastAtMost,axiom,
% 12.45/7.00 ! [A: nat,B2: nat] :
% 12.45/7.00 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 12.45/7.00 = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9518_image__int__atLeastLessThan,axiom,
% 12.45/7.00 ! [A: nat,B2: nat] :
% 12.45/7.00 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B2 ) )
% 12.45/7.00 = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9519_and__not__num_Osimps_I7_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 12.45/7.00 = ( some_num @ ( bit0 @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9520_and__not__num__eq__Some__iff,axiom,
% 12.45/7.00 ! [M: num,N: num,Q2: num] :
% 12.45/7.00 ( ( ( bit_and_not_num @ M @ N )
% 12.45/7.00 = ( some_num @ Q2 ) )
% 12.45/7.00 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( numeral_numeral_int @ Q2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9521_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 12.45/7.00 ! [N: nat,M: num] :
% 12.45/7.00 ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 12.45/7.00 = ( case_nat_option_num @ none_num
% 12.45/7.00 @ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
% 12.45/7.00 @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9522_image__add__int__atLeastLessThan,axiom,
% 12.45/7.00 ! [L2: int,U: int] :
% 12.45/7.00 ( ( image_int_int
% 12.45/7.00 @ ^ [X: int] : ( plus_plus_int @ X @ L2 )
% 12.45/7.00 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
% 12.45/7.00 = ( set_or4662586982721622107an_int @ L2 @ U ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9523_and__not__num__eq__None__iff,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( ( bit_and_not_num @ M @ N )
% 12.45/7.00 = none_num )
% 12.45/7.00 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = zero_zero_int ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9524_image__atLeastZeroLessThan__int,axiom,
% 12.45/7.00 ! [U: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 12.45/7.00 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 12.45/7.00 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9525_int__numeral__and__not__num,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9526_int__numeral__not__and__num,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9527_positive__rat,axiom,
% 12.45/7.00 ! [A: int,B2: int] :
% 12.45/7.00 ( ( positive @ ( fract @ A @ B2 ) )
% 12.45/7.00 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9528_Rat_Opositive__add,axiom,
% 12.45/7.00 ! [X4: rat,Y3: rat] :
% 12.45/7.00 ( ( positive @ X4 )
% 12.45/7.00 => ( ( positive @ Y3 )
% 12.45/7.00 => ( positive @ ( plus_plus_rat @ X4 @ Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9529_Rat_Opositive__mult,axiom,
% 12.45/7.00 ! [X4: rat,Y3: rat] :
% 12.45/7.00 ( ( positive @ X4 )
% 12.45/7.00 => ( ( positive @ Y3 )
% 12.45/7.00 => ( positive @ ( times_times_rat @ X4 @ Y3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9530_less__rat__def,axiom,
% 12.45/7.00 ( ord_less_rat
% 12.45/7.00 = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9531_Rat_Opositive_Orep__eq,axiom,
% 12.45/7.00 ( positive
% 12.45/7.00 = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9532_and__not__num_Oelims,axiom,
% 12.45/7.00 ! [X4: num,Xa: num,Y3: option_num] :
% 12.45/7.00 ( ( ( bit_and_not_num @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3 != none_num ) ) )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ( ? [N4: num] :
% 12.45/7.00 ( Xa
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ one ) ) ) )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ( ? [N4: num] :
% 12.45/7.00 ( Xa
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ( Y3 != none_num ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.45/7.00 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.45/7.00 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 12.45/7.00 => ~ ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9533_Bit__Operations_Otake__bit__num__code,axiom,
% 12.45/7.00 ( bit_take_bit_num
% 12.45/7.00 = ( ^ [N2: nat,M6: num] :
% 12.45/7.00 ( produc478579273971653890on_num
% 12.45/7.00 @ ^ [A4: nat,X: num] :
% 12.45/7.00 ( case_nat_option_num @ none_num
% 12.45/7.00 @ ^ [O: nat] :
% 12.45/7.00 ( case_num_option_num @ ( some_num @ one )
% 12.45/7.00 @ ^ [P3: num] :
% 12.45/7.00 ( case_o6005452278849405969um_num @ none_num
% 12.45/7.00 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 12.45/7.00 @ ( bit_take_bit_num @ O @ P3 ) )
% 12.45/7.00 @ ^ [P3: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P3 ) ) )
% 12.45/7.00 @ X )
% 12.45/7.00 @ A4 )
% 12.45/7.00 @ ( product_Pair_nat_num @ N2 @ M6 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9534_and__not__num_Osimps_I5_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.45/7.00 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9535_and__not__num_Osimps_I9_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9536_and__not__num_Osimps_I6_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9537_and__num_Oelims,axiom,
% 12.45/7.00 ! [X4: num,Xa: num,Y3: option_num] :
% 12.45/7.00 ( ( ( bit_un7362597486090784418nd_num @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ one ) ) ) )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ( ? [N4: num] :
% 12.45/7.00 ( Xa
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ( Y3 != none_num ) ) )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ( ? [N4: num] :
% 12.45/7.00 ( Xa
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ one ) ) ) )
% 12.45/7.00 => ( ( ? [M4: num] :
% 12.45/7.00 ( X4
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3 != none_num ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 12.45/7.00 => ( ( ? [M4: num] :
% 12.45/7.00 ( X4
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ one ) ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 12.45/7.00 => ~ ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.45/7.00 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.45/7.00 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9538_xor__num_Oelims,axiom,
% 12.45/7.00 ! [X4: num,Xa: num,Y3: option_num] :
% 12.45/7.00 ( ( ( bit_un2480387367778600638or_num @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3 != none_num ) ) )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ ( bit1 @ N4 ) ) ) ) )
% 12.45/7.00 => ( ( ( X4 = one )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ ( bit0 @ N4 ) ) ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ ( bit1 @ M4 ) ) ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit0 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ( ( Xa = one )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 12.45/7.00 => ( ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit0 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) )
% 12.45/7.00 => ~ ! [M4: num] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( bit1 @ M4 ) )
% 12.45/7.00 => ! [N4: num] :
% 12.45/7.00 ( ( Xa
% 12.45/7.00 = ( bit1 @ N4 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9539_xor__num_Osimps_I8_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.45/7.00 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9540_and__num_Osimps_I1_J,axiom,
% 12.45/7.00 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 12.45/7.00 = ( some_num @ one ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9541_xor__num_Osimps_I1_J,axiom,
% 12.45/7.00 ( ( bit_un2480387367778600638or_num @ one @ one )
% 12.45/7.00 = none_num ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9542_and__num_Osimps_I5_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.45/7.00 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9543_xor__num_Osimps_I5_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.45/7.00 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9544_and__num_Osimps_I7_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 12.45/7.00 = ( some_num @ one ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9545_and__num_Osimps_I3_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 12.45/7.00 = ( some_num @ one ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9546_and__num_Osimps_I2_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 12.45/7.00 = none_num ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9547_and__num_Osimps_I4_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 12.45/7.00 = none_num ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9548_xor__num_Osimps_I9_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9549_and__num_Osimps_I6_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9550_and__num_Osimps_I8_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.45/7.00 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9551_xor__num_Osimps_I7_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 12.45/7.00 = ( some_num @ ( bit0 @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9552_xor__num_Osimps_I4_J,axiom,
% 12.45/7.00 ! [M: num] :
% 12.45/7.00 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 12.45/7.00 = ( some_num @ ( bit1 @ M ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9553_xor__num_Osimps_I3_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 12.45/7.00 = ( some_num @ ( bit0 @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9554_xor__num_Osimps_I2_J,axiom,
% 12.45/7.00 ! [N: num] :
% 12.45/7.00 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 12.45/7.00 = ( some_num @ ( bit1 @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9555_and__num_Osimps_I9_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.45/7.00 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.45/7.00 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9556_xor__num_Osimps_I6_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.45/7.00 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9557_and__num__dict,axiom,
% 12.45/7.00 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9558_xor__num__dict,axiom,
% 12.45/7.00 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9559_num__of__integer__code,axiom,
% 12.45/7.00 ( code_num_of_integer
% 12.45/7.00 = ( ^ [K3: code_integer] :
% 12.45/7.00 ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
% 12.45/7.00 @ ( produc7336495610019696514er_num
% 12.45/7.00 @ ^ [L: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
% 12.45/7.00 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9560_min__Suc__Suc,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.45/7.00 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9561_min__0R,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_min_nat @ N @ zero_zero_nat )
% 12.45/7.00 = zero_zero_nat ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9562_min__0L,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_min_nat @ zero_zero_nat @ N )
% 12.45/7.00 = zero_zero_nat ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9563_min__numeral__Suc,axiom,
% 12.45/7.00 ! [K: num,N: nat] :
% 12.45/7.00 ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 12.45/7.00 = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9564_min__Suc__numeral,axiom,
% 12.45/7.00 ! [N: nat,K: num] :
% 12.45/7.00 ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.45/7.00 = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9565_nat__mult__min__right,axiom,
% 12.45/7.00 ! [M: nat,N: nat,Q2: nat] :
% 12.45/7.00 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
% 12.45/7.00 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9566_nat__mult__min__left,axiom,
% 12.45/7.00 ! [M: nat,N: nat,Q2: nat] :
% 12.45/7.00 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
% 12.45/7.00 = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9567_min__diff,axiom,
% 12.45/7.00 ! [M: nat,I: nat,N: nat] :
% 12.45/7.00 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 12.45/7.00 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9568_concat__bit__assoc__sym,axiom,
% 12.45/7.00 ! [M: nat,N: nat,K: int,L2: int,R2: int] :
% 12.45/7.00 ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L2 ) @ R2 )
% 12.45/7.00 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L2 @ R2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9569_take__bit__concat__bit__eq,axiom,
% 12.45/7.00 ! [M: nat,N: nat,K: int,L2: int] :
% 12.45/7.00 ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L2 ) )
% 12.45/7.00 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L2 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9570_min__Suc1,axiom,
% 12.45/7.00 ! [N: nat,M: nat] :
% 12.45/7.00 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 12.45/7.00 = ( case_nat_nat @ zero_zero_nat
% 12.45/7.00 @ ^ [M2: nat] : ( suc @ ( ord_min_nat @ N @ M2 ) )
% 12.45/7.00 @ M ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9571_min__Suc2,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 12.45/7.00 = ( case_nat_nat @ zero_zero_nat
% 12.45/7.00 @ ^ [M2: nat] : ( suc @ ( ord_min_nat @ M2 @ N ) )
% 12.45/7.00 @ M ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9572_card__le__Suc__Max,axiom,
% 12.45/7.00 ! [S2: set_nat] :
% 12.45/7.00 ( ( finite_finite_nat @ S2 )
% 12.45/7.00 => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9573_divide__nat__def,axiom,
% 12.45/7.00 ( divide_divide_nat
% 12.45/7.00 = ( ^ [M6: nat,N2: nat] :
% 12.45/7.00 ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 12.45/7.00 @ ( lattic8265883725875713057ax_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M6 ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9574_gcd__is__Max__divisors__nat,axiom,
% 12.45/7.00 ! [N: nat,M: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( ( gcd_gcd_nat @ M @ N )
% 12.45/7.00 = ( lattic8265883725875713057ax_nat
% 12.45/7.00 @ ( collect_nat
% 12.45/7.00 @ ^ [D2: nat] :
% 12.45/7.00 ( ( dvd_dvd_nat @ D2 @ M )
% 12.45/7.00 & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9575_min__enat__simps_I2_J,axiom,
% 12.45/7.00 ! [Q2: extended_enat] :
% 12.45/7.00 ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 12.45/7.00 = zero_z5237406670263579293d_enat ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9576_min__enat__simps_I3_J,axiom,
% 12.45/7.00 ! [Q2: extended_enat] :
% 12.45/7.00 ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 12.45/7.00 = zero_z5237406670263579293d_enat ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9577_sorted__list__of__set__greaterThanAtMost,axiom,
% 12.45/7.00 ! [I: nat,J: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 12.45/7.00 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 12.45/7.00 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9578_sorted__list__of__set__greaterThanLessThan,axiom,
% 12.45/7.00 ! [I: nat,J: nat] :
% 12.45/7.00 ( ( ord_less_nat @ ( suc @ I ) @ J )
% 12.45/7.00 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 12.45/7.00 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9579_upto__aux__rec,axiom,
% 12.45/7.00 ( upto_aux
% 12.45/7.00 = ( ^ [I2: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9580_list__encode_Osimps_I2_J,axiom,
% 12.45/7.00 ! [X4: nat,Xs2: list_nat] :
% 12.45/7.00 ( ( nat_list_encode @ ( cons_nat @ X4 @ Xs2 ) )
% 12.45/7.00 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9581_list__encode_Oelims,axiom,
% 12.45/7.00 ! [X4: list_nat,Y3: nat] :
% 12.45/7.00 ( ( ( nat_list_encode @ X4 )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ( X4 = nil_nat )
% 12.45/7.00 => ( Y3 != zero_zero_nat ) )
% 12.45/7.00 => ~ ! [X3: nat,Xs3: list_nat] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( cons_nat @ X3 @ Xs3 ) )
% 12.45/7.00 => ( Y3
% 12.45/7.00 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9582_quotient__of__def,axiom,
% 12.45/7.00 ( quotient_of
% 12.45/7.00 = ( ^ [X: rat] :
% 12.45/7.00 ( the_Pr4378521158711661632nt_int
% 12.45/7.00 @ ^ [Pair: product_prod_int_int] :
% 12.45/7.00 ( ( X
% 12.45/7.00 = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 12.45/7.00 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 12.45/7.00 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9583_coprime__abs__right__iff,axiom,
% 12.45/7.00 ! [K: int,L2: int] :
% 12.45/7.00 ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L2 ) )
% 12.45/7.00 = ( algebr932160517623751201me_int @ K @ L2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9584_coprime__abs__left__iff,axiom,
% 12.45/7.00 ! [K: int,L2: int] :
% 12.45/7.00 ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L2 )
% 12.45/7.00 = ( algebr932160517623751201me_int @ K @ L2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9585_normalize__stable,axiom,
% 12.45/7.00 ! [Q2: int,P2: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ Q2 )
% 12.45/7.00 => ( ( algebr932160517623751201me_int @ P2 @ Q2 )
% 12.45/7.00 => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 12.45/7.00 = ( product_Pair_int_int @ P2 @ Q2 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9586_coprime__crossproduct__int,axiom,
% 12.45/7.00 ! [A: int,D: int,B2: int,C: int] :
% 12.45/7.00 ( ( algebr932160517623751201me_int @ A @ D )
% 12.45/7.00 => ( ( algebr932160517623751201me_int @ B2 @ C )
% 12.45/7.00 => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
% 12.45/7.00 = ( times_times_int @ ( abs_abs_int @ B2 ) @ ( abs_abs_int @ D ) ) )
% 12.45/7.00 = ( ( ( abs_abs_int @ A )
% 12.45/7.00 = ( abs_abs_int @ B2 ) )
% 12.45/7.00 & ( ( abs_abs_int @ C )
% 12.45/7.00 = ( abs_abs_int @ D ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9587_Rat__cases,axiom,
% 12.45/7.00 ! [Q2: rat] :
% 12.45/7.00 ~ ! [A5: int,B5: int] :
% 12.45/7.00 ( ( Q2
% 12.45/7.00 = ( fract @ A5 @ B5 ) )
% 12.45/7.00 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 12.45/7.00 => ~ ( algebr932160517623751201me_int @ A5 @ B5 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9588_Rat__induct,axiom,
% 12.45/7.00 ! [P: rat > $o,Q2: rat] :
% 12.45/7.00 ( ! [A5: int,B5: int] :
% 12.45/7.00 ( ( ord_less_int @ zero_zero_int @ B5 )
% 12.45/7.00 => ( ( algebr932160517623751201me_int @ A5 @ B5 )
% 12.45/7.00 => ( P @ ( fract @ A5 @ B5 ) ) ) )
% 12.45/7.00 => ( P @ Q2 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9589_coprime__common__divisor__int,axiom,
% 12.45/7.00 ! [A: int,B2: int,X4: int] :
% 12.45/7.00 ( ( algebr932160517623751201me_int @ A @ B2 )
% 12.45/7.00 => ( ( dvd_dvd_int @ X4 @ A )
% 12.45/7.00 => ( ( dvd_dvd_int @ X4 @ B2 )
% 12.45/7.00 => ( ( abs_abs_int @ X4 )
% 12.45/7.00 = one_one_int ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9590_Rat__cases__nonzero,axiom,
% 12.45/7.00 ! [Q2: rat] :
% 12.45/7.00 ( ! [A5: int,B5: int] :
% 12.45/7.00 ( ( Q2
% 12.45/7.00 = ( fract @ A5 @ B5 ) )
% 12.45/7.00 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 12.45/7.00 => ( ( A5 != zero_zero_int )
% 12.45/7.00 => ~ ( algebr932160517623751201me_int @ A5 @ B5 ) ) ) )
% 12.45/7.00 => ( Q2 = zero_zero_rat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9591_card__length__sum__list__rec,axiom,
% 12.45/7.00 ! [M: nat,N3: nat] :
% 12.45/7.00 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 12.45/7.00 => ( ( finite_card_list_nat
% 12.45/7.00 @ ( collect_list_nat
% 12.45/7.00 @ ^ [L: list_nat] :
% 12.45/7.00 ( ( ( size_size_list_nat @ L )
% 12.45/7.00 = M )
% 12.45/7.00 & ( ( groups4561878855575611511st_nat @ L )
% 12.45/7.00 = N3 ) ) ) )
% 12.45/7.00 = ( plus_plus_nat
% 12.45/7.00 @ ( finite_card_list_nat
% 12.45/7.00 @ ( collect_list_nat
% 12.45/7.00 @ ^ [L: list_nat] :
% 12.45/7.00 ( ( ( size_size_list_nat @ L )
% 12.45/7.00 = ( minus_minus_nat @ M @ one_one_nat ) )
% 12.45/7.00 & ( ( groups4561878855575611511st_nat @ L )
% 12.45/7.00 = N3 ) ) ) )
% 12.45/7.00 @ ( finite_card_list_nat
% 12.45/7.00 @ ( collect_list_nat
% 12.45/7.00 @ ^ [L: list_nat] :
% 12.45/7.00 ( ( ( size_size_list_nat @ L )
% 12.45/7.00 = M )
% 12.45/7.00 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L ) @ one_one_nat )
% 12.45/7.00 = N3 ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9592_card__length__sum__list,axiom,
% 12.45/7.00 ! [M: nat,N3: nat] :
% 12.45/7.00 ( ( finite_card_list_nat
% 12.45/7.00 @ ( collect_list_nat
% 12.45/7.00 @ ^ [L: list_nat] :
% 12.45/7.00 ( ( ( size_size_list_nat @ L )
% 12.45/7.00 = M )
% 12.45/7.00 & ( ( groups4561878855575611511st_nat @ L )
% 12.45/7.00 = N3 ) ) ) )
% 12.45/7.00 = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M ) @ one_one_nat ) @ N3 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9593_quotient__of__unique,axiom,
% 12.45/7.00 ! [R2: rat] :
% 12.45/7.00 ? [X3: product_prod_int_int] :
% 12.45/7.00 ( ( R2
% 12.45/7.00 = ( fract @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 12.45/7.00 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X3 ) )
% 12.45/7.00 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) )
% 12.45/7.00 & ! [Y5: product_prod_int_int] :
% 12.45/7.00 ( ( ( R2
% 12.45/7.00 = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 12.45/7.00 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
% 12.45/7.00 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 12.45/7.00 => ( Y5 = X3 ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9594_sorted__list__of__set__atMost__Suc,axiom,
% 12.45/7.00 ! [K: nat] :
% 12.45/7.00 ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 12.45/7.00 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9595_sorted__list__of__set__lessThan__Suc,axiom,
% 12.45/7.00 ! [K: nat] :
% 12.45/7.00 ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 12.45/7.00 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9596_coprime__int__iff,axiom,
% 12.45/7.00 ! [M: nat,N: nat] :
% 12.45/7.00 ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.45/7.00 = ( algebr934650988132801477me_nat @ M @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9597_coprime__nat__abs__right__iff,axiom,
% 12.45/7.00 ! [N: nat,K: int] :
% 12.45/7.00 ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 12.45/7.00 = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9598_coprime__nat__abs__left__iff,axiom,
% 12.45/7.00 ! [K: int,N: nat] :
% 12.45/7.00 ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 12.45/7.00 = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9599_coprime__common__divisor__nat,axiom,
% 12.45/7.00 ! [A: nat,B2: nat,X4: nat] :
% 12.45/7.00 ( ( algebr934650988132801477me_nat @ A @ B2 )
% 12.45/7.00 => ( ( dvd_dvd_nat @ X4 @ A )
% 12.45/7.00 => ( ( dvd_dvd_nat @ X4 @ B2 )
% 12.45/7.00 => ( X4 = one_one_nat ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9600_coprime__Suc__0__right,axiom,
% 12.45/7.00 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9601_coprime__Suc__0__left,axiom,
% 12.45/7.00 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9602_coprime__Suc__left__nat,axiom,
% 12.45/7.00 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9603_coprime__Suc__right__nat,axiom,
% 12.45/7.00 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9604_coprime__crossproduct__nat,axiom,
% 12.45/7.00 ! [A: nat,D: nat,B2: nat,C: nat] :
% 12.45/7.00 ( ( algebr934650988132801477me_nat @ A @ D )
% 12.45/7.00 => ( ( algebr934650988132801477me_nat @ B2 @ C )
% 12.45/7.00 => ( ( ( times_times_nat @ A @ C )
% 12.45/7.00 = ( times_times_nat @ B2 @ D ) )
% 12.45/7.00 = ( ( A = B2 )
% 12.45/7.00 & ( C = D ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9605_coprime__diff__one__right__nat,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9606_coprime__diff__one__left__nat,axiom,
% 12.45/7.00 ! [N: nat] :
% 12.45/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.00 => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9607_upto_Opsimps,axiom,
% 12.45/7.00 ! [I: int,J: int] :
% 12.45/7.00 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 12.45/7.00 => ( ( ( ord_less_eq_int @ I @ J )
% 12.45/7.00 => ( ( upto @ I @ J )
% 12.45/7.00 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_int @ I @ J )
% 12.45/7.00 => ( ( upto @ I @ J )
% 12.45/7.00 = nil_int ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9608_upto__Nil,axiom,
% 12.45/7.00 ! [I: int,J: int] :
% 12.45/7.00 ( ( ( upto @ I @ J )
% 12.45/7.00 = nil_int )
% 12.45/7.00 = ( ord_less_int @ J @ I ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9609_upto__Nil2,axiom,
% 12.45/7.00 ! [I: int,J: int] :
% 12.45/7.00 ( ( nil_int
% 12.45/7.00 = ( upto @ I @ J ) )
% 12.45/7.00 = ( ord_less_int @ J @ I ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9610_upto__empty,axiom,
% 12.45/7.00 ! [J: int,I: int] :
% 12.45/7.00 ( ( ord_less_int @ J @ I )
% 12.45/7.00 => ( ( upto @ I @ J )
% 12.45/7.00 = nil_int ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9611_nth__upto,axiom,
% 12.45/7.00 ! [I: int,K: nat,J: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 12.45/7.00 => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 12.45/7.00 = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9612_length__upto,axiom,
% 12.45/7.00 ! [I: int,J: int] :
% 12.45/7.00 ( ( size_size_list_int @ ( upto @ I @ J ) )
% 12.45/7.00 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9613_upto__rec__numeral_I1_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 = nil_int ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9614_upto__rec__numeral_I2_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = nil_int ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9615_upto__rec__numeral_I3_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 = ( 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 ) ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.45/7.00 = nil_int ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9616_upto__rec__numeral_I4_J,axiom,
% 12.45/7.00 ! [M: num,N: num] :
% 12.45/7.00 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = ( 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 ) ) ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.45/7.00 = nil_int ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9617_upto__split2,axiom,
% 12.45/7.00 ! [I: int,J: int,K: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ I @ J )
% 12.45/7.00 => ( ( ord_less_eq_int @ J @ K )
% 12.45/7.00 => ( ( upto @ I @ K )
% 12.45/7.00 = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9618_upto__split1,axiom,
% 12.45/7.00 ! [I: int,J: int,K: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ I @ J )
% 12.45/7.00 => ( ( ord_less_eq_int @ J @ K )
% 12.45/7.00 => ( ( upto @ I @ K )
% 12.45/7.00 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9619_atLeastLessThan__upto,axiom,
% 12.45/7.00 ( set_or4662586982721622107an_int
% 12.45/7.00 = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9620_greaterThanAtMost__upto,axiom,
% 12.45/7.00 ( set_or6656581121297822940st_int
% 12.45/7.00 = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9621_upto__rec1,axiom,
% 12.45/7.00 ! [I: int,J: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ I @ J )
% 12.45/7.00 => ( ( upto @ I @ J )
% 12.45/7.00 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9622_upto_Oelims,axiom,
% 12.45/7.00 ! [X4: int,Xa: int,Y3: list_int] :
% 12.45/7.00 ( ( ( upto @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( ( ord_less_eq_int @ X4 @ Xa )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_int @ X4 @ Xa )
% 12.45/7.00 => ( Y3 = nil_int ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9623_upto_Osimps,axiom,
% 12.45/7.00 ( upto
% 12.45/7.00 = ( ^ [I2: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I2 @ J3 ) @ ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9624_upto__rec2,axiom,
% 12.45/7.00 ! [I: int,J: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ I @ J )
% 12.45/7.00 => ( ( upto @ I @ J )
% 12.45/7.00 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9625_greaterThanLessThan__upto,axiom,
% 12.45/7.00 ( set_or5832277885323065728an_int
% 12.45/7.00 = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9626_upto__split3,axiom,
% 12.45/7.00 ! [I: int,J: int,K: int] :
% 12.45/7.00 ( ( ord_less_eq_int @ I @ J )
% 12.45/7.00 => ( ( ord_less_eq_int @ J @ K )
% 12.45/7.00 => ( ( upto @ I @ K )
% 12.45/7.00 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9627_upto_Opelims,axiom,
% 12.45/7.00 ! [X4: int,Xa: int,Y3: list_int] :
% 12.45/7.00 ( ( ( upto @ X4 @ Xa )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) )
% 12.45/7.00 => ~ ( ( ( ( ord_less_eq_int @ X4 @ Xa )
% 12.45/7.00 => ( Y3
% 12.45/7.00 = ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
% 12.45/7.00 & ( ~ ( ord_less_eq_int @ X4 @ Xa )
% 12.45/7.00 => ( Y3 = nil_int ) ) )
% 12.45/7.00 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9628_list__encode_Opelims,axiom,
% 12.45/7.00 ! [X4: list_nat,Y3: nat] :
% 12.45/7.00 ( ( ( nat_list_encode @ X4 )
% 12.45/7.00 = Y3 )
% 12.45/7.00 => ( ( accp_list_nat @ nat_list_encode_rel @ X4 )
% 12.45/7.00 => ( ( ( X4 = nil_nat )
% 12.45/7.00 => ( ( Y3 = zero_zero_nat )
% 12.45/7.00 => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 12.45/7.00 => ~ ! [X3: nat,Xs3: list_nat] :
% 12.45/7.00 ( ( X4
% 12.45/7.00 = ( cons_nat @ X3 @ Xs3 ) )
% 12.45/7.00 => ( ( Y3
% 12.45/7.00 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) )
% 12.45/7.00 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ) ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9629_Rats__abs__iff,axiom,
% 12.45/7.00 ! [X4: real] :
% 12.45/7.00 ( ( member_real @ ( abs_abs_real @ X4 ) @ field_5140801741446780682s_real )
% 12.45/7.00 = ( member_real @ X4 @ field_5140801741446780682s_real ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9630_Rats__no__top__le,axiom,
% 12.45/7.00 ! [X4: real] :
% 12.45/7.00 ? [X3: real] :
% 12.45/7.00 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 12.45/7.00 & ( ord_less_eq_real @ X4 @ X3 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9631_Rats__no__bot__less,axiom,
% 12.45/7.00 ! [X4: real] :
% 12.45/7.00 ? [X3: real] :
% 12.45/7.00 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 12.45/7.00 & ( ord_less_real @ X3 @ X4 ) ) ).
% 12.45/7.00
% 12.45/7.00 thf(fact_9632_Rats__dense__in__real,axiom,
% 12.45/7.00 ! [X4: real,Y3: real] :
% 12.45/7.01 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/7.01 => ? [X3: real] :
% 12.45/7.01 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 12.45/7.01 & ( ord_less_real @ X4 @ X3 )
% 12.45/7.01 & ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9633_Rats__abs__nat__div__natE,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 12.45/7.01 => ~ ! [M4: nat,N4: nat] :
% 12.45/7.01 ( ( N4 != zero_zero_nat )
% 12.45/7.01 => ( ( ( abs_abs_real @ X4 )
% 12.45/7.01 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N4 ) ) )
% 12.45/7.01 => ~ ( algebr934650988132801477me_nat @ M4 @ N4 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9634_upt__rec__numeral,axiom,
% 12.45/7.01 ! [M: num,N: num] :
% 12.45/7.01 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.45/7.01 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.45/7.01 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 12.45/7.01 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.45/7.01 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.45/7.01 = nil_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9635_hd__upt,axiom,
% 12.45/7.01 ! [I: nat,J: nat] :
% 12.45/7.01 ( ( ord_less_nat @ I @ J )
% 12.45/7.01 => ( ( hd_nat @ ( upt @ I @ J ) )
% 12.45/7.01 = I ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9636_drop__upt,axiom,
% 12.45/7.01 ! [M: nat,I: nat,J: nat] :
% 12.45/7.01 ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 12.45/7.01 = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9637_take__upt,axiom,
% 12.45/7.01 ! [I: nat,M: nat,N: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 12.45/7.01 => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 12.45/7.01 = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9638_upt__conv__Nil,axiom,
% 12.45/7.01 ! [J: nat,I: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ J @ I )
% 12.45/7.01 => ( ( upt @ I @ J )
% 12.45/7.01 = nil_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9639_length__upt,axiom,
% 12.45/7.01 ! [I: nat,J: nat] :
% 12.45/7.01 ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 12.45/7.01 = ( minus_minus_nat @ J @ I ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9640_upt__eq__Nil__conv,axiom,
% 12.45/7.01 ! [I: nat,J: nat] :
% 12.45/7.01 ( ( ( upt @ I @ J )
% 12.45/7.01 = nil_nat )
% 12.45/7.01 = ( ( J = zero_zero_nat )
% 12.45/7.01 | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9641_nth__upt,axiom,
% 12.45/7.01 ! [I: nat,K: nat,J: nat] :
% 12.45/7.01 ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 12.45/7.01 => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 12.45/7.01 = ( plus_plus_nat @ I @ K ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9642_sum__list__upt,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ M @ N )
% 12.45/7.01 => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 12.45/7.01 = ( groups3542108847815614940at_nat
% 12.45/7.01 @ ^ [X: nat] : X
% 12.45/7.01 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9643_atLeastAtMost__upt,axiom,
% 12.45/7.01 ( set_or1269000886237332187st_nat
% 12.45/7.01 = ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M6 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9644_upt__conv__Cons__Cons,axiom,
% 12.45/7.01 ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
% 12.45/7.01 ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
% 12.45/7.01 = ( upt @ M @ Q2 ) )
% 12.45/7.01 = ( ( cons_nat @ N @ Ns )
% 12.45/7.01 = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9645_greaterThanAtMost__upt,axiom,
% 12.45/7.01 ( set_or6659071591806873216st_nat
% 12.45/7.01 = ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M6 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9646_greaterThanLessThan__upt,axiom,
% 12.45/7.01 ( set_or5834768355832116004an_nat
% 12.45/7.01 = ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M6 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9647_atMost__upto,axiom,
% 12.45/7.01 ( set_ord_atMost_nat
% 12.45/7.01 = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9648_upt__conv__Cons,axiom,
% 12.45/7.01 ! [I: nat,J: nat] :
% 12.45/7.01 ( ( ord_less_nat @ I @ J )
% 12.45/7.01 => ( ( upt @ I @ J )
% 12.45/7.01 = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9649_upt__add__eq__append,axiom,
% 12.45/7.01 ! [I: nat,J: nat,K: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ I @ J )
% 12.45/7.01 => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 12.45/7.01 = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9650_upt__eq__Cons__conv,axiom,
% 12.45/7.01 ! [I: nat,J: nat,X4: nat,Xs2: list_nat] :
% 12.45/7.01 ( ( ( upt @ I @ J )
% 12.45/7.01 = ( cons_nat @ X4 @ Xs2 ) )
% 12.45/7.01 = ( ( ord_less_nat @ I @ J )
% 12.45/7.01 & ( I = X4 )
% 12.45/7.01 & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 12.45/7.01 = Xs2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9651_upt__rec,axiom,
% 12.45/7.01 ( upt
% 12.45/7.01 = ( ^ [I2: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J3 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9652_upt__Suc__append,axiom,
% 12.45/7.01 ! [I: nat,J: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ I @ J )
% 12.45/7.01 => ( ( upt @ I @ ( suc @ J ) )
% 12.45/7.01 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9653_upt__Suc,axiom,
% 12.45/7.01 ! [I: nat,J: nat] :
% 12.45/7.01 ( ( ( ord_less_eq_nat @ I @ J )
% 12.45/7.01 => ( ( upt @ I @ ( suc @ J ) )
% 12.45/7.01 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 12.45/7.01 & ( ~ ( ord_less_eq_nat @ I @ J )
% 12.45/7.01 => ( ( upt @ I @ ( suc @ J ) )
% 12.45/7.01 = nil_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9654_mono__Suc,axiom,
% 12.45/7.01 order_mono_nat_nat @ suc ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9655_mono__times__nat,axiom,
% 12.45/7.01 ! [N: nat] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9656_mono__ge2__power__minus__self,axiom,
% 12.45/7.01 ! [K: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 12.45/7.01 => ( order_mono_nat_nat
% 12.45/7.01 @ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M6 ) @ M6 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9657_map__Suc__upt,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
% 12.45/7.01 = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9658_sorted__upt,axiom,
% 12.45/7.01 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9659_sorted__wrt__upt,axiom,
% 12.45/7.01 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9660_map__add__upt,axiom,
% 12.45/7.01 ! [N: nat,M: nat] :
% 12.45/7.01 ( ( map_nat_nat
% 12.45/7.01 @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N )
% 12.45/7.01 @ ( upt @ zero_zero_nat @ M ) )
% 12.45/7.01 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9661_map__decr__upt,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( map_nat_nat
% 12.45/7.01 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 12.45/7.01 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.45/7.01 = ( upt @ M @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9662_sorted__wrt__less__idx,axiom,
% 12.45/7.01 ! [Ns: list_nat,I: nat] :
% 12.45/7.01 ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 12.45/7.01 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 12.45/7.01 => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9663_sorted__wrt__upto,axiom,
% 12.45/7.01 ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9664_Rats__eq__int__div__int,axiom,
% 12.45/7.01 ( field_5140801741446780682s_real
% 12.45/7.01 = ( collect_real
% 12.45/7.01 @ ^ [Uu: real] :
% 12.45/7.01 ? [I2: int,J3: int] :
% 12.45/7.01 ( ( Uu
% 12.45/7.01 = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 12.45/7.01 & ( J3 != zero_zero_int ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9665_Rats__eq__int__div__nat,axiom,
% 12.45/7.01 ( field_5140801741446780682s_real
% 12.45/7.01 = ( collect_real
% 12.45/7.01 @ ^ [Uu: real] :
% 12.45/7.01 ? [I2: int,N2: nat] :
% 12.45/7.01 ( ( Uu
% 12.45/7.01 = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 12.45/7.01 & ( N2 != zero_zero_nat ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9666_range__mod,axiom,
% 12.45/7.01 ! [N: nat] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( image_nat_nat
% 12.45/7.01 @ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N )
% 12.45/7.01 @ top_top_set_nat )
% 12.45/7.01 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9667_UNIV__nat__eq,axiom,
% 12.45/7.01 ( top_top_set_nat
% 12.45/7.01 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9668_card__UNIV__unit,axiom,
% 12.45/7.01 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 12.45/7.01 = one_one_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9669_range__mult,axiom,
% 12.45/7.01 ! [A: real] :
% 12.45/7.01 ( ( ( A = zero_zero_real )
% 12.45/7.01 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 12.45/7.01 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 12.45/7.01 & ( ( A != zero_zero_real )
% 12.45/7.01 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 12.45/7.01 = top_top_set_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9670_card__UNIV__bool,axiom,
% 12.45/7.01 ( ( finite_card_o @ top_top_set_o )
% 12.45/7.01 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9671_root__def,axiom,
% 12.45/7.01 ( root
% 12.45/7.01 = ( ^ [N2: nat,X: real] :
% 12.45/7.01 ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 12.45/7.01 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 12.45/7.01 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 12.45/7.01 @ X ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9672_card__UNIV__char,axiom,
% 12.45/7.01 ( ( finite_card_char @ top_top_set_char )
% 12.45/7.01 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9673_UNIV__char__of__nat,axiom,
% 12.45/7.01 ( top_top_set_char
% 12.45/7.01 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9674_nat__of__char__less__256,axiom,
% 12.45/7.01 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9675_range__nat__of__char,axiom,
% 12.45/7.01 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 12.45/7.01 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9676_integer__of__char__code,axiom,
% 12.45/7.01 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 12.45/7.01 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 12.45/7.01 = ( 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 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9677_String_Ochar__of__ascii__of,axiom,
% 12.45/7.01 ! [C: char] :
% 12.45/7.01 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 12.45/7.01 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9678_DERIV__even__real__root,axiom,
% 12.45/7.01 ! [N: nat,X4: real] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9679_DERIV__real__root__generic,axiom,
% 12.45/7.01 ! [N: nat,X4: real,D4: real] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( X4 != zero_zero_real )
% 12.45/7.01 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.01 => ( D4
% 12.45/7.01 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 12.45/7.01 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ zero_zero_real )
% 12.45/7.01 => ( D4
% 12.45/7.01 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 12.45/7.01 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/7.01 => ( D4
% 12.45/7.01 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9680_DERIV__neg__dec__right,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,X4: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 12.45/7.01 => ? [D3: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ D3 )
% 12.45/7.01 & ! [H4: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ H4 )
% 12.45/7.01 => ( ( ord_less_real @ H4 @ D3 )
% 12.45/7.01 => ( ord_less_real @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9681_DERIV__pos__inc__right,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,X4: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 12.45/7.01 => ? [D3: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ D3 )
% 12.45/7.01 & ! [H4: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ H4 )
% 12.45/7.01 => ( ( ord_less_real @ H4 @ D3 )
% 12.45/7.01 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9682_DERIV__pos__imp__increasing,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_eq_real @ X3 @ B2 )
% 12.45/7.01 => ? [Y5: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 12.45/7.01 => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9683_DERIV__neg__imp__decreasing,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_eq_real @ X3 @ B2 )
% 12.45/7.01 => ? [Y5: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 12.45/7.01 => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9684_DERIV__nonpos__imp__nonincreasing,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_eq_real @ X3 @ B2 )
% 12.45/7.01 => ? [Y5: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 12.45/7.01 => ( ord_less_eq_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9685_DERIV__nonneg__imp__nondecreasing,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_eq_real @ X3 @ B2 )
% 12.45/7.01 => ? [Y5: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 12.45/7.01 => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9686_deriv__nonneg__imp__mono,axiom,
% 12.45/7.01 ! [A: real,B2: real,G: real > real,G2: real > real] :
% 12.45/7.01 ( ! [X3: real] :
% 12.45/7.01 ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 12.45/7.01 => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
% 12.45/7.01 => ( ( ord_less_eq_real @ A @ B2 )
% 12.45/7.01 => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9687_DERIV__isconst__all,axiom,
% 12.45/7.01 ! [F: real > real,X4: real,Y3: real] :
% 12.45/7.01 ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 => ( ( F @ X4 )
% 12.45/7.01 = ( F @ Y3 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9688_DERIV__mirror,axiom,
% 12.45/7.01 ! [F: real > real,Y3: real,X4: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X4 ) @ top_top_set_real ) )
% 12.45/7.01 = ( has_fi5821293074295781190e_real
% 12.45/7.01 @ ^ [X: real] : ( F @ ( uminus_uminus_real @ X ) )
% 12.45/7.01 @ ( uminus_uminus_real @ Y3 )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9689_DERIV__const__ratio__const2,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real,K: real] :
% 12.45/7.01 ( ( A != B2 )
% 12.45/7.01 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ ( minus_minus_real @ B2 @ A ) )
% 12.45/7.01 = K ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9690_DERIV__const__ratio__const,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real,K: real] :
% 12.45/7.01 ( ( A != B2 )
% 12.45/7.01 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 12.45/7.01 = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ K ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9691_DERIV__pos__inc__left,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,X4: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 12.45/7.01 => ? [D3: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ D3 )
% 12.45/7.01 & ! [H4: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ H4 )
% 12.45/7.01 => ( ( ord_less_real @ H4 @ D3 )
% 12.45/7.01 => ( ord_less_real @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9692_DERIV__neg__dec__left,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,X4: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 12.45/7.01 => ? [D3: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ D3 )
% 12.45/7.01 & ! [H4: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ H4 )
% 12.45/7.01 => ( ( ord_less_real @ H4 @ D3 )
% 12.45/7.01 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9693_DERIV__isconst3,axiom,
% 12.45/7.01 ! [A: real,B2: real,X4: real,Y3: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 12.45/7.01 => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 12.45/7.01 => ( ( F @ X4 )
% 12.45/7.01 = ( F @ Y3 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9694_has__real__derivative__pos__inc__left,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,X4: real,S2: set_real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 12.45/7.01 => ? [D3: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ D3 )
% 12.45/7.01 & ! [H4: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ H4 )
% 12.45/7.01 => ( ( member_real @ ( minus_minus_real @ X4 @ H4 ) @ S2 )
% 12.45/7.01 => ( ( ord_less_real @ H4 @ D3 )
% 12.45/7.01 => ( ord_less_real @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9695_has__real__derivative__neg__dec__left,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,X4: real,S2: set_real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
% 12.45/7.01 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 12.45/7.01 => ? [D3: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ D3 )
% 12.45/7.01 & ! [H4: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ H4 )
% 12.45/7.01 => ( ( member_real @ ( minus_minus_real @ X4 @ H4 ) @ S2 )
% 12.45/7.01 => ( ( ord_less_real @ H4 @ D3 )
% 12.45/7.01 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9696_has__real__derivative__pos__inc__right,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,X4: real,S2: set_real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 12.45/7.01 => ? [D3: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ D3 )
% 12.45/7.01 & ! [H4: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ H4 )
% 12.45/7.01 => ( ( member_real @ ( plus_plus_real @ X4 @ H4 ) @ S2 )
% 12.45/7.01 => ( ( ord_less_real @ H4 @ D3 )
% 12.45/7.01 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9697_has__real__derivative__neg__dec__right,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,X4: real,S2: set_real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
% 12.45/7.01 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 12.45/7.01 => ? [D3: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ D3 )
% 12.45/7.01 & ! [H4: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ H4 )
% 12.45/7.01 => ( ( member_real @ ( plus_plus_real @ X4 @ H4 ) @ S2 )
% 12.45/7.01 => ( ( ord_less_real @ H4 @ D3 )
% 12.45/7.01 => ( ord_less_real @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9698_MVT2,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real,F4: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_eq_real @ X3 @ B2 )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 12.45/7.01 => ? [Z2: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ Z2 )
% 12.45/7.01 & ( ord_less_real @ Z2 @ B2 )
% 12.45/7.01 & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 12.45/7.01 = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ ( F4 @ Z2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9699_DERIV__local__const,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,X4: real,D: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ D )
% 12.45/7.01 => ( ! [Y4: real] :
% 12.45/7.01 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) @ D )
% 12.45/7.01 => ( ( F @ X4 )
% 12.45/7.01 = ( F @ Y4 ) ) )
% 12.45/7.01 => ( L2 = zero_zero_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9700_DERIV__ln,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9701_DERIV__const__average,axiom,
% 12.45/7.01 ! [A: real,B2: real,V: real > real,K: real] :
% 12.45/7.01 ( ( A != B2 )
% 12.45/7.01 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.45/7.01 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9702_DERIV__local__max,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,X4: real,D: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ D )
% 12.45/7.01 => ( ! [Y4: real] :
% 12.45/7.01 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) @ D )
% 12.45/7.01 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 12.45/7.01 => ( L2 = zero_zero_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9703_DERIV__local__min,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,X4: real,D: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ D )
% 12.45/7.01 => ( ! [Y4: real] :
% 12.45/7.01 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) @ D )
% 12.45/7.01 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 12.45/7.01 => ( L2 = zero_zero_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9704_DERIV__ln__divide,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9705_DERIV__pow,axiom,
% 12.45/7.01 ! [N: nat,X4: real,S: set_real] :
% 12.45/7.01 ( has_fi5821293074295781190e_real
% 12.45/7.01 @ ^ [X: real] : ( power_power_real @ X @ N )
% 12.45/7.01 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X4 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ S ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9706_DERIV__fun__pow,axiom,
% 12.45/7.01 ! [G: real > real,M: real,X4: real,N: nat] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real
% 12.45/7.01 @ ^ [X: real] : ( power_power_real @ ( G @ X ) @ N )
% 12.45/7.01 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X4 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9707_has__real__derivative__powr,axiom,
% 12.45/7.01 ! [Z: real,R2: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ Z )
% 12.45/7.01 => ( has_fi5821293074295781190e_real
% 12.45/7.01 @ ^ [Z5: real] : ( powr_real @ Z5 @ R2 )
% 12.45/7.01 @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9708_DERIV__log,axiom,
% 12.45/7.01 ! [X4: real,B2: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( log @ B2 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B2 ) @ X4 ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9709_DERIV__fun__powr,axiom,
% 12.45/7.01 ! [G: real > real,M: real,X4: real,R2: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ ( G @ X4 ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real
% 12.45/7.01 @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R2 )
% 12.45/7.01 @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X4 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9710_DERIV__powr,axiom,
% 12.45/7.01 ! [G: real > real,M: real,X4: real,F: real > real,R2: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ ( G @ X4 ) )
% 12.45/7.01 => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real
% 12.45/7.01 @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
% 12.45/7.01 @ ( times_times_real @ ( powr_real @ ( G @ X4 ) @ ( F @ X4 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X4 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X4 ) ) @ ( G @ X4 ) ) ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9711_DERIV__real__sqrt,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9712_DERIV__series_H,axiom,
% 12.45/7.01 ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B2: real,L5: nat > real] :
% 12.45/7.01 ( ! [N4: nat] :
% 12.45/7.01 ( has_fi5821293074295781190e_real
% 12.45/7.01 @ ^ [X: real] : ( F @ X @ N4 )
% 12.45/7.01 @ ( F4 @ X0 @ N4 )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 12.45/7.01 => ( summable_real @ ( F @ X3 ) ) )
% 12.45/7.01 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 12.45/7.01 => ( ( summable_real @ ( F4 @ X0 ) )
% 12.45/7.01 => ( ( summable_real @ L5 )
% 12.45/7.01 => ( ! [N4: nat,X3: real,Y4: real] :
% 12.45/7.01 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 12.45/7.01 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 12.45/7.01 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N4 ) @ ( F @ Y4 @ N4 ) ) ) @ ( times_times_real @ ( L5 @ N4 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) ) ) ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real
% 12.45/7.01 @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
% 12.45/7.01 @ ( suminf_real @ ( F4 @ X0 ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9713_DERIV__arctan,axiom,
% 12.45/7.01 ! [X4: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9714_arsinh__real__has__field__derivative,axiom,
% 12.45/7.01 ! [X4: real,A3: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A3 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9715_DERIV__real__sqrt__generic,axiom,
% 12.45/7.01 ! [X4: real,D4: real] :
% 12.45/7.01 ( ( X4 != zero_zero_real )
% 12.45/7.01 => ( ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.01 => ( D4
% 12.45/7.01 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.45/7.01 => ( ( ( ord_less_real @ X4 @ zero_zero_real )
% 12.45/7.01 => ( D4
% 12.45/7.01 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9716_arcosh__real__has__field__derivative,axiom,
% 12.45/7.01 ! [X4: real,A3: set_real] :
% 12.45/7.01 ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A3 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9717_artanh__real__has__field__derivative,axiom,
% 12.45/7.01 ! [X4: real,A3: set_real] :
% 12.45/7.01 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A3 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9718_DERIV__power__series_H,axiom,
% 12.45/7.01 ! [R: real,F: nat > real,X0: real] :
% 12.45/7.01 ( ! [X3: real] :
% 12.45/7.01 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 12.45/7.01 => ( summable_real
% 12.45/7.01 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X3 @ N2 ) ) ) )
% 12.45/7.01 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ R )
% 12.45/7.01 => ( has_fi5821293074295781190e_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( suminf_real
% 12.45/7.01 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) )
% 12.45/7.01 @ ( suminf_real
% 12.45/7.01 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9719_DERIV__real__root,axiom,
% 12.45/7.01 ! [N: nat,X4: real] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9720_DERIV__arccos,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9721_DERIV__arcsin,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9722_Maclaurin__all__le__objl,axiom,
% 12.45/7.01 ! [Diff: nat > real > real,F: real > real,X4: real,N: nat] :
% 12.45/7.01 ( ( ( ( Diff @ zero_zero_nat )
% 12.45/7.01 = F )
% 12.45/7.01 & ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 12.45/7.01 => ? [T2: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 12.45/7.01 & ( ( F @ X4 )
% 12.45/7.01 = ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.01 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9723_Maclaurin__all__le,axiom,
% 12.45/7.01 ! [Diff: nat > real > real,F: real > real,X4: real,N: nat] :
% 12.45/7.01 ( ( ( Diff @ zero_zero_nat )
% 12.45/7.01 = F )
% 12.45/7.01 => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 => ? [T2: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 12.45/7.01 & ( ( F @ X4 )
% 12.45/7.01 = ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.01 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9724_DERIV__odd__real__root,axiom,
% 12.45/7.01 ! [N: nat,X4: real] :
% 12.45/7.01 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/7.01 => ( ( X4 != zero_zero_real )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X4 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9725_Maclaurin,axiom,
% 12.45/7.01 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ H2 )
% 12.45/7.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( ( Diff @ zero_zero_nat )
% 12.45/7.01 = F )
% 12.45/7.01 => ( ! [M4: nat,T2: real] :
% 12.45/7.01 ( ( ( ord_less_nat @ M4 @ N )
% 12.45/7.01 & ( ord_less_eq_real @ zero_zero_real @ T2 )
% 12.45/7.01 & ( ord_less_eq_real @ T2 @ H2 ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 12.45/7.01 => ? [T2: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ T2 )
% 12.45/7.01 & ( ord_less_real @ T2 @ H2 )
% 12.45/7.01 & ( ( F @ H2 )
% 12.45/7.01 = ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.01 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9726_Maclaurin2,axiom,
% 12.45/7.01 ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ H2 )
% 12.45/7.01 => ( ( ( Diff @ zero_zero_nat )
% 12.45/7.01 = F )
% 12.45/7.01 => ( ! [M4: nat,T2: real] :
% 12.45/7.01 ( ( ( ord_less_nat @ M4 @ N )
% 12.45/7.01 & ( ord_less_eq_real @ zero_zero_real @ T2 )
% 12.45/7.01 & ( ord_less_eq_real @ T2 @ H2 ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 12.45/7.01 => ? [T2: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ T2 )
% 12.45/7.01 & ( ord_less_eq_real @ T2 @ H2 )
% 12.45/7.01 & ( ( F @ H2 )
% 12.45/7.01 = ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.01 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9727_Maclaurin__minus,axiom,
% 12.45/7.01 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ H2 @ zero_zero_real )
% 12.45/7.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( ( Diff @ zero_zero_nat )
% 12.45/7.01 = F )
% 12.45/7.01 => ( ! [M4: nat,T2: real] :
% 12.45/7.01 ( ( ( ord_less_nat @ M4 @ N )
% 12.45/7.01 & ( ord_less_eq_real @ H2 @ T2 )
% 12.45/7.01 & ( ord_less_eq_real @ T2 @ zero_zero_real ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 12.45/7.01 => ? [T2: real] :
% 12.45/7.01 ( ( ord_less_real @ H2 @ T2 )
% 12.45/7.01 & ( ord_less_real @ T2 @ zero_zero_real )
% 12.45/7.01 & ( ( F @ H2 )
% 12.45/7.01 = ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.01 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9728_Maclaurin__all__lt,axiom,
% 12.45/7.01 ! [Diff: nat > real > real,F: real > real,N: nat,X4: real] :
% 12.45/7.01 ( ( ( Diff @ zero_zero_nat )
% 12.45/7.01 = F )
% 12.45/7.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( X4 != zero_zero_real )
% 12.45/7.01 => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 => ? [T2: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T2 ) )
% 12.45/7.01 & ( ord_less_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 12.45/7.01 & ( ( F @ X4 )
% 12.45/7.01 = ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.01 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9729_Maclaurin__bi__le,axiom,
% 12.45/7.01 ! [Diff: nat > real > real,F: real > real,N: nat,X4: real] :
% 12.45/7.01 ( ( ( Diff @ zero_zero_nat )
% 12.45/7.01 = F )
% 12.45/7.01 => ( ! [M4: nat,T2: real] :
% 12.45/7.01 ( ( ( ord_less_nat @ M4 @ N )
% 12.45/7.01 & ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 12.45/7.01 => ? [T2: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ T2 ) @ ( abs_abs_real @ X4 ) )
% 12.45/7.01 & ( ( F @ X4 )
% 12.45/7.01 = ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.01 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9730_Taylor,axiom,
% 12.45/7.01 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real,X4: real] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( ( Diff @ zero_zero_nat )
% 12.45/7.01 = F )
% 12.45/7.01 => ( ! [M4: nat,T2: real] :
% 12.45/7.01 ( ( ( ord_less_nat @ M4 @ N )
% 12.45/7.01 & ( ord_less_eq_real @ A @ T2 )
% 12.45/7.01 & ( ord_less_eq_real @ T2 @ B2 ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 12.45/7.01 => ( ( ord_less_eq_real @ A @ C )
% 12.45/7.01 => ( ( ord_less_eq_real @ C @ B2 )
% 12.45/7.01 => ( ( ord_less_eq_real @ A @ X4 )
% 12.45/7.01 => ( ( ord_less_eq_real @ X4 @ B2 )
% 12.45/7.01 => ( ( X4 != C )
% 12.45/7.01 => ? [T2: real] :
% 12.45/7.01 ( ( ( ord_less_real @ X4 @ C )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ T2 )
% 12.45/7.01 & ( ord_less_real @ T2 @ C ) ) )
% 12.45/7.01 & ( ~ ( ord_less_real @ X4 @ C )
% 12.45/7.01 => ( ( ord_less_real @ C @ T2 )
% 12.45/7.01 & ( ord_less_real @ T2 @ X4 ) ) )
% 12.45/7.01 & ( ( F @ X4 )
% 12.45/7.01 = ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ C ) @ M6 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.01 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9731_Taylor__up,axiom,
% 12.45/7.01 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( ( Diff @ zero_zero_nat )
% 12.45/7.01 = F )
% 12.45/7.01 => ( ! [M4: nat,T2: real] :
% 12.45/7.01 ( ( ( ord_less_nat @ M4 @ N )
% 12.45/7.01 & ( ord_less_eq_real @ A @ T2 )
% 12.45/7.01 & ( ord_less_eq_real @ T2 @ B2 ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 12.45/7.01 => ( ( ord_less_eq_real @ A @ C )
% 12.45/7.01 => ( ( ord_less_real @ C @ B2 )
% 12.45/7.01 => ? [T2: real] :
% 12.45/7.01 ( ( ord_less_real @ C @ T2 )
% 12.45/7.01 & ( ord_less_real @ T2 @ B2 )
% 12.45/7.01 & ( ( F @ B2 )
% 12.45/7.01 = ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ M6 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.01 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9732_Taylor__down,axiom,
% 12.45/7.01 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( ( Diff @ zero_zero_nat )
% 12.45/7.01 = F )
% 12.45/7.01 => ( ! [M4: nat,T2: real] :
% 12.45/7.01 ( ( ( ord_less_nat @ M4 @ N )
% 12.45/7.01 & ( ord_less_eq_real @ A @ T2 )
% 12.45/7.01 & ( ord_less_eq_real @ T2 @ B2 ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 12.45/7.01 => ( ( ord_less_real @ A @ C )
% 12.45/7.01 => ( ( ord_less_eq_real @ C @ B2 )
% 12.45/7.01 => ? [T2: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ T2 )
% 12.45/7.01 & ( ord_less_real @ T2 @ C )
% 12.45/7.01 & ( ( F @ A )
% 12.45/7.01 = ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M6 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ N ) )
% 12.45/7.01 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9733_Maclaurin__lemma2,axiom,
% 12.45/7.01 ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B4: real] :
% 12.45/7.01 ( ! [M4: nat,T2: real] :
% 12.45/7.01 ( ( ( ord_less_nat @ M4 @ N )
% 12.45/7.01 & ( ord_less_eq_real @ zero_zero_real @ T2 )
% 12.45/7.01 & ( ord_less_eq_real @ T2 @ H2 ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T2 ) @ ( topolo2177554685111907308n_real @ T2 @ top_top_set_real ) ) )
% 12.45/7.01 => ( ( N
% 12.45/7.01 = ( suc @ K ) )
% 12.45/7.01 => ! [M5: nat,T4: real] :
% 12.45/7.01 ( ( ( ord_less_nat @ M5 @ N )
% 12.45/7.01 & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 12.45/7.01 & ( ord_less_eq_real @ T4 @ H2 ) )
% 12.45/7.01 => ( has_fi5821293074295781190e_real
% 12.45/7.01 @ ^ [U4: real] :
% 12.45/7.01 ( minus_minus_real @ ( Diff @ M5 @ U4 )
% 12.45/7.01 @ ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M5 @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ U4 @ P3 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M5 ) ) )
% 12.45/7.01 @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ U4 @ ( minus_minus_nat @ N @ M5 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M5 ) ) ) ) ) )
% 12.45/7.01 @ ( minus_minus_real @ ( Diff @ ( suc @ M5 ) @ T4 )
% 12.45/7.01 @ ( plus_plus_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M5 ) @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ T4 @ P3 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) )
% 12.45/7.01 @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) ) ) ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9734_DERIV__arctan__series,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/7.01 => ( has_fi5821293074295781190e_real
% 12.45/7.01 @ ^ [X9: real] :
% 12.45/7.01 ( suminf_real
% 12.45/7.01 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 12.45/7.01 @ ( suminf_real
% 12.45/7.01 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X4 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9735_isCont__Lb__Ub,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ( ord_less_eq_real @ A @ X3 )
% 12.45/7.01 & ( ord_less_eq_real @ X3 @ B2 ) )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 12.45/7.01 => ? [L6: real,M9: real] :
% 12.45/7.01 ( ! [X5: real] :
% 12.45/7.01 ( ( ( ord_less_eq_real @ A @ X5 )
% 12.45/7.01 & ( ord_less_eq_real @ X5 @ B2 ) )
% 12.45/7.01 => ( ( ord_less_eq_real @ L6 @ ( F @ X5 ) )
% 12.45/7.01 & ( ord_less_eq_real @ ( F @ X5 ) @ M9 ) ) )
% 12.45/7.01 & ! [Y5: real] :
% 12.45/7.01 ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 12.45/7.01 & ( ord_less_eq_real @ Y5 @ M9 ) )
% 12.45/7.01 => ? [X3: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ X3 )
% 12.45/7.01 & ( ord_less_eq_real @ X3 @ B2 )
% 12.45/7.01 & ( ( F @ X3 )
% 12.45/7.01 = Y5 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9736_LIM__fun__less__zero,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,C: real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 12.45/7.01 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 12.45/7.01 => ? [R3: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ R3 )
% 12.45/7.01 & ! [X5: real] :
% 12.45/7.01 ( ( ( X5 != C )
% 12.45/7.01 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 12.45/7.01 => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9737_LIM__fun__not__zero,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,C: real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 12.45/7.01 => ( ( L2 != zero_zero_real )
% 12.45/7.01 => ? [R3: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ R3 )
% 12.45/7.01 & ! [X5: real] :
% 12.45/7.01 ( ( ( X5 != C )
% 12.45/7.01 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 12.45/7.01 => ( ( F @ X5 )
% 12.45/7.01 != zero_zero_real ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9738_LIM__fun__gt__zero,axiom,
% 12.45/7.01 ! [F: real > real,L2: real,C: real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 12.45/7.01 => ? [R3: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ R3 )
% 12.45/7.01 & ! [X5: real] :
% 12.45/7.01 ( ( ( X5 != C )
% 12.45/7.01 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 12.45/7.01 => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9739_isCont__real__sqrt,axiom,
% 12.45/7.01 ! [X4: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ sqrt ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9740_isCont__real__root,axiom,
% 12.45/7.01 ! [X4: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ ( root @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9741_continuous__frac,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ~ ( member_real @ X4 @ ring_1_Ints_real )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ archim2898591450579166408c_real ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9742_isCont__inverse__function2,axiom,
% 12.45/7.01 ! [A: real,X4: real,B2: real,G: real > real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ X4 )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ B2 )
% 12.45/7.01 => ( ! [Z2: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ Z2 )
% 12.45/7.01 => ( ( ord_less_eq_real @ Z2 @ B2 )
% 12.45/7.01 => ( ( G @ ( F @ Z2 ) )
% 12.45/7.01 = Z2 ) ) )
% 12.45/7.01 => ( ! [Z2: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ Z2 )
% 12.45/7.01 => ( ( ord_less_eq_real @ Z2 @ B2 )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X4 ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9743_isCont__arcosh,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arcosh_real ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9744_LIM__cos__div__sin,axiom,
% 12.45/7.01 ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9745_continuous__floor,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ~ ( member_real @ X4 @ ring_1_Ints_real )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ ( comp_int_real_real @ ring_1_of_int_real @ archim6058952711729229775r_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9746_DERIV__inverse__function,axiom,
% 12.45/7.01 ! [F: real > real,D4: real,G: real > real,X4: real,A: real,B2: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X4 ) @ top_top_set_real ) )
% 12.45/7.01 => ( ( D4 != zero_zero_real )
% 12.45/7.01 => ( ( ord_less_real @ A @ X4 )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ B2 )
% 12.45/7.01 => ( ! [Y4: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ Y4 )
% 12.45/7.01 => ( ( ord_less_real @ Y4 @ B2 )
% 12.45/7.01 => ( ( F @ ( G @ Y4 ) )
% 12.45/7.01 = Y4 ) ) )
% 12.45/7.01 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9747_isCont__arccos,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arccos ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9748_isCont__arcsin,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arcsin ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9749_LIM__less__bound,axiom,
% 12.45/7.01 ! [B2: real,X4: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ B2 @ X4 )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B2 @ X4 ) )
% 12.45/7.01 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 12.45/7.01 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F )
% 12.45/7.01 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9750_isCont__artanh,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ artanh_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9751_isCont__inverse__function,axiom,
% 12.45/7.01 ! [D: real,X4: real,G: real > real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ D )
% 12.45/7.01 => ( ! [Z2: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X4 ) ) @ D )
% 12.45/7.01 => ( ( G @ ( F @ Z2 ) )
% 12.45/7.01 = Z2 ) )
% 12.45/7.01 => ( ! [Z2: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X4 ) ) @ D )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X4 ) @ top_top_set_real ) @ G ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9752_GMVT_H,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ! [Z2: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ Z2 )
% 12.45/7.01 => ( ( ord_less_eq_real @ Z2 @ B2 )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 12.45/7.01 => ( ! [Z2: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ Z2 )
% 12.45/7.01 => ( ( ord_less_eq_real @ Z2 @ B2 )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
% 12.45/7.01 => ( ! [Z2: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ Z2 )
% 12.45/7.01 => ( ( ord_less_real @ Z2 @ B2 )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 12.45/7.01 => ( ! [Z2: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ Z2 )
% 12.45/7.01 => ( ( ord_less_real @ Z2 @ B2 )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 12.45/7.01 => ? [C3: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ C3 )
% 12.45/7.01 & ( ord_less_real @ C3 @ B2 )
% 12.45/7.01 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 12.45/7.01 = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ ( F4 @ C3 ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9753_summable__Leibniz_I2_J,axiom,
% 12.45/7.01 ! [A: nat > real] :
% 12.45/7.01 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.45/7.01 => ( ( topolo6980174941875973593q_real @ A )
% 12.45/7.01 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 12.45/7.01 => ! [N6: nat] :
% 12.45/7.01 ( member_real
% 12.45/7.01 @ ( suminf_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 12.45/7.01 @ ( set_or1222579329274155063t_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9754_summable__Leibniz_I3_J,axiom,
% 12.45/7.01 ! [A: nat > real] :
% 12.45/7.01 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.45/7.01 => ( ( topolo6980174941875973593q_real @ A )
% 12.45/7.01 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 12.45/7.01 => ! [N6: nat] :
% 12.45/7.01 ( member_real
% 12.45/7.01 @ ( suminf_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 12.45/7.01 @ ( set_or1222579329274155063t_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9755_incseq__convergent,axiom,
% 12.45/7.01 ! [X8: nat > real,B4: real] :
% 12.45/7.01 ( ( order_mono_nat_real @ X8 )
% 12.45/7.01 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X8 @ I3 ) @ B4 )
% 12.45/7.01 => ~ ! [L6: real] :
% 12.45/7.01 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 12.45/7.01 => ~ ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ L6 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9756_mult__nat__left__at__top,axiom,
% 12.45/7.01 ! [C: nat] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.45/7.01 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9757_mult__nat__right__at__top,axiom,
% 12.45/7.01 ! [C: nat] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.45/7.01 => ( filterlim_nat_nat
% 12.45/7.01 @ ^ [X: nat] : ( times_times_nat @ X @ C )
% 12.45/7.01 @ at_top_nat
% 12.45/7.01 @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9758_monoseq__convergent,axiom,
% 12.45/7.01 ! [X8: nat > real,B4: real] :
% 12.45/7.01 ( ( topolo6980174941875973593q_real @ X8 )
% 12.45/7.01 => ( ! [I3: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I3 ) ) @ B4 )
% 12.45/7.01 => ~ ! [L6: real] :
% 12.45/7.01 ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9759_LIMSEQ__root,axiom,
% 12.45/7.01 ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ one_one_real )
% 12.45/7.01 @ at_top_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9760_nested__sequence__unique,axiom,
% 12.45/7.01 ! [F: nat > real,G: nat > real] :
% 12.45/7.01 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N4 ) ) @ ( G @ N4 ) )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 12.45/7.01 => ( ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.45/7.01 @ at_top_nat )
% 12.45/7.01 => ? [L3: real] :
% 12.45/7.01 ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L3 )
% 12.45/7.01 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat )
% 12.45/7.01 & ! [N6: nat] : ( ord_less_eq_real @ L3 @ ( G @ N6 ) )
% 12.45/7.01 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9761_LIMSEQ__inverse__zero,axiom,
% 12.45/7.01 ! [X8: nat > real] :
% 12.45/7.01 ( ! [R3: real] :
% 12.45/7.01 ? [N7: nat] :
% 12.45/7.01 ! [N4: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ N7 @ N4 )
% 12.45/7.01 => ( ord_less_real @ R3 @ ( X8 @ N4 ) ) )
% 12.45/7.01 => ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( inverse_inverse_real @ ( X8 @ N2 ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.45/7.01 @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9762_lim__inverse__n_H,axiom,
% 12.45/7.01 ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.45/7.01 @ at_top_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9763_LIMSEQ__root__const,axiom,
% 12.45/7.01 ! [C: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ C )
% 12.45/7.01 => ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( root @ N2 @ C )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ one_one_real )
% 12.45/7.01 @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9764_LIMSEQ__inverse__real__of__nat,axiom,
% 12.45/7.01 ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.45/7.01 @ at_top_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9765_LIMSEQ__inverse__real__of__nat__add,axiom,
% 12.45/7.01 ! [R2: real] :
% 12.45/7.01 ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ R2 )
% 12.45/7.01 @ at_top_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9766_increasing__LIMSEQ,axiom,
% 12.45/7.01 ! [F: nat > real,L2: real] :
% 12.45/7.01 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ L2 )
% 12.45/7.01 => ( ! [E2: real] :
% 12.45/7.01 ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.45/7.01 => ? [N6: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N6 ) @ E2 ) ) )
% 12.45/7.01 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9767_LIMSEQ__realpow__zero,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ one_one_real )
% 12.45/7.01 => ( filterlim_nat_real @ ( power_power_real @ X4 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9768_LIMSEQ__divide__realpow__zero,axiom,
% 12.45/7.01 ! [X4: real,A: real] :
% 12.45/7.01 ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/7.01 => ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X4 @ N2 ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.45/7.01 @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9769_LIMSEQ__abs__realpow__zero,axiom,
% 12.45/7.01 ! [C: real] :
% 12.45/7.01 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 12.45/7.01 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9770_LIMSEQ__abs__realpow__zero2,axiom,
% 12.45/7.01 ! [C: real] :
% 12.45/7.01 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 12.45/7.01 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9771_LIMSEQ__inverse__realpow__zero,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_real @ one_one_real @ X4 )
% 12.45/7.01 => ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X4 @ N2 ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.45/7.01 @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9772_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 12.45/7.01 ! [R2: real] :
% 12.45/7.01 ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ R2 )
% 12.45/7.01 @ at_top_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9773_tendsto__exp__limit__sequentially,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
% 12.45/7.01 @ at_top_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9774_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 12.45/7.01 ! [R2: real] :
% 12.45/7.01 ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ R2 )
% 12.45/7.01 @ at_top_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9775_summable__Leibniz_I1_J,axiom,
% 12.45/7.01 ! [A: nat > real] :
% 12.45/7.01 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.45/7.01 => ( ( topolo6980174941875973593q_real @ A )
% 12.45/7.01 => ( summable_real
% 12.45/7.01 @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9776_summable,axiom,
% 12.45/7.01 ! [A: nat > real] :
% 12.45/7.01 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 12.45/7.01 => ( summable_real
% 12.45/7.01 @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9777_cos__diff__limit__1,axiom,
% 12.45/7.01 ! [Theta: nat > real,Theta2: real] :
% 12.45/7.01 ( ( filterlim_nat_real
% 12.45/7.01 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ one_one_real )
% 12.45/7.01 @ at_top_nat )
% 12.45/7.01 => ~ ! [K2: nat > int] :
% 12.45/7.01 ~ ( filterlim_nat_real
% 12.45/7.01 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ Theta2 )
% 12.45/7.01 @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9778_cos__limit__1,axiom,
% 12.45/7.01 ! [Theta: nat > real] :
% 12.45/7.01 ( ( filterlim_nat_real
% 12.45/7.01 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ one_one_real )
% 12.45/7.01 @ at_top_nat )
% 12.45/7.01 => ? [K2: nat > int] :
% 12.45/7.01 ( filterlim_nat_real
% 12.45/7.01 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.45/7.01 @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9779_summable__Leibniz_I4_J,axiom,
% 12.45/7.01 ! [A: nat > real] :
% 12.45/7.01 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.45/7.01 => ( ( topolo6980174941875973593q_real @ A )
% 12.45/7.01 => ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] :
% 12.45/7.01 ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real
% 12.45/7.01 @ ( suminf_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 12.45/7.01 @ at_top_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9780_zeroseq__arctan__series,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
% 12.45/7.01 => ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.45/7.01 @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9781_summable__Leibniz_H_I2_J,axiom,
% 12.45/7.01 ! [A: nat > real,N: nat] :
% 12.45/7.01 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 12.45/7.01 => ( ord_less_eq_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/7.01 @ ( suminf_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9782_summable__Leibniz_H_I3_J,axiom,
% 12.45/7.01 ! [A: nat > real] :
% 12.45/7.01 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 12.45/7.01 => ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] :
% 12.45/7.01 ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real
% 12.45/7.01 @ ( suminf_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 12.45/7.01 @ at_top_nat ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9783_sums__alternating__upper__lower,axiom,
% 12.45/7.01 ! [A: nat > real] :
% 12.45/7.01 ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 12.45/7.01 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.45/7.01 => ? [L3: real] :
% 12.45/7.01 ( ! [N6: nat] :
% 12.45/7.01 ( ord_less_eq_real
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 12.45/7.01 @ L3 )
% 12.45/7.01 & ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] :
% 12.45/7.01 ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ L3 )
% 12.45/7.01 @ at_top_nat )
% 12.45/7.01 & ! [N6: nat] :
% 12.45/7.01 ( ord_less_eq_real @ L3
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
% 12.45/7.01 & ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] :
% 12.45/7.01 ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ L3 )
% 12.45/7.01 @ at_top_nat ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9784_summable__Leibniz_I5_J,axiom,
% 12.45/7.01 ! [A: nat > real] :
% 12.45/7.01 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.45/7.01 => ( ( topolo6980174941875973593q_real @ A )
% 12.45/7.01 => ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] :
% 12.45/7.01 ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real
% 12.45/7.01 @ ( suminf_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 12.45/7.01 @ at_top_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9785_summable__Leibniz_H_I5_J,axiom,
% 12.45/7.01 ! [A: nat > real] :
% 12.45/7.01 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 12.45/7.01 => ( filterlim_nat_real
% 12.45/7.01 @ ^ [N2: nat] :
% 12.45/7.01 ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real
% 12.45/7.01 @ ( suminf_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 12.45/7.01 @ at_top_nat ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9786_summable__Leibniz_H_I4_J,axiom,
% 12.45/7.01 ! [A: nat > real,N: nat] :
% 12.45/7.01 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 12.45/7.01 => ( ord_less_eq_real
% 12.45/7.01 @ ( suminf_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 12.45/7.01 @ ( groups6591440286371151544t_real
% 12.45/7.01 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.45/7.01 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9787_eventually__sequentially__Suc,axiom,
% 12.45/7.01 ! [P: nat > $o] :
% 12.45/7.01 ( ( eventually_nat
% 12.45/7.01 @ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
% 12.45/7.01 @ at_top_nat )
% 12.45/7.01 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9788_eventually__sequentially__seg,axiom,
% 12.45/7.01 ! [P: nat > $o,K: nat] :
% 12.45/7.01 ( ( eventually_nat
% 12.45/7.01 @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
% 12.45/7.01 @ at_top_nat )
% 12.45/7.01 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9789_sequentially__offset,axiom,
% 12.45/7.01 ! [P: nat > $o,K: nat] :
% 12.45/7.01 ( ( eventually_nat @ P @ at_top_nat )
% 12.45/7.01 => ( eventually_nat
% 12.45/7.01 @ ^ [I2: nat] : ( P @ ( plus_plus_nat @ I2 @ K ) )
% 12.45/7.01 @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9790_eventually__sequentially,axiom,
% 12.45/7.01 ! [P: nat > $o] :
% 12.45/7.01 ( ( eventually_nat @ P @ at_top_nat )
% 12.45/7.01 = ( ? [N8: nat] :
% 12.45/7.01 ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ N8 @ N2 )
% 12.45/7.01 => ( P @ N2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9791_eventually__sequentiallyI,axiom,
% 12.45/7.01 ! [C: nat,P: nat > $o] :
% 12.45/7.01 ( ! [X3: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ C @ X3 )
% 12.45/7.01 => ( P @ X3 ) )
% 12.45/7.01 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9792_le__sequentially,axiom,
% 12.45/7.01 ! [F5: filter_nat] :
% 12.45/7.01 ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
% 12.45/7.01 = ( ! [N8: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N8 ) @ F5 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9793_filterlim__Suc,axiom,
% 12.45/7.01 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9794_real__bounded__linear,axiom,
% 12.45/7.01 ( real_V5970128139526366754l_real
% 12.45/7.01 = ( ^ [F2: real > real] :
% 12.45/7.01 ? [C2: real] :
% 12.45/7.01 ( F2
% 12.45/7.01 = ( ^ [X: real] : ( times_times_real @ X @ C2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9795_sqrt__at__top,axiom,
% 12.45/7.01 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9796_lhopital__left__at__top__at__top,axiom,
% 12.45/7.01 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.45/7.01 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ at_top_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ at_top_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9797_lhopital__at__top__at__top,axiom,
% 12.45/7.01 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.45/7.01 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ at_top_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ at_top_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9798_eventually__at__left__real,axiom,
% 12.45/7.01 ! [B2: real,A: real] :
% 12.45/7.01 ( ( ord_less_real @ B2 @ A )
% 12.45/7.01 => ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B2 @ A ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9799_lhopital__left__at__top,axiom,
% 12.45/7.01 ! [G: real > real,X4: real,G2: real > real,F: real > real,F4: real > real,Y3: real] :
% 12.45/7.01 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G2 @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ Y3 )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ Y3 )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9800_lhospital__at__top__at__top,axiom,
% 12.45/7.01 ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X4: real] :
% 12.45/7.01 ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G2 @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ at_top_real )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ at_top_real )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ at_top_real )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ X4 )
% 12.45/7.01 @ at_top_real )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ X4 )
% 12.45/7.01 @ at_top_real ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9801_lhopital__at__top,axiom,
% 12.45/7.01 ! [G: real > real,X4: real,G2: real > real,F: real > real,F4: real > real,Y3: real] :
% 12.45/7.01 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G2 @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ Y3 )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ Y3 )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9802_tanh__real__at__top,axiom,
% 12.45/7.01 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9803_artanh__real__at__left__1,axiom,
% 12.45/7.01 filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9804_ln__x__over__x__tendsto__0,axiom,
% 12.45/7.01 ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ X )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.45/7.01 @ at_top_real ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9805_tendsto__power__div__exp__0,axiom,
% 12.45/7.01 ! [K: nat] :
% 12.45/7.01 ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( power_power_real @ X @ K ) @ ( exp_real @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.45/7.01 @ at_top_real ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9806_lhopital__left,axiom,
% 12.45/7.01 ! [F: real > real,X4: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G2 @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ F5
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ F5
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5984915006950818249n_real @ X4 ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9807_lhopital,axiom,
% 12.45/7.01 ! [F: real > real,X4: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G2 @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ F5
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ F5
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9808_tendsto__exp__limit__at__top,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( filterlim_real_real
% 12.45/7.01 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ Y ) ) @ Y )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
% 12.45/7.01 @ at_top_real ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9809_filterlim__tan__at__left,axiom,
% 12.45/7.01 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 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9810_DERIV__neg__imp__decreasing__at__top,axiom,
% 12.45/7.01 ! [B2: real,F: real > real,Flim: real] :
% 12.45/7.01 ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ B2 @ X3 )
% 12.45/7.01 => ? [Y5: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 12.45/7.01 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 12.45/7.01 => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9811_tendsto__arctan__at__top,axiom,
% 12.45/7.01 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9812_filterlim__pow__at__bot__even,axiom,
% 12.45/7.01 ! [N: nat,F: real > real,F5: filter_real] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 12.45/7.01 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 12.45/7.01 @ at_top_real
% 12.45/7.01 @ F5 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9813_dist__real__def,axiom,
% 12.45/7.01 ( real_V975177566351809787t_real
% 12.45/7.01 = ( ^ [X: real,Y: real] : ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9814_dist__complex__def,axiom,
% 12.45/7.01 ( real_V3694042436643373181omplex
% 12.45/7.01 = ( ^ [X: complex,Y: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9815_tanh__real__at__bot,axiom,
% 12.45/7.01 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9816_lhopital__at__top__at__bot,axiom,
% 12.45/7.01 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.45/7.01 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ at_bot_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ at_bot_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9817_DERIV__pos__imp__increasing__at__bot,axiom,
% 12.45/7.01 ! [B2: real,F: real > real,Flim: real] :
% 12.45/7.01 ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ X3 @ B2 )
% 12.45/7.01 => ? [Y5: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 12.45/7.01 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 12.45/7.01 => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9818_filterlim__pow__at__bot__odd,axiom,
% 12.45/7.01 ! [N: nat,F: real > real,F5: filter_real] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 12.45/7.01 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 12.45/7.01 @ at_bot_real
% 12.45/7.01 @ F5 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9819_lhopital__left__at__top__at__bot,axiom,
% 12.45/7.01 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.45/7.01 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ at_bot_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ at_bot_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9820_tendsto__arctan__at__bot,axiom,
% 12.45/7.01 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9821_incseq__bounded,axiom,
% 12.45/7.01 ! [X8: nat > real,B4: real] :
% 12.45/7.01 ( ( order_mono_nat_real @ X8 )
% 12.45/7.01 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X8 @ I3 ) @ B4 )
% 12.45/7.01 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9822_Bseq__realpow,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/7.01 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/7.01 => ( bfun_nat_real @ ( power_power_real @ X4 ) @ at_top_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9823_tendsto__exp__limit__at__right,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( filterlim_real_real
% 12.45/7.01 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X4 @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9824_filterlim__tan__at__right,axiom,
% 12.45/7.01 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 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9825_eventually__at__right__to__0,axiom,
% 12.45/7.01 ! [P: real > $o,A: real] :
% 12.45/7.01 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 = ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( P @ ( plus_plus_real @ X @ A ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9826_eventually__at__right__real,axiom,
% 12.45/7.01 ! [A: real,B2: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9827_tendsto__arcosh__at__left__1,axiom,
% 12.45/7.01 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9828_artanh__real__at__right__1,axiom,
% 12.45/7.01 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 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9829_lhopital__right__at__top__at__top,axiom,
% 12.45/7.01 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ at_top_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ at_top_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9830_lhopital__right__0,axiom,
% 12.45/7.01 ! [F0: real > real,G0: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 12.45/7.01 ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G0 @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G2 @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ F5
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F0 @ X ) @ ( G0 @ X ) )
% 12.45/7.01 @ F5
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9831_lhopital__right,axiom,
% 12.45/7.01 ! [F: real > real,X4: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G2 @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ F5
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ F5
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9832_lhopital__right__at__top__at__bot,axiom,
% 12.45/7.01 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 12.45/7.01 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ at_bot_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ at_bot_real
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9833_lhopital__right__at__top,axiom,
% 12.45/7.01 ! [G: real > real,X4: real,G2: real > real,F: real > real,F4: real > real,Y3: real] :
% 12.45/7.01 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G2 @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ Y3 )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ Y3 )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ X4 @ ( set_or5849166863359141190n_real @ X4 ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9834_lhopital__right__0__at__top,axiom,
% 12.45/7.01 ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X4: real] :
% 12.45/7.01 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] :
% 12.45/7.01 ( ( G2 @ X )
% 12.45/7.01 != zero_zero_real )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( ( eventually_real
% 12.45/7.01 @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ X4 )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.45/7.01 => ( filterlim_real_real
% 12.45/7.01 @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
% 12.45/7.01 @ ( topolo2815343760600316023s_real @ X4 )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9835_atLeast__0,axiom,
% 12.45/7.01 ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 12.45/7.01 = top_top_set_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9836_atLeast__Suc__greaterThan,axiom,
% 12.45/7.01 ! [K: nat] :
% 12.45/7.01 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 12.45/7.01 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9837_decseq__bounded,axiom,
% 12.45/7.01 ! [X8: nat > real,B4: real] :
% 12.45/7.01 ( ( order_9091379641038594480t_real @ X8 )
% 12.45/7.01 => ( ! [I3: nat] : ( ord_less_eq_real @ B4 @ ( X8 @ I3 ) )
% 12.45/7.01 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9838_greaterThan__0,axiom,
% 12.45/7.01 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 12.45/7.01 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9839_greaterThan__Suc,axiom,
% 12.45/7.01 ! [K: nat] :
% 12.45/7.01 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 12.45/7.01 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9840_decseq__convergent,axiom,
% 12.45/7.01 ! [X8: nat > real,B4: real] :
% 12.45/7.01 ( ( order_9091379641038594480t_real @ X8 )
% 12.45/7.01 => ( ! [I3: nat] : ( ord_less_eq_real @ B4 @ ( X8 @ I3 ) )
% 12.45/7.01 => ~ ! [L6: real] :
% 12.45/7.01 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 12.45/7.01 => ~ ! [I4: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I4 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9841_atLeast__Suc,axiom,
% 12.45/7.01 ! [K: nat] :
% 12.45/7.01 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 12.45/7.01 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9842_GMVT,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real,G: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ( ord_less_eq_real @ A @ X3 )
% 12.45/7.01 & ( ord_less_eq_real @ X3 @ B2 ) )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ( ord_less_real @ A @ X3 )
% 12.45/7.01 & ( ord_less_real @ X3 @ B2 ) )
% 12.45/7.01 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ( ord_less_eq_real @ A @ X3 )
% 12.45/7.01 & ( ord_less_eq_real @ X3 @ B2 ) )
% 12.45/7.01 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ( ord_less_real @ A @ X3 )
% 12.45/7.01 & ( ord_less_real @ X3 @ B2 ) )
% 12.45/7.01 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 12.45/7.01 => ? [G_c: real,F_c: real,C3: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 12.45/7.01 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 12.45/7.01 & ( ord_less_real @ A @ C3 )
% 12.45/7.01 & ( ord_less_real @ C3 @ B2 )
% 12.45/7.01 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ G_c )
% 12.45/7.01 = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9843_real__differentiableE,axiom,
% 12.45/7.01 ! [F: real > real,X4: real,S: set_real] :
% 12.45/7.01 ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ S ) )
% 12.45/7.01 => ~ ! [Df: real] :
% 12.45/7.01 ~ ( has_fi5821293074295781190e_real @ F @ Df @ ( topolo2177554685111907308n_real @ X4 @ S ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9844_real__differentiable__def,axiom,
% 12.45/7.01 ! [F: real > real,X4: real,S: set_real] :
% 12.45/7.01 ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ S ) )
% 12.45/7.01 = ( ? [D6: real] : ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ X4 @ S ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9845_MVT,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_real @ X3 @ B2 )
% 12.45/7.01 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 12.45/7.01 => ? [L3: real,Z2: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ Z2 )
% 12.45/7.01 & ( ord_less_real @ Z2 @ B2 )
% 12.45/7.01 & ( has_fi5821293074295781190e_real @ F @ L3 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
% 12.45/7.01 & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 12.45/7.01 = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ L3 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9846_suminf__eq__SUP__real,axiom,
% 12.45/7.01 ! [X8: nat > real] :
% 12.45/7.01 ( ( summable_real @ X8 )
% 12.45/7.01 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I3 ) )
% 12.45/7.01 => ( ( suminf_real @ X8 )
% 12.45/7.01 = ( comple1385675409528146559p_real
% 12.45/7.01 @ ( image_nat_real
% 12.45/7.01 @ ^ [I2: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I2 ) )
% 12.45/7.01 @ top_top_set_nat ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9847_continuous__on__arcosh,axiom,
% 12.45/7.01 ! [A3: set_real] :
% 12.45/7.01 ( ( ord_less_eq_set_real @ A3 @ ( set_ord_atLeast_real @ one_one_real ) )
% 12.45/7.01 => ( topolo5044208981011980120l_real @ A3 @ arcosh_real ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9848_continuous__on__arcosh_H,axiom,
% 12.45/7.01 ! [A3: set_real,F: real > real] :
% 12.45/7.01 ( ( topolo5044208981011980120l_real @ A3 @ F )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( member_real @ X3 @ A3 )
% 12.45/7.01 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 12.45/7.01 => ( topolo5044208981011980120l_real @ A3
% 12.45/7.01 @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9849_continuous__image__closed__interval,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_eq_real @ A @ B2 )
% 12.45/7.01 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 12.45/7.01 => ? [C3: real,D3: real] :
% 12.45/7.01 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 12.45/7.01 = ( set_or1222579329274155063t_real @ C3 @ D3 ) )
% 12.45/7.01 & ( ord_less_eq_real @ C3 @ D3 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9850_continuous__on__arccos_H,axiom,
% 12.45/7.01 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9851_continuous__on__arcsin_H,axiom,
% 12.45/7.01 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9852_continuous__on__artanh,axiom,
% 12.45/7.01 ! [A3: set_real] :
% 12.45/7.01 ( ( ord_less_eq_set_real @ A3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 12.45/7.01 => ( topolo5044208981011980120l_real @ A3 @ artanh_real ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9853_continuous__on__artanh_H,axiom,
% 12.45/7.01 ! [A3: set_real,F: real > real] :
% 12.45/7.01 ( ( topolo5044208981011980120l_real @ A3 @ F )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( member_real @ X3 @ A3 )
% 12.45/7.01 => ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 12.45/7.01 => ( topolo5044208981011980120l_real @ A3
% 12.45/7.01 @ ^ [X: real] : ( artanh_real @ ( F @ X ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9854_Rolle__deriv,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real,F4: real > real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ( ( F @ A )
% 12.45/7.01 = ( F @ B2 ) )
% 12.45/7.01 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_real @ X3 @ B2 )
% 12.45/7.01 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 12.45/7.01 => ? [Z2: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ Z2 )
% 12.45/7.01 & ( ord_less_real @ Z2 @ B2 )
% 12.45/7.01 & ( ( F4 @ Z2 )
% 12.45/7.01 = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9855_mvt,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real,F4: real > real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_real @ X3 @ B2 )
% 12.45/7.01 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 12.45/7.01 => ~ ! [Xi: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ Xi )
% 12.45/7.01 => ( ( ord_less_real @ Xi @ B2 )
% 12.45/7.01 => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 12.45/7.01 != ( F4 @ Xi @ ( minus_minus_real @ B2 @ A ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9856_DERIV__pos__imp__increasing__open,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_real @ X3 @ B2 )
% 12.45/7.01 => ? [Y5: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 12.45/7.01 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 12.45/7.01 => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9857_DERIV__neg__imp__decreasing__open,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_real @ X3 @ B2 )
% 12.45/7.01 => ? [Y5: real] :
% 12.45/7.01 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 12.45/7.01 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 12.45/7.01 => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9858_DERIV__isconst__end,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_real @ X3 @ B2 )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 12.45/7.01 => ( ( F @ B2 )
% 12.45/7.01 = ( F @ A ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9859_DERIV__isconst2,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real,X4: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_real @ X3 @ B2 )
% 12.45/7.01 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 12.45/7.01 => ( ( ord_less_eq_real @ A @ X4 )
% 12.45/7.01 => ( ( ord_less_eq_real @ X4 @ B2 )
% 12.45/7.01 => ( ( F @ X4 )
% 12.45/7.01 = ( F @ A ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9860_Rolle,axiom,
% 12.45/7.01 ! [A: real,B2: real,F: real > real] :
% 12.45/7.01 ( ( ord_less_real @ A @ B2 )
% 12.45/7.01 => ( ( ( F @ A )
% 12.45/7.01 = ( F @ B2 ) )
% 12.45/7.01 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 12.45/7.01 => ( ! [X3: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ X3 )
% 12.45/7.01 => ( ( ord_less_real @ X3 @ B2 )
% 12.45/7.01 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 12.45/7.01 => ? [Z2: real] :
% 12.45/7.01 ( ( ord_less_real @ A @ Z2 )
% 12.45/7.01 & ( ord_less_real @ Z2 @ B2 )
% 12.45/7.01 & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9861_UN__lessThan__UNIV,axiom,
% 12.45/7.01 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
% 12.45/7.01 = top_top_set_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9862_UN__atMost__UNIV,axiom,
% 12.45/7.01 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
% 12.45/7.01 = top_top_set_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9863_UN__atLeast__UNIV,axiom,
% 12.45/7.01 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
% 12.45/7.01 = top_top_set_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9864_INT__greaterThan__UNIV,axiom,
% 12.45/7.01 ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 12.45/7.01 = bot_bot_set_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9865_Inf__real__def,axiom,
% 12.45/7.01 ( comple4887499456419720421f_real
% 12.45/7.01 = ( ^ [X6: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X6 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9866_inf__enat__def,axiom,
% 12.45/7.01 inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9867_inf__nat__def,axiom,
% 12.45/7.01 inf_inf_nat = ord_min_nat ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9868_rat__less__code,axiom,
% 12.45/7.01 ( ord_less_rat
% 12.45/7.01 = ( ^ [P3: rat,Q4: rat] :
% 12.45/7.01 ( produc4947309494688390418_int_o
% 12.45/7.01 @ ^ [A4: int,C2: int] :
% 12.45/7.01 ( produc4947309494688390418_int_o
% 12.45/7.01 @ ^ [B3: int,D2: int] : ( ord_less_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) )
% 12.45/7.01 @ ( quotient_of @ Q4 ) )
% 12.45/7.01 @ ( quotient_of @ P3 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9869_rat__less__eq__code,axiom,
% 12.45/7.01 ( ord_less_eq_rat
% 12.45/7.01 = ( ^ [P3: rat,Q4: rat] :
% 12.45/7.01 ( produc4947309494688390418_int_o
% 12.45/7.01 @ ^ [A4: int,C2: int] :
% 12.45/7.01 ( produc4947309494688390418_int_o
% 12.45/7.01 @ ^ [B3: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) )
% 12.45/7.01 @ ( quotient_of @ Q4 ) )
% 12.45/7.01 @ ( quotient_of @ P3 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9870_int__ge__less__than2__def,axiom,
% 12.45/7.01 ( int_ge_less_than2
% 12.45/7.01 = ( ^ [D2: int] :
% 12.45/7.01 ( collec213857154873943460nt_int
% 12.45/7.01 @ ( produc4947309494688390418_int_o
% 12.45/7.01 @ ^ [Z7: int,Z5: int] :
% 12.45/7.01 ( ( ord_less_eq_int @ D2 @ Z5 )
% 12.45/7.01 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9871_int__ge__less__than__def,axiom,
% 12.45/7.01 ( int_ge_less_than
% 12.45/7.01 = ( ^ [D2: int] :
% 12.45/7.01 ( collec213857154873943460nt_int
% 12.45/7.01 @ ( produc4947309494688390418_int_o
% 12.45/7.01 @ ^ [Z7: int,Z5: int] :
% 12.45/7.01 ( ( ord_less_eq_int @ D2 @ Z7 )
% 12.45/7.01 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9872_uniformity__real__def,axiom,
% 12.45/7.01 ( topolo1511823702728130853y_real
% 12.45/7.01 = ( comple2936214249959783750l_real
% 12.45/7.01 @ ( image_2178119161166701260l_real
% 12.45/7.01 @ ^ [E3: real] :
% 12.45/7.01 ( princi6114159922880469582l_real
% 12.45/7.01 @ ( collec3799799289383736868l_real
% 12.45/7.01 @ ( produc5414030515140494994real_o
% 12.45/7.01 @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
% 12.45/7.01 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9873_uniformity__complex__def,axiom,
% 12.45/7.01 ( topolo896644834953643431omplex
% 12.45/7.01 = ( comple8358262395181532106omplex
% 12.45/7.01 @ ( image_5971271580939081552omplex
% 12.45/7.01 @ ^ [E3: real] :
% 12.45/7.01 ( princi3496590319149328850omplex
% 12.45/7.01 @ ( collec8663557070575231912omplex
% 12.45/7.01 @ ( produc6771430404735790350plex_o
% 12.45/7.01 @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
% 12.45/7.01 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9874_nonneg__incseq__Bseq__subseq__iff,axiom,
% 12.45/7.01 ! [F: nat > real,G: nat > nat] :
% 12.45/7.01 ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 12.45/7.01 => ( ( order_mono_nat_real @ F )
% 12.45/7.01 => ( ( order_5726023648592871131at_nat @ G )
% 12.45/7.01 => ( ( bfun_nat_real
% 12.45/7.01 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 12.45/7.01 @ at_top_nat )
% 12.45/7.01 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9875_inj__sgn__power,axiom,
% 12.45/7.01 ! [N: nat] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( inj_on_real_real
% 12.45/7.01 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 12.45/7.01 @ top_top_set_real ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9876_strict__mono__imp__increasing,axiom,
% 12.45/7.01 ! [F: nat > nat,N: nat] :
% 12.45/7.01 ( ( order_5726023648592871131at_nat @ F )
% 12.45/7.01 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9877_log__inj,axiom,
% 12.45/7.01 ! [B2: real] :
% 12.45/7.01 ( ( ord_less_real @ one_one_real @ B2 )
% 12.45/7.01 => ( inj_on_real_real @ ( log @ B2 ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9878_inj__Suc,axiom,
% 12.45/7.01 ! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9879_inj__on__diff__nat,axiom,
% 12.45/7.01 ! [N3: set_nat,K: nat] :
% 12.45/7.01 ( ! [N4: nat] :
% 12.45/7.01 ( ( member_nat @ N4 @ N3 )
% 12.45/7.01 => ( ord_less_eq_nat @ K @ N4 ) )
% 12.45/7.01 => ( inj_on_nat_nat
% 12.45/7.01 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
% 12.45/7.01 @ N3 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9880_summable__reindex,axiom,
% 12.45/7.01 ! [F: nat > real,G: nat > nat] :
% 12.45/7.01 ( ( summable_real @ F )
% 12.45/7.01 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 12.45/7.01 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 12.45/7.01 => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9881_suminf__reindex__mono,axiom,
% 12.45/7.01 ! [F: nat > real,G: nat > nat] :
% 12.45/7.01 ( ( summable_real @ F )
% 12.45/7.01 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 12.45/7.01 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 12.45/7.01 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9882_powr__real__of__int_H,axiom,
% 12.45/7.01 ! [X4: real,N: int] :
% 12.45/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/7.01 => ( ( ( X4 != zero_zero_real )
% 12.45/7.01 | ( ord_less_int @ zero_zero_int @ N ) )
% 12.45/7.01 => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N ) )
% 12.45/7.01 = ( power_int_real @ X4 @ N ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9883_inj__on__char__of__nat,axiom,
% 12.45/7.01 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9884_suminf__reindex,axiom,
% 12.45/7.01 ! [F: nat > real,G: nat > nat] :
% 12.45/7.01 ( ( summable_real @ F )
% 12.45/7.01 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 12.45/7.01 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 12.45/7.01 => ( ! [X3: nat] :
% 12.45/7.01 ( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 12.45/7.01 => ( ( F @ X3 )
% 12.45/7.01 = zero_zero_real ) )
% 12.45/7.01 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 12.45/7.01 = ( suminf_real @ F ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9885_pos__deriv__imp__strict__mono,axiom,
% 12.45/7.01 ! [F: real > real,F4: real > real] :
% 12.45/7.01 ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.45/7.01 => ( ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( F4 @ X3 ) )
% 12.45/7.01 => ( order_7092887310737990675l_real @ F ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9886_pred__nat__def,axiom,
% 12.45/7.01 ( pred_nat
% 12.45/7.01 = ( collec3392354462482085612at_nat
% 12.45/7.01 @ ( produc6081775807080527818_nat_o
% 12.45/7.01 @ ^ [M6: nat,N2: nat] :
% 12.45/7.01 ( N2
% 12.45/7.01 = ( suc @ M6 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9887_atLeastLessThan__add__Un,axiom,
% 12.45/7.01 ! [I: nat,J: nat,K: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ I @ J )
% 12.45/7.01 => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 12.45/7.01 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9888_sup__enat__def,axiom,
% 12.45/7.01 sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9889_sup__nat__def,axiom,
% 12.45/7.01 sup_sup_nat = ord_max_nat ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9890_VEBT_Osize_I3_J,axiom,
% 12.45/7.01 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 12.45/7.01 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 12.45/7.01 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9891_VEBT_Osize__gen_I1_J,axiom,
% 12.45/7.01 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 12.45/7.01 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 12.45/7.01 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9892_tl__upt,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( tl_nat @ ( upt @ M @ N ) )
% 12.45/7.01 = ( upt @ ( suc @ M ) @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9893_vanishes__mult__bounded,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ? [A7: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ A7 )
% 12.45/7.01 & ! [N4: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N4 ) ) @ A7 ) )
% 12.45/7.01 => ( ( vanishes @ Y7 )
% 12.45/7.01 => ( vanishes
% 12.45/7.01 @ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9894_vanishes__const,axiom,
% 12.45/7.01 ! [C: rat] :
% 12.45/7.01 ( ( vanishes
% 12.45/7.01 @ ^ [N2: nat] : C )
% 12.45/7.01 = ( C = zero_zero_rat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9895_vanishes__minus,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ( vanishes @ X8 )
% 12.45/7.01 => ( vanishes
% 12.45/7.01 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9896_vanishes__add,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( vanishes @ X8 )
% 12.45/7.01 => ( ( vanishes @ Y7 )
% 12.45/7.01 => ( vanishes
% 12.45/7.01 @ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9897_vanishes__diff,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( vanishes @ X8 )
% 12.45/7.01 => ( ( vanishes @ Y7 )
% 12.45/7.01 => ( vanishes
% 12.45/7.01 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9898_vanishesD,axiom,
% 12.45/7.01 ! [X8: nat > rat,R2: rat] :
% 12.45/7.01 ( ( vanishes @ X8 )
% 12.45/7.01 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 12.45/7.01 => ? [K2: nat] :
% 12.45/7.01 ! [N6: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K2 @ N6 )
% 12.45/7.01 => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N6 ) ) @ R2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9899_vanishesI,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ! [R3: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 12.45/7.01 => ? [K4: nat] :
% 12.45/7.01 ! [N4: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K4 @ N4 )
% 12.45/7.01 => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N4 ) ) @ R3 ) ) )
% 12.45/7.01 => ( vanishes @ X8 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9900_vanishes__def,axiom,
% 12.45/7.01 ( vanishes
% 12.45/7.01 = ( ^ [X6: nat > rat] :
% 12.45/7.01 ! [R5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.45/7.01 => ? [K3: nat] :
% 12.45/7.01 ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ N2 )
% 12.45/7.01 => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N2 ) ) @ R5 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9901_and__not__num_Opelims,axiom,
% 12.45/7.01 ! [X4: num,Xa: num,Y3: option_num] :
% 12.45/7.01 ( ( ( bit_and_not_num @ X4 @ Xa )
% 12.45/7.01 = Y3 )
% 12.45/7.01 => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3 = none_num )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ one ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ( ( Y3 = none_num )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ ( bit0 @ M4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ ( bit0 @ M4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.45/7.01 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.45/7.01 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ~ ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9902_and__num_Opelims,axiom,
% 12.45/7.01 ! [X4: num,Xa: num,Y3: option_num] :
% 12.45/7.01 ( ( ( bit_un7362597486090784418nd_num @ X4 @ Xa )
% 12.45/7.01 = Y3 )
% 12.45/7.01 => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ one ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ( ( Y3 = none_num )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ one ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3 = none_num )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ one ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ~ ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.45/7.01 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.45/7.01 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9903_xor__num_Opelims,axiom,
% 12.45/7.01 ! [X4: num,Xa: num,Y3: option_num] :
% 12.45/7.01 ( ( ( bit_un2480387367778600638or_num @ X4 @ Xa )
% 12.45/7.01 = Y3 )
% 12.45/7.01 => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3 = none_num )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ ( bit1 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ ( bit0 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ ( bit1 @ M4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ ( bit0 @ M4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 12.45/7.01 => ( ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 12.45/7.01 => ~ ! [M4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ! [N4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9904_or__not__num__neg_Opelims,axiom,
% 12.45/7.01 ! [X4: num,Xa: num,Y3: num] :
% 12.45/7.01 ( ( ( bit_or_not_num_neg @ X4 @ Xa )
% 12.45/7.01 = Y3 )
% 12.45/7.01 => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3 = one )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ! [M4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
% 12.45/7.01 => ( ( ( X4 = one )
% 12.45/7.01 => ! [M4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [N4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( bit0 @ one ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ one ) ) ) ) )
% 12.45/7.01 => ( ! [N4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ! [M4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [N4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit0 @ N4 ) )
% 12.45/7.01 => ! [M4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( bit0 @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit1 @ M4 ) ) ) ) ) )
% 12.45/7.01 => ( ! [N4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ( ( Xa = one )
% 12.45/7.01 => ( ( Y3 = one )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ one ) ) ) ) )
% 12.45/7.01 => ( ! [N4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ! [M4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit0 @ M4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 12.45/7.01 => ~ ! [N4: num] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( bit1 @ N4 ) )
% 12.45/7.01 => ! [M4: num] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( bit1 @ M4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 12.45/7.01 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9905_xor__num__rel__dict,axiom,
% 12.45/7.01 bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9906_and__num__rel__dict,axiom,
% 12.45/7.01 bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9907_rcis__inverse,axiom,
% 12.45/7.01 ! [R2: real,A: real] :
% 12.45/7.01 ( ( invers8013647133539491842omplex @ ( rcis @ R2 @ A ) )
% 12.45/7.01 = ( rcis @ ( divide_divide_real @ one_one_real @ R2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9908_Re__rcis,axiom,
% 12.45/7.01 ! [R2: real,A: real] :
% 12.45/7.01 ( ( re @ ( rcis @ R2 @ A ) )
% 12.45/7.01 = ( times_times_real @ R2 @ ( cos_real @ A ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9909_Im__rcis,axiom,
% 12.45/7.01 ! [R2: real,A: real] :
% 12.45/7.01 ( ( im @ ( rcis @ R2 @ A ) )
% 12.45/7.01 = ( times_times_real @ R2 @ ( sin_real @ A ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9910_cis__rcis__eq,axiom,
% 12.45/7.01 ( cis
% 12.45/7.01 = ( rcis @ one_one_real ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9911_rcis__mult,axiom,
% 12.45/7.01 ! [R1: real,A: real,R22: real,B2: real] :
% 12.45/7.01 ( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B2 ) )
% 12.45/7.01 = ( rcis @ ( times_times_real @ R1 @ R22 ) @ ( plus_plus_real @ A @ B2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9912_rcis__divide,axiom,
% 12.45/7.01 ! [R1: real,A: real,R22: real,B2: real] :
% 12.45/7.01 ( ( divide1717551699836669952omplex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B2 ) )
% 12.45/7.01 = ( rcis @ ( divide_divide_real @ R1 @ R22 ) @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9913_rcis__def,axiom,
% 12.45/7.01 ( rcis
% 12.45/7.01 = ( ^ [R5: real,A4: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) @ ( cis @ A4 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9914_DeMoivre2,axiom,
% 12.45/7.01 ! [R2: real,A: real,N: nat] :
% 12.45/7.01 ( ( power_power_complex @ ( rcis @ R2 @ A ) @ N )
% 12.45/7.01 = ( rcis @ ( power_power_real @ R2 @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9915_nat__of__integer__code,axiom,
% 12.45/7.01 ( code_nat_of_integer
% 12.45/7.01 = ( ^ [K3: code_integer] :
% 12.45/7.01 ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 12.45/7.01 @ ( produc1555791787009142072er_nat
% 12.45/7.01 @ ^ [L: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
% 12.45/7.01 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9916_nat__of__integer__code__post_I3_J,axiom,
% 12.45/7.01 ! [K: num] :
% 12.45/7.01 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 12.45/7.01 = ( numeral_numeral_nat @ K ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9917_nat__of__integer__code__post_I2_J,axiom,
% 12.45/7.01 ( ( code_nat_of_integer @ one_one_Code_integer )
% 12.45/7.01 = one_one_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9918_cauchy__def,axiom,
% 12.45/7.01 ( cauchy
% 12.45/7.01 = ( ^ [X6: nat > rat] :
% 12.45/7.01 ! [R5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.45/7.01 => ? [K3: nat] :
% 12.45/7.01 ! [M6: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ M6 )
% 12.45/7.01 => ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ N2 )
% 12.45/7.01 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M6 ) @ ( X6 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9919_cauchyI,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ! [R3: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 12.45/7.01 => ? [K4: nat] :
% 12.45/7.01 ! [M4: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K4 @ M4 )
% 12.45/7.01 => ! [N4: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K4 @ N4 )
% 12.45/7.01 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ R3 ) ) ) )
% 12.45/7.01 => ( cauchy @ X8 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9920_cauchy__inverse,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ~ ( vanishes @ X8 )
% 12.45/7.01 => ( cauchy
% 12.45/7.01 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X8 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9921_cauchy__diff,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( cauchy @ Y7 )
% 12.45/7.01 => ( cauchy
% 12.45/7.01 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9922_cauchy__mult,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( cauchy @ Y7 )
% 12.45/7.01 => ( cauchy
% 12.45/7.01 @ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9923_cauchy__const,axiom,
% 12.45/7.01 ! [X4: rat] :
% 12.45/7.01 ( cauchy
% 12.45/7.01 @ ^ [N2: nat] : X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9924_cauchy__minus,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( cauchy
% 12.45/7.01 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9925_cauchy__add,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( cauchy @ Y7 )
% 12.45/7.01 => ( cauchy
% 12.45/7.01 @ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9926_cauchy__imp__bounded,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ? [B5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ B5 )
% 12.45/7.01 & ! [N6: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N6 ) ) @ B5 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9927_vanishes__diff__inverse,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ~ ( vanishes @ X8 )
% 12.45/7.01 => ( ( cauchy @ Y7 )
% 12.45/7.01 => ( ~ ( vanishes @ Y7 )
% 12.45/7.01 => ( ( vanishes
% 12.45/7.01 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) )
% 12.45/7.01 => ( vanishes
% 12.45/7.01 @ ^ [N2: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X8 @ N2 ) ) @ ( inverse_inverse_rat @ ( Y7 @ N2 ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9928_cauchy__not__vanishes__cases,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ~ ( vanishes @ X8 )
% 12.45/7.01 => ? [B5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ B5 )
% 12.45/7.01 & ? [K2: nat] :
% 12.45/7.01 ( ! [N6: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K2 @ N6 )
% 12.45/7.01 => ( ord_less_rat @ B5 @ ( uminus_uminus_rat @ ( X8 @ N6 ) ) ) )
% 12.45/7.01 | ! [N6: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K2 @ N6 )
% 12.45/7.01 => ( ord_less_rat @ B5 @ ( X8 @ N6 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9929_cauchy__not__vanishes,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ~ ( vanishes @ X8 )
% 12.45/7.01 => ? [B5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ B5 )
% 12.45/7.01 & ? [K2: nat] :
% 12.45/7.01 ! [N6: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K2 @ N6 )
% 12.45/7.01 => ( ord_less_rat @ B5 @ ( abs_abs_rat @ ( X8 @ N6 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9930_cauchyD,axiom,
% 12.45/7.01 ! [X8: nat > rat,R2: rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 12.45/7.01 => ? [K2: nat] :
% 12.45/7.01 ! [M5: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K2 @ M5 )
% 12.45/7.01 => ! [N6: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K2 @ N6 )
% 12.45/7.01 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ R2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9931_le__Real,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( cauchy @ Y7 )
% 12.45/7.01 => ( ( ord_less_eq_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
% 12.45/7.01 = ( ! [R5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.45/7.01 => ? [K3: nat] :
% 12.45/7.01 ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ N2 )
% 12.45/7.01 => ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9932_inverse__Real,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( ( vanishes @ X8 )
% 12.45/7.01 => ( ( inverse_inverse_real @ ( real2 @ X8 ) )
% 12.45/7.01 = zero_zero_real ) )
% 12.45/7.01 & ( ~ ( vanishes @ X8 )
% 12.45/7.01 => ( ( inverse_inverse_real @ ( real2 @ X8 ) )
% 12.45/7.01 = ( real2
% 12.45/7.01 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X8 @ N2 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9933_Real__induct,axiom,
% 12.45/7.01 ! [P: real > $o,X4: real] :
% 12.45/7.01 ( ! [X10: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X10 )
% 12.45/7.01 => ( P @ ( real2 @ X10 ) ) )
% 12.45/7.01 => ( P @ X4 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9934_one__real__def,axiom,
% 12.45/7.01 ( one_one_real
% 12.45/7.01 = ( real2
% 12.45/7.01 @ ^ [N2: nat] : one_one_rat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9935_of__nat__Real,axiom,
% 12.45/7.01 ( semiri5074537144036343181t_real
% 12.45/7.01 = ( ^ [X: nat] :
% 12.45/7.01 ( real2
% 12.45/7.01 @ ^ [N2: nat] : ( semiri681578069525770553at_rat @ X ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9936_zero__real__def,axiom,
% 12.45/7.01 ( zero_zero_real
% 12.45/7.01 = ( real2
% 12.45/7.01 @ ^ [N2: nat] : zero_zero_rat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9937_of__int__Real,axiom,
% 12.45/7.01 ( ring_1_of_int_real
% 12.45/7.01 = ( ^ [X: int] :
% 12.45/7.01 ( real2
% 12.45/7.01 @ ^ [N2: nat] : ( ring_1_of_int_rat @ X ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9938_minus__Real,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( uminus_uminus_real @ ( real2 @ X8 ) )
% 12.45/7.01 = ( real2
% 12.45/7.01 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X8 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9939_add__Real,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( cauchy @ Y7 )
% 12.45/7.01 => ( ( plus_plus_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
% 12.45/7.01 = ( real2
% 12.45/7.01 @ ^ [N2: nat] : ( plus_plus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9940_mult__Real,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( cauchy @ Y7 )
% 12.45/7.01 => ( ( times_times_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
% 12.45/7.01 = ( real2
% 12.45/7.01 @ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9941_diff__Real,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( cauchy @ Y7 )
% 12.45/7.01 => ( ( minus_minus_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
% 12.45/7.01 = ( real2
% 12.45/7.01 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9942_eq__Real,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( cauchy @ Y7 )
% 12.45/7.01 => ( ( ( real2 @ X8 )
% 12.45/7.01 = ( real2 @ Y7 ) )
% 12.45/7.01 = ( vanishes
% 12.45/7.01 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9943_not__positive__Real,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( ~ ( positive2 @ ( real2 @ X8 ) ) )
% 12.45/7.01 = ( ! [R5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.45/7.01 => ? [K3: nat] :
% 12.45/7.01 ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ N2 )
% 12.45/7.01 => ( ord_less_eq_rat @ ( X8 @ N2 ) @ R5 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9944_positive__Real,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( positive2 @ ( real2 @ X8 ) )
% 12.45/7.01 = ( ? [R5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.45/7.01 & ? [K3: nat] :
% 12.45/7.01 ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ N2 )
% 12.45/7.01 => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9945_Real_Opositive__mult,axiom,
% 12.45/7.01 ! [X4: real,Y3: real] :
% 12.45/7.01 ( ( positive2 @ X4 )
% 12.45/7.01 => ( ( positive2 @ Y3 )
% 12.45/7.01 => ( positive2 @ ( times_times_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9946_Real_Opositive__add,axiom,
% 12.45/7.01 ! [X4: real,Y3: real] :
% 12.45/7.01 ( ( positive2 @ X4 )
% 12.45/7.01 => ( ( positive2 @ Y3 )
% 12.45/7.01 => ( positive2 @ ( plus_plus_real @ X4 @ Y3 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9947_Real_Opositive__zero,axiom,
% 12.45/7.01 ~ ( positive2 @ zero_zero_real ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9948_Real_Opositive__minus,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ~ ( positive2 @ X4 )
% 12.45/7.01 => ( ( X4 != zero_zero_real )
% 12.45/7.01 => ( positive2 @ ( uminus_uminus_real @ X4 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9949_less__real__def,axiom,
% 12.45/7.01 ( ord_less_real
% 12.45/7.01 = ( ^ [X: real,Y: real] : ( positive2 @ ( minus_minus_real @ Y @ X ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9950_Real_Opositive_Orep__eq,axiom,
% 12.45/7.01 ( positive2
% 12.45/7.01 = ( ^ [X: real] :
% 12.45/7.01 ? [R5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.45/7.01 & ? [K3: nat] :
% 12.45/7.01 ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ N2 )
% 12.45/7.01 => ( ord_less_rat @ R5 @ ( rep_real @ X @ N2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9951_inverse__real_Oabs__eq,axiom,
% 12.45/7.01 ! [X4: nat > rat] :
% 12.45/7.01 ( ( realrel @ X4 @ X4 )
% 12.45/7.01 => ( ( inverse_inverse_real @ ( real2 @ X4 ) )
% 12.45/7.01 = ( real2
% 12.45/7.01 @ ( if_nat_rat @ ( vanishes @ X4 )
% 12.45/7.01 @ ^ [N2: nat] : zero_zero_rat
% 12.45/7.01 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X4 @ N2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9952_realrel__refl,axiom,
% 12.45/7.01 ! [X8: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( realrel @ X8 @ X8 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9953_one__real_Orsp,axiom,
% 12.45/7.01 ( realrel
% 12.45/7.01 @ ^ [N2: nat] : one_one_rat
% 12.45/7.01 @ ^ [N2: nat] : one_one_rat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9954_zero__real_Orsp,axiom,
% 12.45/7.01 ( realrel
% 12.45/7.01 @ ^ [N2: nat] : zero_zero_rat
% 12.45/7.01 @ ^ [N2: nat] : zero_zero_rat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9955_real_Oabs__induct,axiom,
% 12.45/7.01 ! [P: real > $o,X4: real] :
% 12.45/7.01 ( ! [Y4: nat > rat] :
% 12.45/7.01 ( ( realrel @ Y4 @ Y4 )
% 12.45/7.01 => ( P @ ( real2 @ Y4 ) ) )
% 12.45/7.01 => ( P @ X4 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9956_uminus__real_Oabs__eq,axiom,
% 12.45/7.01 ! [X4: nat > rat] :
% 12.45/7.01 ( ( realrel @ X4 @ X4 )
% 12.45/7.01 => ( ( uminus_uminus_real @ ( real2 @ X4 ) )
% 12.45/7.01 = ( real2
% 12.45/7.01 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X4 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9957_plus__real_Oabs__eq,axiom,
% 12.45/7.01 ! [Xa: nat > rat,X4: nat > rat] :
% 12.45/7.01 ( ( realrel @ Xa @ Xa )
% 12.45/7.01 => ( ( realrel @ X4 @ X4 )
% 12.45/7.01 => ( ( plus_plus_real @ ( real2 @ Xa ) @ ( real2 @ X4 ) )
% 12.45/7.01 = ( real2
% 12.45/7.01 @ ^ [N2: nat] : ( plus_plus_rat @ ( Xa @ N2 ) @ ( X4 @ N2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9958_times__real_Oabs__eq,axiom,
% 12.45/7.01 ! [Xa: nat > rat,X4: nat > rat] :
% 12.45/7.01 ( ( realrel @ Xa @ Xa )
% 12.45/7.01 => ( ( realrel @ X4 @ X4 )
% 12.45/7.01 => ( ( times_times_real @ ( real2 @ Xa ) @ ( real2 @ X4 ) )
% 12.45/7.01 = ( real2
% 12.45/7.01 @ ^ [N2: nat] : ( times_times_rat @ ( Xa @ N2 ) @ ( X4 @ N2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9959_realrelI,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y7: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ( cauchy @ Y7 )
% 12.45/7.01 => ( ( vanishes
% 12.45/7.01 @ ^ [N2: nat] : ( minus_minus_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) )
% 12.45/7.01 => ( realrel @ X8 @ Y7 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9960_realrel__def,axiom,
% 12.45/7.01 ( realrel
% 12.45/7.01 = ( ^ [X6: nat > rat,Y8: nat > rat] :
% 12.45/7.01 ( ( cauchy @ X6 )
% 12.45/7.01 & ( cauchy @ Y8 )
% 12.45/7.01 & ( vanishes
% 12.45/7.01 @ ^ [N2: nat] : ( minus_minus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9961_Real_Opositive_Oabs__eq,axiom,
% 12.45/7.01 ! [X4: nat > rat] :
% 12.45/7.01 ( ( realrel @ X4 @ X4 )
% 12.45/7.01 => ( ( positive2 @ ( real2 @ X4 ) )
% 12.45/7.01 = ( ? [R5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.45/7.01 & ? [K3: nat] :
% 12.45/7.01 ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ N2 )
% 12.45/7.01 => ( ord_less_rat @ R5 @ ( X4 @ N2 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9962_inverse__real__def,axiom,
% 12.45/7.01 ( inverse_inverse_real
% 12.45/7.01 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 12.45/7.01 @ ^ [X6: nat > rat] :
% 12.45/7.01 ( if_nat_rat @ ( vanishes @ X6 )
% 12.45/7.01 @ ^ [N2: nat] : zero_zero_rat
% 12.45/7.01 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9963_cr__real__def,axiom,
% 12.45/7.01 ( cr_real
% 12.45/7.01 = ( ^ [X: nat > rat,Y: real] :
% 12.45/7.01 ( ( realrel @ X @ X )
% 12.45/7.01 & ( ( real2 @ X )
% 12.45/7.01 = Y ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9964_uminus__real__def,axiom,
% 12.45/7.01 ( uminus_uminus_real
% 12.45/7.01 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 12.45/7.01 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9965_times__real__def,axiom,
% 12.45/7.01 ( times_times_real
% 12.45/7.01 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 12.45/7.01 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9966_plus__real__def,axiom,
% 12.45/7.01 ( plus_plus_real
% 12.45/7.01 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 12.45/7.01 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9967_Real_Opositive_Orsp,axiom,
% 12.45/7.01 ( bNF_re728719798268516973at_o_o @ realrel
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [X6: nat > rat] :
% 12.45/7.01 ? [R5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.45/7.01 & ? [K3: nat] :
% 12.45/7.01 ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ N2 )
% 12.45/7.01 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 12.45/7.01 @ ^ [X6: nat > rat] :
% 12.45/7.01 ? [R5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.45/7.01 & ? [K3: nat] :
% 12.45/7.01 ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ N2 )
% 12.45/7.01 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9968_plus__real_Orsp,axiom,
% 12.45/7.01 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 12.45/7.01 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 12.45/7.01 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9969_uminus__real_Orsp,axiom,
% 12.45/7.01 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 12.45/7.01 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
% 12.45/7.01 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9970_times__real_Orsp,axiom,
% 12.45/7.01 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 12.45/7.01 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 12.45/7.01 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9971_sub_Orsp,axiom,
% 12.45/7.01 ( bNF_re8402795839162346335um_int
% 12.45/7.01 @ ^ [Y6: num,Z4: num] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re1822329894187522285nt_int
% 12.45/7.01 @ ^ [Y6: num,Z4: num] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 12.45/7.01 @ ^ [M6: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) )
% 12.45/7.01 @ ^ [M6: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9972_times__natural_Orsp,axiom,
% 12.45/7.01 ( bNF_re1345281282404953727at_nat
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re5653821019739307937at_nat
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 12.45/7.01 @ times_times_nat
% 12.45/7.01 @ times_times_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9973_times__integer_Orsp,axiom,
% 12.45/7.01 ( bNF_re711492959462206631nt_int
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re4712519889275205905nt_int
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 12.45/7.01 @ times_times_int
% 12.45/7.01 @ times_times_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9974_minus__integer_Orsp,axiom,
% 12.45/7.01 ( bNF_re711492959462206631nt_int
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re4712519889275205905nt_int
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 12.45/7.01 @ minus_minus_int
% 12.45/7.01 @ minus_minus_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9975_minus__natural_Orsp,axiom,
% 12.45/7.01 ( bNF_re1345281282404953727at_nat
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re5653821019739307937at_nat
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 12.45/7.01 @ minus_minus_nat
% 12.45/7.01 @ minus_minus_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9976_divide__natural_Orsp,axiom,
% 12.45/7.01 ( bNF_re1345281282404953727at_nat
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re5653821019739307937at_nat
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 12.45/7.01 @ divide_divide_nat
% 12.45/7.01 @ divide_divide_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9977_divide__integer_Orsp,axiom,
% 12.45/7.01 ( bNF_re711492959462206631nt_int
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re4712519889275205905nt_int
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 12.45/7.01 @ divide_divide_int
% 12.45/7.01 @ divide_divide_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9978_modulo__natural_Orsp,axiom,
% 12.45/7.01 ( bNF_re1345281282404953727at_nat
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re5653821019739307937at_nat
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 12.45/7.01 @ modulo_modulo_nat
% 12.45/7.01 @ modulo_modulo_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9979_modulo__integer_Orsp,axiom,
% 12.45/7.01 ( bNF_re711492959462206631nt_int
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re4712519889275205905nt_int
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 12.45/7.01 @ modulo_modulo_int
% 12.45/7.01 @ modulo_modulo_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9980_Suc_Orsp,axiom,
% 12.45/7.01 ( bNF_re5653821019739307937at_nat
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ suc
% 12.45/7.01 @ suc ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9981_dup_Orsp,axiom,
% 12.45/7.01 ( bNF_re4712519889275205905nt_int
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
% 12.45/7.01 @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9982_plus__natural_Orsp,axiom,
% 12.45/7.01 ( bNF_re1345281282404953727at_nat
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re5653821019739307937at_nat
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 12.45/7.01 @ plus_plus_nat
% 12.45/7.01 @ plus_plus_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9983_plus__integer_Orsp,axiom,
% 12.45/7.01 ( bNF_re711492959462206631nt_int
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re4712519889275205905nt_int
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 12.45/7.01 @ plus_plus_int
% 12.45/7.01 @ plus_plus_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9984_less__integer_Orsp,axiom,
% 12.45/7.01 ( bNF_re3403563459893282935_int_o
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re5089333283451836215nt_o_o
% 12.45/7.01 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 12.45/7.01 @ ord_less_int
% 12.45/7.01 @ ord_less_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9985_less__natural_Orsp,axiom,
% 12.45/7.01 ( bNF_re578469030762574527_nat_o
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re4705727531993890431at_o_o
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 12.45/7.01 @ ord_less_nat
% 12.45/7.01 @ ord_less_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9986_less__eq__natural_Orsp,axiom,
% 12.45/7.01 ( bNF_re578469030762574527_nat_o
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ( bNF_re4705727531993890431at_o_o
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 12.45/7.01 @ ord_less_eq_nat
% 12.45/7.01 @ ord_less_eq_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9987_le__enumerate,axiom,
% 12.45/7.01 ! [S2: set_nat,N: nat] :
% 12.45/7.01 ( ~ ( finite_finite_nat @ S2 )
% 12.45/7.01 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9988_inverse__real_Orsp,axiom,
% 12.45/7.01 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 12.45/7.01 @ ^ [X6: nat > rat] :
% 12.45/7.01 ( if_nat_rat @ ( vanishes @ X6 )
% 12.45/7.01 @ ^ [N2: nat] : zero_zero_rat
% 12.45/7.01 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
% 12.45/7.01 @ ^ [X6: nat > rat] :
% 12.45/7.01 ( if_nat_rat @ ( vanishes @ X6 )
% 12.45/7.01 @ ^ [N2: nat] : zero_zero_rat
% 12.45/7.01 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9989_finite__le__enumerate,axiom,
% 12.45/7.01 ! [S2: set_nat,N: nat] :
% 12.45/7.01 ( ( finite_finite_nat @ S2 )
% 12.45/7.01 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
% 12.45/7.01 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9990_Real_Opositive_Otransfer,axiom,
% 12.45/7.01 ( bNF_re4297313714947099218al_o_o @ pcr_real
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [X6: nat > rat] :
% 12.45/7.01 ? [R5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.45/7.01 & ? [K3: nat] :
% 12.45/7.01 ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ N2 )
% 12.45/7.01 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 12.45/7.01 @ positive2 ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9991_plus__rat_Otransfer,axiom,
% 12.45/7.01 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 12.45/7.01 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 12.45/7.01 @ plus_plus_rat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9992_real_Orel__eq__transfer,axiom,
% 12.45/7.01 ( bNF_re4521903465945308077real_o @ pcr_real
% 12.45/7.01 @ ( bNF_re4297313714947099218al_o_o @ pcr_real
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 12.45/7.01 @ realrel
% 12.45/7.01 @ ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9993_real_Opcr__cr__eq,axiom,
% 12.45/7.01 pcr_real = cr_real ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9994_zero__real_Otransfer,axiom,
% 12.45/7.01 ( pcr_real
% 12.45/7.01 @ ^ [N2: nat] : zero_zero_rat
% 12.45/7.01 @ zero_zero_real ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9995_one__real_Otransfer,axiom,
% 12.45/7.01 ( pcr_real
% 12.45/7.01 @ ^ [N2: nat] : one_one_rat
% 12.45/7.01 @ one_one_real ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9996_cr__real__eq,axiom,
% 12.45/7.01 ( pcr_real
% 12.45/7.01 = ( ^ [X: nat > rat,Y: real] :
% 12.45/7.01 ( ( cauchy @ X )
% 12.45/7.01 & ( ( real2 @ X )
% 12.45/7.01 = Y ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9997_uminus__real_Otransfer,axiom,
% 12.45/7.01 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 12.45/7.01 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
% 12.45/7.01 @ uminus_uminus_real ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9998_plus__real_Otransfer,axiom,
% 12.45/7.01 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 12.45/7.01 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 12.45/7.01 @ plus_plus_real ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_9999_times__real_Otransfer,axiom,
% 12.45/7.01 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 12.45/7.01 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 12.45/7.01 @ times_times_real ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10000_inverse__real_Otransfer,axiom,
% 12.45/7.01 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 12.45/7.01 @ ^ [X6: nat > rat] :
% 12.45/7.01 ( if_nat_rat @ ( vanishes @ X6 )
% 12.45/7.01 @ ^ [N2: nat] : zero_zero_rat
% 12.45/7.01 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
% 12.45/7.01 @ inverse_inverse_real ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10001_times__rat_Otransfer,axiom,
% 12.45/7.01 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 12.45/7.01 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 12.45/7.01 @ times_times_rat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10002_Rat_Opositive_Otransfer,axiom,
% 12.45/7.01 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 12.45/7.01 @ positive ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10003_times__int_Otransfer,axiom,
% 12.45/7.01 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) )
% 12.45/7.01 @ times_times_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10004_Least__eq__0,axiom,
% 12.45/7.01 ! [P: nat > $o] :
% 12.45/7.01 ( ( P @ zero_zero_nat )
% 12.45/7.01 => ( ( ord_Least_nat @ P )
% 12.45/7.01 = zero_zero_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10005_Least__Suc,axiom,
% 12.45/7.01 ! [P: nat > $o,N: nat] :
% 12.45/7.01 ( ( P @ N )
% 12.45/7.01 => ( ~ ( P @ zero_zero_nat )
% 12.45/7.01 => ( ( ord_Least_nat @ P )
% 12.45/7.01 = ( suc
% 12.45/7.01 @ ( ord_Least_nat
% 12.45/7.01 @ ^ [M6: nat] : ( P @ ( suc @ M6 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10006_Least__Suc2,axiom,
% 12.45/7.01 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 12.45/7.01 ( ( P @ N )
% 12.45/7.01 => ( ( Q @ M )
% 12.45/7.01 => ( ~ ( P @ zero_zero_nat )
% 12.45/7.01 => ( ! [K2: nat] :
% 12.45/7.01 ( ( P @ ( suc @ K2 ) )
% 12.45/7.01 = ( Q @ K2 ) )
% 12.45/7.01 => ( ( ord_Least_nat @ P )
% 12.45/7.01 = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10007_Sup__real__def,axiom,
% 12.45/7.01 ( comple1385675409528146559p_real
% 12.45/7.01 = ( ^ [X6: set_real] :
% 12.45/7.01 ( ord_Least_real
% 12.45/7.01 @ ^ [Z5: real] :
% 12.45/7.01 ! [X: real] :
% 12.45/7.01 ( ( member_real @ X @ X6 )
% 12.45/7.01 => ( ord_less_eq_real @ X @ Z5 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10008_nat_Otransfer,axiom,
% 12.45/7.01 ( bNF_re4555766996558763186at_nat @ pcr_int
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 12.45/7.01 @ nat2 ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10009_one__int_Otransfer,axiom,
% 12.45/7.01 pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10010_less__int_Otransfer,axiom,
% 12.45/7.01 ( bNF_re717283939379294677_int_o @ pcr_int
% 12.45/7.01 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 12.45/7.01 @ ( produc8739625826339149834_nat_o
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc6081775807080527818_nat_o
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
% 12.45/7.01 @ ord_less_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10011_less__eq__int_Otransfer,axiom,
% 12.45/7.01 ( bNF_re717283939379294677_int_o @ pcr_int
% 12.45/7.01 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 12.45/7.01 @ ( produc8739625826339149834_nat_o
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc6081775807080527818_nat_o
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
% 12.45/7.01 @ ord_less_eq_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10012_plus__int_Otransfer,axiom,
% 12.45/7.01 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 12.45/7.01 @ plus_plus_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10013_minus__int_Otransfer,axiom,
% 12.45/7.01 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) )
% 12.45/7.01 @ minus_minus_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10014_of__rat__Real,axiom,
% 12.45/7.01 ( field_7254667332652039916t_real
% 12.45/7.01 = ( ^ [X: rat] :
% 12.45/7.01 ( real2
% 12.45/7.01 @ ^ [N2: nat] : X ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10015_of__rat__dense,axiom,
% 12.45/7.01 ! [X4: real,Y3: real] :
% 12.45/7.01 ( ( ord_less_real @ X4 @ Y3 )
% 12.45/7.01 => ? [Q3: rat] :
% 12.45/7.01 ( ( ord_less_real @ X4 @ ( field_7254667332652039916t_real @ Q3 ) )
% 12.45/7.01 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q3 ) @ Y3 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10016_less__RealD,axiom,
% 12.45/7.01 ! [Y7: nat > rat,X4: real] :
% 12.45/7.01 ( ( cauchy @ Y7 )
% 12.45/7.01 => ( ( ord_less_real @ X4 @ ( real2 @ Y7 ) )
% 12.45/7.01 => ? [N4: nat] : ( ord_less_real @ X4 @ ( field_7254667332652039916t_real @ ( Y7 @ N4 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10017_le__RealI,axiom,
% 12.45/7.01 ! [Y7: nat > rat,X4: real] :
% 12.45/7.01 ( ( cauchy @ Y7 )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ X4 @ ( field_7254667332652039916t_real @ ( Y7 @ N4 ) ) )
% 12.45/7.01 => ( ord_less_eq_real @ X4 @ ( real2 @ Y7 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10018_Real__leI,axiom,
% 12.45/7.01 ! [X8: nat > rat,Y3: real] :
% 12.45/7.01 ( ( cauchy @ X8 )
% 12.45/7.01 => ( ! [N4: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X8 @ N4 ) ) @ Y3 )
% 12.45/7.01 => ( ord_less_eq_real @ ( real2 @ X8 ) @ Y3 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10019_times__int_Orsp,axiom,
% 12.45/7.01 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10020_intrel__iff,axiom,
% 12.45/7.01 ! [X4: nat,Y3: nat,U: nat,V: nat] :
% 12.45/7.01 ( ( intrel @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( product_Pair_nat_nat @ U @ V ) )
% 12.45/7.01 = ( ( plus_plus_nat @ X4 @ V )
% 12.45/7.01 = ( plus_plus_nat @ U @ Y3 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10021_nat_Orsp,axiom,
% 12.45/7.01 ( bNF_re8246922863344978751at_nat @ intrel
% 12.45/7.01 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 12.45/7.01 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 12.45/7.01 @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10022_one__int_Orsp,axiom,
% 12.45/7.01 intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10023_intrel__def,axiom,
% 12.45/7.01 ( intrel
% 12.45/7.01 = ( produc8739625826339149834_nat_o
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc6081775807080527818_nat_o
% 12.45/7.01 @ ^ [U4: nat,V4: nat] :
% 12.45/7.01 ( ( plus_plus_nat @ X @ V4 )
% 12.45/7.01 = ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10024_less__int_Orsp,axiom,
% 12.45/7.01 ( bNF_re4202695980764964119_nat_o @ intrel
% 12.45/7.01 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 12.45/7.01 @ ( produc8739625826339149834_nat_o
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc6081775807080527818_nat_o
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
% 12.45/7.01 @ ( produc8739625826339149834_nat_o
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc6081775807080527818_nat_o
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10025_less__eq__int_Orsp,axiom,
% 12.45/7.01 ( bNF_re4202695980764964119_nat_o @ intrel
% 12.45/7.01 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 12.45/7.01 @ ( produc8739625826339149834_nat_o
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc6081775807080527818_nat_o
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) )
% 12.45/7.01 @ ( produc8739625826339149834_nat_o
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc6081775807080527818_nat_o
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10026_plus__int_Orsp,axiom,
% 12.45/7.01 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10027_minus__int_Orsp,axiom,
% 12.45/7.01 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10028_has__vector__derivative__id,axiom,
% 12.45/7.01 ! [Net: filter_real] :
% 12.45/7.01 ( has_ve631408500373753343e_real
% 12.45/7.01 @ ^ [X: real] : X
% 12.45/7.01 @ one_one_real
% 12.45/7.01 @ Net ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10029_has__field__derivative__iff__has__vector__derivative,axiom,
% 12.45/7.01 has_fi5821293074295781190e_real = has_ve631408500373753343e_real ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10030_plus__rat_Orsp,axiom,
% 12.45/7.01 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 12.45/7.01 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 12.45/7.01 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10031_ratrel__iff,axiom,
% 12.45/7.01 ( ratrel
% 12.45/7.01 = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 12.45/7.01 ( ( ( product_snd_int_int @ X )
% 12.45/7.01 != zero_zero_int )
% 12.45/7.01 & ( ( product_snd_int_int @ Y )
% 12.45/7.01 != zero_zero_int )
% 12.45/7.01 & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 12.45/7.01 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10032_ratrel__def,axiom,
% 12.45/7.01 ( ratrel
% 12.45/7.01 = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 12.45/7.01 ( ( ( product_snd_int_int @ X )
% 12.45/7.01 != zero_zero_int )
% 12.45/7.01 & ( ( product_snd_int_int @ Y )
% 12.45/7.01 != zero_zero_int )
% 12.45/7.01 & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 12.45/7.01 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10033_times__rat_Orsp,axiom,
% 12.45/7.01 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 12.45/7.01 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 12.45/7.01 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10034_Rat_Opositive_Orsp,axiom,
% 12.45/7.01 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 12.45/7.01 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 12.45/7.01 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 12.45/7.01 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10035_plus__rat_Oabs__eq,axiom,
% 12.45/7.01 ! [Xa: product_prod_int_int,X4: product_prod_int_int] :
% 12.45/7.01 ( ( ratrel @ Xa @ Xa )
% 12.45/7.01 => ( ( ratrel @ X4 @ X4 )
% 12.45/7.01 => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X4 ) )
% 12.45/7.01 = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X4 ) ) @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10036_Rat_Opositive_Oabs__eq,axiom,
% 12.45/7.01 ! [X4: product_prod_int_int] :
% 12.45/7.01 ( ( ratrel @ X4 @ X4 )
% 12.45/7.01 => ( ( positive @ ( abs_Rat @ X4 ) )
% 12.45/7.01 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10037_times__rat_Oabs__eq,axiom,
% 12.45/7.01 ! [Xa: product_prod_int_int,X4: product_prod_int_int] :
% 12.45/7.01 ( ( ratrel @ Xa @ Xa )
% 12.45/7.01 => ( ( ratrel @ X4 @ X4 )
% 12.45/7.01 => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X4 ) )
% 12.45/7.01 = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X4 ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10038_last__upt,axiom,
% 12.45/7.01 ! [I: nat,J: nat] :
% 12.45/7.01 ( ( ord_less_nat @ I @ J )
% 12.45/7.01 => ( ( last_nat @ ( upt @ I @ J ) )
% 12.45/7.01 = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10039_Bseq__monoseq__convergent_H__dec,axiom,
% 12.45/7.01 ! [F: nat > real,M7: nat] :
% 12.45/7.01 ( ( bfun_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M7 ) )
% 12.45/7.01 @ at_top_nat )
% 12.45/7.01 => ( ! [M4: nat,N4: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ M7 @ M4 )
% 12.45/7.01 => ( ( ord_less_eq_nat @ M4 @ N4 )
% 12.45/7.01 => ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ M4 ) ) ) )
% 12.45/7.01 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10040_Bseq__monoseq__convergent_H__inc,axiom,
% 12.45/7.01 ! [F: nat > real,M7: nat] :
% 12.45/7.01 ( ( bfun_nat_real
% 12.45/7.01 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M7 ) )
% 12.45/7.01 @ at_top_nat )
% 12.45/7.01 => ( ! [M4: nat,N4: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ M7 @ M4 )
% 12.45/7.01 => ( ( ord_less_eq_nat @ M4 @ N4 )
% 12.45/7.01 => ( ord_less_eq_real @ ( F @ M4 ) @ ( F @ N4 ) ) ) )
% 12.45/7.01 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10041_Bseq__mono__convergent,axiom,
% 12.45/7.01 ! [X8: nat > real] :
% 12.45/7.01 ( ( bfun_nat_real @ X8 @ at_top_nat )
% 12.45/7.01 => ( ! [M4: nat,N4: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ M4 @ N4 )
% 12.45/7.01 => ( ord_less_eq_real @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) )
% 12.45/7.01 => ( topolo7531315842566124627t_real @ X8 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10042_convergent__realpow,axiom,
% 12.45/7.01 ! [X4: real] :
% 12.45/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.45/7.01 => ( ( ord_less_eq_real @ X4 @ one_one_real )
% 12.45/7.01 => ( topolo7531315842566124627t_real @ ( power_power_real @ X4 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10043_filtermap__at__right__shift,axiom,
% 12.45/7.01 ! [D: real,A: real] :
% 12.45/7.01 ( ( filtermap_real_real
% 12.45/7.01 @ ^ [X: real] : ( minus_minus_real @ X @ D )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.45/7.01 = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10044_at__right__to__0,axiom,
% 12.45/7.01 ! [A: real] :
% 12.45/7.01 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 12.45/7.01 = ( filtermap_real_real
% 12.45/7.01 @ ^ [X: real] : ( plus_plus_real @ X @ A )
% 12.45/7.01 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10045_pair__lessI2,axiom,
% 12.45/7.01 ! [A: nat,B2: nat,S: nat,T: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/7.01 => ( ( ord_less_nat @ S @ T )
% 12.45/7.01 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10046_pair__less__iff1,axiom,
% 12.45/7.01 ! [X4: nat,Y3: nat,Z: nat] :
% 12.45/7.01 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( product_Pair_nat_nat @ X4 @ Z ) ) @ fun_pair_less )
% 12.45/7.01 = ( ord_less_nat @ Y3 @ Z ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10047_pair__lessI1,axiom,
% 12.45/7.01 ! [A: nat,B2: nat,S: nat,T: nat] :
% 12.45/7.01 ( ( ord_less_nat @ A @ B2 )
% 12.45/7.01 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10048_pair__leqI2,axiom,
% 12.45/7.01 ! [A: nat,B2: nat,S: nat,T: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ A @ B2 )
% 12.45/7.01 => ( ( ord_less_eq_nat @ S @ T )
% 12.45/7.01 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10049_pair__leqI1,axiom,
% 12.45/7.01 ! [A: nat,B2: nat,S: nat,T: nat] :
% 12.45/7.01 ( ( ord_less_nat @ A @ B2 )
% 12.45/7.01 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10050_bot__nat__0_Oordering__top__axioms,axiom,
% 12.45/7.01 ( ordering_top_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 12.45/7.01 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 12.45/7.01 @ zero_zero_nat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10051_set__encode__vimage__Suc,axiom,
% 12.45/7.01 ! [A3: set_nat] :
% 12.45/7.01 ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A3 ) )
% 12.45/7.01 = ( divide_divide_nat @ ( nat_set_encode @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10052_euclidean__size__int__def,axiom,
% 12.45/7.01 ( euclid4774559944035922753ze_int
% 12.45/7.01 = ( comp_int_nat_int @ nat2 @ abs_abs_int ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10053_finite__vimage__Suc__iff,axiom,
% 12.45/7.01 ! [F5: set_nat] :
% 12.45/7.01 ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F5 ) )
% 12.45/7.01 = ( finite_finite_nat @ F5 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10054_vimage__Suc__insert__Suc,axiom,
% 12.45/7.01 ! [N: nat,A3: set_nat] :
% 12.45/7.01 ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A3 ) )
% 12.45/7.01 = ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A3 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10055_vimage__Suc__insert__0,axiom,
% 12.45/7.01 ! [A3: set_nat] :
% 12.45/7.01 ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A3 ) )
% 12.45/7.01 = ( vimage_nat_nat @ suc @ A3 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10056_set__decode__div__2,axiom,
% 12.45/7.01 ! [X4: nat] :
% 12.45/7.01 ( ( nat_set_decode @ ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/7.01 = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X4 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10057_abs__division__segment,axiom,
% 12.45/7.01 ! [K: int] :
% 12.45/7.01 ( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
% 12.45/7.01 = one_one_int ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10058_division__segment__nat__def,axiom,
% 12.45/7.01 ( euclid3398187327856392827nt_nat
% 12.45/7.01 = ( ^ [N2: nat] : one_one_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10059_division__segment__eq__sgn,axiom,
% 12.45/7.01 ! [K: int] :
% 12.45/7.01 ( ( K != zero_zero_int )
% 12.45/7.01 => ( ( euclid3395696857347342551nt_int @ K )
% 12.45/7.01 = ( sgn_sgn_int @ K ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10060_division__segment__int__def,axiom,
% 12.45/7.01 ( euclid3395696857347342551nt_int
% 12.45/7.01 = ( ^ [K3: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10061_less__eq__enat__def,axiom,
% 12.45/7.01 ( ord_le2932123472753598470d_enat
% 12.45/7.01 = ( ^ [M6: extended_enat] :
% 12.45/7.01 ( extended_case_enat_o
% 12.45/7.01 @ ^ [N1: nat] :
% 12.45/7.01 ( extended_case_enat_o
% 12.45/7.01 @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
% 12.45/7.01 @ $false
% 12.45/7.01 @ M6 )
% 12.45/7.01 @ $true ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10062_transp__realrel,axiom,
% 12.45/7.01 transp_nat_rat @ realrel ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10063_less__enat__def,axiom,
% 12.45/7.01 ( ord_le72135733267957522d_enat
% 12.45/7.01 = ( ^ [M6: extended_enat,N2: extended_enat] :
% 12.45/7.01 ( extended_case_enat_o
% 12.45/7.01 @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N2 )
% 12.45/7.01 @ $false
% 12.45/7.01 @ M6 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10064_Divides_Oadjust__div__def,axiom,
% 12.45/7.01 ( adjust_div
% 12.45/7.01 = ( produc8211389475949308722nt_int
% 12.45/7.01 @ ^ [Q4: int,R5: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10065_rat__floor__code,axiom,
% 12.45/7.01 ( archim3151403230148437115or_rat
% 12.45/7.01 = ( ^ [P3: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P3 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10066_prod__decode__triangle__add,axiom,
% 12.45/7.01 ! [K: nat,M: nat] :
% 12.45/7.01 ( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) )
% 12.45/7.01 = ( nat_prod_decode_aux @ K @ M ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10067_list__decode_Opinduct,axiom,
% 12.45/7.01 ! [A0: nat,P: nat > $o] :
% 12.45/7.01 ( ( accp_nat @ nat_list_decode_rel @ A0 )
% 12.45/7.01 => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
% 12.45/7.01 => ( P @ zero_zero_nat ) )
% 12.45/7.01 => ( ! [N4: nat] :
% 12.45/7.01 ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N4 ) )
% 12.45/7.01 => ( ! [X5: nat,Y5: nat] :
% 12.45/7.01 ( ( ( product_Pair_nat_nat @ X5 @ Y5 )
% 12.45/7.01 = ( nat_prod_decode @ N4 ) )
% 12.45/7.01 => ( P @ Y5 ) )
% 12.45/7.01 => ( P @ ( suc @ N4 ) ) ) )
% 12.45/7.01 => ( P @ A0 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10068_list__decode_Oelims,axiom,
% 12.45/7.01 ! [X4: nat,Y3: list_nat] :
% 12.45/7.01 ( ( ( nat_list_decode @ X4 )
% 12.45/7.01 = Y3 )
% 12.45/7.01 => ( ( ( X4 = zero_zero_nat )
% 12.45/7.01 => ( Y3 != nil_nat ) )
% 12.45/7.01 => ~ ! [N4: nat] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( suc @ N4 ) )
% 12.45/7.01 => ( Y3
% 12.45/7.01 != ( produc2761476792215241774st_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 12.45/7.01 @ ( nat_prod_decode @ N4 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10069_list__decode_Opsimps_I2_J,axiom,
% 12.45/7.01 ! [N: nat] :
% 12.45/7.01 ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) )
% 12.45/7.01 => ( ( nat_list_decode @ ( suc @ N ) )
% 12.45/7.01 = ( produc2761476792215241774st_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 12.45/7.01 @ ( nat_prod_decode @ N ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10070_list__decode_Osimps_I2_J,axiom,
% 12.45/7.01 ! [N: nat] :
% 12.45/7.01 ( ( nat_list_decode @ ( suc @ N ) )
% 12.45/7.01 = ( produc2761476792215241774st_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 12.45/7.01 @ ( nat_prod_decode @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10071_list__decode_Opelims,axiom,
% 12.45/7.01 ! [X4: nat,Y3: list_nat] :
% 12.45/7.01 ( ( ( nat_list_decode @ X4 )
% 12.45/7.01 = Y3 )
% 12.45/7.01 => ( ( accp_nat @ nat_list_decode_rel @ X4 )
% 12.45/7.01 => ( ( ( X4 = zero_zero_nat )
% 12.45/7.01 => ( ( Y3 = nil_nat )
% 12.45/7.01 => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
% 12.45/7.01 => ~ ! [N4: nat] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( suc @ N4 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( produc2761476792215241774st_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 12.45/7.01 @ ( nat_prod_decode @ N4 ) ) )
% 12.45/7.01 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N4 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10072_compute__powr__real,axiom,
% 12.45/7.01 ( powr_real2
% 12.45/7.01 = ( ^ [B3: real,I2: real] :
% 12.45/7.01 ( if_real @ ( ord_less_eq_real @ B3 @ zero_zero_real )
% 12.45/7.01 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 12.45/7.01 @ ^ [Uu: product_unit] : ( powr_real2 @ B3 @ I2 ) )
% 12.45/7.01 @ ( if_real
% 12.45/7.01 @ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I2 ) )
% 12.45/7.01 = I2 )
% 12.45/7.01 @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I2 ) @ ( power_power_real @ B3 @ ( nat2 @ ( archim6058952711729229775r_real @ I2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B3 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I2 ) ) ) ) ) )
% 12.45/7.01 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 12.45/7.01 @ ^ [Uu: product_unit] : ( powr_real2 @ B3 @ I2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10073_pairs__le__eq__Sigma,axiom,
% 12.45/7.01 ! [M: nat] :
% 12.45/7.01 ( ( collec3392354462482085612at_nat
% 12.45/7.01 @ ( produc6081775807080527818_nat_o
% 12.45/7.01 @ ^ [I2: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ J3 ) @ M ) ) )
% 12.45/7.01 = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 12.45/7.01 @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10074_product__atMost__eq__Un,axiom,
% 12.45/7.01 ! [A3: set_nat,M: nat] :
% 12.45/7.01 ( ( produc457027306803732586at_nat @ A3
% 12.45/7.01 @ ^ [Uu: nat] : ( set_ord_atMost_nat @ M ) )
% 12.45/7.01 = ( sup_su6327502436637775413at_nat
% 12.45/7.01 @ ( produc457027306803732586at_nat @ A3
% 12.45/7.01 @ ^ [I2: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ I2 ) ) )
% 12.45/7.01 @ ( produc457027306803732586at_nat @ A3
% 12.45/7.01 @ ^ [I2: nat] : ( set_or6659071591806873216st_nat @ ( minus_minus_nat @ M @ I2 ) @ M ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10075_of__nat__eq__id,axiom,
% 12.45/7.01 semiri1316708129612266289at_nat = id_nat ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10076_Real_Opositive__def,axiom,
% 12.45/7.01 ( positive2
% 12.45/7.01 = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
% 12.45/7.01 @ ^ [X6: nat > rat] :
% 12.45/7.01 ? [R5: rat] :
% 12.45/7.01 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.45/7.01 & ? [K3: nat] :
% 12.45/7.01 ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ K3 @ N2 )
% 12.45/7.01 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10077_euclidean__size__nat__def,axiom,
% 12.45/7.01 euclid4777050414544973029ze_nat = id_nat ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10078_times__rat__def,axiom,
% 12.45/7.01 ( times_times_rat
% 12.45/7.01 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 12.45/7.01 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10079_plus__rat__def,axiom,
% 12.45/7.01 ( plus_plus_rat
% 12.45/7.01 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 12.45/7.01 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10080_Rat_Opositive__def,axiom,
% 12.45/7.01 ( positive
% 12.45/7.01 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 12.45/7.01 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10081_nat__def,axiom,
% 12.45/7.01 ( nat2
% 12.45/7.01 = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10082_less__int__def,axiom,
% 12.45/7.01 ( ord_less_int
% 12.45/7.01 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 12.45/7.01 @ ( produc8739625826339149834_nat_o
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc6081775807080527818_nat_o
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10083_less__eq__int__def,axiom,
% 12.45/7.01 ( ord_less_eq_int
% 12.45/7.01 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 12.45/7.01 @ ( produc8739625826339149834_nat_o
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc6081775807080527818_nat_o
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U4 @ Y ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10084_plus__int__def,axiom,
% 12.45/7.01 ( plus_plus_int
% 12.45/7.01 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U4 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10085_minus__int__def,axiom,
% 12.45/7.01 ( minus_minus_int
% 12.45/7.01 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U4 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10086_times__int__def,axiom,
% 12.45/7.01 ( times_times_int
% 12.45/7.01 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 12.45/7.01 @ ( produc27273713700761075at_nat
% 12.45/7.01 @ ^ [X: nat,Y: nat] :
% 12.45/7.01 ( produc2626176000494625587at_nat
% 12.45/7.01 @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U4 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U4 ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10087_eventually__prod__sequentially,axiom,
% 12.45/7.01 ! [P: product_prod_nat_nat > $o] :
% 12.45/7.01 ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 12.45/7.01 = ( ? [N8: nat] :
% 12.45/7.01 ! [M6: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ N8 @ M6 )
% 12.45/7.01 => ! [N2: nat] :
% 12.45/7.01 ( ( ord_less_eq_nat @ N8 @ N2 )
% 12.45/7.01 => ( P @ ( product_Pair_nat_nat @ N2 @ M6 ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10088_less__eq,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 12.45/7.01 = ( ord_less_nat @ M @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10089_numeral__le__enat__iff,axiom,
% 12.45/7.01 ! [M: num,N: nat] :
% 12.45/7.01 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 12.45/7.01 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10090_enat_Oinject,axiom,
% 12.45/7.01 ! [Nat: nat,Nat2: nat] :
% 12.45/7.01 ( ( ( extended_enat2 @ Nat )
% 12.45/7.01 = ( extended_enat2 @ Nat2 ) )
% 12.45/7.01 = ( Nat = Nat2 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10091_enat__ord__simps_I2_J,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 12.45/7.01 = ( ord_less_nat @ M @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10092_enat__ord__simps_I1_J,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 12.45/7.01 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10093_plus__enat__simps_I1_J,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 12.45/7.01 = ( extended_enat2 @ ( plus_plus_nat @ M @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10094_idiff__enat__0__right,axiom,
% 12.45/7.01 ! [N: extended_enat] :
% 12.45/7.01 ( ( minus_3235023915231533773d_enat @ N @ ( extended_enat2 @ zero_zero_nat ) )
% 12.45/7.01 = N ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10095_idiff__enat__0,axiom,
% 12.45/7.01 ! [N: extended_enat] :
% 12.45/7.01 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N )
% 12.45/7.01 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10096_idiff__enat__enat,axiom,
% 12.45/7.01 ! [A: nat,B2: nat] :
% 12.45/7.01 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ ( extended_enat2 @ B2 ) )
% 12.45/7.01 = ( extended_enat2 @ ( minus_minus_nat @ A @ B2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10097_times__enat__simps_I1_J,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 12.45/7.01 = ( extended_enat2 @ ( times_times_nat @ M @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10098_max__enat__simps_I1_J,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( ord_ma741700101516333627d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 12.45/7.01 = ( extended_enat2 @ ( ord_max_nat @ M @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10099_min__enat__simps_I1_J,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( ord_mi8085742599997312461d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 12.45/7.01 = ( extended_enat2 @ ( ord_min_nat @ M @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10100_numeral__less__enat__iff,axiom,
% 12.45/7.01 ! [M: num,N: nat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 12.45/7.01 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10101_zero__enat__def,axiom,
% 12.45/7.01 ( zero_z5237406670263579293d_enat
% 12.45/7.01 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10102_enat__0__iff_I1_J,axiom,
% 12.45/7.01 ! [X4: nat] :
% 12.45/7.01 ( ( ( extended_enat2 @ X4 )
% 12.45/7.01 = zero_z5237406670263579293d_enat )
% 12.45/7.01 = ( X4 = zero_zero_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10103_enat__0__iff_I2_J,axiom,
% 12.45/7.01 ! [X4: nat] :
% 12.45/7.01 ( ( zero_z5237406670263579293d_enat
% 12.45/7.01 = ( extended_enat2 @ X4 ) )
% 12.45/7.01 = ( X4 = zero_zero_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10104_of__nat__eq__enat,axiom,
% 12.45/7.01 semiri4216267220026989637d_enat = extended_enat2 ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10105_enat__iless,axiom,
% 12.45/7.01 ! [N: extended_enat,M: nat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
% 12.45/7.01 => ? [K2: nat] :
% 12.45/7.01 ( N
% 12.45/7.01 = ( extended_enat2 @ K2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10106_less__enatE,axiom,
% 12.45/7.01 ! [N: extended_enat,M: nat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
% 12.45/7.01 => ~ ! [K2: nat] :
% 12.45/7.01 ( ( N
% 12.45/7.01 = ( extended_enat2 @ K2 ) )
% 12.45/7.01 => ~ ( ord_less_nat @ K2 @ M ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10107_one__enat__def,axiom,
% 12.45/7.01 ( one_on7984719198319812577d_enat
% 12.45/7.01 = ( extended_enat2 @ one_one_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10108_enat__1__iff_I1_J,axiom,
% 12.45/7.01 ! [X4: nat] :
% 12.45/7.01 ( ( ( extended_enat2 @ X4 )
% 12.45/7.01 = one_on7984719198319812577d_enat )
% 12.45/7.01 = ( X4 = one_one_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10109_enat__1__iff_I2_J,axiom,
% 12.45/7.01 ! [X4: nat] :
% 12.45/7.01 ( ( one_on7984719198319812577d_enat
% 12.45/7.01 = ( extended_enat2 @ X4 ) )
% 12.45/7.01 = ( X4 = one_one_nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10110_numeral__eq__enat,axiom,
% 12.45/7.01 ( numera1916890842035813515d_enat
% 12.45/7.01 = ( ^ [K3: num] : ( extended_enat2 @ ( numeral_numeral_nat @ K3 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10111_Suc__ile__eq,axiom,
% 12.45/7.01 ! [M: nat,N: extended_enat] :
% 12.45/7.01 ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
% 12.45/7.01 = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10112_finite__enat__bounded,axiom,
% 12.45/7.01 ! [A3: set_Extended_enat,N: nat] :
% 12.45/7.01 ( ! [Y4: extended_enat] :
% 12.45/7.01 ( ( member_Extended_enat @ Y4 @ A3 )
% 12.45/7.01 => ( ord_le2932123472753598470d_enat @ Y4 @ ( extended_enat2 @ N ) ) )
% 12.45/7.01 => ( finite4001608067531595151d_enat @ A3 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10113_enat__ile,axiom,
% 12.45/7.01 ! [N: extended_enat,M: nat] :
% 12.45/7.01 ( ( ord_le2932123472753598470d_enat @ N @ ( extended_enat2 @ M ) )
% 12.45/7.01 => ? [K2: nat] :
% 12.45/7.01 ( N
% 12.45/7.01 = ( extended_enat2 @ K2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10114_pred__nat__trancl__eq__le,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 12.45/7.01 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10115_iadd__le__enat__iff,axiom,
% 12.45/7.01 ! [X4: extended_enat,Y3: extended_enat,N: nat] :
% 12.45/7.01 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y3 ) @ ( extended_enat2 @ N ) )
% 12.45/7.01 = ( ? [Y9: nat,X9: nat] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( extended_enat2 @ X9 ) )
% 12.45/7.01 & ( Y3
% 12.45/7.01 = ( extended_enat2 @ Y9 ) )
% 12.45/7.01 & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y9 ) @ N ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10116_elimnum,axiom,
% 12.45/7.01 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 12.45/7.01 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 12.45/7.01 => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.45/7.01 = ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10117_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
% 12.45/7.01 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,L2: nat] :
% 12.45/7.01 ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ L2 ) )
% 12.45/7.01 = ( vEBT_Node @ Info @ Deg
% 12.45/7.01 @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.01 @ ( map_VE8901447254227204932T_VEBT
% 12.45/7.01 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.01 @ TreeList2 ) )
% 12.45/7.01 @ ( vEBT_VEBT_elim_dead @ Summary @ ( extended_enat2 @ ( divide_divide_nat @ L2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10118_VEBT__internal_Oelim__dead_Oelims,axiom,
% 12.45/7.01 ! [X4: vEBT_VEBT,Xa: extended_enat,Y3: vEBT_VEBT] :
% 12.45/7.01 ( ( ( vEBT_VEBT_elim_dead @ X4 @ Xa )
% 12.45/7.01 = Y3 )
% 12.45/7.01 => ( ! [A5: $o,B5: $o] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.01 => ( Y3
% 12.45/7.01 != ( vEBT_Leaf @ A5 @ B5 ) ) )
% 12.45/7.01 => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/7.01 => ( ( Xa = extend5688581933313929465d_enat )
% 12.45/7.01 => ( Y3
% 12.45/7.01 != ( vEBT_Node @ Info2 @ Deg2
% 12.45/7.01 @ ( map_VE8901447254227204932T_VEBT
% 12.45/7.01 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.01 @ TreeList3 )
% 12.45/7.01 @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) ) ) )
% 12.45/7.01 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/7.01 => ! [L3: nat] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( extended_enat2 @ L3 ) )
% 12.45/7.01 => ( Y3
% 12.45/7.01 != ( vEBT_Node @ Info2 @ Deg2
% 12.45/7.01 @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.01 @ ( map_VE8901447254227204932T_VEBT
% 12.45/7.01 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.01 @ TreeList3 ) )
% 12.45/7.01 @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10119_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
% 12.45/7.01 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 12.45/7.01 ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
% 12.45/7.01 = ( vEBT_Node @ Info @ Deg
% 12.45/7.01 @ ( map_VE8901447254227204932T_VEBT
% 12.45/7.01 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.01 @ TreeList2 )
% 12.45/7.01 @ ( vEBT_VEBT_elim_dead @ Summary @ extend5688581933313929465d_enat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10120_elimcomplete,axiom,
% 12.45/7.01 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 12.45/7.01 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 12.45/7.01 => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
% 12.45/7.01 = ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10121_not__infinity__eq,axiom,
% 12.45/7.01 ! [X4: extended_enat] :
% 12.45/7.01 ( ( X4 != extend5688581933313929465d_enat )
% 12.45/7.01 = ( ? [I2: nat] :
% 12.45/7.01 ( X4
% 12.45/7.01 = ( extended_enat2 @ I2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10122_not__enat__eq,axiom,
% 12.45/7.01 ! [X4: extended_enat] :
% 12.45/7.01 ( ( ! [Y: nat] :
% 12.45/7.01 ( X4
% 12.45/7.01 != ( extended_enat2 @ Y ) ) )
% 12.45/7.01 = ( X4 = extend5688581933313929465d_enat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10123_enat__ord__simps_I4_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ Q2 @ extend5688581933313929465d_enat )
% 12.45/7.01 = ( Q2 != extend5688581933313929465d_enat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10124_enat__ord__simps_I6_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] :
% 12.45/7.01 ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ Q2 ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10125_plus__enat__simps_I2_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] :
% 12.45/7.01 ( ( plus_p3455044024723400733d_enat @ extend5688581933313929465d_enat @ Q2 )
% 12.45/7.01 = extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10126_plus__enat__simps_I3_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] :
% 12.45/7.01 ( ( plus_p3455044024723400733d_enat @ Q2 @ extend5688581933313929465d_enat )
% 12.45/7.01 = extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10127_enat__ord__code_I3_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] : ( ord_le2932123472753598470d_enat @ Q2 @ extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10128_enat__ord__simps_I5_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] :
% 12.45/7.01 ( ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ Q2 )
% 12.45/7.01 = ( Q2 = extend5688581933313929465d_enat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10129_idiff__infinity,axiom,
% 12.45/7.01 ! [N: extended_enat] :
% 12.45/7.01 ( ( minus_3235023915231533773d_enat @ extend5688581933313929465d_enat @ N )
% 12.45/7.01 = extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10130_times__enat__simps_I2_J,axiom,
% 12.45/7.01 ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ extend5688581933313929465d_enat )
% 12.45/7.01 = extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10131_max__enat__simps_I4_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] :
% 12.45/7.01 ( ( ord_ma741700101516333627d_enat @ Q2 @ extend5688581933313929465d_enat )
% 12.45/7.01 = extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10132_max__enat__simps_I5_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] :
% 12.45/7.01 ( ( ord_ma741700101516333627d_enat @ extend5688581933313929465d_enat @ Q2 )
% 12.45/7.01 = extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10133_min__enat__simps_I4_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] :
% 12.45/7.01 ( ( ord_mi8085742599997312461d_enat @ Q2 @ extend5688581933313929465d_enat )
% 12.45/7.01 = Q2 ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10134_min__enat__simps_I5_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] :
% 12.45/7.01 ( ( ord_mi8085742599997312461d_enat @ extend5688581933313929465d_enat @ Q2 )
% 12.45/7.01 = Q2 ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10135_idiff__self,axiom,
% 12.45/7.01 ! [N: extended_enat] :
% 12.45/7.01 ( ( N != extend5688581933313929465d_enat )
% 12.45/7.01 => ( ( minus_3235023915231533773d_enat @ N @ N )
% 12.45/7.01 = zero_z5237406670263579293d_enat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10136_add__diff__cancel__enat,axiom,
% 12.45/7.01 ! [X4: extended_enat,Y3: extended_enat] :
% 12.45/7.01 ( ( X4 != extend5688581933313929465d_enat )
% 12.45/7.01 => ( ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y3 ) @ X4 )
% 12.45/7.01 = Y3 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10137_idiff__infinity__right,axiom,
% 12.45/7.01 ! [A: nat] :
% 12.45/7.01 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ extend5688581933313929465d_enat )
% 12.45/7.01 = zero_z5237406670263579293d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10138_times__enat__simps_I3_J,axiom,
% 12.45/7.01 ! [N: nat] :
% 12.45/7.01 ( ( ( N = zero_zero_nat )
% 12.45/7.01 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 12.45/7.01 = zero_z5237406670263579293d_enat ) )
% 12.45/7.01 & ( ( N != zero_zero_nat )
% 12.45/7.01 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 12.45/7.01 = extend5688581933313929465d_enat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10139_times__enat__simps_I4_J,axiom,
% 12.45/7.01 ! [M: nat] :
% 12.45/7.01 ( ( ( M = zero_zero_nat )
% 12.45/7.01 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 12.45/7.01 = zero_z5237406670263579293d_enat ) )
% 12.45/7.01 & ( ( M != zero_zero_nat )
% 12.45/7.01 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 12.45/7.01 = extend5688581933313929465d_enat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10140_Sup__enat__def,axiom,
% 12.45/7.01 ( comple4398354569131411667d_enat
% 12.45/7.01 = ( ^ [A6: set_Extended_enat] : ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ zero_z5237406670263579293d_enat @ ( if_Extended_enat @ ( finite4001608067531595151d_enat @ A6 ) @ ( lattic921264341876707157d_enat @ A6 ) @ extend5688581933313929465d_enat ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10141_Inf__enat__def,axiom,
% 12.45/7.01 ( comple2295165028678016749d_enat
% 12.45/7.01 = ( ^ [A6: set_Extended_enat] :
% 12.45/7.01 ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ extend5688581933313929465d_enat
% 12.45/7.01 @ ( ord_Le1955565732374568822d_enat
% 12.45/7.01 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10142_imult__is__infinity,axiom,
% 12.45/7.01 ! [A: extended_enat,B2: extended_enat] :
% 12.45/7.01 ( ( ( times_7803423173614009249d_enat @ A @ B2 )
% 12.45/7.01 = extend5688581933313929465d_enat )
% 12.45/7.01 = ( ( ( A = extend5688581933313929465d_enat )
% 12.45/7.01 & ( B2 != zero_z5237406670263579293d_enat ) )
% 12.45/7.01 | ( ( B2 = extend5688581933313929465d_enat )
% 12.45/7.01 & ( A != zero_z5237406670263579293d_enat ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10143_infinity__ne__i0,axiom,
% 12.45/7.01 extend5688581933313929465d_enat != zero_z5237406670263579293d_enat ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10144_enat__add__left__cancel__less,axiom,
% 12.45/7.01 ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
% 12.45/7.01 = ( ( A != extend5688581933313929465d_enat )
% 12.45/7.01 & ( ord_le72135733267957522d_enat @ B2 @ C ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10145_infinity__ne__i1,axiom,
% 12.45/7.01 extend5688581933313929465d_enat != one_on7984719198319812577d_enat ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10146_plus__eq__infty__iff__enat,axiom,
% 12.45/7.01 ! [M: extended_enat,N: extended_enat] :
% 12.45/7.01 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 12.45/7.01 = extend5688581933313929465d_enat )
% 12.45/7.01 = ( ( M = extend5688581933313929465d_enat )
% 12.45/7.01 | ( N = extend5688581933313929465d_enat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10147_enat__add__left__cancel,axiom,
% 12.45/7.01 ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
% 12.45/7.01 ( ( ( plus_p3455044024723400733d_enat @ A @ B2 )
% 12.45/7.01 = ( plus_p3455044024723400733d_enat @ A @ C ) )
% 12.45/7.01 = ( ( A = extend5688581933313929465d_enat )
% 12.45/7.01 | ( B2 = C ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10148_numeral__ne__infinity,axiom,
% 12.45/7.01 ! [K: num] :
% 12.45/7.01 ( ( numera1916890842035813515d_enat @ K )
% 12.45/7.01 != extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10149_top__enat__def,axiom,
% 12.45/7.01 top_to3028658606643905974d_enat = extend5688581933313929465d_enat ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10150_enat__ord__simps_I3_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] : ( ord_le2932123472753598470d_enat @ Q2 @ extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10151_enat__add__left__cancel__le,axiom,
% 12.45/7.01 ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
% 12.45/7.01 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
% 12.45/7.01 = ( ( A = extend5688581933313929465d_enat )
% 12.45/7.01 | ( ord_le2932123472753598470d_enat @ B2 @ C ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10152_enat__ord__code_I5_J,axiom,
% 12.45/7.01 ! [N: nat] :
% 12.45/7.01 ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10153_infinity__ileE,axiom,
% 12.45/7.01 ! [M: nat] :
% 12.45/7.01 ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10154_enat__ex__split,axiom,
% 12.45/7.01 ( ( ^ [P5: extended_enat > $o] :
% 12.45/7.01 ? [X7: extended_enat] : ( P5 @ X7 ) )
% 12.45/7.01 = ( ^ [P6: extended_enat > $o] :
% 12.45/7.01 ( ( P6 @ extend5688581933313929465d_enat )
% 12.45/7.01 | ? [X: nat] : ( P6 @ ( extended_enat2 @ X ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10155_enat3__cases,axiom,
% 12.45/7.01 ! [Y3: extended_enat,Ya: extended_enat,Yb: extended_enat] :
% 12.45/7.01 ( ( ? [Nat3: nat] :
% 12.45/7.01 ( Y3
% 12.45/7.01 = ( extended_enat2 @ Nat3 ) )
% 12.45/7.01 => ( ? [Nata: nat] :
% 12.45/7.01 ( Ya
% 12.45/7.01 = ( extended_enat2 @ Nata ) )
% 12.45/7.01 => ! [Natb: nat] :
% 12.45/7.01 ( Yb
% 12.45/7.01 != ( extended_enat2 @ Natb ) ) ) )
% 12.45/7.01 => ( ( ? [Nat3: nat] :
% 12.45/7.01 ( Y3
% 12.45/7.01 = ( extended_enat2 @ Nat3 ) )
% 12.45/7.01 => ( ? [Nata: nat] :
% 12.45/7.01 ( Ya
% 12.45/7.01 = ( extended_enat2 @ Nata ) )
% 12.45/7.01 => ( Yb != extend5688581933313929465d_enat ) ) )
% 12.45/7.01 => ( ( ? [Nat3: nat] :
% 12.45/7.01 ( Y3
% 12.45/7.01 = ( extended_enat2 @ Nat3 ) )
% 12.45/7.01 => ( ( Ya = extend5688581933313929465d_enat )
% 12.45/7.01 => ! [Nata: nat] :
% 12.45/7.01 ( Yb
% 12.45/7.01 != ( extended_enat2 @ Nata ) ) ) )
% 12.45/7.01 => ( ( ? [Nat3: nat] :
% 12.45/7.01 ( Y3
% 12.45/7.01 = ( extended_enat2 @ Nat3 ) )
% 12.45/7.01 => ( ( Ya = extend5688581933313929465d_enat )
% 12.45/7.01 => ( Yb != extend5688581933313929465d_enat ) ) )
% 12.45/7.01 => ( ( ( Y3 = extend5688581933313929465d_enat )
% 12.45/7.01 => ( ? [Nat3: nat] :
% 12.45/7.01 ( Ya
% 12.45/7.01 = ( extended_enat2 @ Nat3 ) )
% 12.45/7.01 => ! [Nata: nat] :
% 12.45/7.01 ( Yb
% 12.45/7.01 != ( extended_enat2 @ Nata ) ) ) )
% 12.45/7.01 => ( ( ( Y3 = extend5688581933313929465d_enat )
% 12.45/7.01 => ( ? [Nat3: nat] :
% 12.45/7.01 ( Ya
% 12.45/7.01 = ( extended_enat2 @ Nat3 ) )
% 12.45/7.01 => ( Yb != extend5688581933313929465d_enat ) ) )
% 12.45/7.01 => ( ( ( Y3 = extend5688581933313929465d_enat )
% 12.45/7.01 => ( ( Ya = extend5688581933313929465d_enat )
% 12.45/7.01 => ! [Nat3: nat] :
% 12.45/7.01 ( Yb
% 12.45/7.01 != ( extended_enat2 @ Nat3 ) ) ) )
% 12.45/7.01 => ~ ( ( Y3 = extend5688581933313929465d_enat )
% 12.45/7.01 => ( ( Ya = extend5688581933313929465d_enat )
% 12.45/7.01 => ( Yb != extend5688581933313929465d_enat ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10156_enat2__cases,axiom,
% 12.45/7.01 ! [Y3: extended_enat,Ya: extended_enat] :
% 12.45/7.01 ( ( ? [Nat3: nat] :
% 12.45/7.01 ( Y3
% 12.45/7.01 = ( extended_enat2 @ Nat3 ) )
% 12.45/7.01 => ! [Nata: nat] :
% 12.45/7.01 ( Ya
% 12.45/7.01 != ( extended_enat2 @ Nata ) ) )
% 12.45/7.01 => ( ( ? [Nat3: nat] :
% 12.45/7.01 ( Y3
% 12.45/7.01 = ( extended_enat2 @ Nat3 ) )
% 12.45/7.01 => ( Ya != extend5688581933313929465d_enat ) )
% 12.45/7.01 => ( ( ( Y3 = extend5688581933313929465d_enat )
% 12.45/7.01 => ! [Nat3: nat] :
% 12.45/7.01 ( Ya
% 12.45/7.01 != ( extended_enat2 @ Nat3 ) ) )
% 12.45/7.01 => ~ ( ( Y3 = extend5688581933313929465d_enat )
% 12.45/7.01 => ( Ya != extend5688581933313929465d_enat ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10157_enat_Oexhaust,axiom,
% 12.45/7.01 ! [Y3: extended_enat] :
% 12.45/7.01 ( ! [Nat3: nat] :
% 12.45/7.01 ( Y3
% 12.45/7.01 != ( extended_enat2 @ Nat3 ) )
% 12.45/7.01 => ( Y3 = extend5688581933313929465d_enat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10158_enat_Odistinct_I1_J,axiom,
% 12.45/7.01 ! [Nat: nat] :
% 12.45/7.01 ( ( extended_enat2 @ Nat )
% 12.45/7.01 != extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10159_enat_Odistinct_I2_J,axiom,
% 12.45/7.01 ! [Nat: nat] :
% 12.45/7.01 ( extend5688581933313929465d_enat
% 12.45/7.01 != ( extended_enat2 @ Nat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10160_infinity__ilessE,axiom,
% 12.45/7.01 ! [M: nat] :
% 12.45/7.01 ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10161_less__infinityE,axiom,
% 12.45/7.01 ! [N: extended_enat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ N @ extend5688581933313929465d_enat )
% 12.45/7.01 => ~ ! [K2: nat] :
% 12.45/7.01 ( N
% 12.45/7.01 != ( extended_enat2 @ K2 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10162_enat__ord__code_I4_J,axiom,
% 12.45/7.01 ! [M: nat] : ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10163_imult__infinity,axiom,
% 12.45/7.01 ! [N: extended_enat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 12.45/7.01 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ N )
% 12.45/7.01 = extend5688581933313929465d_enat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10164_imult__infinity__right,axiom,
% 12.45/7.01 ! [N: extended_enat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 12.45/7.01 => ( ( times_7803423173614009249d_enat @ N @ extend5688581933313929465d_enat )
% 12.45/7.01 = extend5688581933313929465d_enat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10165_plus__enat__def,axiom,
% 12.45/7.01 ( plus_p3455044024723400733d_enat
% 12.45/7.01 = ( ^ [M6: extended_enat,N2: extended_enat] :
% 12.45/7.01 ( extend3600170679010898289d_enat
% 12.45/7.01 @ ^ [O: nat] :
% 12.45/7.01 ( extend3600170679010898289d_enat
% 12.45/7.01 @ ^ [P3: nat] : ( extended_enat2 @ ( plus_plus_nat @ O @ P3 ) )
% 12.45/7.01 @ extend5688581933313929465d_enat
% 12.45/7.01 @ N2 )
% 12.45/7.01 @ extend5688581933313929465d_enat
% 12.45/7.01 @ M6 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10166_diff__enat__def,axiom,
% 12.45/7.01 ( minus_3235023915231533773d_enat
% 12.45/7.01 = ( ^ [A4: extended_enat,B3: extended_enat] :
% 12.45/7.01 ( extend3600170679010898289d_enat
% 12.45/7.01 @ ^ [X: nat] :
% 12.45/7.01 ( extend3600170679010898289d_enat
% 12.45/7.01 @ ^ [Y: nat] : ( extended_enat2 @ ( minus_minus_nat @ X @ Y ) )
% 12.45/7.01 @ zero_z5237406670263579293d_enat
% 12.45/7.01 @ B3 )
% 12.45/7.01 @ extend5688581933313929465d_enat
% 12.45/7.01 @ A4 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10167_times__enat__def,axiom,
% 12.45/7.01 ( times_7803423173614009249d_enat
% 12.45/7.01 = ( ^ [M6: extended_enat,N2: extended_enat] :
% 12.45/7.01 ( extend3600170679010898289d_enat
% 12.45/7.01 @ ^ [O: nat] :
% 12.45/7.01 ( extend3600170679010898289d_enat
% 12.45/7.01 @ ^ [P3: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P3 ) )
% 12.45/7.01 @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 12.45/7.01 @ N2 )
% 12.45/7.01 @ ( if_Extended_enat @ ( N2 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 12.45/7.01 @ M6 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10168_VEBT__internal_Oelim__dead_Opelims,axiom,
% 12.45/7.01 ! [X4: vEBT_VEBT,Xa: extended_enat,Y3: vEBT_VEBT] :
% 12.45/7.01 ( ( ( vEBT_VEBT_elim_dead @ X4 @ Xa )
% 12.45/7.01 = Y3 )
% 12.45/7.01 => ( ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ X4 @ Xa ) )
% 12.45/7.01 => ( ! [A5: $o,B5: $o] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( vEBT_Leaf @ A5 @ B5 ) )
% 12.45/7.01 => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
% 12.45/7.01 => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/7.01 => ( ( Xa = extend5688581933313929465d_enat )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( vEBT_Node @ Info2 @ Deg2
% 12.45/7.01 @ ( map_VE8901447254227204932T_VEBT
% 12.45/7.01 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.01 @ TreeList3 )
% 12.45/7.01 @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) )
% 12.45/7.01 => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ extend5688581933313929465d_enat ) ) ) ) )
% 12.45/7.01 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.45/7.01 ( ( X4
% 12.45/7.01 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
% 12.45/7.01 => ! [L3: nat] :
% 12.45/7.01 ( ( Xa
% 12.45/7.01 = ( extended_enat2 @ L3 ) )
% 12.45/7.01 => ( ( Y3
% 12.45/7.01 = ( vEBT_Node @ Info2 @ Deg2
% 12.45/7.01 @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.45/7.01 @ ( map_VE8901447254227204932T_VEBT
% 12.45/7.01 @ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.45/7.01 @ TreeList3 ) )
% 12.45/7.01 @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 12.45/7.01 => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ ( extended_enat2 @ L3 ) ) ) ) ) ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10169_the__enat_Osimps,axiom,
% 12.45/7.01 ! [N: nat] :
% 12.45/7.01 ( ( extended_the_enat @ ( extended_enat2 @ N ) )
% 12.45/7.01 = N ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10170_eSuc__Max,axiom,
% 12.45/7.01 ! [A3: set_Extended_enat] :
% 12.45/7.01 ( ( finite4001608067531595151d_enat @ A3 )
% 12.45/7.01 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 12.45/7.01 => ( ( extended_eSuc @ ( lattic921264341876707157d_enat @ A3 ) )
% 12.45/7.01 = ( lattic921264341876707157d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10171_eSuc__inject,axiom,
% 12.45/7.01 ! [M: extended_enat,N: extended_enat] :
% 12.45/7.01 ( ( ( extended_eSuc @ M )
% 12.45/7.01 = ( extended_eSuc @ N ) )
% 12.45/7.01 = ( M = N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10172_eSuc__infinity,axiom,
% 12.45/7.01 ( ( extended_eSuc @ extend5688581933313929465d_enat )
% 12.45/7.01 = extend5688581933313929465d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10173_eSuc__mono,axiom,
% 12.45/7.01 ! [N: extended_enat,M: extended_enat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 12.45/7.01 = ( ord_le72135733267957522d_enat @ N @ M ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10174_eSuc__ile__mono,axiom,
% 12.45/7.01 ! [N: extended_enat,M: extended_enat] :
% 12.45/7.01 ( ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 12.45/7.01 = ( ord_le2932123472753598470d_enat @ N @ M ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10175_eSuc__minus__eSuc,axiom,
% 12.45/7.01 ! [N: extended_enat,M: extended_enat] :
% 12.45/7.01 ( ( minus_3235023915231533773d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 12.45/7.01 = ( minus_3235023915231533773d_enat @ N @ M ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10176_iless__eSuc0,axiom,
% 12.45/7.01 ! [N: extended_enat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ N @ ( extended_eSuc @ zero_z5237406670263579293d_enat ) )
% 12.45/7.01 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10177_eSuc__minus__1,axiom,
% 12.45/7.01 ! [N: extended_enat] :
% 12.45/7.01 ( ( minus_3235023915231533773d_enat @ ( extended_eSuc @ N ) @ one_on7984719198319812577d_enat )
% 12.45/7.01 = N ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10178_eSuc__numeral,axiom,
% 12.45/7.01 ! [K: num] :
% 12.45/7.01 ( ( extended_eSuc @ ( numera1916890842035813515d_enat @ K ) )
% 12.45/7.01 = ( numera1916890842035813515d_enat @ ( plus_plus_num @ K @ one ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10179_iless__Suc__eq,axiom,
% 12.45/7.01 ! [M: nat,N: extended_enat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N ) )
% 12.45/7.01 = ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10180_one__eSuc,axiom,
% 12.45/7.01 ( one_on7984719198319812577d_enat
% 12.45/7.01 = ( extended_eSuc @ zero_z5237406670263579293d_enat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10181_i0__iless__eSuc,axiom,
% 12.45/7.01 ! [N: extended_enat] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( extended_eSuc @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10182_zero__ne__eSuc,axiom,
% 12.45/7.01 ! [N: extended_enat] :
% 12.45/7.01 ( zero_z5237406670263579293d_enat
% 12.45/7.01 != ( extended_eSuc @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10183_eSuc__plus__1,axiom,
% 12.45/7.01 ( extended_eSuc
% 12.45/7.01 = ( ^ [N2: extended_enat] : ( plus_p3455044024723400733d_enat @ N2 @ one_on7984719198319812577d_enat ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10184_plus__1__eSuc_I1_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] :
% 12.45/7.01 ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ Q2 )
% 12.45/7.01 = ( extended_eSuc @ Q2 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10185_plus__1__eSuc_I2_J,axiom,
% 12.45/7.01 ! [Q2: extended_enat] :
% 12.45/7.01 ( ( plus_p3455044024723400733d_enat @ Q2 @ one_on7984719198319812577d_enat )
% 12.45/7.01 = ( extended_eSuc @ Q2 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10186_mono__eSuc,axiom,
% 12.45/7.01 order_4130057895858720880d_enat @ extended_eSuc ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10187_iadd__Suc__right,axiom,
% 12.45/7.01 ! [M: extended_enat,N: extended_enat] :
% 12.45/7.01 ( ( plus_p3455044024723400733d_enat @ M @ ( extended_eSuc @ N ) )
% 12.45/7.01 = ( extended_eSuc @ ( plus_p3455044024723400733d_enat @ M @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10188_iadd__Suc,axiom,
% 12.45/7.01 ! [M: extended_enat,N: extended_enat] :
% 12.45/7.01 ( ( plus_p3455044024723400733d_enat @ ( extended_eSuc @ M ) @ N )
% 12.45/7.01 = ( extended_eSuc @ ( plus_p3455044024723400733d_enat @ M @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10189_eSuc__max,axiom,
% 12.45/7.01 ! [X4: extended_enat,Y3: extended_enat] :
% 12.45/7.01 ( ( extended_eSuc @ ( ord_ma741700101516333627d_enat @ X4 @ Y3 ) )
% 12.45/7.01 = ( ord_ma741700101516333627d_enat @ ( extended_eSuc @ X4 ) @ ( extended_eSuc @ Y3 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10190_mult__eSuc__right,axiom,
% 12.45/7.01 ! [M: extended_enat,N: extended_enat] :
% 12.45/7.01 ( ( times_7803423173614009249d_enat @ M @ ( extended_eSuc @ N ) )
% 12.45/7.01 = ( plus_p3455044024723400733d_enat @ M @ ( times_7803423173614009249d_enat @ M @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10191_mult__eSuc,axiom,
% 12.45/7.01 ! [M: extended_enat,N: extended_enat] :
% 12.45/7.01 ( ( times_7803423173614009249d_enat @ ( extended_eSuc @ M ) @ N )
% 12.45/7.01 = ( plus_p3455044024723400733d_enat @ N @ ( times_7803423173614009249d_enat @ M @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10192_ileI1,axiom,
% 12.45/7.01 ! [M: extended_enat,N: extended_enat] :
% 12.45/7.01 ( ( ord_le72135733267957522d_enat @ M @ N )
% 12.45/7.01 => ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ M ) @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10193_ile__eSuc,axiom,
% 12.45/7.01 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ N @ ( extended_eSuc @ N ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10194_not__eSuc__ilei0,axiom,
% 12.45/7.01 ! [N: extended_enat] :
% 12.45/7.01 ~ ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ zero_z5237406670263579293d_enat ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10195_eSuc__enat,axiom,
% 12.45/7.01 ! [N: nat] :
% 12.45/7.01 ( ( extended_eSuc @ ( extended_enat2 @ N ) )
% 12.45/7.01 = ( extended_enat2 @ ( suc @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10196_eSuc__enat__iff,axiom,
% 12.45/7.01 ! [X4: extended_enat,Y3: nat] :
% 12.45/7.01 ( ( ( extended_eSuc @ X4 )
% 12.45/7.01 = ( extended_enat2 @ Y3 ) )
% 12.45/7.01 = ( ? [N2: nat] :
% 12.45/7.01 ( ( Y3
% 12.45/7.01 = ( suc @ N2 ) )
% 12.45/7.01 & ( X4
% 12.45/7.01 = ( extended_enat2 @ N2 ) ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10197_enat__eSuc__iff,axiom,
% 12.45/7.01 ! [Y3: nat,X4: extended_enat] :
% 12.45/7.01 ( ( ( extended_enat2 @ Y3 )
% 12.45/7.01 = ( extended_eSuc @ X4 ) )
% 12.45/7.01 = ( ? [N2: nat] :
% 12.45/7.01 ( ( Y3
% 12.45/7.01 = ( suc @ N2 ) )
% 12.45/7.01 & ( ( extended_enat2 @ N2 )
% 12.45/7.01 = X4 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10198_eSuc__Sup,axiom,
% 12.45/7.01 ! [A3: set_Extended_enat] :
% 12.45/7.01 ( ( A3 != bot_bo7653980558646680370d_enat )
% 12.45/7.01 => ( ( extended_eSuc @ ( comple4398354569131411667d_enat @ A3 ) )
% 12.45/7.01 = ( comple4398354569131411667d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10199_eSuc__def,axiom,
% 12.45/7.01 ( extended_eSuc
% 12.45/7.01 = ( extend3600170679010898289d_enat
% 12.45/7.01 @ ^ [N2: nat] : ( extended_enat2 @ ( suc @ N2 ) )
% 12.45/7.01 @ extend5688581933313929465d_enat ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10200_less__than__iff,axiom,
% 12.45/7.01 ! [X4: nat,Y3: nat] :
% 12.45/7.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ less_than )
% 12.45/7.01 = ( ord_less_nat @ X4 @ Y3 ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10201_Quotient__real,axiom,
% 12.45/7.01 quotie3684837364556693515t_real @ realrel @ real2 @ rep_real @ cr_real ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10202_lcm__code__integer,axiom,
% 12.45/7.01 ( gcd_lcm_Code_integer
% 12.45/7.01 = ( ^ [A4: code_integer,B3: code_integer] : ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A4 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( gcd_gcd_Code_integer @ A4 @ B3 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10203_lcm__1__iff__nat,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( ( gcd_lcm_nat @ M @ N )
% 12.45/7.01 = ( suc @ zero_zero_nat ) )
% 12.45/7.01 = ( ( M
% 12.45/7.01 = ( suc @ zero_zero_nat ) )
% 12.45/7.01 & ( N
% 12.45/7.01 = ( suc @ zero_zero_nat ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10204_prod__gcd__lcm__int,axiom,
% 12.45/7.01 ! [M: int,N: int] :
% 12.45/7.01 ( ( times_times_int @ ( abs_abs_int @ M ) @ ( abs_abs_int @ N ) )
% 12.45/7.01 = ( times_times_int @ ( gcd_gcd_int @ M @ N ) @ ( gcd_lcm_int @ M @ N ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10205_prod__gcd__lcm__nat,axiom,
% 12.45/7.01 ( times_times_nat
% 12.45/7.01 = ( ^ [M6: nat,N2: nat] : ( times_times_nat @ ( gcd_gcd_nat @ M6 @ N2 ) @ ( gcd_lcm_nat @ M6 @ N2 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10206_lcm__nat__def,axiom,
% 12.45/7.01 ( gcd_lcm_nat
% 12.45/7.01 = ( ^ [X: nat,Y: nat] : ( divide_divide_nat @ ( times_times_nat @ X @ Y ) @ ( gcd_gcd_nat @ X @ Y ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10207_lcm__pos__nat,axiom,
% 12.45/7.01 ! [M: nat,N: nat] :
% 12.45/7.01 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.45/7.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.45/7.01 => ( ord_less_nat @ zero_zero_nat @ ( gcd_lcm_nat @ M @ N ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10208_lcm__pos__int,axiom,
% 12.45/7.01 ! [M: int,N: int] :
% 12.45/7.01 ( ( M != zero_zero_int )
% 12.45/7.01 => ( ( N != zero_zero_int )
% 12.45/7.01 => ( ord_less_int @ zero_zero_int @ ( gcd_lcm_int @ M @ N ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(fact_10209_lcm__altdef__int,axiom,
% 12.45/7.01 ( gcd_lcm_int
% 12.45/7.01 = ( ^ [A4: int,B3: int] : ( divide_divide_int @ ( times_times_int @ ( abs_abs_int @ A4 ) @ ( abs_abs_int @ B3 ) ) @ ( gcd_gcd_int @ A4 @ B3 ) ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 12.45/7.01 ! [X4: int,Y3: int] :
% 12.45/7.01 ( ( if_int @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 12.45/7.01 ! [X4: int,Y3: int] :
% 12.45/7.01 ( ( if_int @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 12.45/7.01 ! [X4: nat,Y3: nat] :
% 12.45/7.01 ( ( if_nat @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 12.45/7.01 ! [X4: nat,Y3: nat] :
% 12.45/7.01 ( ( if_nat @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 12.45/7.01 ! [X4: num,Y3: num] :
% 12.45/7.01 ( ( if_num @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 12.45/7.01 ! [X4: num,Y3: num] :
% 12.45/7.01 ( ( if_num @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 12.45/7.01 ! [X4: rat,Y3: rat] :
% 12.45/7.01 ( ( if_rat @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 12.45/7.01 ! [X4: rat,Y3: rat] :
% 12.45/7.01 ( ( if_rat @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 12.45/7.01 ! [X4: real,Y3: real] :
% 12.45/7.01 ( ( if_real @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 12.45/7.01 ! [X4: real,Y3: real] :
% 12.45/7.01 ( ( if_real @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 12.45/7.01 ! [P: real > $o] :
% 12.45/7.01 ( ( P @ ( fChoice_real @ P ) )
% 12.45/7.01 = ( ? [X6: real] : ( P @ X6 ) ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 12.45/7.01 ! [X4: complex,Y3: complex] :
% 12.45/7.01 ( ( if_complex @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 12.45/7.01 ! [X4: complex,Y3: complex] :
% 12.45/7.01 ( ( if_complex @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 12.45/7.01 ! [X4: extended_enat,Y3: extended_enat] :
% 12.45/7.01 ( ( if_Extended_enat @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 12.45/7.01 ! [X4: extended_enat,Y3: extended_enat] :
% 12.45/7.01 ( ( if_Extended_enat @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 12.45/7.01 ! [X4: code_integer,Y3: code_integer] :
% 12.45/7.01 ( ( if_Code_integer @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 12.45/7.01 ! [X4: code_integer,Y3: code_integer] :
% 12.45/7.01 ( ( if_Code_integer @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 12.45/7.01 ! [X4: set_int,Y3: set_int] :
% 12.45/7.01 ( ( if_set_int @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 12.45/7.01 ! [X4: set_int,Y3: set_int] :
% 12.45/7.01 ( ( if_set_int @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 12.45/7.01 ! [X4: vEBT_VEBT,Y3: vEBT_VEBT] :
% 12.45/7.01 ( ( if_VEBT_VEBT @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 12.45/7.01 ! [X4: vEBT_VEBT,Y3: vEBT_VEBT] :
% 12.45/7.01 ( ( if_VEBT_VEBT @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 12.45/7.01 ! [X4: list_int,Y3: list_int] :
% 12.45/7.01 ( ( if_list_int @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 12.45/7.01 ! [X4: list_int,Y3: list_int] :
% 12.45/7.01 ( ( if_list_int @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 12.45/7.01 ! [X4: list_nat,Y3: list_nat] :
% 12.45/7.01 ( ( if_list_nat @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 12.45/7.01 ! [X4: list_nat,Y3: list_nat] :
% 12.45/7.01 ( ( if_list_nat @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 12.45/7.01 ! [X4: int > int,Y3: int > int] :
% 12.45/7.01 ( ( if_int_int @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 12.45/7.01 ! [X4: int > int,Y3: int > int] :
% 12.45/7.01 ( ( if_int_int @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 12.45/7.01 ! [X4: nat > rat,Y3: nat > rat] :
% 12.45/7.01 ( ( if_nat_rat @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 12.45/7.01 ! [X4: nat > rat,Y3: nat > rat] :
% 12.45/7.01 ( ( if_nat_rat @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 12.45/7.01 ! [X4: option_nat,Y3: option_nat] :
% 12.45/7.01 ( ( if_option_nat @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 12.45/7.01 ! [X4: option_nat,Y3: option_nat] :
% 12.45/7.01 ( ( if_option_nat @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 12.45/7.01 ! [X4: option_num,Y3: option_num] :
% 12.45/7.01 ( ( if_option_num @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 12.45/7.01 ! [X4: option_num,Y3: option_num] :
% 12.45/7.01 ( ( if_option_num @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 12.45/7.01 ! [X4: product_prod_int_int,Y3: product_prod_int_int] :
% 12.45/7.01 ( ( if_Pro3027730157355071871nt_int @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 12.45/7.01 ! [X4: product_prod_int_int,Y3: product_prod_int_int] :
% 12.45/7.01 ( ( if_Pro3027730157355071871nt_int @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 12.45/7.01 ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 12.45/7.01 ( ( if_Pro6206227464963214023at_nat @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 12.45/7.01 ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 12.45/7.01 ( ( if_Pro6206227464963214023at_nat @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 12.45/7.01 ! [X4: nat > int > int,Y3: nat > int > int] :
% 12.45/7.01 ( ( if_nat_int_int @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 12.45/7.01 ! [X4: nat > int > int,Y3: nat > int > int] :
% 12.45/7.01 ( ( if_nat_int_int @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 12.45/7.01 ! [X4: nat > nat > nat,Y3: nat > nat > nat] :
% 12.45/7.01 ( ( if_nat_nat_nat @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 12.45/7.01 ! [X4: nat > nat > nat,Y3: nat > nat > nat] :
% 12.45/7.01 ( ( if_nat_nat_nat @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 12.45/7.01 ! [X4: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 12.45/7.01 ( ( if_Pro5737122678794959658eger_o @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 12.45/7.01 ! [X4: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 12.45/7.01 ( ( if_Pro5737122678794959658eger_o @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 12.45/7.01 ! [P: $o] :
% 12.45/7.01 ( ( P = $true )
% 12.45/7.01 | ( P = $false ) ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 12.45/7.01 ! [X4: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 12.45/7.01 ( ( if_Pro6119634080678213985nteger @ $false @ X4 @ Y3 )
% 12.45/7.01 = Y3 ) ).
% 12.45/7.01
% 12.45/7.01 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 12.45/7.01 ! [X4: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 12.45/7.01 ( ( if_Pro6119634080678213985nteger @ $true @ X4 @ Y3 )
% 12.45/7.01 = X4 ) ).
% 12.45/7.01
% 12.45/7.01 thf(conj_0,conjecture,
% 12.45/7.01 ( ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.45/7.01 = ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.45/7.01 ------------------------
% 12.46/7.43 % (21712)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_21372 for (2996ds/27Mi)
% 12.46/7.43 % (21714)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_21372 for (2996ds/2Mi)
% 12.46/7.43 % (21713)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_21372 for (2996ds/2Mi)
% 12.46/7.43 % (21710)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_21372 for (2996ds/183Mi)
% 12.46/7.43 % (21711)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_21372 for (2996ds/4Mi)
% 12.46/7.43 % (21715)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_21372 for (2996ds/275Mi)
% 12.46/7.43 % (21716)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_21372 for (2996ds/18Mi)
% 12.46/7.43 % (21714)Instruction limit reached!
% 12.46/7.43 % (21714)------------------------------
% 12.46/7.43 % (21714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.43 % (21714)Termination reason: Unknown
% 12.46/7.43 % (21714)Termination phase: shuffling
% 12.46/7.43
% 12.46/7.43 % (21714)Memory used [KB]: 12025
% 12.46/7.43 % (21714)Time elapsed: 0.003 s
% 12.46/7.43 % (21714)Instructions burned: 2 (million)
% 12.46/7.43 % (21714)------------------------------
% 12.46/7.43 % (21714)------------------------------
% 12.46/7.43 % (21713)Instruction limit reached!
% 12.46/7.43 % (21713)------------------------------
% 12.46/7.43 % (21713)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.43 % (21713)Termination reason: Unknown
% 12.46/7.43 % (21713)Termination phase: shuffling
% 12.46/7.43
% 12.46/7.43 % (21713)Memory used [KB]: 12025
% 12.46/7.43 % (21713)Time elapsed: 0.004 s
% 12.46/7.43 % (21713)Instructions burned: 2 (million)
% 12.46/7.43 % (21713)------------------------------
% 12.46/7.43 % (21713)------------------------------
% 12.46/7.43 % (21711)Instruction limit reached!
% 12.46/7.43 % (21711)------------------------------
% 12.46/7.43 % (21711)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.43 % (21711)Termination reason: Unknown
% 12.46/7.43 % (21711)Termination phase: shuffling
% 12.46/7.43
% 12.46/7.43 % (21711)Memory used [KB]: 12153
% 12.46/7.43 % (21711)Time elapsed: 0.007 s
% 12.46/7.43 % (21711)Instructions burned: 4 (million)
% 12.46/7.43 % (21711)------------------------------
% 12.46/7.43 % (21711)------------------------------
% 12.46/7.45 % (21717)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_21372 for (2995ds/3Mi)
% 12.46/7.45 % (21718)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_21372 for (2995ds/37Mi)
% 12.46/7.45 % (21716)Instruction limit reached!
% 12.46/7.45 % (21716)------------------------------
% 12.46/7.45 % (21716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.45 % (21716)Termination reason: Unknown
% 12.46/7.45 % (21716)Termination phase: shuffling
% 12.46/7.45
% 12.46/7.45 % (21716)Memory used [KB]: 12792
% 12.46/7.45 % (21716)Time elapsed: 0.023 s
% 12.46/7.45 % (21716)Instructions burned: 18 (million)
% 12.46/7.45 % (21716)------------------------------
% 12.46/7.45 % (21716)------------------------------
% 12.46/7.45 % (21717)Instruction limit reached!
% 12.46/7.45 % (21717)------------------------------
% 12.46/7.45 % (21717)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.45 % (21717)Termination reason: Unknown
% 12.46/7.45 % (21717)Termination phase: shuffling
% 12.46/7.45
% 12.46/7.45 % (21717)Memory used [KB]: 12025
% 12.46/7.45 % (21717)Time elapsed: 0.005 s
% 12.46/7.45 % (21717)Instructions burned: 3 (million)
% 12.46/7.45 % (21717)------------------------------
% 12.46/7.45 % (21717)------------------------------
% 12.46/7.45 % (21719)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_21372 for (2995ds/15Mi)
% 12.46/7.46 % (21712)Instruction limit reached!
% 12.46/7.46 % (21712)------------------------------
% 12.46/7.46 % (21712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.46 % (21712)Termination reason: Unknown
% 12.46/7.46 % (21712)Termination phase: shuffling
% 12.46/7.46
% 12.46/7.46 % (21712)Memory used [KB]: 12920
% 12.46/7.46 % (21712)Time elapsed: 0.029 s
% 12.46/7.46 % (21712)Instructions burned: 27 (million)
% 12.46/7.46 % (21712)------------------------------
% 12.46/7.46 % (21712)------------------------------
% 12.46/7.47 % (21720)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_21372 for (2995ds/3Mi)
% 12.46/7.47 % (21719)Instruction limit reached!
% 12.46/7.47 % (21719)------------------------------
% 12.46/7.47 % (21719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.47 % (21719)Termination reason: Unknown
% 12.46/7.47 % (21719)Termination phase: shuffling
% 12.46/7.47
% 12.46/7.47 % (21719)Memory used [KB]: 12792
% 12.46/7.47 % (21719)Time elapsed: 0.020 s
% 12.46/7.47 % (21719)Instructions burned: 15 (million)
% 12.46/7.47 % (21719)------------------------------
% 12.46/7.47 % (21719)------------------------------
% 12.46/7.47 % (21721)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_21372 for (2995ds/1041Mi)
% 12.46/7.47 % (21720)Instruction limit reached!
% 12.46/7.47 % (21720)------------------------------
% 12.46/7.47 % (21720)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.47 % (21720)Termination reason: Unknown
% 12.46/7.47 % (21720)Termination phase: shuffling
% 12.46/7.47
% 12.46/7.47 % (21720)Memory used [KB]: 12025
% 12.46/7.47 % (21720)Time elapsed: 0.005 s
% 12.46/7.47 % (21720)Instructions burned: 3 (million)
% 12.46/7.47 % (21720)------------------------------
% 12.46/7.47 % (21720)------------------------------
% 12.46/7.48 % (21722)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_21372 for (2995ds/7Mi)
% 12.46/7.48 % (21718)Instruction limit reached!
% 12.46/7.48 % (21718)------------------------------
% 12.46/7.48 % (21718)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.48 % (21718)Termination reason: Unknown
% 12.46/7.48 % (21718)Termination phase: shuffling
% 12.46/7.48
% 12.46/7.48 % (21718)Memory used [KB]: 13048
% 12.46/7.48 % (21718)Time elapsed: 0.033 s
% 12.46/7.48 % (21718)Instructions burned: 37 (million)
% 12.46/7.48 % (21718)------------------------------
% 12.46/7.48 % (21718)------------------------------
% 12.46/7.49 % (21722)Instruction limit reached!
% 12.46/7.49 % (21722)------------------------------
% 12.46/7.49 % (21722)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.49 % (21722)Termination reason: Unknown
% 12.46/7.49 % (21722)Termination phase: shuffling
% 12.46/7.49
% 12.46/7.49 % (21722)Memory used [KB]: 12409
% 12.46/7.49 % (21722)Time elapsed: 0.011 s
% 12.46/7.49 % (21722)Instructions burned: 7 (million)
% 12.46/7.49 % (21722)------------------------------
% 12.46/7.49 % (21722)------------------------------
% 12.46/7.49 % (21723)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_21372 for (2995ds/16Mi)
% 12.46/7.49 % (21724)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_21372 for (2995ds/3Mi)
% 12.46/7.50 % (21724)Instruction limit reached!
% 12.46/7.50 % (21724)------------------------------
% 12.46/7.50 % (21724)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.50 % (21724)Termination reason: Unknown
% 12.46/7.50 % (21724)Termination phase: shuffling
% 12.46/7.50
% 12.46/7.50 % (21724)Memory used [KB]: 12153
% 12.46/7.50 % (21724)Time elapsed: 0.006 s
% 12.46/7.50 % (21724)Instructions burned: 3 (million)
% 12.46/7.50 % (21724)------------------------------
% 12.46/7.50 % (21724)------------------------------
% 12.46/7.50 % (21725)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_21372 for (2995ds/3Mi)
% 12.46/7.50 % (21725)Instruction limit reached!
% 12.46/7.50 % (21725)------------------------------
% 12.46/7.50 % (21725)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.50 % (21725)Termination reason: Unknown
% 12.46/7.50 % (21725)Termination phase: shuffling
% 12.46/7.50
% 12.46/7.50 % (21725)Memory used [KB]: 12153
% 12.46/7.50 % (21725)Time elapsed: 0.005 s
% 12.46/7.50 % (21725)Instructions burned: 3 (million)
% 12.46/7.50 % (21725)------------------------------
% 12.46/7.50 % (21725)------------------------------
% 12.46/7.50 % (21726)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_21372 for (2995ds/7Mi)
% 12.46/7.51 % (21723)Instruction limit reached!
% 12.46/7.51 % (21723)------------------------------
% 12.46/7.51 % (21723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.51 % (21723)Termination reason: Unknown
% 12.46/7.51 % (21723)Termination phase: shuffling
% 12.46/7.51
% 12.46/7.51 % (21723)Memory used [KB]: 12792
% 12.46/7.51 % (21723)Time elapsed: 0.022 s
% 12.46/7.51 % (21723)Instructions burned: 17 (million)
% 12.46/7.51 % (21723)------------------------------
% 12.46/7.51 % (21723)------------------------------
% 12.46/7.51 % (21726)Instruction limit reached!
% 12.46/7.51 % (21726)------------------------------
% 12.46/7.51 % (21726)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.51 % (21726)Termination reason: Unknown
% 12.46/7.51 % (21726)Termination phase: shuffling
% 12.46/7.51
% 12.46/7.51 % (21726)Memory used [KB]: 12409
% 12.46/7.51 % (21726)Time elapsed: 0.012 s
% 12.46/7.51 % (21726)Instructions burned: 7 (million)
% 12.46/7.51 % (21726)------------------------------
% 12.46/7.51 % (21726)------------------------------
% 12.46/7.51 % (21727)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_21372 for (2995ds/3Mi)
% 12.46/7.52 % (21727)Instruction limit reached!
% 12.46/7.52 % (21727)------------------------------
% 12.46/7.52 % (21727)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.52 % (21727)Termination reason: Unknown
% 12.46/7.52 % (21727)Termination phase: shuffling
% 12.46/7.52
% 12.46/7.52 % (21727)Memory used [KB]: 12153
% 12.46/7.52 % (21727)Time elapsed: 0.005 s
% 12.46/7.52 % (21727)Instructions burned: 3 (million)
% 12.46/7.52 % (21727)------------------------------
% 12.46/7.52 % (21727)------------------------------
% 12.46/7.52 % (21728)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_21372 for (2995ds/4Mi)
% 12.46/7.52 % (21729)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_21372 for (2995ds/18Mi)
% 12.46/7.53 % (21728)Instruction limit reached!
% 12.46/7.53 % (21728)------------------------------
% 12.46/7.53 % (21728)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.53 % (21728)Termination reason: Unknown
% 12.46/7.53 % (21728)Termination phase: shuffling
% 12.46/7.53
% 12.46/7.53 % (21728)Memory used [KB]: 12153
% 12.46/7.53 % (21728)Time elapsed: 0.007 s
% 12.46/7.53 % (21728)Instructions burned: 4 (million)
% 12.46/7.53 % (21728)------------------------------
% 12.46/7.53 % (21728)------------------------------
% 12.46/7.53 % (21731)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_21372 for (2995ds/6Mi)
% 12.46/7.53 % (21730)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_21372 for (2995ds/710Mi)
% 12.46/7.53 % (21731)Instruction limit reached!
% 12.46/7.53 % (21731)------------------------------
% 12.46/7.53 % (21731)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.53 % (21731)Termination reason: Unknown
% 12.46/7.53 % (21731)Termination phase: shuffling
% 12.46/7.53
% 12.46/7.53 % (21731)Memory used [KB]: 12281
% 12.46/7.53 % (21731)Time elapsed: 0.007 s
% 12.46/7.53 % (21731)Instructions burned: 6 (million)
% 12.46/7.53 % (21731)------------------------------
% 12.46/7.53 % (21731)------------------------------
% 12.46/7.54 % (21729)Instruction limit reached!
% 12.46/7.54 % (21729)------------------------------
% 12.46/7.54 % (21729)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.54 % (21729)Termination reason: Unknown
% 12.46/7.54 % (21729)Termination phase: shuffling
% 12.46/7.54
% 12.46/7.54 % (21729)Memory used [KB]: 12792
% 12.46/7.54 % (21729)Time elapsed: 0.015 s
% 12.46/7.54 % (21729)Instructions burned: 18 (million)
% 12.46/7.54 % (21729)------------------------------
% 12.46/7.54 % (21729)------------------------------
% 12.46/7.54 % (21732)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_21372 for (2995ds/902Mi)
% 12.46/7.54 % (21710)Instruction limit reached!
% 12.46/7.54 % (21710)------------------------------
% 12.46/7.54 % (21710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.54 % (21710)Termination reason: Unknown
% 12.46/7.54 % (21710)Termination phase: shuffling
% 12.46/7.54
% 12.46/7.54 % (21710)Memory used [KB]: 15095
% 12.46/7.54 % (21710)Time elapsed: 0.115 s
% 12.46/7.54 % (21710)Instructions burned: 183 (million)
% 12.46/7.54 % (21710)------------------------------
% 12.46/7.54 % (21710)------------------------------
% 12.46/7.54 % (21733)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_21372 for (2994ds/21Mi)
% 12.46/7.55 % (21734)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_21372 for (2994ds/5Mi)
% 12.46/7.55 % (21734)Instruction limit reached!
% 12.46/7.55 % (21734)------------------------------
% 12.46/7.55 % (21734)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.55 % (21734)Termination reason: Unknown
% 12.46/7.55 % (21734)Termination phase: shuffling
% 12.46/7.55
% 12.46/7.55 % (21734)Memory used [KB]: 12281
% 12.46/7.55 % (21734)Time elapsed: 0.006 s
% 12.46/7.55 % (21734)Instructions burned: 5 (million)
% 12.46/7.55 % (21734)------------------------------
% 12.46/7.55 % (21734)------------------------------
% 12.46/7.55 % (21735)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_21372 for (2994ds/6Mi)
% 12.46/7.56 % (21733)Instruction limit reached!
% 12.46/7.56 % (21733)------------------------------
% 12.46/7.56 % (21733)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.56 % (21733)Termination reason: Unknown
% 12.46/7.56 % (21733)Termination phase: shuffling
% 12.46/7.56
% 12.46/7.56 % (21733)Memory used [KB]: 12920
% 12.46/7.56 % (21733)Time elapsed: 0.016 s
% 12.46/7.56 % (21733)Instructions burned: 21 (million)
% 12.46/7.56 % (21733)------------------------------
% 12.46/7.56 % (21733)------------------------------
% 12.46/7.56 % (21735)Instruction limit reached!
% 12.46/7.56 % (21735)------------------------------
% 12.46/7.56 % (21735)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.56 % (21735)Termination reason: Unknown
% 12.46/7.56 % (21735)Termination phase: shuffling
% 12.46/7.56
% 12.46/7.56 % (21735)Memory used [KB]: 12281
% 12.46/7.56 % (21735)Time elapsed: 0.007 s
% 12.46/7.56 % (21735)Instructions burned: 6 (million)
% 12.46/7.56 % (21735)------------------------------
% 12.46/7.56 % (21735)------------------------------
% 12.46/7.56 % (21736)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_21372 for (2994ds/377Mi)
% 12.46/7.57 % (21738)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_21372 for (2994ds/19Mi)
% 12.46/7.57 % (21737)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_21372 for (2994ds/779Mi)
% 12.46/7.58 % (21715)Instruction limit reached!
% 12.46/7.58 % (21715)------------------------------
% 12.46/7.58 % (21715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.58 % (21715)Termination reason: Unknown
% 12.46/7.58 % (21715)Termination phase: shuffling
% 12.46/7.58
% 12.46/7.58 % (21715)Memory used [KB]: 16375
% 12.46/7.58 % (21715)Time elapsed: 0.148 s
% 12.46/7.58 % (21715)Instructions burned: 275 (million)
% 12.46/7.58 % (21715)------------------------------
% 12.46/7.58 % (21715)------------------------------
% 12.46/7.58 % (21738)Instruction limit reached!
% 12.46/7.58 % (21738)------------------------------
% 12.46/7.58 % (21738)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.58 % (21738)Termination reason: Unknown
% 12.46/7.58 % (21738)Termination phase: shuffling
% 12.46/7.58
% 12.46/7.58 % (21738)Memory used [KB]: 12920
% 12.46/7.58 % (21738)Time elapsed: 0.015 s
% 12.46/7.58 % (21738)Instructions burned: 21 (million)
% 12.46/7.58 % (21738)------------------------------
% 12.46/7.58 % (21738)------------------------------
% 12.46/7.59 % (21739)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_21372 for (2994ds/879Mi)
% 12.46/7.59 % (21740)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_21372 for (2994ds/17Mi)
% 12.46/7.61 % (21740)Instruction limit reached!
% 12.46/7.61 % (21740)------------------------------
% 12.46/7.61 % (21740)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.61 % (21740)Termination reason: Unknown
% 12.46/7.61 % (21740)Termination phase: shuffling
% 12.46/7.61
% 12.46/7.61 % (21740)Memory used [KB]: 12792
% 12.46/7.61 % (21740)Time elapsed: 0.014 s
% 12.46/7.61 % (21740)Instructions burned: 17 (million)
% 12.46/7.61 % (21740)------------------------------
% 12.46/7.61 % (21740)------------------------------
% 12.46/7.62 % (21741)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_21372 for (2994ds/3Mi)
% 12.46/7.62 % (21741)Instruction limit reached!
% 12.46/7.62 % (21741)------------------------------
% 12.46/7.62 % (21741)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.62 % (21741)Termination reason: Unknown
% 12.46/7.62 % (21741)Termination phase: shuffling
% 12.46/7.62
% 12.46/7.62 % (21741)Memory used [KB]: 12025
% 12.46/7.62 % (21741)Time elapsed: 0.003 s
% 12.46/7.62 % (21741)Instructions burned: 3 (million)
% 12.46/7.62 % (21741)------------------------------
% 12.46/7.62 % (21741)------------------------------
% 12.46/7.63 % (21742)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_21372 for (2994ds/30Mi)
% 12.46/7.65 % (21742)Instruction limit reached!
% 12.46/7.65 % (21742)------------------------------
% 12.46/7.65 % (21742)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.65 % (21742)Termination reason: Unknown
% 12.46/7.65 % (21742)Termination phase: shuffling
% 12.46/7.65
% 12.46/7.65 % (21742)Memory used [KB]: 13048
% 12.46/7.65 % (21742)Time elapsed: 0.019 s
% 12.46/7.65 % (21742)Instructions burned: 32 (million)
% 12.46/7.65 % (21742)------------------------------
% 12.46/7.65 % (21742)------------------------------
% 12.46/7.66 % (21743)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_21372 for (2993ds/127Mi)
% 12.46/7.70 % (21736)Instruction limit reached!
% 12.46/7.70 % (21736)------------------------------
% 12.46/7.70 % (21736)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.70 % (21736)Termination reason: Unknown
% 12.46/7.70 % (21736)Termination phase: shuffling
% 12.46/7.70
% 12.46/7.70 % (21736)Memory used [KB]: 17654
% 12.46/7.70 % (21736)Time elapsed: 0.135 s
% 12.46/7.70 % (21736)Instructions burned: 379 (million)
% 12.46/7.70 % (21736)------------------------------
% 12.46/7.70 % (21736)------------------------------
% 12.46/7.71 % (21743)Instruction limit reached!
% 12.46/7.71 % (21743)------------------------------
% 12.46/7.71 % (21743)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.71 % (21743)Termination reason: Unknown
% 12.46/7.71 % (21743)Termination phase: shuffling
% 12.46/7.71
% 12.46/7.71 % (21743)Memory used [KB]: 14456
% 12.46/7.71 % (21743)Time elapsed: 0.051 s
% 12.46/7.71 % (21743)Instructions burned: 129 (million)
% 12.46/7.71 % (21743)------------------------------
% 12.46/7.71 % (21743)------------------------------
% 12.46/7.71 % (21744)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_21372 for (2993ds/100Mi)
% 12.46/7.72 % (21745)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_21372 for (2993ds/3Mi)
% 12.46/7.72 % (21745)Instruction limit reached!
% 12.46/7.72 % (21745)------------------------------
% 12.46/7.72 % (21745)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.72 % (21745)Termination reason: Unknown
% 12.46/7.72 % (21745)Termination phase: shuffling
% 12.46/7.72
% 12.46/7.72 % (21745)Memory used [KB]: 12153
% 12.46/7.72 % (21745)Time elapsed: 0.004 s
% 12.46/7.72 % (21745)Instructions burned: 3 (million)
% 12.46/7.72 % (21745)------------------------------
% 12.46/7.72 % (21745)------------------------------
% 12.46/7.73 % (21746)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_21372 for (2992ds/20Mi)
% 12.46/7.75 % (21744)Instruction limit reached!
% 12.46/7.75 % (21744)------------------------------
% 12.46/7.75 % (21744)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.75 % (21744)Termination reason: Unknown
% 12.46/7.75 % (21744)Termination phase: shuffling
% 12.46/7.75
% 12.46/7.75 % (21744)Memory used [KB]: 14072
% 12.46/7.75 % (21744)Time elapsed: 0.037 s
% 12.46/7.75 % (21744)Instructions burned: 101 (million)
% 12.46/7.75 % (21744)------------------------------
% 12.46/7.75 % (21744)------------------------------
% 12.46/7.75 % (21746)Instruction limit reached!
% 12.46/7.75 % (21746)------------------------------
% 12.46/7.75 % (21746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.75 % (21746)Termination reason: Unknown
% 12.46/7.75 % (21746)Termination phase: shuffling
% 12.46/7.75
% 12.46/7.75 % (21746)Memory used [KB]: 12792
% 12.46/7.75 % (21746)Time elapsed: 0.015 s
% 12.46/7.75 % (21746)Instructions burned: 20 (million)
% 12.46/7.75 % (21746)------------------------------
% 12.46/7.75 % (21746)------------------------------
% 12.46/7.76 % (21747)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_21372 for (2992ds/86Mi)
% 12.46/7.76 % (21748)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_21372 for (2992ds/107Mi)
% 12.46/7.78 % (21730)Instruction limit reached!
% 12.46/7.78 % (21730)------------------------------
% 12.46/7.78 % (21730)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.78 % (21730)Termination reason: Unknown
% 12.46/7.78 % (21730)Termination phase: Property scanning
% 12.46/7.78
% 12.46/7.78 % (21730)Memory used [KB]: 21875
% 12.46/7.78 % (21730)Time elapsed: 0.254 s
% 12.46/7.78 % (21730)Instructions burned: 713 (million)
% 12.46/7.78 % (21730)------------------------------
% 12.46/7.78 % (21730)------------------------------
% 12.46/7.79 % (21747)Instruction limit reached!
% 12.46/7.79 % (21747)------------------------------
% 12.46/7.79 % (21747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.79 % (21747)Termination reason: Unknown
% 12.46/7.79 % (21747)Termination phase: shuffling
% 12.46/7.79
% 12.46/7.79 % (21747)Memory used [KB]: 13816
% 12.46/7.79 % (21747)Time elapsed: 0.037 s
% 12.46/7.79 % (21747)Instructions burned: 86 (million)
% 12.46/7.79 % (21747)------------------------------
% 12.46/7.79 % (21747)------------------------------
% 12.46/7.80 % (21749)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_21372 for (2992ds/39Mi)
% 12.46/7.80 % (21748)Instruction limit reached!
% 12.46/7.80 % (21748)------------------------------
% 12.46/7.80 % (21748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.80 % (21748)Termination reason: Unknown
% 12.46/7.80 % (21748)Termination phase: shuffling
% 12.46/7.80
% 12.46/7.80 % (21748)Memory used [KB]: 14072
% 12.46/7.80 % (21748)Time elapsed: 0.044 s
% 12.46/7.80 % (21748)Instructions burned: 107 (million)
% 12.46/7.80 % (21748)------------------------------
% 12.46/7.80 % (21748)------------------------------
% 12.46/7.80 % (21737)Instruction limit reached!
% 12.46/7.80 % (21737)------------------------------
% 12.46/7.80 % (21737)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.80 % (21737)Termination reason: Unknown
% 12.46/7.80 % (21737)Termination phase: shuffling
% 12.46/7.80
% 12.46/7.80 % (21737)Memory used [KB]: 25585
% 12.46/7.80 % (21737)Time elapsed: 0.233 s
% 12.46/7.80 % (21737)Instructions burned: 779 (million)
% 12.46/7.80 % (21737)------------------------------
% 12.46/7.80 % (21737)------------------------------
% 12.46/7.80 % (21750)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_21372 for (2992ds/448Mi)
% 12.46/7.81 % (21721)Instruction limit reached!
% 12.46/7.81 % (21721)------------------------------
% 12.46/7.81 % (21721)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.81 % (21721)Termination reason: Unknown
% 12.46/7.81 % (21721)Termination phase: shuffling
% 12.46/7.81
% 12.46/7.81 % (21721)Memory used [KB]: 33005
% 12.46/7.81 % (21721)Time elapsed: 0.337 s
% 12.46/7.81 % (21721)Instructions burned: 1041 (million)
% 12.46/7.81 % (21721)------------------------------
% 12.46/7.81 % (21721)------------------------------
% 12.46/7.81 % (21751)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_21372 for (2992ds/46Mi)
% 12.46/7.81 % (21752)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_21372 for (2992ds/98Mi)
% 12.46/7.81 % (21749)Instruction limit reached!
% 12.46/7.81 % (21749)------------------------------
% 12.46/7.81 % (21749)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.81 % (21749)Termination reason: Unknown
% 12.46/7.81 % (21749)Termination phase: shuffling
% 12.46/7.81
% 12.46/7.81 % (21749)Memory used [KB]: 13176
% 12.46/7.81 % (21749)Time elapsed: 0.020 s
% 12.46/7.81 % (21749)Instructions burned: 40 (million)
% 12.46/7.81 % (21749)------------------------------
% 12.46/7.81 % (21749)------------------------------
% 12.46/7.82 % (21753)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_21372 for (2992ds/507Mi)
% 12.46/7.83 % (21754)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_21372 for (2992ds/149Mi)
% 12.46/7.84 % (21751)Instruction limit reached!
% 12.46/7.84 % (21751)------------------------------
% 12.46/7.84 % (21751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.84 % (21751)Termination reason: Unknown
% 12.46/7.84 % (21751)Termination phase: shuffling
% 12.46/7.84
% 12.46/7.84 % (21751)Memory used [KB]: 13176
% 12.46/7.84 % (21751)Time elapsed: 0.024 s
% 12.46/7.84 % (21751)Instructions burned: 47 (million)
% 12.46/7.84 % (21751)------------------------------
% 12.46/7.84 % (21751)------------------------------
% 12.46/7.84 % (21732)Instruction limit reached!
% 12.46/7.84 % (21732)------------------------------
% 12.46/7.84 % (21732)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.84 % (21732)Termination reason: Unknown
% 12.46/7.84 % (21732)Termination phase: Property scanning
% 12.46/7.84
% 12.46/7.84 % (21732)Memory used [KB]: 21875
% 12.46/7.84 % (21732)Time elapsed: 0.304 s
% 12.46/7.84 % (21732)Instructions burned: 903 (million)
% 12.46/7.84 % (21732)------------------------------
% 12.46/7.84 % (21732)------------------------------
% 12.46/7.85 % (21755)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_21372 for (2991ds/39Mi)
% 12.46/7.85 % (21752)Instruction limit reached!
% 12.46/7.85 % (21752)------------------------------
% 12.46/7.85 % (21752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.85 % (21752)Termination reason: Unknown
% 12.46/7.85 % (21752)Termination phase: shuffling
% 12.46/7.85
% 12.46/7.85 % (21752)Memory used [KB]: 14072
% 12.46/7.85 % (21752)Time elapsed: 0.037 s
% 12.46/7.85 % (21752)Instructions burned: 100 (million)
% 12.46/7.85 % (21752)------------------------------
% 12.46/7.85 % (21752)------------------------------
% 12.46/7.85 % (21756)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_21372 for (2991ds/6Mi)
% 12.46/7.86 % (21756)Instruction limit reached!
% 12.46/7.86 % (21756)------------------------------
% 12.46/7.86 % (21756)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.86 % (21756)Termination reason: Unknown
% 12.46/7.86 % (21756)Termination phase: shuffling
% 12.46/7.86
% 12.46/7.86 % (21756)Memory used [KB]: 12281
% 12.46/7.86 % (21756)Time elapsed: 0.007 s
% 12.46/7.86 % (21756)Instructions burned: 6 (million)
% 12.46/7.86 % (21756)------------------------------
% 12.46/7.86 % (21756)------------------------------
% 12.46/7.86 % (21757)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_21372 for (2991ds/1087Mi)
% 12.46/7.87 % (21755)Instruction limit reached!
% 12.46/7.87 % (21755)------------------------------
% 12.46/7.87 % (21755)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.87 % (21755)Termination reason: Unknown
% 12.46/7.87 % (21755)Termination phase: shuffling
% 12.46/7.87
% 12.46/7.87 % (21755)Memory used [KB]: 13176
% 12.46/7.87 % (21755)Time elapsed: 0.022 s
% 12.46/7.87 % (21755)Instructions burned: 40 (million)
% 12.46/7.87 % (21755)------------------------------
% 12.46/7.87 % (21755)------------------------------
% 12.46/7.87 % (21758)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_21372 for (2991ds/47Mi)
% 12.46/7.88 % (21759)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_21372 for (2991ds/44Mi)
% 12.46/7.88 % (21739)Instruction limit reached!
% 12.46/7.88 % (21739)------------------------------
% 12.46/7.88 % (21739)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.88 % (21739)Termination reason: Unknown
% 12.46/7.88 % (21739)Termination phase: Property scanning
% 12.46/7.88
% 12.46/7.88 % (21739)Memory used [KB]: 23794
% 12.46/7.88 % (21739)Time elapsed: 0.295 s
% 12.46/7.88 % (21739)Instructions burned: 879 (million)
% 12.46/7.88 % (21739)------------------------------
% 12.46/7.88 % (21739)------------------------------
% 12.46/7.88 % (21754)Instruction limit reached!
% 12.46/7.88 % (21754)------------------------------
% 12.46/7.88 % (21754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.88 % (21754)Termination reason: Unknown
% 12.46/7.88 % (21754)Termination phase: shuffling
% 12.46/7.88
% 12.46/7.88 % (21754)Memory used [KB]: 14583
% 12.46/7.88 % (21754)Time elapsed: 0.058 s
% 12.46/7.88 % (21754)Instructions burned: 149 (million)
% 12.46/7.88 % (21754)------------------------------
% 12.46/7.88 % (21754)------------------------------
% 12.46/7.89 % (21758)Instruction limit reached!
% 12.46/7.89 % (21758)------------------------------
% 12.46/7.89 % (21758)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.89 % (21758)Termination reason: Unknown
% 12.46/7.89 % (21758)Termination phase: shuffling
% 12.46/7.89
% 12.46/7.89 % (21758)Memory used [KB]: 13176
% 12.46/7.89 % (21758)Time elapsed: 0.025 s
% 12.46/7.89 % (21758)Instructions burned: 48 (million)
% 12.46/7.89 % (21758)------------------------------
% 12.46/7.89 % (21758)------------------------------
% 12.46/7.89 % (21761)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_21372 for (2991ds/62Mi)
% 12.46/7.89 % (21760)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_21372 for (2991ds/432Mi)
% 12.46/7.90 % (21759)Instruction limit reached!
% 12.46/7.90 % (21759)------------------------------
% 12.46/7.90 % (21759)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.90 % (21759)Termination reason: Unknown
% 12.46/7.90 % (21759)Termination phase: shuffling
% 12.46/7.90
% 12.46/7.90 % (21759)Memory used [KB]: 13304
% 12.46/7.90 % (21759)Time elapsed: 0.022 s
% 12.46/7.90 % (21759)Instructions burned: 46 (million)
% 12.46/7.90 % (21759)------------------------------
% 12.46/7.90 % (21759)------------------------------
% 12.46/7.91 % (21762)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_21372 for (2991ds/31Mi)
% 12.46/7.91 % (21763)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_21372 for (2991ds/96Mi)
% 12.46/7.92 % (21761)Instruction limit reached!
% 12.46/7.92 % (21761)------------------------------
% 12.46/7.92 % (21761)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.92 % (21761)Termination reason: Unknown
% 12.46/7.92 % (21761)Termination phase: shuffling
% 12.46/7.92
% 12.46/7.92 % (21761)Memory used [KB]: 13560
% 12.46/7.92 % (21761)Time elapsed: 0.027 s
% 12.46/7.92 % (21761)Instructions burned: 63 (million)
% 12.46/7.92 % (21761)------------------------------
% 12.46/7.92 % (21761)------------------------------
% 12.46/7.92 % (21762)Instruction limit reached!
% 12.46/7.92 % (21762)------------------------------
% 12.46/7.92 % (21762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.92 % (21762)Termination reason: Unknown
% 12.46/7.92 % (21762)Termination phase: shuffling
% 12.46/7.92
% 12.46/7.92 % (21762)Memory used [KB]: 13048
% 12.46/7.92 % (21762)Time elapsed: 0.020 s
% 12.46/7.92 % (21762)Instructions burned: 33 (million)
% 12.46/7.92 % (21762)------------------------------
% 12.46/7.92 % (21762)------------------------------
% 12.46/7.93 % (21764)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_21372 for (2991ds/453Mi)
% 12.46/7.93 % (21765)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_21372 for (2991ds/71Mi)
% 12.46/7.94 % (21750)Instruction limit reached!
% 12.46/7.94 % (21750)------------------------------
% 12.46/7.94 % (21750)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.94 % (21750)Termination reason: Unknown
% 12.46/7.94 % (21750)Termination phase: shuffling
% 12.46/7.94
% 12.46/7.94 % (21750)Memory used [KB]: 20980
% 12.46/7.94 % (21750)Time elapsed: 0.140 s
% 12.46/7.94 % (21750)Instructions burned: 450 (million)
% 12.46/7.94 % (21750)------------------------------
% 12.46/7.94 % (21750)------------------------------
% 12.46/7.95 % (21763)Instruction limit reached!
% 12.46/7.95 % (21763)------------------------------
% 12.46/7.95 % (21763)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.95 % (21763)Termination reason: Unknown
% 12.46/7.95 % (21763)Termination phase: shuffling
% 12.46/7.95
% 12.46/7.95 % (21763)Memory used [KB]: 13944
% 12.46/7.95 % (21763)Time elapsed: 0.040 s
% 12.46/7.95 % (21763)Instructions burned: 96 (million)
% 12.46/7.95 % (21763)------------------------------
% 12.46/7.95 % (21763)------------------------------
% 12.46/7.96 % (21766)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_21372 for (2990ds/83Mi)
% 12.46/7.96 % (21767)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_21372 for (2990ds/11Mi)
% 12.46/7.96 % (21765)Instruction limit reached!
% 12.46/7.96 % (21765)------------------------------
% 12.46/7.96 % (21765)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.96 % (21765)Termination reason: Unknown
% 12.46/7.96 % (21765)Termination phase: shuffling
% 12.46/7.96
% 12.46/7.96 % (21765)Memory used [KB]: 13560
% 12.46/7.96 % (21765)Time elapsed: 0.031 s
% 12.46/7.96 % (21765)Instructions burned: 71 (million)
% 12.46/7.96 % (21765)------------------------------
% 12.46/7.96 % (21765)------------------------------
% 12.46/7.97 % (21767)Instruction limit reached!
% 12.46/7.97 % (21767)------------------------------
% 12.46/7.97 % (21767)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.97 % (21767)Termination reason: Unknown
% 12.46/7.97 % (21767)Termination phase: shuffling
% 12.46/7.97
% 12.46/7.97 % (21767)Memory used [KB]: 12792
% 12.46/7.97 % (21767)Time elapsed: 0.011 s
% 12.46/7.97 % (21767)Instructions burned: 11 (million)
% 12.46/7.97 % (21767)------------------------------
% 12.46/7.97 % (21767)------------------------------
% 12.46/7.98 % (21768)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_21372 for (2990ds/41Mi)
% 12.46/7.98 % (21769)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_21372 for (2990ds/6Mi)
% 12.46/7.99 % (21769)Instruction limit reached!
% 12.46/7.99 % (21769)------------------------------
% 12.46/7.99 % (21769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.99 % (21769)Termination reason: Unknown
% 12.46/7.99 % (21769)Termination phase: shuffling
% 12.46/7.99
% 12.46/7.99 % (21769)Memory used [KB]: 12281
% 12.46/7.99 % (21769)Time elapsed: 0.007 s
% 12.46/7.99 % (21769)Instructions burned: 6 (million)
% 12.46/7.99 % (21769)------------------------------
% 12.46/7.99 % (21769)------------------------------
% 12.46/7.99 % (21766)Instruction limit reached!
% 12.46/7.99 % (21766)------------------------------
% 12.46/7.99 % (21766)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.99 % (21766)Termination reason: Unknown
% 12.46/7.99 % (21766)Termination phase: shuffling
% 12.46/7.99
% 12.46/7.99 % (21766)Memory used [KB]: 13816
% 12.46/7.99 % (21766)Time elapsed: 0.036 s
% 12.46/7.99 % (21766)Instructions burned: 83 (million)
% 12.46/7.99 % (21766)------------------------------
% 12.46/7.99 % (21766)------------------------------
% 12.46/7.99 % (21768)Instruction limit reached!
% 12.46/7.99 % (21768)------------------------------
% 12.46/7.99 % (21768)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/7.99 % (21768)Termination reason: Unknown
% 12.46/7.99 % (21768)Termination phase: shuffling
% 12.46/7.99
% 12.46/7.99 % (21768)Memory used [KB]: 13176
% 12.46/7.99 % (21768)Time elapsed: 0.021 s
% 12.46/7.99 % (21768)Instructions burned: 42 (million)
% 12.46/7.99 % (21768)------------------------------
% 12.46/7.99 % (21768)------------------------------
% 12.46/8.00 % (21770)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_21372 for (2990ds/150Mi)
% 12.46/8.00 % (21753)Instruction limit reached!
% 12.46/8.00 % (21753)------------------------------
% 12.46/8.00 % (21753)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/8.00 % (21753)Termination reason: Unknown
% 12.46/8.00 % (21753)Termination phase: shuffling
% 12.46/8.00
% 12.46/8.00 % (21753)Memory used [KB]: 19061
% 12.46/8.00 % (21753)Time elapsed: 0.181 s
% 12.46/8.00 % (21753)Instructions burned: 508 (million)
% 12.46/8.00 % (21753)------------------------------
% 12.46/8.00 % (21753)------------------------------
% 12.46/8.00 % (21771)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_21372 for (2990ds/343Mi)
% 12.46/8.01 % (21772)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_21372 for (2990ds/109Mi)
% 16.99/8.02 % (21773)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_21372 for (2990ds/253Mi)
% 16.99/8.03 % (21760)Instruction limit reached!
% 16.99/8.03 % (21760)------------------------------
% 16.99/8.03 % (21760)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.99/8.03 % (21760)Termination reason: Unknown
% 16.99/8.03 % (21760)Termination phase: shuffling
% 16.99/8.03
% 16.99/8.03 % (21760)Memory used [KB]: 20852
% 16.99/8.03 % (21760)Time elapsed: 0.134 s
% 16.99/8.03 % (21760)Instructions burned: 434 (million)
% 16.99/8.03 % (21760)------------------------------
% 16.99/8.03 % (21760)------------------------------
% 16.99/8.04 % (21774)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_21372 for (2989ds/140Mi)
% 16.99/8.05 % (21772)Instruction limit reached!
% 16.99/8.05 % (21772)------------------------------
% 16.99/8.05 % (21772)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.99/8.05 % (21772)Termination reason: Unknown
% 16.99/8.05 % (21772)Termination phase: shuffling
% 16.99/8.05
% 16.99/8.05 % (21772)Memory used [KB]: 14200
% 16.99/8.05 % (21772)Time elapsed: 0.045 s
% 16.99/8.05 % (21772)Instructions burned: 111 (million)
% 16.99/8.05 % (21772)------------------------------
% 16.99/8.05 % (21772)------------------------------
% 16.99/8.05 % (21770)Instruction limit reached!
% 16.99/8.05 % (21770)------------------------------
% 16.99/8.05 % (21770)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.99/8.05 % (21770)Termination reason: Unknown
% 16.99/8.05 % (21770)Termination phase: shuffling
% 16.99/8.05
% 16.99/8.05 % (21770)Memory used [KB]: 14711
% 16.99/8.05 % (21770)Time elapsed: 0.058 s
% 16.99/8.05 % (21770)Instructions burned: 152 (million)
% 16.99/8.05 % (21770)------------------------------
% 16.99/8.05 % (21770)------------------------------
% 16.99/8.06 % (21775)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_21372 for (2989ds/614Mi)
% 16.99/8.07 % (21776)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_21372 for (2989ds/202Mi)
% 16.99/8.07 % (21764)Instruction limit reached!
% 16.99/8.07 % (21764)------------------------------
% 16.99/8.07 % (21764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.99/8.07 % (21764)Termination reason: Unknown
% 16.99/8.07 % (21764)Termination phase: shuffling
% 16.99/8.07
% 16.99/8.07 % (21764)Memory used [KB]: 21108
% 16.99/8.07 % (21764)Time elapsed: 0.140 s
% 16.99/8.07 % (21764)Instructions burned: 454 (million)
% 16.99/8.07 % (21764)------------------------------
% 16.99/8.07 % (21764)------------------------------
% 16.99/8.08 % (21777)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_21372 for (2989ds/339Mi)
% 16.99/8.09 % (21774)Instruction limit reached!
% 16.99/8.09 % (21774)------------------------------
% 16.99/8.09 % (21774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.99/8.09 % (21774)Termination reason: Unknown
% 16.99/8.09 % (21774)Termination phase: shuffling
% 16.99/8.09
% 16.99/8.09 % (21774)Memory used [KB]: 14456
% 16.99/8.09 % (21774)Time elapsed: 0.055 s
% 16.99/8.09 % (21774)Instructions burned: 141 (million)
% 16.99/8.09 % (21774)------------------------------
% 16.99/8.09 % (21774)------------------------------
% 16.99/8.10 % (21778)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_21372 for (2989ds/283Mi)
% 17.45/8.12 % (21771)Instruction limit reached!
% 17.45/8.12 % (21771)------------------------------
% 17.45/8.12 % (21771)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.45/8.12 % (21771)Termination reason: Unknown
% 17.45/8.12 % (21771)Termination phase: shuffling
% 17.45/8.12
% 17.45/8.12 % (21771)Memory used [KB]: 17270
% 17.45/8.12 % (21771)Time elapsed: 0.123 s
% 17.45/8.12 % (21771)Instructions burned: 345 (million)
% 17.45/8.12 % (21771)------------------------------
% 17.45/8.12 % (21771)------------------------------
% 17.45/8.14 % (21779)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_21372 for (2988ds/134Mi)
% 17.45/8.14 % (21776)Instruction limit reached!
% 17.45/8.14 % (21776)------------------------------
% 17.45/8.14 % (21776)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.45/8.14 % (21776)Termination reason: Unknown
% 17.45/8.14 % (21776)Termination phase: shuffling
% 17.45/8.14
% 17.45/8.14 % (21776)Memory used [KB]: 15479
% 17.45/8.14 % (21776)Time elapsed: 0.076 s
% 17.45/8.14 % (21776)Instructions burned: 202 (million)
% 17.45/8.14 % (21776)------------------------------
% 17.45/8.14 % (21776)------------------------------
% 17.45/8.14 % (21773)Instruction limit reached!
% 17.45/8.14 % (21773)------------------------------
% 17.45/8.14 % (21773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.45/8.14 % (21773)Termination reason: Unknown
% 17.45/8.14 % (21773)Termination phase: shuffling
% 17.45/8.14
% 17.45/8.14 % (21773)Memory used [KB]: 16119
% 17.45/8.14 % (21773)Time elapsed: 0.126 s
% 17.45/8.14 % (21773)Instructions burned: 255 (million)
% 17.45/8.14 % (21773)------------------------------
% 17.45/8.14 % (21773)------------------------------
% 17.45/8.15 % (21782)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_21372 for (2988ds/298Mi)
% 17.45/8.16 % (21784)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_21372 for (2988ds/192Mi)
% 17.45/8.19 % (21779)Instruction limit reached!
% 17.45/8.19 % (21779)------------------------------
% 17.45/8.19 % (21779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.45/8.19 % (21779)Termination reason: Unknown
% 17.45/8.19 % (21779)Termination phase: shuffling
% 17.45/8.19
% 17.45/8.19 % (21779)Memory used [KB]: 14328
% 17.45/8.19 % (21779)Time elapsed: 0.052 s
% 17.45/8.19 % (21779)Instructions burned: 134 (million)
% 17.45/8.19 % (21779)------------------------------
% 17.45/8.19 % (21779)------------------------------
% 17.45/8.19 % (21778)Instruction limit reached!
% 17.45/8.19 % (21778)------------------------------
% 17.45/8.19 % (21778)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.45/8.19 % (21778)Termination reason: Unknown
% 17.45/8.19 % (21778)Termination phase: shuffling
% 17.45/8.19
% 17.45/8.19 % (21778)Memory used [KB]: 18677
% 17.45/8.19 % (21778)Time elapsed: 0.091 s
% 17.45/8.19 % (21778)Instructions burned: 283 (million)
% 17.45/8.19 % (21778)------------------------------
% 17.45/8.19 % (21778)------------------------------
% 17.96/8.20 % (21798)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_21372 for (2988ds/258Mi)
% 17.96/8.20 % (21777)Instruction limit reached!
% 17.96/8.20 % (21777)------------------------------
% 17.96/8.20 % (21777)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.96/8.20 % (21777)Termination reason: Unknown
% 17.96/8.20 % (21777)Termination phase: shuffling
% 17.96/8.20
% 17.96/8.20 % (21777)Memory used [KB]: 17270
% 17.96/8.20 % (21777)Time elapsed: 0.122 s
% 17.96/8.20 % (21777)Instructions burned: 341 (million)
% 17.96/8.20 % (21777)------------------------------
% 17.96/8.20 % (21777)------------------------------
% 17.96/8.20 % (21799)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_21372 for (2988ds/90Mi)
% 17.96/8.21 % (21757)Instruction limit reached!
% 17.96/8.21 % (21757)------------------------------
% 17.96/8.21 % (21757)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.96/8.21 % (21757)Termination reason: Unknown
% 17.96/8.21 % (21757)Termination phase: Property scanning
% 17.96/8.21
% 17.96/8.21 % (21757)Memory used [KB]: 21875
% 17.96/8.21 % (21757)Time elapsed: 0.350 s
% 17.96/8.21 % (21757)Instructions burned: 1089 (million)
% 17.96/8.21 % (21757)------------------------------
% 17.96/8.21 % (21757)------------------------------
% 17.96/8.22 % (21800)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_21372 for (2988ds/231Mi)
% 17.96/8.22 % (21801)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_21372 for (2988ds/458Mi)
% 17.96/8.23 % (21784)Instruction limit reached!
% 17.96/8.23 % (21784)------------------------------
% 17.96/8.23 % (21784)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.96/8.23 % (21784)Termination reason: Unknown
% 17.96/8.23 % (21784)Termination phase: shuffling
% 17.96/8.23
% 17.96/8.23 % (21784)Memory used [KB]: 15223
% 17.96/8.23 % (21784)Time elapsed: 0.072 s
% 17.96/8.23 % (21784)Instructions burned: 192 (million)
% 17.96/8.23 % (21784)------------------------------
% 17.96/8.23 % (21784)------------------------------
% 17.96/8.24 % (21802)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_21372 for (2987ds/31Mi)
% 17.96/8.24 % (21799)Instruction limit reached!
% 17.96/8.24 % (21799)------------------------------
% 17.96/8.24 % (21799)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.96/8.24 % (21799)Termination reason: Unknown
% 17.96/8.24 % (21799)Termination phase: shuffling
% 17.96/8.24
% 17.96/8.24 % (21799)Memory used [KB]: 13816
% 17.96/8.24 % (21799)Time elapsed: 0.040 s
% 17.96/8.24 % (21799)Instructions burned: 91 (million)
% 17.96/8.24 % (21799)------------------------------
% 17.96/8.24 % (21799)------------------------------
% 17.96/8.25 % (21782)Instruction limit reached!
% 17.96/8.25 % (21782)------------------------------
% 17.96/8.25 % (21782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.96/8.25 % (21782)Termination reason: Unknown
% 17.96/8.25 % (21782)Termination phase: shuffling
% 17.96/8.25
% 17.96/8.25 % (21782)Memory used [KB]: 18805
% 17.96/8.25 % (21782)Time elapsed: 0.096 s
% 17.96/8.25 % (21782)Instructions burned: 300 (million)
% 17.96/8.25 % (21782)------------------------------
% 17.96/8.25 % (21782)------------------------------
% 17.96/8.25 % (21803)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_21372 for (2987ds/297Mi)
% 17.96/8.26 % (21802)Instruction limit reached!
% 17.96/8.26 % (21802)------------------------------
% 17.96/8.26 % (21802)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.96/8.26 % (21802)Termination reason: Unknown
% 17.96/8.26 % (21802)Termination phase: shuffling
% 17.96/8.26
% 17.96/8.26 % (21802)Memory used [KB]: 13048
% 17.96/8.26 % (21802)Time elapsed: 0.019 s
% 17.96/8.26 % (21802)Instructions burned: 32 (million)
% 17.96/8.26 % (21802)------------------------------
% 17.96/8.26 % (21802)------------------------------
% 17.96/8.26 % (21804)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_21372 for (2987ds/987Mi)
% 17.96/8.27 % (21805)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_21372 for (2987ds/182Mi)
% 17.96/8.27 % (21775)Instruction limit reached!
% 17.96/8.27 % (21775)------------------------------
% 17.96/8.27 % (21775)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.96/8.27 % (21775)Termination reason: Unknown
% 17.96/8.27 % (21775)Termination phase: shuffling
% 17.96/8.27
% 17.96/8.27 % (21775)Memory used [KB]: 20596
% 17.96/8.27 % (21775)Time elapsed: 0.216 s
% 17.96/8.27 % (21775)Instructions burned: 615 (million)
% 17.96/8.27 % (21775)------------------------------
% 17.96/8.27 % (21775)------------------------------
% 17.96/8.29 % (21806)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_21372 for (2987ds/182Mi)
% 17.96/8.29 % (21800)Instruction limit reached!
% 17.96/8.29 % (21800)------------------------------
% 17.96/8.29 % (21800)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.96/8.29 % (21800)Termination reason: Unknown
% 17.96/8.29 % (21800)Termination phase: shuffling
% 17.96/8.29
% 17.96/8.29 % (21800)Memory used [KB]: 15991
% 17.96/8.29 % (21800)Time elapsed: 0.077 s
% 17.96/8.29 % (21800)Instructions burned: 234 (million)
% 17.96/8.29 % (21800)------------------------------
% 17.96/8.29 % (21800)------------------------------
% 17.96/8.29 % (21798)Instruction limit reached!
% 17.96/8.29 % (21798)------------------------------
% 17.96/8.29 % (21798)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.96/8.29 % (21798)Termination reason: Unknown
% 17.96/8.29 % (21798)Termination phase: shuffling
% 17.96/8.29
% 17.96/8.29 % (21798)Memory used [KB]: 16119
% 17.96/8.29 % (21798)Time elapsed: 0.096 s
% 17.96/8.29 % (21798)Instructions burned: 259 (million)
% 17.96/8.29 % (21798)------------------------------
% 17.96/8.29 % (21798)------------------------------
% 18.80/8.30 % (21807)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_21372 for (2987ds/154Mi)
% 18.80/8.30 % (21808)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_21372 for (2987ds/79Mi)
% 18.80/8.33 % (21805)Instruction limit reached!
% 18.80/8.33 % (21805)------------------------------
% 18.80/8.33 % (21805)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.80/8.33 % (21805)Termination reason: Unknown
% 18.80/8.33 % (21805)Termination phase: shuffling
% 18.80/8.33
% 18.80/8.33 % (21805)Memory used [KB]: 14967
% 18.80/8.33 % (21805)Time elapsed: 0.069 s
% 18.80/8.33 % (21805)Instructions burned: 184 (million)
% 18.80/8.33 % (21805)------------------------------
% 18.80/8.33 % (21805)------------------------------
% 18.80/8.34 % (21808)Instruction limit reached!
% 18.80/8.34 % (21808)------------------------------
% 18.80/8.34 % (21808)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.80/8.34 % (21808)Termination reason: Unknown
% 18.80/8.34 % (21808)Termination phase: shuffling
% 18.80/8.34
% 18.80/8.34 % (21808)Memory used [KB]: 13688
% 18.80/8.34 % (21808)Time elapsed: 0.034 s
% 18.80/8.34 % (21808)Instructions burned: 81 (million)
% 18.80/8.34 % (21808)------------------------------
% 18.80/8.34 % (21808)------------------------------
% 18.80/8.35 % (21809)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_21372 for (2986ds/216Mi)
% 18.80/8.35 % (21803)Instruction limit reached!
% 18.80/8.35 % (21803)------------------------------
% 18.80/8.35 % (21803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.80/8.35 % (21803)Termination reason: Unknown
% 18.80/8.35 % (21803)Termination phase: shuffling
% 18.80/8.35
% 18.80/8.35 % (21803)Memory used [KB]: 18933
% 18.80/8.35 % (21803)Time elapsed: 0.094 s
% 18.80/8.35 % (21803)Instructions burned: 299 (million)
% 18.80/8.35 % (21803)------------------------------
% 18.80/8.35 % (21803)------------------------------
% 18.80/8.35 % (21810)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_21372 for (2986ds/114Mi)
% 18.80/8.35 % (21807)Instruction limit reached!
% 18.80/8.35 % (21807)------------------------------
% 18.80/8.35 % (21807)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.80/8.35 % (21807)Termination reason: Unknown
% 18.80/8.35 % (21807)Termination phase: shuffling
% 18.80/8.35
% 18.80/8.35 % (21807)Memory used [KB]: 14839
% 18.80/8.35 % (21807)Time elapsed: 0.053 s
% 18.80/8.35 % (21807)Instructions burned: 157 (million)
% 18.80/8.35 % (21807)------------------------------
% 18.80/8.35 % (21807)------------------------------
% 18.80/8.36 % (21806)Instruction limit reached!
% 18.80/8.36 % (21806)------------------------------
% 18.80/8.36 % (21806)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.80/8.36 % (21806)Termination reason: Unknown
% 18.80/8.36 % (21806)Termination phase: shuffling
% 18.80/8.36
% 18.80/8.36 % (21806)Memory used [KB]: 15095
% 18.80/8.36 % (21806)Time elapsed: 0.070 s
% 18.80/8.36 % (21806)Instructions burned: 182 (million)
% 18.80/8.36 % (21806)------------------------------
% 18.80/8.36 % (21806)------------------------------
% 18.80/8.36 % (21811)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_21372 for (2986ds/922Mi)
% 18.80/8.37 % (21812)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_21372 for (2986ds/182Mi)
% 18.80/8.37 % (21813)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_21372 for (2986ds/164Mi)
% 18.80/8.38 % (21801)Instruction limit reached!
% 18.80/8.38 % (21801)------------------------------
% 18.80/8.38 % (21801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.80/8.38 % (21801)Termination reason: Unknown
% 18.80/8.38 % (21801)Termination phase: shuffling
% 18.80/8.38
% 18.80/8.38 % (21801)Memory used [KB]: 18677
% 18.80/8.38 % (21801)Time elapsed: 0.164 s
% 18.80/8.38 % (21801)Instructions burned: 458 (million)
% 18.80/8.38 % (21801)------------------------------
% 18.80/8.38 % (21801)------------------------------
% 18.80/8.39 % (21810)Instruction limit reached!
% 18.80/8.39 % (21810)------------------------------
% 18.80/8.39 % (21810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.80/8.39 % (21810)Termination reason: Unknown
% 18.80/8.39 % (21810)Termination phase: shuffling
% 18.80/8.39
% 18.80/8.39 % (21810)Memory used [KB]: 14200
% 18.80/8.39 % (21810)Time elapsed: 0.047 s
% 18.80/8.39 % (21810)Instructions burned: 117 (million)
% 18.80/8.39 % (21810)------------------------------
% 18.80/8.39 % (21810)------------------------------
% 18.80/8.40 % (21814)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_21372 for (2986ds/50Mi)
% 19.40/8.41 % (21815)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_21372 for (2986ds/368Mi)
% 19.40/8.42 % (21809)Instruction limit reached!
% 19.40/8.42 % (21809)------------------------------
% 19.40/8.42 % (21809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.40/8.42 % (21809)Termination reason: Unknown
% 19.40/8.42 % (21809)Termination phase: shuffling
% 19.40/8.42
% 19.40/8.42 % (21809)Memory used [KB]: 15735
% 19.40/8.42 % (21809)Time elapsed: 0.072 s
% 19.40/8.42 % (21809)Instructions burned: 218 (million)
% 19.40/8.42 % (21809)------------------------------
% 19.40/8.42 % (21809)------------------------------
% 19.40/8.42 % (21814)Instruction limit reached!
% 19.40/8.42 % (21814)------------------------------
% 19.40/8.42 % (21814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.40/8.42 % (21814)Termination reason: Unknown
% 19.40/8.42 % (21814)Termination phase: shuffling
% 19.40/8.42
% 19.40/8.42 % (21814)Memory used [KB]: 13304
% 19.40/8.42 % (21814)Time elapsed: 0.024 s
% 19.40/8.42 % (21814)Instructions burned: 52 (million)
% 19.40/8.42 % (21814)------------------------------
% 19.40/8.42 % (21814)------------------------------
% 19.40/8.42 % (21813)Instruction limit reached!
% 19.40/8.42 % (21813)------------------------------
% 19.40/8.42 % (21813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.40/8.42 % (21813)Termination reason: Unknown
% 19.40/8.42 % (21813)Termination phase: shuffling
% 19.40/8.42
% 19.40/8.42 % (21813)Memory used [KB]: 14967
% 19.40/8.42 % (21813)Time elapsed: 0.056 s
% 19.40/8.42 % (21813)Instructions burned: 165 (million)
% 19.40/8.42 % (21813)------------------------------
% 19.40/8.42 % (21813)------------------------------
% 19.40/8.42 % (21812)Instruction limit reached!
% 19.40/8.42 % (21812)------------------------------
% 19.40/8.42 % (21812)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.40/8.42 % (21812)Termination reason: Unknown
% 19.40/8.42 % (21812)Termination phase: shuffling
% 19.40/8.42
% 19.40/8.42 % (21812)Memory used [KB]: 15223
% 19.40/8.42 % (21812)Time elapsed: 0.061 s
% 19.40/8.42 % (21812)Instructions burned: 184 (million)
% 19.40/8.42 % (21812)------------------------------
% 19.40/8.42 % (21812)------------------------------
% 19.40/8.43 % (21816)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_21372 for (2986ds/1021Mi)
% 19.40/8.43 % (21817)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_21372 for (2986ds/267Mi)
% 19.40/8.43 % (21818)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_21372 for (2986ds/204Mi)
% 19.40/8.44 % (21819)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_21372 for (2986ds/26Mi)
% 19.40/8.45 % (21819)Instruction limit reached!
% 19.40/8.45 % (21819)------------------------------
% 19.40/8.45 % (21819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.40/8.45 % (21819)Termination reason: Unknown
% 19.40/8.45 % (21819)Termination phase: shuffling
% 19.40/8.45
% 19.40/8.45 % (21819)Memory used [KB]: 12920
% 19.40/8.45 % (21819)Time elapsed: 0.017 s
% 19.40/8.45 % (21819)Instructions burned: 27 (million)
% 19.40/8.45 % (21819)------------------------------
% 19.40/8.45 % (21819)------------------------------
% 19.40/8.46 % (21820)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_21372 for (2985ds/53Mi)
% 19.40/8.49 % (21820)Instruction limit reached!
% 19.40/8.49 % (21820)------------------------------
% 19.40/8.49 % (21820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.40/8.49 % (21820)Termination reason: Unknown
% 19.40/8.49 % (21820)Termination phase: shuffling
% 19.40/8.49
% 19.40/8.49 % (21820)Memory used [KB]: 13304
% 19.40/8.49 % (21820)Time elapsed: 0.024 s
% 19.40/8.49 % (21820)Instructions burned: 53 (million)
% 19.40/8.49 % (21820)------------------------------
% 19.40/8.49 % (21820)------------------------------
% 19.72/8.50 % (21821)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_21372 for (2985ds/4436Mi)
% 19.72/8.51 % (21818)Instruction limit reached!
% 19.72/8.51 % (21818)------------------------------
% 19.72/8.51 % (21818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.72/8.51 % (21818)Termination reason: Unknown
% 19.72/8.51 % (21818)Termination phase: shuffling
% 19.72/8.51
% 19.72/8.51 % (21818)Memory used [KB]: 15351
% 19.72/8.51 % (21818)Time elapsed: 0.078 s
% 19.72/8.51 % (21818)Instructions burned: 205 (million)
% 19.72/8.51 % (21818)------------------------------
% 19.72/8.51 % (21818)------------------------------
% 19.72/8.51 % (21817)Instruction limit reached!
% 19.72/8.51 % (21817)------------------------------
% 19.72/8.51 % (21817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.72/8.51 % (21817)Termination reason: Unknown
% 19.72/8.51 % (21817)Termination phase: shuffling
% 19.72/8.51
% 19.72/8.51 % (21817)Memory used [KB]: 18421
% 19.72/8.51 % (21817)Time elapsed: 0.085 s
% 19.72/8.51 % (21817)Instructions burned: 267 (million)
% 19.72/8.51 % (21817)------------------------------
% 19.72/8.51 % (21817)------------------------------
% 19.72/8.52 % (21822)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_21372 for (2985ds/1539Mi)
% 19.72/8.52 % (21823)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_21372 for (2985ds/408Mi)
% 19.72/8.54 % (21815)Instruction limit reached!
% 19.72/8.54 % (21815)------------------------------
% 19.72/8.54 % (21815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.72/8.54 % (21815)Termination reason: Unknown
% 19.72/8.54 % (21815)Termination phase: shuffling
% 19.72/8.54
% 19.72/8.54 % (21815)Memory used [KB]: 17526
% 19.72/8.54 % (21815)Time elapsed: 0.134 s
% 19.72/8.54 % (21815)Instructions burned: 370 (million)
% 19.72/8.54 % (21815)------------------------------
% 19.72/8.54 % (21815)------------------------------
% 19.72/8.55 % (21824)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_21372 for (2984ds/5205Mi)
% 19.72/8.55 % (21804)Instruction limit reached!
% 19.72/8.55 % (21804)------------------------------
% 19.72/8.55 % (21804)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.72/8.55 % (21804)Termination reason: Unknown
% 19.72/8.55 % (21804)Termination phase: shuffling
% 19.72/8.55
% 19.72/8.55 % (21804)Memory used [KB]: 32366
% 19.72/8.55 % (21804)Time elapsed: 0.298 s
% 19.72/8.55 % (21804)Instructions burned: 988 (million)
% 19.72/8.55 % (21804)------------------------------
% 19.72/8.55 % (21804)------------------------------
% 19.72/8.57 % (21825)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_21372 for (2984ds/1913Mi)
% 20.37/8.67 % (21811)Instruction limit reached!
% 20.37/8.67 % (21811)------------------------------
% 20.37/8.67 % (21811)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.37/8.67 % (21811)Termination reason: Unknown
% 20.37/8.67 % (21811)Termination phase: Property scanning
% 20.37/8.67
% 20.37/8.67 % (21811)Memory used [KB]: 21875
% 20.37/8.67 % (21811)Time elapsed: 0.308 s
% 20.37/8.67 % (21811)Instructions burned: 923 (million)
% 20.37/8.67 % (21811)------------------------------
% 20.37/8.67 % (21811)------------------------------
% 20.37/8.67 % (21823)Instruction limit reached!
% 20.37/8.67 % (21823)------------------------------
% 20.37/8.67 % (21823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.37/8.67 % (21823)Termination reason: Unknown
% 20.37/8.67 % (21823)Termination phase: shuffling
% 20.37/8.67
% 20.37/8.67 % (21823)Memory used [KB]: 18166
% 20.37/8.67 % (21823)Time elapsed: 0.146 s
% 20.37/8.67 % (21823)Instructions burned: 410 (million)
% 20.37/8.67 % (21823)------------------------------
% 20.37/8.67 % (21823)------------------------------
% 20.37/8.68 % (21826)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_21372 for (2983ds/4350Mi)
% 20.37/8.68 % (21827)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_21372 for (2983ds/68Mi)
% 20.84/8.71 % (21827)Instruction limit reached!
% 20.84/8.71 % (21827)------------------------------
% 20.84/8.71 % (21827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.84/8.71 % (21827)Termination reason: Unknown
% 20.84/8.71 % (21827)Termination phase: shuffling
% 20.84/8.71
% 20.84/8.71 % (21827)Memory used [KB]: 13560
% 20.84/8.71 % (21827)Time elapsed: 0.031 s
% 20.84/8.71 % (21827)Instructions burned: 70 (million)
% 20.84/8.71 % (21827)------------------------------
% 20.84/8.71 % (21827)------------------------------
% 20.84/8.72 % (21828)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_21372 for (2983ds/358Mi)
% 20.84/8.76 % (21816)Instruction limit reached!
% 20.84/8.76 % (21816)------------------------------
% 20.84/8.76 % (21816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.84/8.76 % (21816)Termination reason: Unknown
% 20.84/8.76 % (21816)Termination phase: Property scanning
% 20.84/8.76
% 20.84/8.76 % (21816)Memory used [KB]: 21875
% 20.84/8.76 % (21816)Time elapsed: 0.330 s
% 20.84/8.76 % (21816)Instructions burned: 1021 (million)
% 20.84/8.76 % (21816)------------------------------
% 20.84/8.76 % (21816)------------------------------
% 20.84/8.77 % (21829)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_21372 for (2982ds/31Mi)
% 20.84/8.79 % (21829)Instruction limit reached!
% 20.84/8.79 % (21829)------------------------------
% 20.84/8.79 % (21829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.84/8.79 % (21829)Termination reason: Unknown
% 20.84/8.79 % (21829)Termination phase: shuffling
% 20.84/8.79
% 20.84/8.79 % (21829)Memory used [KB]: 13048
% 20.84/8.79 % (21829)Time elapsed: 0.019 s
% 20.84/8.79 % (21829)Instructions burned: 31 (million)
% 20.84/8.79 % (21829)------------------------------
% 20.84/8.79 % (21829)------------------------------
% 21.25/8.80 % (21830)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_21372 for (2982ds/279Mi)
% 21.25/8.85 % (21828)Instruction limit reached!
% 21.25/8.85 % (21828)------------------------------
% 21.25/8.85 % (21828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.25/8.85 % (21828)Termination reason: Unknown
% 21.25/8.85 % (21828)Termination phase: shuffling
% 21.25/8.85
% 21.25/8.85 % (21828)Memory used [KB]: 17398
% 21.25/8.85 % (21828)Time elapsed: 0.130 s
% 21.25/8.85 % (21828)Instructions burned: 360 (million)
% 21.25/8.85 % (21828)------------------------------
% 21.25/8.85 % (21828)------------------------------
% 21.25/8.86 % (21831)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_21372 for (2981ds/258Mi)
% 21.25/8.89 % (21830)Instruction limit reached!
% 21.25/8.89 % (21830)------------------------------
% 21.25/8.89 % (21830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.25/8.89 % (21830)Termination reason: Unknown
% 21.25/8.89 % (21830)Termination phase: shuffling
% 21.25/8.89
% 21.25/8.89 % (21830)Memory used [KB]: 18549
% 21.25/8.89 % (21830)Time elapsed: 0.089 s
% 21.25/8.89 % (21830)Instructions burned: 280 (million)
% 21.25/8.89 % (21830)------------------------------
% 21.25/8.89 % (21830)------------------------------
% 21.49/8.90 % (21832)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_21372 for (2981ds/26Mi)
% 21.49/8.92 % (21832)Instruction limit reached!
% 21.49/8.92 % (21832)------------------------------
% 21.49/8.92 % (21832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.49/8.92 % (21832)Termination reason: Unknown
% 21.49/8.92 % (21832)Termination phase: shuffling
% 21.49/8.92
% 21.49/8.92 % (21832)Memory used [KB]: 12920
% 21.49/8.92 % (21832)Time elapsed: 0.017 s
% 21.49/8.92 % (21832)Instructions burned: 28 (million)
% 21.49/8.92 % (21832)------------------------------
% 21.49/8.92 % (21832)------------------------------
% 21.49/8.93 % (21833)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_21372 for (2981ds/995Mi)
% 21.49/8.95 % (21831)Instruction limit reached!
% 21.49/8.95 % (21831)------------------------------
% 21.49/8.95 % (21831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.49/8.95 % (21831)Termination reason: Unknown
% 21.49/8.95 % (21831)Termination phase: shuffling
% 21.49/8.95
% 21.49/8.95 % (21831)Memory used [KB]: 16119
% 21.49/8.95 % (21831)Time elapsed: 0.094 s
% 21.49/8.95 % (21831)Instructions burned: 260 (million)
% 21.49/8.95 % (21831)------------------------------
% 21.49/8.95 % (21831)------------------------------
% 21.49/8.97 % (21834)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_21372 for (2980ds/373Mi)
% 21.49/9.00 % (21822)Instruction limit reached!
% 21.49/9.00 % (21822)------------------------------
% 21.49/9.00 % (21822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.49/9.00 % (21822)Termination reason: Unknown
% 21.49/9.00 % (21822)Termination phase: Preprocessing 1
% 21.49/9.00
% 21.49/9.00 % (21822)Memory used [KB]: 24818
% 21.49/9.00 % (21822)Time elapsed: 0.484 s
% 21.49/9.00 % (21822)Instructions burned: 1539 (million)
% 21.49/9.00 % (21822)------------------------------
% 21.49/9.00 % (21822)------------------------------
% 21.49/9.02 % (21835)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_21372 for (2980ds/253Mi)
% 24.06/9.09 % (21835)Instruction limit reached!
% 24.06/9.09 % (21835)------------------------------
% 24.06/9.09 % (21835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.06/9.09 % (21835)Termination reason: Unknown
% 24.06/9.09 % (21835)Termination phase: shuffling
% 24.06/9.09
% 24.06/9.09 % (21835)Memory used [KB]: 16247
% 24.06/9.09 % (21835)Time elapsed: 0.079 s
% 24.06/9.09 % (21835)Instructions burned: 256 (million)
% 24.06/9.09 % (21835)------------------------------
% 24.06/9.09 % (21835)------------------------------
% 24.06/9.10 % (21834)Instruction limit reached!
% 24.06/9.10 % (21834)------------------------------
% 24.06/9.10 % (21834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.06/9.10 % (21834)Termination reason: Unknown
% 24.06/9.10 % (21834)Termination phase: shuffling
% 24.06/9.10
% 24.06/9.10 % (21834)Memory used [KB]: 17526
% 24.06/9.10 % (21834)Time elapsed: 0.131 s
% 24.06/9.10 % (21834)Instructions burned: 375 (million)
% 24.06/9.10 % (21834)------------------------------
% 24.06/9.10 % (21834)------------------------------
% 24.06/9.11 % (21836)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_21372 for (2979ds/832Mi)
% 24.06/9.11 % (21837)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_21372 for (2979ds/668Mi)
% 24.77/9.16 % (21825)Instruction limit reached!
% 24.77/9.16 % (21825)------------------------------
% 24.77/9.16 % (21825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.77/9.16 % (21825)Termination reason: Unknown
% 24.77/9.16 % (21825)Termination phase: Preprocessing 1
% 24.77/9.16
% 24.77/9.16 % (21825)Memory used [KB]: 25074
% 24.77/9.16 % (21825)Time elapsed: 0.596 s
% 24.77/9.16 % (21825)Instructions burned: 1913 (million)
% 24.77/9.16 % (21825)------------------------------
% 24.77/9.16 % (21825)------------------------------
% 24.77/9.17 % (21838)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_21372 for (2978ds/40Mi)
% 24.77/9.19 % (21838)Instruction limit reached!
% 24.77/9.19 % (21838)------------------------------
% 24.77/9.19 % (21838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.77/9.19 % (21838)Termination reason: Unknown
% 24.77/9.19 % (21838)Termination phase: shuffling
% 24.77/9.19
% 24.77/9.19 % (21838)Memory used [KB]: 13176
% 24.77/9.19 % (21838)Time elapsed: 0.021 s
% 24.77/9.19 % (21838)Instructions burned: 43 (million)
% 24.77/9.19 % (21838)------------------------------
% 24.77/9.19 % (21838)------------------------------
% 25.13/9.20 % (21839)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_21372 for (2978ds/294Mi)
% 25.13/9.22 % (21833)Instruction limit reached!
% 25.13/9.22 % (21833)------------------------------
% 25.13/9.22 % (21833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.13/9.22 % (21833)Termination reason: Unknown
% 25.13/9.22 % (21833)Termination phase: shuffling
% 25.13/9.22
% 25.13/9.22 % (21833)Memory used [KB]: 32494
% 25.13/9.22 % (21833)Time elapsed: 0.292 s
% 25.13/9.22 % (21833)Instructions burned: 997 (million)
% 25.13/9.22 % (21833)------------------------------
% 25.13/9.22 % (21833)------------------------------
% 25.13/9.23 % (21840)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_21372 for (2977ds/187Mi)
% 25.13/9.29 % (21839)Instruction limit reached!
% 25.13/9.29 % (21839)------------------------------
% 25.13/9.29 % (21839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.13/9.29 % (21839)Termination reason: Unknown
% 25.13/9.29 % (21839)Termination phase: shuffling
% 25.13/9.29
% 25.13/9.29 % (21839)Memory used [KB]: 18805
% 25.13/9.29 % (21839)Time elapsed: 0.091 s
% 25.13/9.29 % (21839)Instructions burned: 295 (million)
% 25.13/9.29 % (21839)------------------------------
% 25.13/9.29 % (21839)------------------------------
% 25.57/9.30 % (21840)Instruction limit reached!
% 25.57/9.30 % (21840)------------------------------
% 25.57/9.30 % (21840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.57/9.30 % (21840)Termination reason: Unknown
% 25.57/9.30 % (21840)Termination phase: shuffling
% 25.57/9.30
% 25.57/9.30 % (21840)Memory used [KB]: 15223
% 25.57/9.30 % (21840)Time elapsed: 0.070 s
% 25.57/9.30 % (21840)Instructions burned: 189 (million)
% 25.57/9.30 % (21840)------------------------------
% 25.57/9.30 % (21840)------------------------------
% 25.57/9.30 % (21837)Instruction limit reached!
% 25.57/9.30 % (21837)------------------------------
% 25.57/9.30 % (21837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.57/9.30 % (21837)Termination reason: Unknown
% 25.57/9.30 % (21837)Termination phase: shuffling
% 25.57/9.30
% 25.57/9.30 % (21837)Memory used [KB]: 23922
% 25.57/9.30 % (21837)Time elapsed: 0.196 s
% 25.57/9.30 % (21837)Instructions burned: 669 (million)
% 25.57/9.30 % (21837)------------------------------
% 25.57/9.30 % (21837)------------------------------
% 25.57/9.31 % (21841)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_21372 for (2977ds/371Mi)
% 25.57/9.31 % (21842)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_21372 for (2977ds/1827Mi)
% 25.57/9.31 % (21843)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_21372 for (2977ds/70Mi)
% 25.57/9.34 % (21843)Instruction limit reached!
% 25.57/9.34 % (21843)------------------------------
% 25.57/9.34 % (21843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.57/9.34 % (21843)Termination reason: Unknown
% 25.57/9.34 % (21843)Termination phase: shuffling
% 25.57/9.34
% 25.57/9.34 % (21843)Memory used [KB]: 13560
% 25.57/9.34 % (21843)Time elapsed: 0.029 s
% 25.57/9.34 % (21843)Instructions burned: 70 (million)
% 25.57/9.34 % (21843)------------------------------
% 25.57/9.34 % (21843)------------------------------
% 25.57/9.35 % (21844)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_21372 for (2976ds/316Mi)
% 25.57/9.39 % (21836)Instruction limit reached!
% 25.57/9.39 % (21836)------------------------------
% 25.57/9.39 % (21836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.57/9.39 % (21836)Termination reason: Unknown
% 25.57/9.39 % (21836)Termination phase: Property scanning
% 25.57/9.39
% 25.57/9.39 % (21836)Memory used [KB]: 21875
% 25.57/9.39 % (21836)Time elapsed: 0.282 s
% 25.57/9.39 % (21836)Instructions burned: 832 (million)
% 25.57/9.39 % (21836)------------------------------
% 25.57/9.39 % (21836)------------------------------
% 26.48/9.40 % (21845)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_21372 for (2976ds/932Mi)
% 26.48/9.44 % (21841)Instruction limit reached!
% 26.48/9.44 % (21841)------------------------------
% 26.48/9.44 % (21841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.48/9.44 % (21841)Termination reason: Unknown
% 26.48/9.44 % (21841)Termination phase: shuffling
% 26.48/9.44
% 26.48/9.44 % (21841)Memory used [KB]: 17526
% 26.48/9.44 % (21841)Time elapsed: 0.134 s
% 26.48/9.44 % (21841)Instructions burned: 373 (million)
% 26.48/9.44 % (21841)------------------------------
% 26.48/9.44 % (21841)------------------------------
% 26.48/9.45 % (21846)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_21372 for (2975ds/4714Mi)
% 26.48/9.45 % (21844)Instruction limit reached!
% 26.48/9.45 % (21844)------------------------------
% 26.48/9.45 % (21844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.48/9.45 % (21844)Termination reason: Unknown
% 26.48/9.45 % (21844)Termination phase: shuffling
% 26.48/9.45
% 26.48/9.45 % (21844)Memory used [KB]: 19189
% 26.48/9.45 % (21844)Time elapsed: 0.099 s
% 26.48/9.45 % (21844)Instructions burned: 318 (million)
% 26.48/9.45 % (21844)------------------------------
% 26.48/9.45 % (21844)------------------------------
% 26.48/9.46 % (21847)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_21372 for (2975ds/8250Mi)
% 27.36/9.67 % (21845)Instruction limit reached!
% 27.36/9.67 % (21845)------------------------------
% 27.36/9.67 % (21845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.36/9.67 % (21845)Termination reason: Unknown
% 27.36/9.67 % (21845)Termination phase: shuffling
% 27.36/9.67
% 27.36/9.67 % (21845)Memory used [KB]: 31598
% 27.36/9.67 % (21845)Time elapsed: 0.273 s
% 27.36/9.67 % (21845)Instructions burned: 935 (million)
% 27.36/9.67 % (21845)------------------------------
% 27.36/9.67 % (21845)------------------------------
% 27.36/9.69 % (21848)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_21372 for (2973ds/770Mi)
% 28.23/9.88 % (21824)Instruction limit reached!
% 28.23/9.88 % (21824)------------------------------
% 28.23/9.88 % (21824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.23/9.88 % (21824)Termination reason: Unknown
% 28.23/9.88 % (21824)Termination phase: Function definition elimination
% 28.23/9.88
% 28.23/9.88 % (21824)Memory used [KB]: 43879
% 28.23/9.88 % (21824)Time elapsed: 1.332 s
% 28.23/9.88 % (21824)Instructions burned: 5207 (million)
% 28.23/9.88 % (21824)------------------------------
% 28.23/9.88 % (21824)------------------------------
% 28.23/9.89 % (21842)Instruction limit reached!
% 28.23/9.89 % (21842)------------------------------
% 28.23/9.89 % (21842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.23/9.89 % (21842)Termination reason: Unknown
% 28.23/9.89 % (21842)Termination phase: Property scanning
% 28.23/9.89
% 28.23/9.89 % (21842)Memory used [KB]: 32877
% 28.23/9.89 % (21842)Time elapsed: 0.582 s
% 28.23/9.89 % (21842)Instructions burned: 1828 (million)
% 28.23/9.89 % (21842)------------------------------
% 28.23/9.89 % (21842)------------------------------
% 28.23/9.89 % (21849)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_21372 for (2971ds/794Mi)
% 29.11/9.91 % (21850)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_21372 for (2971ds/464Mi)
% 29.11/9.94 % (21821)Instruction limit reached!
% 29.11/9.94 % (21821)------------------------------
% 29.11/9.94 % (21821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.11/9.94 % (21821)Termination reason: Unknown
% 29.11/9.94 % (21821)Termination phase: Saturation
% 29.11/9.94
% 29.11/9.94 % (21821)Memory used [KB]: 32110
% 29.11/9.94 % (21821)Time elapsed: 1.464 s
% 29.11/9.94 % (21821)Instructions burned: 4438 (million)
% 29.11/9.94 % (21821)------------------------------
% 29.11/9.94 % (21821)------------------------------
% 29.11/9.95 % (21851)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_21372 for (2970ds/1376Mi)
% 29.11/9.96 % (21848)Instruction limit reached!
% 29.11/9.96 % (21848)------------------------------
% 29.11/9.96 % (21848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.11/9.96 % (21848)Termination reason: Unknown
% 29.11/9.96 % (21848)Termination phase: shuffling
% 29.11/9.96
% 29.11/9.96 % (21848)Memory used [KB]: 24306
% 29.11/9.96 % (21848)Time elapsed: 0.269 s
% 29.11/9.96 % (21848)Instructions burned: 772 (million)
% 29.11/9.96 % (21848)------------------------------
% 29.11/9.96 % (21848)------------------------------
% 29.11/9.97 % (21852)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_21372 for (2970ds/906Mi)
% 29.11/10.02 % (21826)Refutation not found, non-redundant clauses discarded% (21826)------------------------------
% 29.11/10.02 % (21826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.11/10.02 % (21826)Termination reason: Refutation not found, non-redundant clauses discarded
% 29.11/10.02
% 29.11/10.02 % (21826)Memory used [KB]: 30958
% 29.11/10.02 % (21826)Time elapsed: 1.362 s
% 29.11/10.02 % (21826)Instructions burned: 4212 (million)
% 29.11/10.02 % (21826)------------------------------
% 29.11/10.02 % (21826)------------------------------
% 29.73/10.03 % (21853)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_21372 for (2970ds/434Mi)
% 29.73/10.11 % (21850)Instruction limit reached!
% 29.73/10.11 % (21850)------------------------------
% 29.73/10.11 % (21850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.73/10.11 % (21850)Termination reason: Unknown
% 29.73/10.11 % (21850)Termination phase: shuffling
% 29.73/10.11
% 29.73/10.11 % (21850)Memory used [KB]: 18677
% 29.73/10.11 % (21850)Time elapsed: 0.201 s
% 29.73/10.11 % (21850)Instructions burned: 466 (million)
% 29.73/10.11 % (21850)------------------------------
% 29.73/10.11 % (21850)------------------------------
% 31.71/10.13 % (21854)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_21372 for (2969ds/865Mi)
% 31.71/10.16 % (21849)Instruction limit reached!
% 31.71/10.16 % (21849)------------------------------
% 31.71/10.16 % (21849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.71/10.16 % (21849)Termination reason: Unknown
% 31.71/10.16 % (21849)Termination phase: Property scanning
% 31.71/10.16
% 31.71/10.16 % (21849)Memory used [KB]: 23794
% 31.71/10.16 % (21849)Time elapsed: 0.270 s
% 31.71/10.16 % (21849)Instructions burned: 796 (million)
% 31.71/10.16 % (21849)------------------------------
% 31.71/10.16 % (21849)------------------------------
% 31.71/10.18 % (21855)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_21372 for (2968ds/593Mi)
% 31.71/10.19 % (21853)Instruction limit reached!
% 31.71/10.19 % (21853)------------------------------
% 31.71/10.19 % (21853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.71/10.19 % (21853)Termination reason: Unknown
% 31.71/10.19 % (21853)Termination phase: shuffling
% 31.71/10.19
% 31.71/10.19 % (21853)Memory used [KB]: 18549
% 31.71/10.19 % (21853)Time elapsed: 0.158 s
% 31.71/10.19 % (21853)Instructions burned: 436 (million)
% 31.71/10.19 % (21853)------------------------------
% 31.71/10.19 % (21853)------------------------------
% 32.24/10.20 % (21861)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_21372 for (2968ds/443Mi)
% 32.24/10.28 % (21852)Instruction limit reached!
% 32.24/10.28 % (21852)------------------------------
% 32.24/10.28 % (21852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.24/10.28 % (21852)Termination reason: Unknown
% 32.24/10.28 % (21852)Termination phase: Property scanning
% 32.24/10.28
% 32.24/10.28 % (21852)Memory used [KB]: 23794
% 32.24/10.28 % (21852)Time elapsed: 0.303 s
% 32.24/10.28 % (21852)Instructions burned: 909 (million)
% 32.24/10.28 % (21852)------------------------------
% 32.24/10.28 % (21852)------------------------------
% 32.24/10.29 % (21887)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_21372 for (2967ds/594Mi)
% 32.64/10.36 % (21855)Instruction limit reached!
% 32.64/10.36 % (21855)------------------------------
% 32.64/10.36 % (21855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.64/10.36 % (21855)Termination reason: Unknown
% 32.64/10.36 % (21855)Termination phase: shuffling
% 32.64/10.36
% 32.64/10.36 % (21855)Memory used [KB]: 23027
% 32.64/10.36 % (21855)Time elapsed: 0.180 s
% 32.64/10.36 % (21855)Instructions burned: 594 (million)
% 32.64/10.36 % (21855)------------------------------
% 32.64/10.36 % (21855)------------------------------
% 32.64/10.36 % (21861)Instruction limit reached!
% 32.64/10.36 % (21861)------------------------------
% 32.64/10.36 % (21861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.64/10.36 % (21861)Termination reason: Unknown
% 32.64/10.36 % (21861)Termination phase: shuffling
% 32.64/10.36
% 32.64/10.36 % (21861)Memory used [KB]: 18421
% 32.64/10.36 % (21861)Time elapsed: 0.156 s
% 32.64/10.36 % (21861)Instructions burned: 445 (million)
% 32.64/10.36 % (21861)------------------------------
% 32.64/10.36 % (21861)------------------------------
% 32.64/10.36 % (21851)Instruction limit reached!
% 32.64/10.36 % (21851)------------------------------
% 32.64/10.36 % (21851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.64/10.36 % (21851)Termination reason: Unknown
% 32.64/10.36 % (21851)Termination phase: shuffling
% 32.64/10.36
% 32.64/10.36 % (21851)Memory used [KB]: 37483
% 32.64/10.36 % (21851)Time elapsed: 0.411 s
% 32.64/10.36 % (21851)Instructions burned: 1379 (million)
% 32.64/10.36 % (21851)------------------------------
% 32.64/10.36 % (21851)------------------------------
% 32.64/10.37 % (21889)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_21372 for (2966ds/5411Mi)
% 32.64/10.37 % (21888)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_21372 for (2966ds/327Mi)
% 32.64/10.38 % (21890)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_21372 for (2966ds/1762Mi)
% 32.64/10.39 % (21854)Instruction limit reached!
% 32.64/10.39 % (21854)------------------------------
% 32.64/10.39 % (21854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.64/10.39 % (21854)Termination reason: Unknown
% 32.64/10.39 % (21854)Termination phase: shuffling
% 32.64/10.39
% 32.64/10.39 % (21854)Memory used [KB]: 26609
% 32.64/10.39 % (21854)Time elapsed: 0.261 s
% 32.64/10.39 % (21854)Instructions burned: 867 (million)
% 32.64/10.39 % (21854)------------------------------
% 32.64/10.39 % (21854)------------------------------
% 33.54/10.40 % (21891)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_21372 for (2966ds/96Mi)
% 33.54/10.43 % (21891)Instruction limit reached!
% 33.54/10.43 % (21891)------------------------------
% 33.54/10.43 % (21891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.54/10.43 % (21891)Termination reason: Unknown
% 33.54/10.43 % (21891)Termination phase: shuffling
% 33.54/10.43
% 33.54/10.43 % (21891)Memory used [KB]: 13944
% 33.54/10.43 % (21891)Time elapsed: 0.037 s
% 33.54/10.43 % (21891)Instructions burned: 97 (million)
% 33.54/10.43 % (21891)------------------------------
% 33.54/10.43 % (21891)------------------------------
% 33.54/10.45 % (21892)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_21372 for (2965ds/503Mi)
% 33.54/10.49 % (21888)Instruction limit reached!
% 33.54/10.49 % (21888)------------------------------
% 33.54/10.49 % (21888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.54/10.49 % (21888)Termination reason: Unknown
% 33.54/10.49 % (21888)Termination phase: shuffling
% 33.54/10.49
% 33.54/10.49 % (21888)Memory used [KB]: 17142
% 33.54/10.49 % (21888)Time elapsed: 0.124 s
% 33.54/10.49 % (21888)Instructions burned: 329 (million)
% 33.54/10.49 % (21888)------------------------------
% 33.54/10.49 % (21888)------------------------------
% 33.54/10.49 % (21887)Instruction limit reached!
% 33.54/10.49 % (21887)------------------------------
% 33.54/10.49 % (21887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.54/10.49 % (21887)Termination reason: Unknown
% 33.54/10.49 % (21887)Termination phase: shuffling
% 33.54/10.49
% 33.54/10.49 % (21887)Memory used [KB]: 20340
% 33.54/10.49 % (21887)Time elapsed: 0.209 s
% 33.54/10.49 % (21887)Instructions burned: 594 (million)
% 33.54/10.49 % (21887)------------------------------
% 33.54/10.49 % (21887)------------------------------
% 33.91/10.50 % (21893)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_21372 for (2965ds/90Mi)
% 33.91/10.51 % (21894)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_21372 for (2965ds/593Mi)
% 33.91/10.54 % (21893)Instruction limit reached!
% 33.91/10.54 % (21893)------------------------------
% 33.91/10.54 % (21893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.91/10.54 % (21893)Termination reason: Unknown
% 33.91/10.54 % (21893)Termination phase: shuffling
% 33.91/10.54
% 33.91/10.54 % (21893)Memory used [KB]: 13816
% 33.91/10.54 % (21893)Time elapsed: 0.038 s
% 33.91/10.54 % (21893)Instructions burned: 90 (million)
% 33.91/10.54 % (21893)------------------------------
% 33.91/10.54 % (21893)------------------------------
% 33.91/10.55 % (21895)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_21372 for (2964ds/1512Mi)
% 34.38/10.62 % (21892)Instruction limit reached!
% 34.38/10.62 % (21892)------------------------------
% 34.38/10.62 % (21892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.38/10.62 % (21892)Termination reason: Unknown
% 34.38/10.62 % (21892)Termination phase: shuffling
% 34.38/10.62
% 34.38/10.62 % (21892)Memory used [KB]: 19189
% 34.38/10.62 % (21892)Time elapsed: 0.176 s
% 34.38/10.62 % (21892)Instructions burned: 505 (million)
% 34.38/10.62 % (21892)------------------------------
% 34.38/10.62 % (21892)------------------------------
% 34.38/10.63 % (21896)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_21372 for (2963ds/569Mi)
% 34.38/10.66 % (21846)Instruction limit reached!
% 34.38/10.66 % (21846)------------------------------
% 34.38/10.66 % (21846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.38/10.66 % (21846)Termination reason: Unknown
% 34.38/10.66 % (21846)Termination phase: Function definition elimination
% 34.38/10.66
% 34.38/10.66 % (21846)Memory used [KB]: 41321
% 34.38/10.66 % (21846)Time elapsed: 1.209 s
% 34.38/10.66 % (21846)Instructions burned: 4717 (million)
% 34.38/10.66 % (21846)------------------------------
% 34.38/10.66 % (21846)------------------------------
% 34.38/10.67 % (21897)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_21372 for (2963ds/501Mi)
% 34.74/10.71 % (21894)Instruction limit reached!
% 34.74/10.71 % (21894)------------------------------
% 34.74/10.71 % (21894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.74/10.71 % (21894)Termination reason: Unknown
% 34.74/10.71 % (21894)Termination phase: shuffling
% 34.74/10.71
% 34.74/10.71 % (21894)Memory used [KB]: 20340
% 34.74/10.71 % (21894)Time elapsed: 0.206 s
% 34.74/10.71 % (21894)Instructions burned: 594 (million)
% 34.74/10.71 % (21894)------------------------------
% 34.74/10.71 % (21894)------------------------------
% 34.74/10.72 % (21898)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_21372 for (2963ds/562Mi)
% 35.07/10.80 % (21896)Instruction limit reached!
% 35.07/10.80 % (21896)------------------------------
% 35.07/10.80 % (21896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.07/10.80 % (21896)Termination reason: Unknown
% 35.07/10.80 % (21896)Termination phase: shuffling
% 35.07/10.80
% 35.07/10.80 % (21896)Memory used [KB]: 22643
% 35.07/10.80 % (21896)Time elapsed: 0.172 s
% 35.07/10.80 % (21896)Instructions burned: 570 (million)
% 35.07/10.80 % (21896)------------------------------
% 35.07/10.80 % (21896)------------------------------
% 35.07/10.82 % (21899)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_21372 for (2962ds/406Mi)
% 35.07/10.85 % (21897)Instruction limit reached!
% 35.07/10.85 % (21897)------------------------------
% 35.07/10.85 % (21897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.07/10.85 % (21897)Termination reason: Unknown
% 35.07/10.85 % (21897)Termination phase: shuffling
% 35.07/10.85
% 35.07/10.85 % (21897)Memory used [KB]: 19061
% 35.07/10.85 % (21897)Time elapsed: 0.177 s
% 35.07/10.85 % (21897)Instructions burned: 502 (million)
% 35.07/10.85 % (21897)------------------------------
% 35.07/10.85 % (21897)------------------------------
% 35.07/10.86 % (21900)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_21372 for (2961ds/582Mi)
% 35.07/10.89 % (21890)Instruction limit reached!
% 35.07/10.89 % (21890)------------------------------
% 35.07/10.89 % (21890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.07/10.89 % (21890)Termination reason: Unknown
% 35.07/10.89 % (21890)Termination phase: Property scanning
% 35.07/10.89
% 35.07/10.89 % (21890)Memory used [KB]: 38762
% 35.07/10.89 % (21890)Time elapsed: 0.513 s
% 35.07/10.89 % (21890)Instructions burned: 1764 (million)
% 35.07/10.89 % (21890)------------------------------
% 35.07/10.89 % (21890)------------------------------
% 35.07/10.89 % (21898)Instruction limit reached!
% 35.07/10.89 % (21898)------------------------------
% 35.07/10.89 % (21898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.07/10.89 % (21898)Termination reason: Unknown
% 35.07/10.89 % (21898)Termination phase: shuffling
% 35.07/10.89
% 35.07/10.89 % (21898)Memory used [KB]: 22515
% 35.07/10.89 % (21898)Time elapsed: 0.169 s
% 35.07/10.89 % (21898)Instructions burned: 562 (million)
% 35.07/10.89 % (21898)------------------------------
% 35.07/10.89 % (21898)------------------------------
% 36.24/10.90 % (21901)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_21372 for (2961ds/456Mi)
% 36.24/10.90 % (21902)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_21372 for (2961ds/393Mi)
% 36.24/10.96 % (21899)Instruction limit reached!
% 36.24/10.96 % (21899)------------------------------
% 36.24/10.96 % (21899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.24/10.96 % (21899)Termination reason: Unknown
% 36.24/10.96 % (21899)Termination phase: shuffling
% 36.24/10.96
% 36.24/10.96 % (21899)Memory used [KB]: 17782
% 36.24/10.96 % (21899)Time elapsed: 0.142 s
% 36.24/10.96 % (21899)Instructions burned: 406 (million)
% 36.24/10.96 % (21899)------------------------------
% 36.24/10.96 % (21899)------------------------------
% 36.24/10.97 % (21903)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_21372 for (2960ds/1335Mi)
% 36.24/11.02 % (21902)Instruction limit reached!
% 36.24/11.02 % (21902)------------------------------
% 36.24/11.02 % (21902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.24/11.02 % (21902)Termination reason: Unknown
% 36.24/11.02 % (21902)Termination phase: shuffling
% 36.24/11.02
% 36.24/11.02 % (21902)Memory used [KB]: 20340
% 36.24/11.02 % (21902)Time elapsed: 0.119 s
% 36.24/11.02 % (21902)Instructions burned: 395 (million)
% 36.24/11.02 % (21902)------------------------------
% 36.24/11.02 % (21902)------------------------------
% 37.96/11.02 % (21895)Instruction limit reached!
% 37.96/11.02 % (21895)------------------------------
% 37.96/11.02 % (21895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.96/11.02 % (21895)Termination reason: Unknown
% 37.96/11.02 % (21895)Termination phase: SInE selection
% 37.96/11.02
% 37.96/11.02 % (21895)Memory used [KB]: 26865
% 37.96/11.02 % (21895)Time elapsed: 0.476 s
% 37.96/11.02 % (21895)Instructions burned: 1514 (million)
% 37.96/11.02 % (21895)------------------------------
% 37.96/11.02 % (21895)------------------------------
% 37.96/11.03 % (21904)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_21372 for (2960ds/3373Mi)
% 37.96/11.04 % (21905)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_21372 for (2960ds/474Mi)
% 37.96/11.06 % (21901)Instruction limit reached!
% 37.96/11.06 % (21901)------------------------------
% 37.96/11.06 % (21901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.96/11.06 % (21901)Termination reason: Unknown
% 37.96/11.06 % (21901)Termination phase: shuffling
% 37.96/11.06
% 37.96/11.06 % (21901)Memory used [KB]: 18549
% 37.96/11.06 % (21901)Time elapsed: 0.159 s
% 37.96/11.06 % (21901)Instructions burned: 457 (million)
% 37.96/11.06 % (21901)------------------------------
% 37.96/11.06 % (21901)------------------------------
% 37.96/11.06 % (21900)Instruction limit reached!
% 37.96/11.06 % (21900)------------------------------
% 37.96/11.06 % (21900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.96/11.06 % (21900)Termination reason: Unknown
% 37.96/11.06 % (21900)Termination phase: shuffling
% 37.96/11.06
% 37.96/11.06 % (21900)Memory used [KB]: 20468
% 37.96/11.06 % (21900)Time elapsed: 0.205 s
% 37.96/11.06 % (21900)Instructions burned: 582 (million)
% 37.96/11.06 % (21900)------------------------------
% 37.96/11.06 % (21900)------------------------------
% 37.96/11.07 % (21906)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_21372 for (2959ds/157Mi)
% 37.96/11.08 % (21907)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_21372 for (2959ds/590Mi)
% 37.96/11.12 % (21906)Instruction limit reached!
% 37.96/11.12 % (21906)------------------------------
% 37.96/11.12 % (21906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.96/11.12 % (21906)Termination reason: Unknown
% 37.96/11.12 % (21906)Termination phase: shuffling
% 37.96/11.12
% 37.96/11.12 % (21906)Memory used [KB]: 14967
% 37.96/11.12 % (21906)Time elapsed: 0.052 s
% 37.96/11.12 % (21906)Instructions burned: 157 (million)
% 37.96/11.12 % (21906)------------------------------
% 37.96/11.12 % (21906)------------------------------
% 38.59/11.13 % (21908)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=104:si=on:rtra=on_0 on DTF2THF_21372 for (2958ds/104Mi)
% 38.59/11.18 % (21908)Instruction limit reached!
% 38.59/11.18 % (21908)------------------------------
% 38.59/11.18 % (21908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.59/11.18 % (21908)Termination reason: Unknown
% 38.59/11.18 % (21908)Termination phase: shuffling
% 38.59/11.18
% 38.59/11.18 % (21908)Memory used [KB]: 13944
% 38.59/11.18 % (21908)Time elapsed: 0.043 s
% 38.59/11.18 % (21908)Instructions burned: 104 (million)
% 38.59/11.18 % (21908)------------------------------
% 38.59/11.18 % (21908)------------------------------
% 38.59/11.18 % (21905)Instruction limit reached!
% 38.59/11.18 % (21905)------------------------------
% 38.59/11.18 % (21905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.59/11.18 % (21905)Termination reason: Unknown
% 38.59/11.18 % (21905)Termination phase: shuffling
% 38.59/11.18
% 38.59/11.18 % (21905)Memory used [KB]: 21364
% 38.59/11.18 % (21905)Time elapsed: 0.145 s
% 38.59/11.18 % (21905)Instructions burned: 475 (million)
% 38.59/11.18 % (21905)------------------------------
% 38.59/11.18 % (21905)------------------------------
% 38.59/11.19 % (21909)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=414:si=on:rtra=on_0 on DTF2THF_21372 for (2958ds/414Mi)
% 38.59/11.19 % (21910)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_21372 for (2958ds/715Mi)
% 39.02/11.28 % (21907)Instruction limit reached!
% 39.02/11.28 % (21907)------------------------------
% 39.02/11.28 % (21907)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.02/11.28 % (21907)Termination reason: Unknown
% 39.02/11.28 % (21907)Termination phase: shuffling
% 39.02/11.28
% 39.02/11.28 % (21907)Memory used [KB]: 20340
% 39.02/11.28 % (21907)Time elapsed: 0.206 s
% 39.02/11.28 % (21907)Instructions burned: 591 (million)
% 39.02/11.28 % (21907)------------------------------
% 39.02/11.28 % (21907)------------------------------
% 39.02/11.29 % (21911)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_21372 for (2957ds/961Mi)
% 39.55/11.31 % (21909)Instruction limit reached!
% 39.55/11.31 % (21909)------------------------------
% 39.55/11.31 % (21909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.55/11.31 % (21909)Termination reason: Unknown
% 39.55/11.31 % (21909)Termination phase: shuffling
% 39.55/11.31
% 39.55/11.31 % (21909)Memory used [KB]: 20468
% 39.55/11.31 % (21909)Time elapsed: 0.125 s
% 39.55/11.31 % (21909)Instructions burned: 415 (million)
% 39.55/11.31 % (21909)------------------------------
% 39.55/11.31 % (21909)------------------------------
% 39.55/11.32 % (21912)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_21372 for (2957ds/1213Mi)
% 39.55/11.36 % (21903)Instruction limit reached!
% 39.55/11.36 % (21903)------------------------------
% 39.55/11.36 % (21903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.55/11.36 % (21903)Termination reason: Unknown
% 39.55/11.36 % (21903)Termination phase: shuffling
% 39.55/11.36
% 39.55/11.36 % (21903)Memory used [KB]: 36971
% 39.55/11.36 % (21903)Time elapsed: 0.392 s
% 39.55/11.36 % (21903)Instructions burned: 1336 (million)
% 39.55/11.36 % (21903)------------------------------
% 39.55/11.36 % (21903)------------------------------
% 39.55/11.37 % (21913)lrs+1002_1:12_bd=off:hud=1:ntd=on:piset=equals:prag=on:i=1979:si=on:rtra=on_0 on DTF2THF_21372 for (2956ds/1979Mi)
% 39.95/11.44 % (21910)Instruction limit reached!
% 39.95/11.44 % (21910)------------------------------
% 39.95/11.44 % (21910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.95/11.44 % (21910)Termination reason: Unknown
% 39.95/11.44 % (21910)Termination phase: Property scanning
% 39.95/11.44
% 39.95/11.44 % (21910)Memory used [KB]: 23794
% 39.95/11.44 % (21910)Time elapsed: 0.246 s
% 39.95/11.44 % (21910)Instructions burned: 716 (million)
% 39.95/11.44 % (21910)------------------------------
% 39.95/11.44 % (21910)------------------------------
% 39.95/11.45 % (21914)dis+1002_1:1_amm=off:au=on:bsd=on:cbe=off:cnfonf=lazy_gen:sp=unary_frequency:i=481:si=on:rtra=on_0 on DTF2THF_21372 for (2955ds/481Mi)
% 40.41/11.59 % (21914)Instruction limit reached!
% 40.41/11.59 % (21914)------------------------------
% 40.41/11.59 % (21914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.41/11.59 % (21914)Termination reason: Unknown
% 40.41/11.59 % (21914)Termination phase: shuffling
% 40.41/11.59
% 40.41/11.59 % (21914)Memory used [KB]: 21492
% 40.41/11.59 % (21914)Time elapsed: 0.143 s
% 40.41/11.59 % (21914)Instructions burned: 483 (million)
% 40.41/11.59 % (21914)------------------------------
% 40.41/11.59 % (21914)------------------------------
% 40.76/11.60 % (21911)Instruction limit reached!
% 40.76/11.60 % (21911)------------------------------
% 40.76/11.60 % (21911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.76/11.60 % (21911)Termination reason: Unknown
% 40.76/11.60 % (21911)Termination phase: Property scanning
% 40.76/11.60
% 40.76/11.60 % (21911)Memory used [KB]: 23794
% 40.76/11.60 % (21911)Time elapsed: 0.308 s
% 40.76/11.60 % (21911)Instructions burned: 963 (million)
% 40.76/11.60 % (21911)------------------------------
% 40.76/11.60 % (21911)------------------------------
% 40.76/11.60 % (21915)dis+10_8:1_apa=on:cnfonf=off:e2e=on:plsq=on:plsqc=1:plsqr=32,1:ss=axioms:i=783:si=on:rtra=on_0 on DTF2THF_21372 for (2954ds/783Mi)
% 40.76/11.61 % (21916)lrs+1010_1:1_avsq=on:avsqr=1,16:kws=inv_arity:nwc=5.0:sd=1:ss=axioms:i=432:si=on:rtra=on_0 on DTF2THF_21372 for (2954ds/432Mi)
% 41.28/11.70 % (21912)Instruction limit reached!
% 41.28/11.70 % (21912)------------------------------
% 41.28/11.70 % (21912)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.28/11.70 % (21912)Termination reason: Unknown
% 41.28/11.70 % (21912)Termination phase: Property scanning
% 41.28/11.70
% 41.28/11.70 % (21912)Memory used [KB]: 21875
% 41.28/11.70 % (21912)Time elapsed: 0.381 s
% 41.28/11.70 % (21912)Instructions burned: 1215 (million)
% 41.28/11.70 % (21912)------------------------------
% 41.28/11.70 % (21912)------------------------------
% 41.28/11.72 % (21917)lrs+1010_1:1_atotf=0.5:au=on:cbe=off:cnfonf=lazy_not_be_gen:fe=off:nwc=5.0:spb=units:i=1670:si=on:rtra=on_0 on DTF2THF_21372 for (2953ds/1670Mi)
% 41.28/11.76 % (21916)Instruction limit reached!
% 41.28/11.76 % (21916)------------------------------
% 41.28/11.76 % (21916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.28/11.76 % (21916)Termination reason: Unknown
% 41.28/11.76 % (21916)Termination phase: shuffling
% 41.28/11.76
% 41.28/11.76 % (21916)Memory used [KB]: 18293
% 41.28/11.76 % (21916)Time elapsed: 0.152 s
% 41.28/11.76 % (21916)Instructions burned: 432 (million)
% 41.28/11.76 % (21916)------------------------------
% 41.28/11.76 % (21916)------------------------------
% 41.28/11.77 % (21918)dis+2_1:1_au=on:hfaw=0:hfsq=on:sp=weighted_frequency:i=90:si=on:rtra=on_0 on DTF2THF_21372 for (2952ds/90Mi)
% 42.07/11.81 % (21918)Instruction limit reached!
% 42.07/11.81 % (21918)------------------------------
% 42.07/11.81 % (21918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.07/11.81 % (21918)Termination reason: Unknown
% 42.07/11.81 % (21918)Termination phase: shuffling
% 42.07/11.81
% 42.07/11.81 % (21918)Memory used [KB]: 13816
% 42.07/11.81 % (21918)Time elapsed: 0.039 s
% 42.07/11.81 % (21918)Instructions burned: 92 (million)
% 42.07/11.81 % (21918)------------------------------
% 42.07/11.81 % (21918)------------------------------
% 42.07/11.82 % (21919)dis+10_1:1_au=on:cnfonf=lazy_gen:ixr=off:plsq=on:plsqc=1:plsqr=32,1:i=346:si=on:rtra=on_0 on DTF2THF_21372 for (2952ds/346Mi)
% 42.07/11.88 % (21915)Instruction limit reached!
% 42.07/11.88 % (21915)------------------------------
% 42.07/11.88 % (21915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.07/11.88 % (21915)Termination reason: Unknown
% 42.07/11.88 % (21915)Termination phase: shuffling
% 42.07/11.88
% 42.07/11.88 % (21915)Memory used [KB]: 24562
% 42.07/11.88 % (21915)Time elapsed: 0.275 s
% 42.07/11.88 % (21915)Instructions burned: 783 (million)
% 42.07/11.88 % (21915)------------------------------
% 42.07/11.88 % (21915)------------------------------
% 42.07/11.89 % (21920)dis+1002_1:1_cnfonf=lazy_not_gen:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=401:si=on:rtra=on_0 on DTF2THF_21372 for (2951ds/401Mi)
% 42.60/11.93 % (21919)Instruction limit reached!
% 42.60/11.93 % (21919)------------------------------
% 42.60/11.93 % (21919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.60/11.93 % (21919)Termination reason: Unknown
% 42.60/11.93 % (21919)Termination phase: shuffling
% 42.60/11.93
% 42.60/11.93 % (21919)Memory used [KB]: 19573
% 42.60/11.93 % (21919)Time elapsed: 0.110 s
% 42.60/11.93 % (21919)Instructions burned: 348 (million)
% 42.60/11.93 % (21919)------------------------------
% 42.60/11.93 % (21919)------------------------------
% 42.60/11.94 % (21921)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=6454:si=on:rtra=on_0 on DTF2THF_21372 for (2950ds/6454Mi)
% 42.60/11.96 % (21904)Instruction limit reached!
% 42.60/11.96 % (21904)------------------------------
% 42.60/11.96 % (21904)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.60/11.96 % (21904)Termination reason: Unknown
% 42.60/11.96 % (21904)Termination phase: Function definition elimination
% 42.60/11.96
% 42.60/11.96 % (21904)Memory used [KB]: 34029
% 42.60/11.96 % (21904)Time elapsed: 0.928 s
% 42.60/11.96 % (21904)Instructions burned: 3375 (million)
% 42.60/11.96 % (21904)------------------------------
% 42.60/11.96 % (21904)------------------------------
% 42.60/11.97 % (21922)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:kws=arity:prag=on:i=283:si=on:rtra=on_0 on DTF2THF_21372 for (2950ds/283Mi)
% 42.60/11.99 % (21913)Instruction limit reached!
% 42.60/11.99 % (21913)------------------------------
% 42.60/11.99 % (21913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.60/11.99 % (21913)Termination reason: Unknown
% 42.60/11.99 % (21913)Termination phase: Property scanning
% 42.60/11.99
% 42.60/11.99 % (21913)Memory used [KB]: 32877
% 42.60/11.99 % (21913)Time elapsed: 0.622 s
% 42.60/11.99 % (21913)Instructions burned: 1979 (million)
% 42.60/11.99 % (21913)------------------------------
% 42.60/11.99 % (21913)------------------------------
% 42.60/12.01 % (21923)ott+10_1:1_aac=none:apa=on:au=on:cnfonf=off:ntd=on:nwc=12.0:sos=on:i=865:si=on:rtra=on_0 on DTF2THF_21372 for (2950ds/865Mi)
% 42.60/12.01 % (21920)Instruction limit reached!
% 42.60/12.01 % (21920)------------------------------
% 42.60/12.01 % (21920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.60/12.01 % (21920)Termination reason: Unknown
% 42.60/12.01 % (21920)Termination phase: shuffling
% 42.60/12.01
% 42.60/12.01 % (21920)Memory used [KB]: 20212
% 42.60/12.01 % (21920)Time elapsed: 0.124 s
% 42.60/12.01 % (21920)Instructions burned: 401 (million)
% 42.60/12.01 % (21920)------------------------------
% 42.60/12.01 % (21920)------------------------------
% 45.11/12.03 % (21924)lrs+1002_3:4_au=on:av=off:cnfonf=off:fd=off:fsr=off:ins=2:sd=1:sos=on:ss=axioms:i=4528:si=on:rtra=on_0 on DTF2THF_21372 for (2950ds/4528Mi)
% 45.11/12.05 % (21889)Instruction limit reached!
% 45.11/12.05 % (21889)------------------------------
% 45.11/12.05 % (21889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.11/12.05 % (21889)Termination reason: Unknown
% 45.11/12.05 % (21889)Termination phase: Saturation
% 45.11/12.05
% 45.11/12.05 % (21889)Memory used [KB]: 32622
% 45.11/12.05 % (21889)Time elapsed: 1.683 s
% 45.11/12.05 % (21889)Instructions burned: 5412 (million)
% 45.11/12.05 % (21889)------------------------------
% 45.11/12.05 % (21889)------------------------------
% 45.11/12.06 % (21925)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=1001:si=on:rtra=on_0 on DTF2THF_21372 for (2949ds/1001Mi)
% 45.11/12.08 % (21922)Instruction limit reached!
% 45.11/12.08 % (21922)------------------------------
% 45.11/12.08 % (21922)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.11/12.08 % (21922)Termination reason: Unknown
% 45.11/12.08 % (21922)Termination phase: shuffling
% 45.11/12.08
% 45.11/12.08 % (21922)Memory used [KB]: 16375
% 45.11/12.08 % (21922)Time elapsed: 0.103 s
% 45.11/12.08 % (21922)Instructions burned: 283 (million)
% 45.11/12.08 % (21922)------------------------------
% 45.11/12.08 % (21922)------------------------------
% 45.11/12.09 % (21926)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=102:si=on:rtra=on_0 on DTF2THF_21372 for (2949ds/102Mi)
% 45.11/12.12 % (21847)Instruction limit reached!
% 45.11/12.12 % (21847)------------------------------
% 45.11/12.12 % (21847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.11/12.12 % (21847)Termination reason: Unknown
% 45.11/12.12 % (21847)Termination phase: Saturation
% 45.11/12.12
% 45.11/12.12 % (21847)Memory used [KB]: 54625
% 45.11/12.12 % (21847)Time elapsed: 2.654 s
% 45.11/12.12 % (21847)Instructions burned: 8251 (million)
% 45.11/12.12 % (21847)------------------------------
% 45.11/12.12 % (21847)------------------------------
% 45.11/12.12 % (21926)Instruction limit reached!
% 45.11/12.12 % (21926)------------------------------
% 45.11/12.12 % (21926)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.11/12.12 % (21926)Termination reason: Unknown
% 45.11/12.12 % (21926)Termination phase: shuffling
% 45.11/12.12
% 45.11/12.12 % (21926)Memory used [KB]: 14072
% 45.11/12.12 % (21926)Time elapsed: 0.037 s
% 45.11/12.12 % (21926)Instructions burned: 102 (million)
% 45.11/12.12 % (21926)------------------------------
% 45.11/12.12 % (21926)------------------------------
% 45.52/12.13 % (21927)ott+1010_3:1_nm=20:s2a=on:s2at=3.0:sp=const_frequency:i=952:si=on:rtra=on_0 on DTF2THF_21372 for (2949ds/952Mi)
% 45.52/12.14 % (21928)lrs+1010_1:1024_add=large:c=on:cbe=off:cnfonf=lazy_gen:fde=none:nm=30:piset=and:sac=on:uhcvi=on:i=578:si=on:rtra=on_0 on DTF2THF_21372 for (2949ds/578Mi)
% 45.72/12.21 % (21917)Instruction limit reached!
% 45.72/12.21 % (21917)------------------------------
% 45.72/12.21 % (21917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.72/12.21 % (21917)Termination reason: Unknown
% 45.72/12.21 % (21917)Termination phase: Property scanning
% 45.72/12.21
% 45.72/12.21 % (21917)Memory used [KB]: 38634
% 45.72/12.21 % (21917)Time elapsed: 0.491 s
% 45.72/12.21 % (21917)Instructions burned: 1673 (million)
% 45.72/12.21 % (21917)------------------------------
% 45.72/12.21 % (21917)------------------------------
% 45.72/12.22 % (21929)dis+1010_1:4_avsq=on:avsqr=16,1:bet=on:cbe=off:cnfonf=lazy_not_be_gen:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=643:si=on:rtra=on_0 on DTF2THF_21372 for (2948ds/643Mi)
% 45.85/12.30 % (21923)Instruction limit reached!
% 45.85/12.30 % (21923)------------------------------
% 45.85/12.30 % (21923)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.85/12.30 % (21923)Termination reason: Unknown
% 45.85/12.30 % (21923)Termination phase: shuffling
% 45.85/12.30
% 45.85/12.30 % (21923)Memory used [KB]: 25458
% 45.85/12.30 % (21923)Time elapsed: 0.296 s
% 45.85/12.30 % (21923)Instructions burned: 866 (million)
% 45.85/12.30 % (21923)------------------------------
% 45.85/12.30 % (21923)------------------------------
% 45.85/12.31 % (21930)lrs+1010_1:1_cnfonf=off:ntd=on:nwc=3.0:s2a=on:sp=const_frequency:i=865:si=on:rtra=on_0 on DTF2THF_21372 for (2947ds/865Mi)
% 45.85/12.36 % (21928)Instruction limit reached!
% 45.85/12.36 % (21928)------------------------------
% 45.85/12.36 % (21928)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.85/12.36 % (21928)Termination reason: Unknown
% 45.85/12.36 % (21928)Termination phase: shuffling
% 45.85/12.36
% 45.85/12.36 % (21928)Memory used [KB]: 22771
% 45.85/12.36 % (21928)Time elapsed: 0.218 s
% 45.85/12.36 % (21928)Instructions burned: 579 (million)
% 45.85/12.36 % (21928)------------------------------
% 45.85/12.36 % (21928)------------------------------
% 45.85/12.37 % (21931)lrs+10_1:1_au=on:erd=off:s2a=on:s2at=3.0:sos=all:i=1174:si=on:rtra=on_0 on DTF2THF_21372 for (2946ds/1174Mi)
% 45.85/12.38 % (21925)Instruction limit reached!
% 45.85/12.38 % (21925)------------------------------
% 45.85/12.38 % (21925)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.85/12.38 % (21925)Termination reason: Unknown
% 45.85/12.38 % (21925)Termination phase: Property scanning
% 45.85/12.38
% 45.85/12.38 % (21925)Memory used [KB]: 23794
% 45.85/12.38 % (21925)Time elapsed: 0.322 s
% 45.85/12.38 % (21925)Instructions burned: 1004 (million)
% 45.85/12.38 % (21925)------------------------------
% 45.85/12.38 % (21925)------------------------------
% 46.83/12.40 % (21932)ott+1010_1:32_apa=on:atotf=0.3:au=on:c=on:cnfonf=off:fe=off:fsr=off:nm=6:piset=or:sac=on:sos=on:i=644:si=on:rtra=on_0 on DTF2THF_21372 for (2946ds/644Mi)
% 46.83/12.41 % (21929)Instruction limit reached!
% 46.83/12.41 % (21929)------------------------------
% 46.83/12.41 % (21929)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.83/12.41 % (21929)Termination reason: Unknown
% 46.83/12.41 % (21929)Termination phase: shuffling
% 46.83/12.41
% 46.83/12.41 % (21929)Memory used [KB]: 23666
% 46.83/12.41 % (21929)Time elapsed: 0.192 s
% 46.83/12.41 % (21929)Instructions burned: 645 (million)
% 46.83/12.41 % (21929)------------------------------
% 46.83/12.41 % (21929)------------------------------
% 46.83/12.42 % (21933)lrs+10_1:1_alpa=true:au=on:cs=on:tgt=ground:i=1201:si=on:rtra=on_0 on DTF2THF_21372 for (2946ds/1201Mi)
% 46.83/12.44 % (21927)Instruction limit reached!
% 46.83/12.44 % (21927)------------------------------
% 46.83/12.44 % (21927)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.83/12.44 % (21927)Termination reason: Unknown
% 46.83/12.44 % (21927)Termination phase: Property scanning
% 46.83/12.44
% 46.83/12.44 % (21927)Memory used [KB]: 23794
% 46.83/12.44 % (21927)Time elapsed: 0.312 s
% 46.83/12.44 % (21927)Instructions burned: 953 (million)
% 46.83/12.44 % (21927)------------------------------
% 46.83/12.44 % (21927)------------------------------
% 46.83/12.45 % (21934)lrs+1666_1:64_cnfonf=lazy_simp:hud=5:plsq=on:plsql=on:plsqr=152565,524288:tnu=2:i=1046:si=on:rtra=on_0 on DTF2THF_21372 for (2945ds/1046Mi)
% 47.56/12.61 % (21930)Instruction limit reached!
% 47.56/12.61 % (21930)------------------------------
% 47.56/12.61 % (21930)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.56/12.61 % (21930)Termination reason: Unknown
% 47.56/12.61 % (21930)Termination phase: shuffling
% 47.56/12.61
% 47.56/12.61 % (21930)Memory used [KB]: 25458
% 47.56/12.61 % (21930)Time elapsed: 0.302 s
% 47.56/12.61 % (21930)Instructions burned: 865 (million)
% 47.56/12.61 % (21930)------------------------------
% 47.56/12.61 % (21930)------------------------------
% 47.56/12.62 % (21932)Instruction limit reached!
% 47.56/12.62 % (21932)------------------------------
% 47.56/12.62 % (21932)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.56/12.62 % (21932)Termination reason: Unknown
% 47.56/12.62 % (21932)Termination phase: shuffling
% 47.56/12.62
% 47.56/12.62 % (21932)Memory used [KB]: 20724
% 47.56/12.62 % (21932)Time elapsed: 0.228 s
% 47.56/12.62 % (21932)Instructions burned: 646 (million)
% 47.56/12.62 % (21932)------------------------------
% 47.56/12.62 % (21932)------------------------------
% 47.56/12.63 % (21935)lrs+1010_1:1_c=on:cnfonf=lazy_simp:fsr=off:hud=5:kws=arity:nm=32:plsq=on:plsqc=1:plsqr=642,29:prag=on:i=222:si=on:rtra=on_0 on DTF2THF_21372 for (2943ds/222Mi)
% 47.56/12.64 % (21936)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_21372 for (2943ds/68Mi)
% 47.56/12.67 % (21936)Instruction limit reached!
% 47.56/12.67 % (21936)------------------------------
% 47.56/12.67 % (21936)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.56/12.67 % (21936)Termination reason: Unknown
% 47.56/12.67 % (21936)Termination phase: shuffling
% 47.56/12.67
% 47.56/12.67 % (21936)Memory used [KB]: 13560
% 47.56/12.67 % (21936)Time elapsed: 0.032 s
% 47.56/12.67 % (21936)Instructions burned: 68 (million)
% 47.56/12.67 % (21936)------------------------------
% 47.56/12.67 % (21936)------------------------------
% 47.56/12.68 % (21937)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=454:si=on:rtra=on_0 on DTF2THF_21372 for (2943ds/454Mi)
% 48.26/12.70 % (21935)Instruction limit reached!
% 48.26/12.70 % (21935)------------------------------
% 48.26/12.70 % (21935)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.26/12.70 % (21935)Termination reason: Unknown
% 48.26/12.70 % (21935)Termination phase: shuffling
% 48.26/12.70
% 48.26/12.70 % (21935)Memory used [KB]: 15735
% 48.26/12.70 % (21935)Time elapsed: 0.073 s
% 48.26/12.70 % (21935)Instructions burned: 223 (million)
% 48.26/12.70 % (21935)------------------------------
% 48.26/12.70 % (21935)------------------------------
% 48.26/12.71 % (21938)dis+1002_1:1_cnfonf=lazy_simp:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=360:si=on:rtra=on_0 on DTF2THF_21372 for (2943ds/360Mi)
% 48.26/12.76 % (21931)Instruction limit reached!
% 48.26/12.76 % (21931)------------------------------
% 48.26/12.76 % (21931)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.26/12.76 % (21931)Termination reason: Unknown
% 48.26/12.76 % (21931)Termination phase: Property scanning
% 48.26/12.76
% 48.26/12.76 % (21931)Memory used [KB]: 23794
% 48.26/12.76 % (21931)Time elapsed: 0.388 s
% 48.26/12.76 % (21931)Instructions burned: 1176 (million)
% 48.26/12.76 % (21931)------------------------------
% 48.26/12.76 % (21931)------------------------------
% 48.26/12.77 % (21934)Instruction limit reached!
% 48.26/12.77 % (21934)------------------------------
% 48.26/12.77 % (21934)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.26/12.77 % (21934)Termination reason: Unknown
% 48.26/12.77 % (21934)Termination phase: shuffling
% 48.26/12.77
% 48.26/12.77 % (21934)Memory used [KB]: 33133
% 48.26/12.77 % (21934)Time elapsed: 0.316 s
% 48.26/12.77 % (21934)Instructions burned: 1047 (million)
% 48.26/12.77 % (21934)------------------------------
% 48.26/12.77 % (21934)------------------------------
% 48.26/12.77 % (21939)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_21372 for (2942ds/157Mi)
% 48.26/12.78 % (21940)ott+1004_1:24_anc=none:cbe=off:cnfonf=lazy_not_gen:fe=off:piset=not:prag=on:rp=on:i=5061:si=on:rtra=on_0 on DTF2THF_21372 for (2942ds/5061Mi)
% 49.17/12.81 % (21933)Instruction limit reached!
% 49.17/12.81 % (21933)------------------------------
% 49.17/12.81 % (21933)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.17/12.81 % (21933)Termination reason: Unknown
% 49.17/12.81 % (21933)Termination phase: Property scanning
% 49.17/12.81
% 49.17/12.81 % (21933)Memory used [KB]: 23794
% 49.17/12.81 % (21933)Time elapsed: 0.384 s
% 49.17/12.81 % (21933)Instructions burned: 1201 (million)
% 49.17/12.81 % (21933)------------------------------
% 49.17/12.81 % (21933)------------------------------
% 49.17/12.81 % (21937)Instruction limit reached!
% 49.17/12.81 % (21937)------------------------------
% 49.17/12.81 % (21937)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.17/12.81 % (21937)Termination reason: Unknown
% 49.17/12.81 % (21937)Termination phase: shuffling
% 49.17/12.81
% 49.17/12.81 % (21937)Memory used [KB]: 20980
% 49.17/12.81 % (21937)Time elapsed: 0.136 s
% 49.17/12.81 % (21937)Instructions burned: 454 (million)
% 49.17/12.81 % (21937)------------------------------
% 49.17/12.81 % (21937)------------------------------
% 49.17/12.82 % (21938)Instruction limit reached!
% 49.17/12.82 % (21938)------------------------------
% 49.17/12.82 % (21938)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.17/12.82 % (21938)Termination reason: Unknown
% 49.17/12.82 % (21938)Termination phase: shuffling
% 49.17/12.82
% 49.17/12.82 % (21938)Memory used [KB]: 19829
% 49.17/12.82 % (21938)Time elapsed: 0.110 s
% 49.17/12.82 % (21938)Instructions burned: 362 (million)
% 49.17/12.82 % (21938)------------------------------
% 49.17/12.82 % (21938)------------------------------
% 49.17/12.82 % (21941)lrs+1002_1:8_av=off:fd=off:hud=5:tnu=1:i=2401:si=on:rtra=on_0 on DTF2THF_21372 for (2942ds/2401Mi)
% 49.17/12.82 % (21939)Instruction limit reached!
% 49.17/12.82 % (21939)------------------------------
% 49.17/12.82 % (21939)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.17/12.82 % (21939)Termination reason: Unknown
% 49.17/12.82 % (21939)Termination phase: shuffling
% 49.17/12.82
% 49.17/12.82 % (21939)Memory used [KB]: 14839
% 49.17/12.82 % (21939)Time elapsed: 0.054 s
% 49.17/12.82 % (21939)Instructions burned: 160 (million)
% 49.17/12.82 % (21939)------------------------------
% 49.17/12.82 % (21939)------------------------------
% 49.17/12.82 % (21942)dis+10_1:1_av=off:e2e=on:nm=4:s2a=on:i=5798:si=on:rtra=on_0 on DTF2THF_21372 for (2942ds/5798Mi)
% 49.17/12.83 % (21943)dis+10_1:1_au=on:cnfonf=lazy_gen:ixr=off:plsq=on:plsqc=1:plsqr=32,1:i=346:si=on:rtra=on_0 on DTF2THF_21372 for (2941ds/346Mi)
% 49.17/12.83 % (21944)lrs+1002_1:1_av=off:bd=off:bet=on:e2e=on:fde=unused:fe=off:plsq=on:prag=on:sos=on:i=1321:si=on:rtra=on_0 on DTF2THF_21372 for (2941ds/1321Mi)
% 49.99/12.94 % (21943)Instruction limit reached!
% 49.99/12.94 % (21943)------------------------------
% 49.99/12.94 % (21943)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.99/12.94 % (21943)Termination reason: Unknown
% 49.99/12.94 % (21943)Termination phase: shuffling
% 49.99/12.94
% 49.99/12.94 % (21943)Memory used [KB]: 19573
% 49.99/12.94 % (21943)Time elapsed: 0.106 s
% 49.99/12.94 % (21943)Instructions burned: 349 (million)
% 49.99/12.94 % (21943)------------------------------
% 49.99/12.94 % (21943)------------------------------
% 49.99/12.95 % (21945)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:i=1312:si=on:rtra=on_0 on DTF2THF_21372 for (2940ds/1312Mi)
% 54.07/13.32 % (21944)Instruction limit reached!
% 54.07/13.32 % (21944)------------------------------
% 54.07/13.32 % (21944)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.07/13.32 % (21944)Termination reason: Unknown
% 54.07/13.32 % (21944)Termination phase: Preprocessing 1
% 54.07/13.32
% 54.07/13.32 % (21944)Memory used [KB]: 25330
% 54.07/13.32 % (21944)Time elapsed: 0.488 s
% 54.07/13.32 % (21944)Instructions burned: 1323 (million)
% 54.07/13.32 % (21944)------------------------------
% 54.07/13.32 % (21944)------------------------------
% 54.07/13.33 % (21946)ott+2_8:1_add=large:c=on:cnfonf=off:e2e=on:er=filter:hud=5:plsq=on:plsqr=1,1:ss=axioms:i=672:si=on:rtra=on_0 on DTF2THF_21372 for (2937ds/672Mi)
% 54.07/13.34 % (21945)Instruction limit reached!
% 54.07/13.34 % (21945)------------------------------
% 54.07/13.34 % (21945)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.07/13.34 % (21945)Termination reason: Unknown
% 54.07/13.34 % (21945)Termination phase: shuffling
% 54.07/13.34
% 54.07/13.34 % (21945)Memory used [KB]: 36715
% 54.07/13.34 % (21945)Time elapsed: 0.395 s
% 54.07/13.34 % (21945)Instructions burned: 1313 (million)
% 54.07/13.34 % (21945)------------------------------
% 54.07/13.34 % (21945)------------------------------
% 54.07/13.35 % (21947)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=1897:si=on:rtra=on_0 on DTF2THF_21372 for (2936ds/1897Mi)
% 54.66/13.42 % (21924)Instruction limit reached!
% 54.66/13.42 % (21924)------------------------------
% 54.66/13.42 % (21924)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.66/13.42 % (21924)Termination reason: Unknown
% 54.66/13.42 % (21924)Termination phase: Saturation
% 54.66/13.42
% 54.66/13.42 % (21924)Memory used [KB]: 30319
% 54.66/13.42 % (21924)Time elapsed: 1.396 s
% 54.66/13.42 % (21924)Instructions burned: 4528 (million)
% 54.66/13.42 % (21924)------------------------------
% 54.66/13.42 % (21924)------------------------------
% 54.66/13.43 % (21948)lrs+10_1:40_av=off:fde=unused:ntd=on:plsq=on:plsqc=2:plsqr=411915,1048576:sp=unary_frequency:tgt=full:i=1202:si=on:rtra=on_0 on DTF2THF_21372 for (2935ds/1202Mi)
% 55.10/13.56 % (21941)Instruction limit reached!
% 55.10/13.56 % (21941)------------------------------
% 55.10/13.56 % (21941)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.10/13.56 % (21941)Termination reason: Unknown
% 55.10/13.56 % (21941)Termination phase: Function definition elimination
% 55.10/13.56
% 55.10/13.56 % (21941)Memory used [KB]: 32877
% 55.10/13.56 % (21941)Time elapsed: 0.746 s
% 55.10/13.56 % (21941)Instructions burned: 2401 (million)
% 55.10/13.56 % (21941)------------------------------
% 55.10/13.56 % (21941)------------------------------
% 55.10/13.57 % (21946)Instruction limit reached!
% 55.10/13.57 % (21946)------------------------------
% 55.10/13.57 % (21946)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.10/13.57 % (21946)Termination reason: Unknown
% 55.10/13.57 % (21946)Termination phase: shuffling
% 55.10/13.57
% 55.10/13.57 % (21946)Memory used [KB]: 21236
% 55.10/13.57 % (21946)Time elapsed: 0.236 s
% 55.10/13.57 % (21946)Instructions burned: 672 (million)
% 55.10/13.57 % (21946)------------------------------
% 55.10/13.57 % (21946)------------------------------
% 55.10/13.58 % (21949)dis+21_1:1_au=on:cnfonf=lazy_pi_sigma_gen:hfsq=on:hfsqc=40:hfsqr=8,1:ntd=on:sos=on:sp=occurrence:ss=axioms:i=1493:si=on:rtra=on_0 on DTF2THF_21372 for (2934ds/1493Mi)
% 55.10/13.58 % (21950)dis+1010_3:1_av=off:fde=unused:hud=10:nwc=10.0:s2a=on:ss=axioms:i=7679:si=on:rtra=on_0 on DTF2THF_21372 for (2934ds/7679Mi)
% 56.33/13.81 % (21948)Instruction limit reached!
% 56.33/13.81 % (21948)------------------------------
% 56.33/13.81 % (21948)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.33/13.81 % (21948)Termination reason: Unknown
% 56.33/13.81 % (21948)Termination phase: Property scanning
% 56.33/13.81
% 56.33/13.81 % (21948)Memory used [KB]: 23794
% 56.33/13.81 % (21948)Time elapsed: 0.376 s
% 56.33/13.81 % (21948)Instructions burned: 1205 (million)
% 56.33/13.81 % (21948)------------------------------
% 56.33/13.81 % (21948)------------------------------
% 56.33/13.82 % (21951)lrs+2_1:1024_anc=all:cnfonf=off:cs=on:e2e=on:nm=20:s2at=1.5:slsq=on:slsqr=1,2:spb=goal:i=340:si=on:rtra=on_0 on DTF2THF_21372 for (2932ds/340Mi)
% 56.33/13.89 % (21921)Instruction limit reached!
% 56.33/13.89 % (21921)------------------------------
% 56.33/13.89 % (21921)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.33/13.89 % (21921)Termination reason: Unknown
% 56.33/13.89 % (21921)Termination phase: Saturation
% 56.33/13.89
% 56.33/13.89 % (21921)Memory used [KB]: 31086
% 56.33/13.89 % (21921)Time elapsed: 1.947 s
% 56.33/13.89 % (21921)Instructions burned: 6455 (million)
% 56.33/13.89 % (21921)------------------------------
% 56.33/13.89 % (21921)------------------------------
% 57.04/13.90 % (21952)dis+1002_1:1_acc=model:au=on:nwc=5.0:sd=5:ss=axioms:st=2.5:i=3672:si=on:rtra=on_0 on DTF2THF_21372 for (2931ds/3672Mi)
% 57.04/13.94 % (21947)Instruction limit reached!
% 57.04/13.94 % (21947)------------------------------
% 57.04/13.94 % (21947)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 57.04/13.94 % (21947)Termination reason: Unknown
% 57.04/13.94 % (21947)Termination phase: Property scanning
% 57.04/13.94
% 57.04/13.94 % (21947)Memory used [KB]: 27888
% 57.04/13.94 % (21947)Time elapsed: 0.586 s
% 57.04/13.94 % (21947)Instructions burned: 1897 (million)
% 57.04/13.94 % (21947)------------------------------
% 57.04/13.94 % (21947)------------------------------
% 57.04/13.94 % (21951)Instruction limit reached!
% 57.04/13.94 % (21951)------------------------------
% 57.04/13.94 % (21951)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 57.04/13.94 % (21951)Termination reason: Unknown
% 57.04/13.94 % (21951)Termination phase: shuffling
% 57.04/13.94
% 57.04/13.94 % (21951)Memory used [KB]: 17142
% 57.04/13.94 % (21951)Time elapsed: 0.123 s
% 57.04/13.94 % (21951)Instructions burned: 342 (million)
% 57.04/13.94 % (21951)------------------------------
% 57.04/13.94 % (21951)------------------------------
% 57.04/13.95 % (21953)dis-1002_3:1_acc=model:bd=off:bsd=on:bsdmm=1:cnfonf=lazy_not_be_gen:fd=off:fe=off:ntd=on:nwc=10.0:piset=equals:plsq=on:plsqc=1:plsqr=32,1:sp=const_max:uhcvi=on:i=1535:si=on:rtra=on_0 on DTF2THF_21372 for (2930ds/1535Mi)
% 57.04/13.95 % (21954)dis+10_16:1_hud=1:plsq=on:plsqc=1:plsqr=32,1:prag=on:sd=1:sgt=64:ss=axioms:i=4637:si=on:rtra=on_0 on DTF2THF_21372 for (2930ds/4637Mi)
% 57.04/14.01 % (21949)Instruction limit reached!
% 57.04/14.01 % (21949)------------------------------
% 57.04/14.01 % (21949)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 57.04/14.01 % (21949)Termination reason: Unknown
% 57.04/14.01 % (21949)Termination phase: Property scanning
% 57.04/14.01
% 57.04/14.01 % (21949)Memory used [KB]: 38762
% 57.04/14.01 % (21949)Time elapsed: 0.436 s
% 57.04/14.01 % (21949)Instructions burned: 1494 (million)
% 57.04/14.01 % (21949)------------------------------
% 57.04/14.01 % (21949)------------------------------
% 59.16/14.03 % (21955)lrs+10_1:1_fde=unused:prag=on:sd=2:sgt=32:ss=axioms:i=3174:si=on:rtra=on_0 on DTF2THF_21372 for (2930ds/3174Mi)
% 59.16/14.06 % (21940)Instruction limit reached!
% 59.16/14.06 % (21940)------------------------------
% 59.16/14.06 % (21940)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.16/14.06 % (21940)Termination reason: Unknown
% 59.16/14.06 % (21940)Termination phase: Function definition elimination
% 59.16/14.06
% 59.16/14.06 % (21940)Memory used [KB]: 41065
% 59.16/14.06 % (21940)Time elapsed: 1.282 s
% 59.16/14.06 % (21940)Instructions burned: 5065 (million)
% 59.16/14.06 % (21940)------------------------------
% 59.16/14.06 % (21940)------------------------------
% 59.16/14.08 % (21956)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=307:si=on:rtra=on_0 on DTF2THF_21372 for (2929ds/307Mi)
% 59.75/14.17 % (21956)Instruction limit reached!
% 59.75/14.17 % (21956)------------------------------
% 59.75/14.17 % (21956)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.75/14.17 % (21956)Termination reason: Unknown
% 59.75/14.17 % (21956)Termination phase: shuffling
% 59.75/14.17
% 59.75/14.17 % (21956)Memory used [KB]: 19061
% 59.75/14.17 % (21956)Time elapsed: 0.096 s
% 59.75/14.17 % (21956)Instructions burned: 307 (million)
% 59.75/14.17 % (21956)------------------------------
% 59.75/14.17 % (21956)------------------------------
% 59.75/14.18 % (21958)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=3619:si=on:rtra=on_0 on DTF2THF_21372 for (2928ds/3619Mi)
% 60.70/14.38 % (21942)Instruction limit reached!
% 60.70/14.38 % (21942)------------------------------
% 60.70/14.38 % (21942)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.70/14.38 % (21942)Termination reason: Unknown
% 60.70/14.38 % (21942)Termination phase: Property scanning
% 60.70/14.38
% 60.70/14.38 % (21942)Memory used [KB]: 42856
% 60.70/14.38 % (21942)Time elapsed: 1.554 s
% 60.70/14.38 % (21942)Instructions burned: 5799 (million)
% 60.70/14.38 % (21942)------------------------------
% 60.70/14.38 % (21942)------------------------------
% 60.70/14.39 % (22018)lrs+1002_1:1_au=on:cbe=off:fde=none:ins=1:sp=const_max:i=2325:si=on:rtra=on_0 on DTF2THF_21372 for (2926ds/2325Mi)
% 60.70/14.40 % (21953)Instruction limit reached!
% 60.70/14.40 % (21953)------------------------------
% 60.70/14.40 % (21953)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.70/14.40 % (21953)Termination reason: Unknown
% 60.70/14.40 % (21953)Termination phase: Property scanning
% 60.70/14.40
% 60.70/14.40 % (21953)Memory used [KB]: 38634
% 60.70/14.40 % (21953)Time elapsed: 0.445 s
% 60.70/14.40 % (21953)Instructions burned: 1538 (million)
% 60.70/14.40 % (21953)------------------------------
% 60.70/14.40 % (21953)------------------------------
% 60.74/14.41 % (22031)dis+1002_1:1_au=on:sac=on:sd=5:slsq=on:slsqr=1,4:ss=axioms:st=2.0:i=2341:si=on:rtra=on_0 on DTF2THF_21372 for (2926ds/2341Mi)
% 62.42/14.84 % (21950)First to succeed.
% 62.42/14.85 % (21950)Refutation found. Thanks to Tanya!
% 62.42/14.85 % SZS status Theorem for DTF2THF_21372
% 62.42/14.85 % SZS output start Proof for DTF2THF_21372
% 62.42/14.85 thf(type_def_5, type, rat: $tType).
% 62.42/14.85 thf(type_def_6, type, int: $tType).
% 62.42/14.85 thf(type_def_8, type, real: $tType).
% 62.42/14.85 thf(type_def_9, type, nat: $tType).
% 62.42/14.85 thf(type_def_10, type, num: $tType).
% 62.42/14.85 thf(type_def_11, type, product_prod_int_int: $tType).
% 62.42/14.85 thf(type_def_12, type, product_prod_nat_nat: $tType).
% 62.42/14.85 thf(type_def_13, type, complex: $tType).
% 62.42/14.85 thf(type_def_14, type, option_num: $tType).
% 62.42/14.85 thf(type_def_15, type, product_prod_num_num: $tType).
% 62.42/14.85 thf(type_def_16, type, code_integer: $tType).
% 62.42/14.85 thf(type_def_17, type, produc6271795597528267376eger_o: $tType).
% 62.42/14.85 thf(type_def_18, type, produc8923325533196201883nteger: $tType).
% 62.42/14.85 thf(type_def_19, type, set_Extended_enat: $tType).
% 62.42/14.85 thf(type_def_20, type, extended_enat: $tType).
% 62.42/14.85 thf(type_def_21, type, set_fi4554929511873752355omplex: $tType).
% 62.42/14.85 thf(type_def_22, type, filter6041513312241820739omplex: $tType).
% 62.42/14.85 thf(type_def_23, type, set_fi7789364187291644575l_real: $tType).
% 62.42/14.85 thf(type_def_24, type, filter2146258269922977983l_real: $tType).
% 62.42/14.85 thf(type_def_25, type, set_real: $tType).
% 62.42/14.85 thf(type_def_26, type, set_set_nat: $tType).
% 62.42/14.85 thf(type_def_27, type, set_nat: $tType).
% 62.42/14.85 thf(type_def_28, type, filter_real: $tType).
% 62.42/14.85 thf(type_def_29, type, filter_nat: $tType).
% 62.42/14.85 thf(type_def_30, type, filter1242075044329608583at_nat: $tType).
% 62.42/14.85 thf(type_def_31, type, set_Pr5085853215250843933omplex: $tType).
% 62.42/14.85 thf(type_def_32, type, set_Pr6218003697084177305l_real: $tType).
% 62.42/14.85 thf(type_def_33, type, set_o: $tType).
% 62.42/14.85 thf(type_def_34, type, set_complex: $tType).
% 62.42/14.85 thf(type_def_35, type, set_int: $tType).
% 62.42/14.85 thf(type_def_36, type, set_list_nat: $tType).
% 62.42/14.85 thf(type_def_37, type, set_Product_unit: $tType).
% 62.42/14.85 thf(type_def_38, type, set_char: $tType).
% 62.42/14.85 thf(type_def_39, type, char: $tType).
% 62.42/14.85 thf(type_def_40, type, set_Pr8693737435421807431at_nat: $tType).
% 62.42/14.85 thf(type_def_41, type, list_nat: $tType).
% 62.42/14.85 thf(type_def_42, type, literal: $tType).
% 62.42/14.85 thf(type_def_43, type, list_o: $tType).
% 62.42/14.85 thf(type_def_44, type, list_int: $tType).
% 62.42/14.85 thf(type_def_45, type, option_nat: $tType).
% 62.42/14.85 thf(type_def_46, type, vEBT_VEBT: $tType).
% 62.42/14.85 thf(type_def_47, type, set_Pr958786334691620121nt_int: $tType).
% 62.42/14.85 thf(type_def_48, type, set_Code_integer: $tType).
% 62.42/14.85 thf(type_def_49, type, set_rat: $tType).
% 62.42/14.85 thf(type_def_50, type, set_Pr1261947904930325089at_nat: $tType).
% 62.42/14.85 thf(type_def_51, type, list_VEBT_VEBT: $tType).
% 62.42/14.85 thf(type_def_52, type, list_complex: $tType).
% 62.42/14.85 thf(type_def_53, type, list_Extended_enat: $tType).
% 62.42/14.85 thf(type_def_54, type, list_list_nat: $tType).
% 62.42/14.85 thf(type_def_55, type, list_real: $tType).
% 62.42/14.85 thf(type_def_56, type, set_VEBT_VEBT: $tType).
% 62.42/14.85 thf(type_def_57, type, list_num: $tType).
% 62.42/14.85 thf(type_def_58, type, list_P4002435161011370285od_o_o: $tType).
% 62.42/14.85 thf(type_def_59, type, product_prod_o_o: $tType).
% 62.42/14.85 thf(type_def_60, type, list_P3795440434834930179_o_int: $tType).
% 62.42/14.85 thf(type_def_61, type, product_prod_o_int: $tType).
% 62.42/14.85 thf(type_def_62, type, list_P6285523579766656935_o_nat: $tType).
% 62.42/14.85 thf(type_def_63, type, product_prod_o_nat: $tType).
% 62.42/14.85 thf(type_def_64, type, list_P7495141550334521929T_VEBT: $tType).
% 62.42/14.85 thf(type_def_65, type, produc2504756804600209347T_VEBT: $tType).
% 62.42/14.85 thf(type_def_66, type, list_P1726324292696863441at_num: $tType).
% 62.42/14.85 thf(type_def_67, type, product_prod_nat_num: $tType).
% 62.42/14.85 thf(type_def_68, type, list_P3744719386663036955um_num: $tType).
% 62.42/14.85 thf(type_def_69, type, list_P3126845725202233233VEBT_o: $tType).
% 62.42/14.85 thf(type_def_70, type, produc334124729049499915VEBT_o: $tType).
% 62.42/14.85 thf(type_def_71, type, list_P4547456442757143711BT_int: $tType).
% 62.42/14.85 thf(type_def_72, type, produc4894624898956917775BT_int: $tType).
% 62.42/14.85 thf(type_def_73, type, list_P7037539587688870467BT_nat: $tType).
% 62.42/14.85 thf(type_def_74, type, produc9072475918466114483BT_nat: $tType).
% 62.42/14.85 thf(type_def_75, type, list_P7413028617227757229T_VEBT: $tType).
% 62.42/14.85 thf(type_def_76, type, produc8243902056947475879T_VEBT: $tType).
% 62.42/14.85 thf(type_def_77, type, list_P7333126701944960589_nat_o: $tType).
% 62.42/14.85 thf(type_def_78, type, list_P5647936690300460905T_VEBT: $tType).
% 62.42/14.85 thf(type_def_79, type, option4927543243414619207at_nat: $tType).
% 62.42/14.85 thf(type_def_80, type, set_num: $tType).
% 62.42/14.85 thf(type_def_81, type, produc7248412053542808358at_nat: $tType).
% 62.42/14.85 thf(type_def_82, type, produc4471711990508489141at_nat: $tType).
% 62.42/14.85 thf(type_def_83, type, produc2963631642982155120at_num: $tType).
% 62.42/14.85 thf(type_def_84, type, produc3368934014287244435at_num: $tType).
% 62.42/14.85 thf(type_def_85, type, produc859450856879609959at_nat: $tType).
% 62.42/14.85 thf(type_def_86, type, produc7272778201969148633d_enat: $tType).
% 62.42/14.85 thf(type_def_87, type, produc4411394909380815293omplex: $tType).
% 62.42/14.85 thf(type_def_88, type, produc2422161461964618553l_real: $tType).
% 62.42/14.85 thf(type_def_89, type, product_unit: $tType).
% 62.42/14.85 thf(func_def_0, type, produc3368934014287244435at_num: $tType).
% 62.42/14.85 thf(func_def_1, type, produc4471711990508489141at_nat: $tType).
% 62.42/14.85 thf(func_def_2, type, set_Pr8693737435421807431at_nat: $tType).
% 62.42/14.85 thf(func_def_3, type, produc859450856879609959at_nat: $tType).
% 62.42/14.85 thf(func_def_4, type, set_fi4554929511873752355omplex: $tType).
% 62.42/14.85 thf(func_def_5, type, list_P7413028617227757229T_VEBT: $tType).
% 62.42/14.85 thf(func_def_6, type, produc2963631642982155120at_num: $tType).
% 62.42/14.85 thf(func_def_7, type, produc7248412053542808358at_nat: $tType).
% 62.42/14.85 thf(func_def_8, type, set_fi7789364187291644575l_real: $tType).
% 62.42/14.85 thf(func_def_9, type, filter6041513312241820739omplex: $tType).
% 62.42/14.85 thf(func_def_10, type, list_P7037539587688870467BT_nat: $tType).
% 62.42/14.85 thf(func_def_11, type, list_P4547456442757143711BT_int: $tType).
% 62.42/14.85 thf(func_def_12, type, list_P5647936690300460905T_VEBT: $tType).
% 62.42/14.85 thf(func_def_13, type, produc8243902056947475879T_VEBT: $tType).
% 62.42/14.85 thf(func_def_14, type, set_Pr5085853215250843933omplex: $tType).
% 62.42/14.85 thf(func_def_15, type, produc8923325533196201883nteger: $tType).
% 62.42/14.85 thf(func_def_16, type, produc7272778201969148633d_enat: $tType).
% 62.42/14.85 thf(func_def_17, type, list_P3126845725202233233VEBT_o: $tType).
% 62.42/14.85 thf(func_def_18, type, list_P7495141550334521929T_VEBT: $tType).
% 62.42/14.85 thf(func_def_19, type, filter2146258269922977983l_real: $tType).
% 62.42/14.85 thf(func_def_20, type, option4927543243414619207at_nat: $tType).
% 62.42/14.85 thf(func_def_21, type, filter1242075044329608583at_nat: $tType).
% 62.42/14.85 thf(func_def_22, type, set_Pr6218003697084177305l_real: $tType).
% 62.42/14.85 thf(func_def_23, type, list_P3744719386663036955um_num: $tType).
% 62.42/14.85 thf(func_def_24, type, list_P1726324292696863441at_num: $tType).
% 62.42/14.85 thf(func_def_25, type, produc9072475918466114483BT_nat: $tType).
% 62.42/14.85 thf(func_def_26, type, produc4894624898956917775BT_int: $tType).
% 62.42/14.85 thf(func_def_27, type, set_Pr1261947904930325089at_nat: $tType).
% 62.42/14.85 thf(func_def_28, type, set_Pr958786334691620121nt_int: $tType).
% 62.42/14.85 thf(func_def_29, type, produc4411394909380815293omplex: $tType).
% 62.42/14.85 thf(func_def_30, type, list_P7333126701944960589_nat_o: $tType).
% 62.42/14.85 thf(func_def_31, type, list_P6285523579766656935_o_nat: $tType).
% 62.42/14.85 thf(func_def_32, type, list_P3795440434834930179_o_int: $tType).
% 62.42/14.85 thf(func_def_33, type, produc334124729049499915VEBT_o: $tType).
% 62.42/14.85 thf(func_def_34, type, produc2504756804600209347T_VEBT: $tType).
% 62.42/14.85 thf(func_def_35, type, produc6271795597528267376eger_o: $tType).
% 62.42/14.85 thf(func_def_36, type, produc2422161461964618553l_real: $tType).
% 62.42/14.85 thf(func_def_37, type, product_prod_num_num: $tType).
% 62.42/14.85 thf(func_def_38, type, product_prod_nat_num: $tType).
% 62.42/14.85 thf(func_def_39, type, product_prod_nat_nat: $tType).
% 62.42/14.85 thf(func_def_40, type, product_prod_int_int: $tType).
% 62.42/14.85 thf(func_def_41, type, list_P4002435161011370285od_o_o: $tType).
% 62.42/14.85 thf(func_def_42, type, list_list_nat: $tType).
% 62.42/14.85 thf(func_def_43, type, list_VEBT_VEBT: $tType).
% 62.42/14.85 thf(func_def_44, type, set_list_nat: $tType).
% 62.42/14.85 thf(func_def_45, type, product_prod_o_nat: $tType).
% 62.42/14.85 thf(func_def_46, type, product_prod_o_int: $tType).
% 62.42/14.85 thf(func_def_47, type, set_VEBT_VEBT: $tType).
% 62.42/14.85 thf(func_def_48, type, set_set_nat: $tType).
% 62.42/14.85 thf(func_def_49, type, set_Code_integer: $tType).
% 62.42/14.85 thf(func_def_50, type, list_Extended_enat: $tType).
% 62.42/14.85 thf(func_def_51, type, set_Product_unit: $tType).
% 62.42/14.85 thf(func_def_52, type, set_Extended_enat: $tType).
% 62.42/14.85 thf(func_def_53, type, list_complex: $tType).
% 62.42/14.85 thf(func_def_54, type, product_prod_o_o: $tType).
% 62.42/14.85 thf(func_def_55, type, set_complex: $tType).
% 62.42/14.85 thf(func_def_56, type, filter_real: $tType).
% 62.42/14.85 thf(func_def_57, type, option_num: $tType).
% 62.42/14.85 thf(func_def_58, type, option_nat: $tType).
% 62.42/14.85 thf(func_def_59, type, filter_nat: $tType).
% 62.42/14.85 thf(func_def_60, type, set_char: $tType).
% 62.42/14.85 thf(func_def_61, type, list_real: $tType).
% 62.42/14.85 thf(func_def_62, type, set_real: $tType).
% 62.42/14.85 thf(func_def_63, type, list_num: $tType).
% 62.42/14.85 thf(func_def_64, type, list_nat: $tType).
% 62.42/14.85 thf(func_def_65, type, list_int: $tType).
% 62.42/14.85 thf(func_def_66, type, vEBT_VEBT: $tType).
% 62.42/14.85 thf(func_def_67, type, set_rat: $tType).
% 62.42/14.85 thf(func_def_68, type, set_num: $tType).
% 62.42/14.85 thf(func_def_69, type, set_nat: $tType).
% 62.42/14.85 thf(func_def_70, type, set_int: $tType).
% 62.42/14.85 thf(func_def_71, type, code_integer: $tType).
% 62.42/14.85 thf(func_def_72, type, product_unit: $tType).
% 62.42/14.85 thf(func_def_73, type, extended_enat: $tType).
% 62.42/14.85 thf(func_def_74, type, list_o: $tType).
% 62.42/14.85 thf(func_def_75, type, complex: $tType).
% 62.42/14.85 thf(func_def_76, type, literal: $tType).
% 62.42/14.85 thf(func_def_77, type, set_o: $tType).
% 62.42/14.85 thf(func_def_78, type, char: $tType).
% 62.42/14.85 thf(func_def_79, type, real: $tType).
% 62.42/14.85 thf(func_def_80, type, rat: $tType).
% 62.42/14.85 thf(func_def_81, type, num: $tType).
% 62.42/14.85 thf(func_def_82, type, nat: $tType).
% 62.42/14.85 thf(func_def_83, type, int: $tType).
% 62.42/14.85 thf(func_def_84, type, archim2889992004027027881ng_rat: rat > int).
% 62.42/14.85 thf(func_def_85, type, archim7802044766580827645g_real: real > int).
% 62.42/14.85 thf(func_def_86, type, archim3151403230148437115or_rat: rat > int).
% 62.42/14.85 thf(func_def_87, type, archim6058952711729229775r_real: real > int).
% 62.42/14.85 thf(func_def_88, type, archimedean_frac_rat: rat > rat).
% 62.42/14.85 thf(func_def_89, type, archim2898591450579166408c_real: real > real).
% 62.42/14.85 thf(func_def_90, type, archim7778729529865785530nd_rat: rat > int).
% 62.42/14.85 thf(func_def_91, type, archim8280529875227126926d_real: real > int).
% 62.42/14.85 thf(func_def_92, type, bNF_re1962705104956426057at_rat: ((nat > rat) > (nat > rat) > $o) > (((nat > rat) > nat > rat) > ((nat > rat) > nat > rat) > $o) > ((nat > rat) > (nat > rat) > nat > rat) > ((nat > rat) > (nat > rat) > nat > rat) > $o).
% 62.42/14.85 thf(func_def_93, type, bNF_re895249473297799549at_rat: ((nat > rat) > (nat > rat) > $o) > ((nat > rat) > (nat > rat) > $o) > ((nat > rat) > nat > rat) > ((nat > rat) > nat > rat) > $o).
% 62.42/14.85 thf(func_def_94, type, bNF_re728719798268516973at_o_o: ((nat > rat) > (nat > rat) > $o) > ($o > $o > $o) > ((nat > rat) > $o) > ((nat > rat) > $o) > $o).
% 62.42/14.85 thf(func_def_95, type, bNF_re4695409256820837752l_real: ((nat > rat) > real > $o) > (((nat > rat) > nat > rat) > (real > real) > $o) > ((nat > rat) > (nat > rat) > nat > rat) > (real > real > real) > $o).
% 62.42/14.85 thf(func_def_96, type, bNF_re4521903465945308077real_o: ((nat > rat) > real > $o) > (((nat > rat) > $o) > (real > $o) > $o) > ((nat > rat) > (nat > rat) > $o) > (real > real > $o) > $o).
% 62.42/14.85 thf(func_def_97, type, bNF_re3023117138289059399t_real: ((nat > rat) > real > $o) > ((nat > rat) > real > $o) > ((nat > rat) > nat > rat) > (real > real) > $o).
% 62.42/14.85 thf(func_def_98, type, bNF_re4297313714947099218al_o_o: ((nat > rat) > real > $o) > ($o > $o > $o) > ((nat > rat) > $o) > (real > $o) > $o).
% 62.42/14.85 thf(func_def_99, type, bNF_re3403563459893282935_int_o: (int > int > $o) > ((int > $o) > (int > $o) > $o) > (int > int > $o) > (int > int > $o) > $o).
% 62.42/14.85 thf(func_def_100, type, bNF_re711492959462206631nt_int: (int > int > $o) > ((int > int) > (int > int) > $o) > (int > int > int) > (int > int > int) > $o).
% 62.42/14.85 thf(func_def_101, type, bNF_re5089333283451836215nt_o_o: (int > int > $o) > ($o > $o > $o) > (int > $o) > (int > $o) > $o).
% 62.42/14.85 thf(func_def_102, type, bNF_re4712519889275205905nt_int: (int > int > $o) > (int > int > $o) > (int > int) > (int > int) > $o).
% 62.42/14.85 thf(func_def_103, type, bNF_re578469030762574527_nat_o: (nat > nat > $o) > ((nat > $o) > (nat > $o) > $o) > (nat > nat > $o) > (nat > nat > $o) > $o).
% 62.42/14.85 thf(func_def_104, type, bNF_re1345281282404953727at_nat: (nat > nat > $o) > ((nat > nat) > (nat > nat) > $o) > (nat > nat > nat) > (nat > nat > nat) > $o).
% 62.42/14.85 thf(func_def_105, type, bNF_re4705727531993890431at_o_o: (nat > nat > $o) > ($o > $o > $o) > (nat > $o) > (nat > $o) > $o).
% 62.42/14.85 thf(func_def_106, type, bNF_re5653821019739307937at_nat: (nat > nat > $o) > (nat > nat > $o) > (nat > nat) > (nat > nat) > $o).
% 62.42/14.85 thf(func_def_107, type, bNF_re8402795839162346335um_int: (num > num > $o) > ((num > int) > (num > int) > $o) > (num > num > int) > (num > num > int) > $o).
% 62.42/14.85 thf(func_def_108, type, bNF_re1822329894187522285nt_int: (num > num > $o) > (int > int > $o) > (num > int) > (num > int) > $o).
% 62.42/14.85 thf(func_def_109, type, bNF_re5228765855967844073nt_int: (product_prod_int_int > product_prod_int_int > $o) > ((product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int) > $o) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > $o).
% 62.42/14.85 thf(func_def_110, type, bNF_re8699439704749558557nt_o_o: (product_prod_int_int > product_prod_int_int > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (product_prod_int_int > $o) > $o).
% 62.42/14.85 thf(func_def_111, type, bNF_re7145576690424134365nt_int: (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int) > $o).
% 62.42/14.85 thf(func_def_112, type, bNF_re7627151682743391978at_rat: (product_prod_int_int > rat > $o) > ((product_prod_int_int > product_prod_int_int) > (rat > rat) > $o) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > (rat > rat > rat) > $o).
% 62.42/14.85 thf(func_def_113, type, bNF_re1494630372529172596at_o_o: (product_prod_int_int > rat > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (rat > $o) > $o).
% 62.42/14.85 thf(func_def_114, type, bNF_re8279943556446156061nt_rat: (product_prod_int_int > rat > $o) > (product_prod_int_int > rat > $o) > (product_prod_int_int > product_prod_int_int) > (rat > rat) > $o).
% 62.42/14.85 thf(func_def_115, type, bNF_re717283939379294677_int_o: (product_prod_nat_nat > int > $o) > ((product_prod_nat_nat > $o) > (int > $o) > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (int > int > $o) > $o).
% 62.42/14.85 thf(func_def_116, type, bNF_re7408651293131936558nt_int: (product_prod_nat_nat > int > $o) > ((product_prod_nat_nat > product_prod_nat_nat) > (int > int) > $o) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > (int > int > int) > $o).
% 62.42/14.85 thf(func_def_117, type, bNF_re6644619430987730960nt_o_o: (product_prod_nat_nat > int > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (int > $o) > $o).
% 62.42/14.85 thf(func_def_118, type, bNF_re4555766996558763186at_nat: (product_prod_nat_nat > int > $o) > (nat > nat > $o) > (product_prod_nat_nat > nat) > (int > nat) > $o).
% 62.42/14.85 thf(func_def_119, type, bNF_re7400052026677387805at_int: (product_prod_nat_nat > int > $o) > (product_prod_nat_nat > int > $o) > (product_prod_nat_nat > product_prod_nat_nat) > (int > int) > $o).
% 62.42/14.85 thf(func_def_120, type, bNF_re4202695980764964119_nat_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > ((product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > $o).
% 62.42/14.85 thf(func_def_121, type, bNF_re3099431351363272937at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > ((product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat) > $o) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > $o).
% 62.42/14.85 thf(func_def_122, type, bNF_re3666534408544137501at_o_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 62.42/14.85 thf(func_def_123, type, bNF_re8246922863344978751at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > (nat > nat > $o) > (product_prod_nat_nat > nat) > (product_prod_nat_nat > nat) > $o).
% 62.42/14.85 thf(func_def_124, type, bNF_re2241393799969408733at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat) > $o).
% 62.42/14.85 thf(func_def_125, type, binomial: nat > nat > nat).
% 62.42/14.85 thf(func_def_126, type, gbinomial_complex: complex > nat > complex).
% 62.42/14.85 thf(func_def_127, type, gbinomial_int: int > nat > int).
% 62.42/14.85 thf(func_def_128, type, gbinomial_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_129, type, gbinomial_rat: rat > nat > rat).
% 62.42/14.85 thf(func_def_130, type, gbinomial_real: real > nat > real).
% 62.42/14.85 thf(func_def_131, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 62.42/14.85 thf(func_def_132, type, bit_and_not_num: num > num > option_num).
% 62.42/14.85 thf(func_def_133, type, bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o).
% 62.42/14.85 thf(func_def_134, type, bit_concat_bit: nat > int > int > int).
% 62.42/14.85 thf(func_def_135, type, bit_or_not_num_neg: num > num > num).
% 62.42/14.85 thf(func_def_136, type, bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o).
% 62.42/14.85 thf(func_def_137, type, bit_ri7632146776885996613nteger: code_integer > code_integer).
% 62.42/14.85 thf(func_def_138, type, bit_ri7919022796975470100ot_int: int > int).
% 62.42/14.85 thf(func_def_139, type, bit_ri6519982836138164636nteger: nat > code_integer > code_integer).
% 62.42/14.85 thf(func_def_140, type, bit_ri631733984087533419it_int: nat > int > int).
% 62.42/14.85 thf(func_def_141, type, bit_se3949692690581998587nteger: code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_142, type, bit_se725231765392027082nd_int: int > int > int).
% 62.42/14.85 thf(func_def_143, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_144, type, bit_se8568078237143864401it_int: nat > int > int).
% 62.42/14.85 thf(func_def_145, type, bit_se8570568707652914677it_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_146, type, bit_se1345352211410354436nteger: nat > code_integer > code_integer).
% 62.42/14.85 thf(func_def_147, type, bit_se2159334234014336723it_int: nat > int > int).
% 62.42/14.85 thf(func_def_148, type, bit_se2161824704523386999it_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_149, type, bit_se2119862282449309892nteger: nat > code_integer).
% 62.42/14.85 thf(func_def_150, type, bit_se2000444600071755411sk_int: nat > int).
% 62.42/14.85 thf(func_def_151, type, bit_se2002935070580805687sk_nat: nat > nat).
% 62.42/14.85 thf(func_def_152, type, bit_se1409905431419307370or_int: int > int > int).
% 62.42/14.85 thf(func_def_153, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_154, type, bit_se7788150548672797655nteger: nat > code_integer > code_integer).
% 62.42/14.85 thf(func_def_155, type, bit_se545348938243370406it_int: nat > int > int).
% 62.42/14.85 thf(func_def_156, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_157, type, bit_se2793503036327961859nteger: nat > code_integer > code_integer).
% 62.42/14.85 thf(func_def_158, type, bit_se7879613467334960850it_int: nat > int > int).
% 62.42/14.85 thf(func_def_159, type, bit_se7882103937844011126it_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_160, type, bit_se1745604003318907178nteger: nat > code_integer > code_integer).
% 62.42/14.85 thf(func_def_161, type, bit_se2923211474154528505it_int: nat > int > int).
% 62.42/14.85 thf(func_def_162, type, bit_se2925701944663578781it_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_163, type, bit_se8260200283734997820nteger: nat > code_integer > code_integer).
% 62.42/14.85 thf(func_def_164, type, bit_se4203085406695923979it_int: nat > int > int).
% 62.42/14.85 thf(func_def_165, type, bit_se4205575877204974255it_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_166, type, bit_se3222712562003087583nteger: code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_167, type, bit_se6526347334894502574or_int: int > int > int).
% 62.42/14.85 thf(func_def_168, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_169, type, bit_se9216721137139052372nteger: code_integer > nat > $o).
% 62.42/14.85 thf(func_def_170, type, bit_se1146084159140164899it_int: int > nat > $o).
% 62.42/14.85 thf(func_def_171, type, bit_se1148574629649215175it_nat: nat > nat > $o).
% 62.42/14.85 thf(func_def_172, type, bit_take_bit_num: nat > num > option_num).
% 62.42/14.85 thf(func_def_173, type, bit_un1837492267222099188nd_num: num > num > option_num).
% 62.42/14.85 thf(func_def_174, type, bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o).
% 62.42/14.85 thf(func_def_175, type, bit_un6178654185764691216or_num: num > num > option_num).
% 62.42/14.85 thf(func_def_176, type, bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o).
% 62.42/14.85 thf(func_def_177, type, bit_un7362597486090784418nd_num: num > num > option_num).
% 62.42/14.85 thf(func_def_178, type, bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o).
% 62.42/14.85 thf(func_def_179, type, bit_un2480387367778600638or_num: num > num > option_num).
% 62.42/14.85 thf(func_def_180, type, bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o).
% 62.42/14.85 thf(func_def_181, type, code_bit_cut_integer: code_integer > produc6271795597528267376eger_o).
% 62.42/14.85 thf(func_def_182, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 62.42/14.85 thf(func_def_183, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 62.42/14.85 thf(func_def_184, type, code_int_of_integer: code_integer > int).
% 62.42/14.85 thf(func_def_185, type, code_integer_of_int: int > code_integer).
% 62.42/14.85 thf(func_def_186, type, code_integer_of_num: num > code_integer).
% 62.42/14.85 thf(func_def_187, type, code_nat_of_integer: code_integer > nat).
% 62.42/14.85 thf(func_def_188, type, code_num_of_integer: code_integer > num).
% 62.42/14.85 thf(func_def_189, type, comple2295165028678016749d_enat: set_Extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_190, type, comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex).
% 62.42/14.85 thf(func_def_191, type, comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real).
% 62.42/14.85 thf(func_def_192, type, comple4887499456419720421f_real: set_real > real).
% 62.42/14.85 thf(func_def_193, type, comple7806235888213564991et_nat: set_set_nat > set_nat).
% 62.42/14.85 thf(func_def_194, type, comple4398354569131411667d_enat: set_Extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_195, type, comple1385675409528146559p_real: set_real > real).
% 62.42/14.85 thf(func_def_196, type, comple7399068483239264473et_nat: set_set_nat > set_nat).
% 62.42/14.85 thf(func_def_197, type, arg: complex > real).
% 62.42/14.85 thf(func_def_198, type, cis: real > complex).
% 62.42/14.85 thf(func_def_199, type, cnj: complex > complex).
% 62.42/14.85 thf(func_def_200, type, complex2: real > real > complex).
% 62.42/14.85 thf(func_def_201, type, im: complex > real).
% 62.42/14.85 thf(func_def_202, type, re: complex > real).
% 62.42/14.85 thf(func_def_203, type, csqrt: complex > complex).
% 62.42/14.85 thf(func_def_204, type, imaginary_unit: complex).
% 62.42/14.85 thf(func_def_205, type, rcis: real > real > complex).
% 62.42/14.85 thf(func_def_206, type, differ6690327859849518006l_real: (real > real) > filter_real > $o).
% 62.42/14.85 thf(func_def_207, type, has_de1759254742604945161l_real: (real > real) > (real > real) > filter_real > $o).
% 62.42/14.85 thf(func_def_208, type, has_fi5821293074295781190e_real: (real > real) > real > filter_real > $o).
% 62.42/14.85 thf(func_def_209, type, has_ve631408500373753343e_real: (real > real) > real > filter_real > $o).
% 62.42/14.85 thf(func_def_210, type, adjust_div: product_prod_int_int > int).
% 62.42/14.85 thf(func_def_211, type, adjust_mod: int > int > int).
% 62.42/14.85 thf(func_def_212, type, divmod_nat: nat > nat > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_213, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 62.42/14.85 thf(func_def_214, type, unique5706413561485394159nteger: produc8923325533196201883nteger > $o).
% 62.42/14.85 thf(func_def_215, type, unique6319869463603278526ux_int: product_prod_int_int > $o).
% 62.42/14.85 thf(func_def_216, type, unique6322359934112328802ux_nat: product_prod_nat_nat > $o).
% 62.42/14.85 thf(func_def_217, type, unique3479559517661332726nteger: num > num > produc8923325533196201883nteger).
% 62.42/14.85 thf(func_def_218, type, unique5052692396658037445od_int: num > num > product_prod_int_int).
% 62.42/14.85 thf(func_def_219, type, unique5055182867167087721od_nat: num > num > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_220, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 62.42/14.85 thf(func_def_221, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 62.42/14.85 thf(func_def_222, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_223, type, euclid4774559944035922753ze_int: int > nat).
% 62.42/14.85 thf(func_def_224, type, euclid4777050414544973029ze_nat: nat > nat).
% 62.42/14.85 thf(func_def_225, type, euclid3395696857347342551nt_int: int > int).
% 62.42/14.85 thf(func_def_226, type, euclid3398187327856392827nt_nat: nat > nat).
% 62.42/14.85 thf(func_def_227, type, extended_eSuc: extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_228, type, extended_enat2: nat > extended_enat).
% 62.42/14.85 thf(func_def_229, type, extended_case_enat_o: (nat > $o) > $o > extended_enat > $o).
% 62.42/14.85 thf(func_def_230, type, extend3600170679010898289d_enat: (nat > extended_enat) > extended_enat > extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_231, type, extend5688581933313929465d_enat: extended_enat).
% 62.42/14.85 thf(func_def_232, type, extended_the_enat: extended_enat > nat).
% 62.42/14.85 thf(func_def_233, type, comm_s8582702949713902594nteger: code_integer > nat > code_integer).
% 62.42/14.85 thf(func_def_234, type, comm_s2602460028002588243omplex: complex > nat > complex).
% 62.42/14.85 thf(func_def_235, type, comm_s4660882817536571857er_int: int > nat > int).
% 62.42/14.85 thf(func_def_236, type, comm_s4663373288045622133er_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_237, type, comm_s4028243227959126397er_rat: rat > nat > rat).
% 62.42/14.85 thf(func_def_238, type, comm_s7457072308508201937r_real: real > nat > real).
% 62.42/14.85 thf(func_def_239, type, semiri3624122377584611663nteger: nat > code_integer).
% 62.42/14.85 thf(func_def_240, type, semiri5044797733671781792omplex: nat > complex).
% 62.42/14.85 thf(func_def_241, type, semiri1406184849735516958ct_int: nat > int).
% 62.42/14.85 thf(func_def_242, type, semiri1408675320244567234ct_nat: nat > nat).
% 62.42/14.85 thf(func_def_243, type, semiri773545260158071498ct_rat: nat > rat).
% 62.42/14.85 thf(func_def_244, type, semiri2265585572941072030t_real: nat > real).
% 62.42/14.85 thf(func_def_245, type, invers8013647133539491842omplex: complex > complex).
% 62.42/14.85 thf(func_def_246, type, inverse_inverse_rat: rat > rat).
% 62.42/14.85 thf(func_def_247, type, inverse_inverse_real: real > real).
% 62.42/14.85 thf(func_def_248, type, at_bot_real: filter_real).
% 62.42/14.85 thf(func_def_249, type, at_top_nat: filter_nat).
% 62.42/14.85 thf(func_def_250, type, at_top_real: filter_real).
% 62.42/14.85 thf(func_def_251, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 62.42/14.85 thf(func_def_252, type, eventu1038000079068216329at_nat: (product_prod_nat_nat > $o) > filter1242075044329608583at_nat > $o).
% 62.42/14.85 thf(func_def_253, type, eventually_real: (real > $o) > filter_real > $o).
% 62.42/14.85 thf(func_def_254, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 62.42/14.85 thf(func_def_255, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 62.42/14.85 thf(func_def_256, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 62.42/14.85 thf(func_def_257, type, filtermap_real_real: (real > real) > filter_real > filter_real).
% 62.42/14.85 thf(func_def_258, type, princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex).
% 62.42/14.85 thf(func_def_259, type, princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real).
% 62.42/14.85 thf(func_def_260, type, prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat).
% 62.42/14.85 thf(func_def_261, type, finite_card_o: set_o > nat).
% 62.42/14.85 thf(func_def_262, type, finite_card_complex: set_complex > nat).
% 62.42/14.85 thf(func_def_263, type, finite_card_int: set_int > nat).
% 62.42/14.85 thf(func_def_264, type, finite_card_list_nat: set_list_nat > nat).
% 62.42/14.85 thf(func_def_265, type, finite_card_nat: set_nat > nat).
% 62.42/14.85 thf(func_def_266, type, finite410649719033368117t_unit: set_Product_unit > nat).
% 62.42/14.85 thf(func_def_267, type, finite_card_char: set_char > nat).
% 62.42/14.85 thf(func_def_268, type, finite3207457112153483333omplex: set_complex > $o).
% 62.42/14.85 thf(func_def_269, type, finite4001608067531595151d_enat: set_Extended_enat > $o).
% 62.42/14.85 thf(func_def_270, type, finite_finite_int: set_int > $o).
% 62.42/14.85 thf(func_def_271, type, finite_finite_nat: set_nat > $o).
% 62.42/14.85 thf(func_def_272, type, bij_be1856998921033663316omplex: (complex > complex) > set_complex > set_complex > $o).
% 62.42/14.85 thf(func_def_273, type, bij_betw_nat_complex: (nat > complex) > set_nat > set_complex > $o).
% 62.42/14.85 thf(func_def_274, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 62.42/14.85 thf(func_def_275, type, comp_C8797469213163452608nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 62.42/14.85 thf(func_def_276, type, comp_C1593894019821074884nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 62.42/14.85 thf(func_def_277, type, comp_int_nat_int: (int > nat) > (int > int) > int > nat).
% 62.42/14.85 thf(func_def_278, type, comp_int_real_real: (int > real) > (real > int) > real > real).
% 62.42/14.85 thf(func_def_279, type, comp_nat_nat_nat: (nat > nat) > (nat > nat) > nat > nat).
% 62.42/14.85 thf(func_def_280, type, comp_nat_real_nat: (nat > real) > (nat > nat) > nat > real).
% 62.42/14.85 thf(func_def_281, type, id_o: $o > $o).
% 62.42/14.85 thf(func_def_282, type, id_nat: nat > nat).
% 62.42/14.85 thf(func_def_283, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 62.42/14.85 thf(func_def_284, type, inj_on_nat_char: (nat > char) > set_nat > $o).
% 62.42/14.85 thf(func_def_285, type, inj_on_real_real: (real > real) > set_real > $o).
% 62.42/14.85 thf(func_def_286, 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 > $o).
% 62.42/14.85 thf(func_def_287, type, map_fu4960017516451851995nt_int: (int > product_prod_nat_nat) > ((product_prod_nat_nat > product_prod_nat_nat) > int > int) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > int > int > int).
% 62.42/14.85 thf(func_def_288, type, map_fu4826362097070443709at_o_o: (int > product_prod_nat_nat) > ($o > $o) > (product_prod_nat_nat > $o) > int > $o).
% 62.42/14.85 thf(func_def_289, type, map_fu2345160673673942751at_nat: (int > product_prod_nat_nat) > (nat > nat) > (product_prod_nat_nat > nat) > int > nat).
% 62.42/14.85 thf(func_def_290, type, map_fu3667384564859982768at_int: (int > product_prod_nat_nat) > (product_prod_nat_nat > int) > (product_prod_nat_nat > product_prod_nat_nat) > int > int).
% 62.42/14.85 thf(func_def_291, type, map_fu4333342158222067775at_rat: (rat > product_prod_int_int) > ((product_prod_int_int > product_prod_int_int) > rat > rat) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > rat > rat > rat).
% 62.42/14.85 thf(func_def_292, type, map_fu898904425404107465nt_o_o: (rat > product_prod_int_int) > ($o > $o) > (product_prod_int_int > $o) > rat > $o).
% 62.42/14.85 thf(func_def_293, type, map_fu5673905371560938248nt_rat: (rat > product_prod_int_int) > (product_prod_int_int > rat) > (product_prod_int_int > product_prod_int_int) > rat > rat).
% 62.42/14.85 thf(func_def_294, type, map_fu1532550112467129777l_real: (real > nat > rat) > (((nat > rat) > nat > rat) > real > real) > ((nat > rat) > (nat > rat) > nat > rat) > real > real > real).
% 62.42/14.85 thf(func_def_295, type, map_fu7146612038024189824t_real: (real > nat > rat) > ((nat > rat) > real) > ((nat > rat) > nat > rat) > real > real).
% 62.42/14.85 thf(func_def_296, type, map_fu1856342031159181835at_o_o: (real > nat > rat) > ($o > $o) > ((nat > rat) > $o) > real > $o).
% 62.42/14.85 thf(func_def_297, type, strict1292158309912662752at_nat: (nat > nat) > set_nat > $o).
% 62.42/14.85 thf(func_def_298, type, the_in5290026491893676941l_real: set_real > (real > real) > real > real).
% 62.42/14.85 thf(func_def_299, type, fun_pair_leq: set_Pr8693737435421807431at_nat).
% 62.42/14.85 thf(func_def_300, type, fun_pair_less: set_Pr8693737435421807431at_nat).
% 62.42/14.85 thf(func_def_301, type, gcd_Gcd_nat: set_nat > nat).
% 62.42/14.85 thf(func_def_302, type, bezw: nat > nat > product_prod_int_int).
% 62.42/14.85 thf(func_def_303, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 62.42/14.85 thf(func_def_304, type, gcd_gcd_Code_integer: code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_305, type, gcd_gcd_int: int > int > int).
% 62.42/14.85 thf(func_def_306, type, gcd_gcd_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_307, type, gcd_lcm_Code_integer: code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_308, type, gcd_lcm_int: int > int > int).
% 62.42/14.85 thf(func_def_309, type, gcd_lcm_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_310, type, gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 62.42/14.85 thf(func_def_311, type, abs_abs_Code_integer: code_integer > code_integer).
% 62.42/14.85 thf(func_def_312, type, abs_abs_complex: complex > complex).
% 62.42/14.85 thf(func_def_313, type, abs_abs_int: int > int).
% 62.42/14.85 thf(func_def_314, type, abs_abs_rat: rat > rat).
% 62.42/14.85 thf(func_def_315, type, abs_abs_real: real > real).
% 62.42/14.85 thf(func_def_316, type, minus_8727706125548526216plex_o: (complex > $o) > (complex > $o) > complex > $o).
% 62.42/14.85 thf(func_def_317, type, minus_2020553357622893040enat_o: (extended_enat > $o) > (extended_enat > $o) > extended_enat > $o).
% 62.42/14.85 thf(func_def_318, type, minus_minus_int_o: (int > $o) > (int > $o) > int > $o).
% 62.42/14.85 thf(func_def_319, type, minus_1139252259498527702_nat_o: (list_nat > $o) > (list_nat > $o) > list_nat > $o).
% 62.42/14.85 thf(func_def_320, type, minus_minus_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 62.42/14.85 thf(func_def_321, type, minus_minus_real_o: (real > $o) > (real > $o) > real > $o).
% 62.42/14.85 thf(func_def_322, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_323, type, minus_minus_complex: complex > complex > complex).
% 62.42/14.85 thf(func_def_324, type, minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_325, type, minus_minus_int: int > int > int).
% 62.42/14.85 thf(func_def_326, type, minus_minus_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_327, type, minus_minus_rat: rat > rat > rat).
% 62.42/14.85 thf(func_def_328, type, minus_minus_real: real > real > real).
% 62.42/14.85 thf(func_def_329, type, minus_811609699411566653omplex: set_complex > set_complex > set_complex).
% 62.42/14.85 thf(func_def_330, type, minus_925952699566721837d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat).
% 62.42/14.85 thf(func_def_331, type, minus_minus_set_int: set_int > set_int > set_int).
% 62.42/14.85 thf(func_def_332, type, minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat).
% 62.42/14.85 thf(func_def_333, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 62.42/14.85 thf(func_def_334, type, minus_minus_set_real: set_real > set_real > set_real).
% 62.42/14.85 thf(func_def_335, type, one_one_Code_integer: code_integer).
% 62.42/14.85 thf(func_def_336, type, one_one_complex: complex).
% 62.42/14.85 thf(func_def_337, type, one_on7984719198319812577d_enat: extended_enat).
% 62.42/14.85 thf(func_def_338, type, one_one_int: int).
% 62.42/14.85 thf(func_def_339, type, one_one_nat: nat).
% 62.42/14.85 thf(func_def_340, type, one_one_rat: rat).
% 62.42/14.85 thf(func_def_341, type, one_one_real: real).
% 62.42/14.85 thf(func_def_342, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_343, type, plus_plus_complex: complex > complex > complex).
% 62.42/14.85 thf(func_def_344, type, plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_345, type, plus_plus_int: int > int > int).
% 62.42/14.85 thf(func_def_346, type, plus_plus_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_347, type, plus_plus_num: num > num > num).
% 62.42/14.85 thf(func_def_348, type, plus_plus_rat: rat > rat > rat).
% 62.42/14.85 thf(func_def_349, type, plus_plus_real: real > real > real).
% 62.42/14.85 thf(func_def_350, type, plus_plus_literal: literal > literal > literal).
% 62.42/14.85 thf(func_def_351, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 62.42/14.85 thf(func_def_352, type, sgn_sgn_complex: complex > complex).
% 62.42/14.85 thf(func_def_353, type, sgn_sgn_int: int > int).
% 62.42/14.85 thf(func_def_354, type, sgn_sgn_rat: rat > rat).
% 62.42/14.85 thf(func_def_355, type, sgn_sgn_real: real > real).
% 62.42/14.85 thf(func_def_356, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_357, type, times_times_complex: complex > complex > complex).
% 62.42/14.85 thf(func_def_358, type, times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_359, type, times_times_int: int > int > int).
% 62.42/14.85 thf(func_def_360, type, times_times_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_361, type, times_times_num: num > num > num).
% 62.42/14.85 thf(func_def_362, type, times_times_rat: rat > rat > rat).
% 62.42/14.85 thf(func_def_363, type, times_times_real: real > real > real).
% 62.42/14.85 thf(func_def_364, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 62.42/14.85 thf(func_def_365, type, uminus1482373934393186551omplex: complex > complex).
% 62.42/14.85 thf(func_def_366, type, uminus_uminus_int: int > int).
% 62.42/14.85 thf(func_def_367, type, uminus_uminus_rat: rat > rat).
% 62.42/14.85 thf(func_def_368, type, uminus_uminus_real: real > real).
% 62.42/14.85 thf(func_def_369, type, uminus5710092332889474511et_nat: set_nat > set_nat).
% 62.42/14.85 thf(func_def_370, type, zero_z3403309356797280102nteger: code_integer).
% 62.42/14.85 thf(func_def_371, type, zero_zero_complex: complex).
% 62.42/14.85 thf(func_def_372, type, zero_z5237406670263579293d_enat: extended_enat).
% 62.42/14.85 thf(func_def_373, type, zero_zero_int: int).
% 62.42/14.85 thf(func_def_374, type, zero_zero_nat: nat).
% 62.42/14.85 thf(func_def_375, type, zero_zero_rat: rat).
% 62.42/14.85 thf(func_def_376, type, zero_zero_real: real).
% 62.42/14.85 thf(func_def_377, type, zero_zero_literal: literal).
% 62.42/14.85 thf(func_def_378, type, groups7754918857620584856omplex: (complex > complex) > set_complex > complex).
% 62.42/14.85 thf(func_def_379, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 62.42/14.85 thf(func_def_380, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 62.42/14.85 thf(func_def_381, type, groups6591440286371151544t_real: (nat > real) > set_nat > real).
% 62.42/14.85 thf(func_def_382, type, groups3708469109370488835omplex: (complex > complex) > set_complex > complex).
% 62.42/14.85 thf(func_def_383, type, groups225925009352817453ex_rat: (complex > rat) > set_complex > rat).
% 62.42/14.85 thf(func_def_384, type, groups766887009212190081x_real: (complex > real) > set_complex > real).
% 62.42/14.85 thf(func_def_385, type, groups4622424608036095791omplex: (extended_enat > complex) > set_Extended_enat > complex).
% 62.42/14.85 thf(func_def_386, type, groups2878480467620962989at_int: (extended_enat > int) > set_Extended_enat > int).
% 62.42/14.85 thf(func_def_387, type, groups2880970938130013265at_nat: (extended_enat > nat) > set_Extended_enat > nat).
% 62.42/14.85 thf(func_def_388, type, groups2245840878043517529at_rat: (extended_enat > rat) > set_Extended_enat > rat).
% 62.42/14.85 thf(func_def_389, type, groups97031904164794029t_real: (extended_enat > real) > set_Extended_enat > real).
% 62.42/14.85 thf(func_def_390, type, groups7440179247065528705omplex: (int > complex) > set_int > complex).
% 62.42/14.85 thf(func_def_391, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 62.42/14.85 thf(func_def_392, type, groups1072433553688619179nt_rat: (int > rat) > set_int > rat).
% 62.42/14.85 thf(func_def_393, type, groups2316167850115554303t_real: (int > real) > set_int > real).
% 62.42/14.85 thf(func_def_394, type, groups6464643781859351333omplex: (nat > complex) > set_nat > complex).
% 62.42/14.85 thf(func_def_395, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 62.42/14.85 thf(func_def_396, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 62.42/14.85 thf(func_def_397, type, groups73079841787564623at_rat: (nat > rat) > set_nat > rat).
% 62.42/14.85 thf(func_def_398, type, groups129246275422532515t_real: (nat > real) > set_nat > real).
% 62.42/14.85 thf(func_def_399, type, groups713298508707869441omplex: (real > complex) > set_real > complex).
% 62.42/14.85 thf(func_def_400, type, groups4694064378042380927al_int: (real > int) > set_real > int).
% 62.42/14.85 thf(func_def_401, type, groups4696554848551431203al_nat: (real > nat) > set_real > nat).
% 62.42/14.85 thf(func_def_402, type, groups4061424788464935467al_rat: (real > rat) > set_real > rat).
% 62.42/14.85 thf(func_def_403, type, groups1681761925125756287l_real: (real > real) > set_real > real).
% 62.42/14.85 thf(func_def_404, type, groups9116527308978886569_o_int: ($o > int) > int > list_o > int).
% 62.42/14.85 thf(func_def_405, type, groups4561878855575611511st_nat: list_nat > nat).
% 62.42/14.85 thf(func_def_406, type, the_int: (int > $o) > int).
% 62.42/14.85 thf(func_def_407, type, the_Pr4378521158711661632nt_int: (product_prod_int_int > $o) > product_prod_int_int).
% 62.42/14.85 thf(func_def_408, type, the_real: (real > $o) > real).
% 62.42/14.85 thf(func_def_409, type, if_int_int: $o > (int > int) > (int > int) > int > int).
% 62.42/14.85 thf(func_def_410, type, if_nat_int_int: $o > (nat > int > int) > (nat > int > int) > nat > int > int).
% 62.42/14.85 thf(func_def_411, type, if_nat_nat_nat: $o > (nat > nat > nat) > (nat > nat > nat) > nat > nat > nat).
% 62.42/14.85 thf(func_def_412, type, if_nat_rat: $o > (nat > rat) > (nat > rat) > nat > rat).
% 62.42/14.85 thf(func_def_413, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_414, type, if_complex: $o > complex > complex > complex).
% 62.42/14.85 thf(func_def_415, type, if_Extended_enat: $o > extended_enat > extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_416, type, if_int: $o > int > int > int).
% 62.42/14.85 thf(func_def_417, type, if_list_int: $o > list_int > list_int > list_int).
% 62.42/14.85 thf(func_def_418, type, if_list_nat: $o > list_nat > list_nat > list_nat).
% 62.42/14.85 thf(func_def_419, type, if_nat: $o > nat > nat > nat).
% 62.42/14.85 thf(func_def_420, type, if_num: $o > num > num > num).
% 62.42/14.85 thf(func_def_421, type, if_option_nat: $o > option_nat > option_nat > option_nat).
% 62.42/14.85 thf(func_def_422, type, if_option_num: $o > option_num > option_num > option_num).
% 62.42/14.85 thf(func_def_423, type, if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 62.42/14.85 thf(func_def_424, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 62.42/14.85 thf(func_def_425, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 62.42/14.85 thf(func_def_426, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_427, type, if_rat: $o > rat > rat > rat).
% 62.42/14.85 thf(func_def_428, type, if_real: $o > real > real > real).
% 62.42/14.85 thf(func_def_429, type, if_set_int: $o > set_int > set_int > set_int).
% 62.42/14.85 thf(func_def_430, type, if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_431, type, infini8530281810654367211te_nat: set_nat > nat > nat).
% 62.42/14.85 thf(func_def_432, type, abs_Integ: product_prod_nat_nat > int).
% 62.42/14.85 thf(func_def_433, type, rep_Integ: int > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_434, type, int_ge_less_than: int > set_Pr958786334691620121nt_int).
% 62.42/14.85 thf(func_def_435, type, int_ge_less_than2: int > set_Pr958786334691620121nt_int).
% 62.42/14.85 thf(func_def_436, type, intrel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 62.42/14.85 thf(func_def_437, type, nat2: int > nat).
% 62.42/14.85 thf(func_def_438, type, pcr_int: product_prod_nat_nat > int > $o).
% 62.42/14.85 thf(func_def_439, type, power_int_real: real > int > real).
% 62.42/14.85 thf(func_def_440, type, ring_11222124179247155820nteger: set_Code_integer).
% 62.42/14.85 thf(func_def_441, type, ring_1_Ints_complex: set_complex).
% 62.42/14.85 thf(func_def_442, type, ring_1_Ints_int: set_int).
% 62.42/14.85 thf(func_def_443, type, ring_1_Ints_rat: set_rat).
% 62.42/14.85 thf(func_def_444, type, ring_1_Ints_real: set_real).
% 62.42/14.85 thf(func_def_445, type, ring_18347121197199848620nteger: int > code_integer).
% 62.42/14.85 thf(func_def_446, type, ring_17405671764205052669omplex: int > complex).
% 62.42/14.85 thf(func_def_447, type, ring_1_of_int_int: int > int).
% 62.42/14.85 thf(func_def_448, type, ring_1_of_int_rat: int > rat).
% 62.42/14.85 thf(func_def_449, type, ring_1_of_int_real: int > real).
% 62.42/14.85 thf(func_def_450, type, inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_451, type, inf_inf_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_452, type, semila1623282765462674594er_nat: (nat > nat > nat) > nat > (nat > nat > $o) > (nat > nat > $o) > $o).
% 62.42/14.85 thf(func_def_453, type, sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_454, type, sup_sup_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_455, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 62.42/14.85 thf(func_def_456, type, sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 62.42/14.85 thf(func_def_457, type, lattic921264341876707157d_enat: set_Extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_458, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 62.42/14.85 thf(func_def_459, type, quotie3684837364556693515t_real: ((nat > rat) > (nat > rat) > $o) > ((nat > rat) > real) > (real > nat > rat) > ((nat > rat) > real > $o) > $o).
% 62.42/14.85 thf(func_def_460, type, bfun_nat_real: (nat > real) > filter_nat > $o).
% 62.42/14.85 thf(func_def_461, type, append_int: list_int > list_int > list_int).
% 62.42/14.85 thf(func_def_462, type, append_nat: list_nat > list_nat > list_nat).
% 62.42/14.85 thf(func_def_463, type, drop_nat: nat > list_nat > list_nat).
% 62.42/14.85 thf(func_def_464, type, last_nat: list_nat > nat).
% 62.42/14.85 thf(func_def_465, type, linord2614967742042102400et_nat: set_nat > list_nat).
% 62.42/14.85 thf(func_def_466, type, cons_int: int > list_int > list_int).
% 62.42/14.85 thf(func_def_467, type, cons_nat: nat > list_nat > list_nat).
% 62.42/14.85 thf(func_def_468, type, nil_int: list_int).
% 62.42/14.85 thf(func_def_469, type, nil_nat: list_nat).
% 62.42/14.85 thf(func_def_470, type, hd_nat: list_nat > nat).
% 62.42/14.85 thf(func_def_471, type, map_nat_nat: (nat > nat) > list_nat > list_nat).
% 62.42/14.85 thf(func_def_472, type, map_VE8901447254227204932T_VEBT: (vEBT_VEBT > vEBT_VEBT) > list_VEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_473, type, set_o2: list_o > set_o).
% 62.42/14.85 thf(func_def_474, type, set_complex2: list_complex > set_complex).
% 62.42/14.85 thf(func_def_475, type, set_Extended_enat2: list_Extended_enat > set_Extended_enat).
% 62.42/14.85 thf(func_def_476, type, set_int2: list_int > set_int).
% 62.42/14.85 thf(func_def_477, type, set_list_nat2: list_list_nat > set_list_nat).
% 62.42/14.85 thf(func_def_478, type, set_nat2: list_nat > set_nat).
% 62.42/14.85 thf(func_def_479, type, set_real2: list_real > set_real).
% 62.42/14.85 thf(func_def_480, type, set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT).
% 62.42/14.85 thf(func_def_481, type, size_list_VEBT_VEBT: (vEBT_VEBT > nat) > list_VEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_482, type, tl_nat: list_nat > list_nat).
% 62.42/14.85 thf(func_def_483, type, list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_484, type, nth_o: list_o > nat > $o).
% 62.42/14.85 thf(func_def_485, type, nth_complex: list_complex > nat > complex).
% 62.42/14.85 thf(func_def_486, type, nth_Extended_enat: list_Extended_enat > nat > extended_enat).
% 62.42/14.85 thf(func_def_487, type, nth_int: list_int > nat > int).
% 62.42/14.85 thf(func_def_488, type, nth_list_nat: list_list_nat > nat > list_nat).
% 62.42/14.85 thf(func_def_489, type, nth_nat: list_nat > nat > nat).
% 62.42/14.85 thf(func_def_490, type, nth_num: list_num > nat > num).
% 62.42/14.85 thf(func_def_491, type, nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o).
% 62.42/14.85 thf(func_def_492, type, nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int).
% 62.42/14.85 thf(func_def_493, type, nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat).
% 62.42/14.85 thf(func_def_494, type, nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT).
% 62.42/14.85 thf(func_def_495, type, nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num).
% 62.42/14.85 thf(func_def_496, type, nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num).
% 62.42/14.85 thf(func_def_497, type, nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o).
% 62.42/14.85 thf(func_def_498, type, nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int).
% 62.42/14.85 thf(func_def_499, type, nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat).
% 62.42/14.85 thf(func_def_500, type, nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT).
% 62.42/14.85 thf(func_def_501, type, nth_real: list_real > nat > real).
% 62.42/14.85 thf(func_def_502, type, nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT).
% 62.42/14.85 thf(func_def_503, type, product_o_o: list_o > list_o > list_P4002435161011370285od_o_o).
% 62.42/14.85 thf(func_def_504, type, product_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 62.42/14.85 thf(func_def_505, type, product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat).
% 62.42/14.85 thf(func_def_506, type, product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 62.42/14.85 thf(func_def_507, type, product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o).
% 62.42/14.85 thf(func_def_508, type, product_nat_num: list_nat > list_num > list_P1726324292696863441at_num).
% 62.42/14.85 thf(func_def_509, type, produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 62.42/14.85 thf(func_def_510, type, product_num_num: list_num > list_num > list_P3744719386663036955um_num).
% 62.42/14.85 thf(func_def_511, type, product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 62.42/14.85 thf(func_def_512, type, produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 62.42/14.85 thf(func_def_513, type, produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat).
% 62.42/14.85 thf(func_def_514, type, produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 62.42/14.85 thf(func_def_515, type, replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_516, type, sorted_wrt_int: (int > int > $o) > list_int > $o).
% 62.42/14.85 thf(func_def_517, type, sorted_wrt_nat: (nat > nat > $o) > list_nat > $o).
% 62.42/14.85 thf(func_def_518, type, take_nat: nat > list_nat > list_nat).
% 62.42/14.85 thf(func_def_519, type, take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_520, type, upt: nat > nat > list_nat).
% 62.42/14.85 thf(func_def_521, type, upto: int > int > list_int).
% 62.42/14.85 thf(func_def_522, type, upto_aux: int > int > list_int > list_int).
% 62.42/14.85 thf(func_def_523, type, upto_rel: product_prod_int_int > product_prod_int_int > $o).
% 62.42/14.85 thf(func_def_524, type, suc: nat > nat).
% 62.42/14.85 thf(func_def_525, type, compow_nat_nat: nat > (nat > nat) > nat > nat).
% 62.42/14.85 thf(func_def_526, type, case_nat_o: $o > (nat > $o) > nat > $o).
% 62.42/14.85 thf(func_def_527, type, case_nat_nat: nat > (nat > nat) > nat > nat).
% 62.42/14.85 thf(func_def_528, type, case_nat_option_num: option_num > (nat > option_num) > nat > option_num).
% 62.42/14.85 thf(func_def_529, type, pred: nat > nat).
% 62.42/14.85 thf(func_def_530, type, semiri4939895301339042750nteger: nat > code_integer).
% 62.42/14.85 thf(func_def_531, type, semiri8010041392384452111omplex: nat > complex).
% 62.42/14.85 thf(func_def_532, type, semiri4216267220026989637d_enat: nat > extended_enat).
% 62.42/14.85 thf(func_def_533, type, semiri1314217659103216013at_int: nat > int).
% 62.42/14.85 thf(func_def_534, type, semiri1316708129612266289at_nat: nat > nat).
% 62.42/14.85 thf(func_def_535, type, semiri681578069525770553at_rat: nat > rat).
% 62.42/14.85 thf(func_def_536, type, semiri5074537144036343181t_real: nat > real).
% 62.42/14.85 thf(func_def_537, type, size_size_list_o: list_o > nat).
% 62.42/14.85 thf(func_def_538, type, size_s3451745648224563538omplex: list_complex > nat).
% 62.42/14.85 thf(func_def_539, type, size_s3941691890525107288d_enat: list_Extended_enat > nat).
% 62.42/14.85 thf(func_def_540, type, size_size_list_int: list_int > nat).
% 62.42/14.85 thf(func_def_541, type, size_s3023201423986296836st_nat: list_list_nat > nat).
% 62.42/14.85 thf(func_def_542, type, size_size_list_nat: list_nat > nat).
% 62.42/14.85 thf(func_def_543, type, size_size_list_num: list_num > nat).
% 62.42/14.85 thf(func_def_544, type, size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat).
% 62.42/14.85 thf(func_def_545, type, size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat).
% 62.42/14.85 thf(func_def_546, type, size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat).
% 62.42/14.85 thf(func_def_547, type, size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat).
% 62.42/14.85 thf(func_def_548, type, size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat).
% 62.42/14.85 thf(func_def_549, type, size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat).
% 62.42/14.85 thf(func_def_550, type, size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat).
% 62.42/14.85 thf(func_def_551, type, size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat).
% 62.42/14.85 thf(func_def_552, type, size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat).
% 62.42/14.85 thf(func_def_553, type, size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat).
% 62.42/14.85 thf(func_def_554, type, size_size_list_real: list_real > nat).
% 62.42/14.85 thf(func_def_555, type, size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_556, type, size_size_num: num > nat).
% 62.42/14.85 thf(func_def_557, type, size_size_VEBT_VEBT: vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_558, type, nat_list_decode: nat > list_nat).
% 62.42/14.85 thf(func_def_559, type, nat_list_decode_rel: nat > nat > $o).
% 62.42/14.85 thf(func_def_560, type, nat_list_encode: list_nat > nat).
% 62.42/14.85 thf(func_def_561, type, nat_list_encode_rel: list_nat > list_nat > $o).
% 62.42/14.85 thf(func_def_562, type, nat_prod_decode: nat > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_563, type, nat_prod_decode_aux: nat > nat > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_564, type, nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 62.42/14.85 thf(func_def_565, type, nat_prod_encode: product_prod_nat_nat > nat).
% 62.42/14.85 thf(func_def_566, type, nat_set_decode: nat > set_nat).
% 62.42/14.85 thf(func_def_567, type, nat_set_encode: set_nat > nat).
% 62.42/14.85 thf(func_def_568, type, nat_triangle: nat > nat).
% 62.42/14.85 thf(func_def_569, type, root: nat > real > real).
% 62.42/14.85 thf(func_def_570, type, sqrt: real > real).
% 62.42/14.85 thf(func_def_571, type, bitM: num > num).
% 62.42/14.85 thf(func_def_572, type, inc: num > num).
% 62.42/14.85 thf(func_def_573, type, neg_nu8804712462038260780nteger: code_integer > code_integer).
% 62.42/14.85 thf(func_def_574, type, neg_nu7009210354673126013omplex: complex > complex).
% 62.42/14.85 thf(func_def_575, type, neg_numeral_dbl_int: int > int).
% 62.42/14.85 thf(func_def_576, type, neg_numeral_dbl_rat: rat > rat).
% 62.42/14.85 thf(func_def_577, type, neg_numeral_dbl_real: real > real).
% 62.42/14.85 thf(func_def_578, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 62.42/14.85 thf(func_def_579, type, neg_nu6511756317524482435omplex: complex > complex).
% 62.42/14.85 thf(func_def_580, type, neg_nu3811975205180677377ec_int: int > int).
% 62.42/14.85 thf(func_def_581, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 62.42/14.85 thf(func_def_582, type, neg_nu6075765906172075777c_real: real > real).
% 62.42/14.85 thf(func_def_583, type, neg_nu5831290666863070958nteger: code_integer > code_integer).
% 62.42/14.85 thf(func_def_584, type, neg_nu8557863876264182079omplex: complex > complex).
% 62.42/14.85 thf(func_def_585, type, neg_nu5851722552734809277nc_int: int > int).
% 62.42/14.85 thf(func_def_586, type, neg_nu5219082963157363817nc_rat: rat > rat).
% 62.42/14.85 thf(func_def_587, type, neg_nu8295874005876285629c_real: real > real).
% 62.42/14.85 thf(func_def_588, type, neg_numeral_sub_int: num > num > int).
% 62.42/14.85 thf(func_def_589, type, bit0: num > num).
% 62.42/14.85 thf(func_def_590, type, bit1: num > num).
% 62.42/14.85 thf(func_def_591, type, one: num).
% 62.42/14.85 thf(func_def_592, type, case_num_option_num: option_num > (num > option_num) > (num > option_num) > num > option_num).
% 62.42/14.85 thf(func_def_593, type, size_num: num > nat).
% 62.42/14.85 thf(func_def_594, type, num_of_nat: nat > num).
% 62.42/14.85 thf(func_def_595, type, numera6620942414471956472nteger: num > code_integer).
% 62.42/14.85 thf(func_def_596, type, numera6690914467698888265omplex: num > complex).
% 62.42/14.85 thf(func_def_597, type, numera1916890842035813515d_enat: num > extended_enat).
% 62.42/14.85 thf(func_def_598, type, numeral_numeral_int: num > int).
% 62.42/14.85 thf(func_def_599, type, numeral_numeral_nat: num > nat).
% 62.42/14.85 thf(func_def_600, type, numeral_numeral_rat: num > rat).
% 62.42/14.85 thf(func_def_601, type, numeral_numeral_real: num > real).
% 62.42/14.85 thf(func_def_602, type, pow: num > num > num).
% 62.42/14.85 thf(func_def_603, type, pred_numeral: num > nat).
% 62.42/14.85 thf(func_def_604, type, sqr: num > num).
% 62.42/14.85 thf(func_def_605, type, none_nat: option_nat).
% 62.42/14.85 thf(func_def_606, type, none_num: option_num).
% 62.42/14.85 thf(func_def_607, type, none_P5556105721700978146at_nat: option4927543243414619207at_nat).
% 62.42/14.85 thf(func_def_608, type, some_nat: nat > option_nat).
% 62.42/14.85 thf(func_def_609, type, some_num: num > option_num).
% 62.42/14.85 thf(func_def_610, type, some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat).
% 62.42/14.85 thf(func_def_611, type, case_o184042715313410164at_nat: $o > (product_prod_nat_nat > $o) > option4927543243414619207at_nat > $o).
% 62.42/14.85 thf(func_def_612, type, case_option_int_num: int > (num > int) > option_num > int).
% 62.42/14.85 thf(func_def_613, type, case_option_num_num: num > (num > num) > option_num > num).
% 62.42/14.85 thf(func_def_614, type, case_o6005452278849405969um_num: option_num > (num > option_num) > option_num > option_num).
% 62.42/14.85 thf(func_def_615, type, map_option_num_num: (num > num) > option_num > option_num).
% 62.42/14.85 thf(func_def_616, type, the_nat: option_nat > nat).
% 62.42/14.85 thf(func_def_617, type, bot_bot_nat_o: nat > $o).
% 62.42/14.85 thf(func_def_618, type, bot_bo4199563552545308370d_enat: extended_enat).
% 62.42/14.85 thf(func_def_619, type, bot_bot_nat: nat).
% 62.42/14.85 thf(func_def_620, type, bot_bo7653980558646680370d_enat: set_Extended_enat).
% 62.42/14.85 thf(func_def_621, type, bot_bot_set_int: set_int).
% 62.42/14.85 thf(func_def_622, type, bot_bot_set_nat: set_nat).
% 62.42/14.85 thf(func_def_623, type, bot_bot_set_num: set_num).
% 62.42/14.85 thf(func_def_624, type, bot_bot_set_rat: set_rat).
% 62.42/14.85 thf(func_def_625, type, bot_bot_set_real: set_real).
% 62.42/14.85 thf(func_def_626, type, bot_bot_set_set_nat: set_set_nat).
% 62.42/14.85 thf(func_def_627, type, ord_Le1955565732374568822d_enat: (extended_enat > $o) > extended_enat).
% 62.42/14.85 thf(func_def_628, type, ord_Least_nat: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_629, type, ord_Least_real: (real > $o) > real).
% 62.42/14.85 thf(func_def_630, type, ord_less_complex_o: (complex > $o) > (complex > $o) > $o).
% 62.42/14.85 thf(func_def_631, type, ord_le8499522857272258027enat_o: (extended_enat > $o) > (extended_enat > $o) > $o).
% 62.42/14.85 thf(func_def_632, type, ord_less_int_o: (int > $o) > (int > $o) > $o).
% 62.42/14.85 thf(func_def_633, type, ord_less_nat_o: (nat > $o) > (nat > $o) > $o).
% 62.42/14.85 thf(func_def_634, type, ord_less_real_o: (real > $o) > (real > $o) > $o).
% 62.42/14.85 thf(func_def_635, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 62.42/14.85 thf(func_def_636, type, ord_le72135733267957522d_enat: extended_enat > extended_enat > $o).
% 62.42/14.85 thf(func_def_637, type, ord_less_int: int > int > $o).
% 62.42/14.85 thf(func_def_638, type, ord_less_nat: nat > nat > $o).
% 62.42/14.85 thf(func_def_639, type, ord_less_num: num > num > $o).
% 62.42/14.85 thf(func_def_640, type, ord_less_rat: rat > rat > $o).
% 62.42/14.85 thf(func_def_641, type, ord_less_real: real > real > $o).
% 62.42/14.85 thf(func_def_642, type, ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o).
% 62.42/14.85 thf(func_def_643, type, ord_less_set_complex: set_complex > set_complex > $o).
% 62.42/14.85 thf(func_def_644, type, ord_le2529575680413868914d_enat: set_Extended_enat > set_Extended_enat > $o).
% 62.42/14.85 thf(func_def_645, type, ord_less_set_int: set_int > set_int > $o).
% 62.42/14.85 thf(func_def_646, type, ord_less_set_nat: set_nat > set_nat > $o).
% 62.42/14.85 thf(func_def_647, type, ord_less_set_num: set_num > set_num > $o).
% 62.42/14.85 thf(func_def_648, type, ord_less_set_rat: set_rat > set_rat > $o).
% 62.42/14.85 thf(func_def_649, type, ord_less_set_real: set_real > set_real > $o).
% 62.42/14.85 thf(func_def_650, type, ord_less_set_set_nat: set_set_nat > set_set_nat > $o).
% 62.42/14.85 thf(func_def_651, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 62.42/14.85 thf(func_def_652, type, ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o).
% 62.42/14.85 thf(func_def_653, type, ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o).
% 62.42/14.85 thf(func_def_654, type, ord_less_eq_int: int > int > $o).
% 62.42/14.85 thf(func_def_655, type, ord_less_eq_nat: nat > nat > $o).
% 62.42/14.85 thf(func_def_656, type, ord_less_eq_num: num > num > $o).
% 62.42/14.85 thf(func_def_657, type, ord_less_eq_rat: rat > rat > $o).
% 62.42/14.85 thf(func_def_658, type, ord_less_eq_real: real > real > $o).
% 62.42/14.85 thf(func_def_659, type, ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o).
% 62.42/14.85 thf(func_def_660, type, ord_le211207098394363844omplex: set_complex > set_complex > $o).
% 62.42/14.85 thf(func_def_661, type, ord_le7203529160286727270d_enat: set_Extended_enat > set_Extended_enat > $o).
% 62.42/14.85 thf(func_def_662, type, ord_less_eq_set_int: set_int > set_int > $o).
% 62.42/14.85 thf(func_def_663, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 62.42/14.85 thf(func_def_664, type, ord_less_eq_set_num: set_num > set_num > $o).
% 62.42/14.85 thf(func_def_665, type, ord_less_eq_set_rat: set_rat > set_rat > $o).
% 62.42/14.85 thf(func_def_666, type, ord_less_eq_set_real: set_real > set_real > $o).
% 62.42/14.85 thf(func_def_667, type, ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o).
% 62.42/14.85 thf(func_def_668, type, ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_669, type, ord_max_int: int > int > int).
% 62.42/14.85 thf(func_def_670, type, ord_max_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_671, type, ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat).
% 62.42/14.85 thf(func_def_672, type, ord_min_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_673, type, order_Greatest_nat: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_674, type, order_9091379641038594480t_real: (nat > real) > $o).
% 62.42/14.85 thf(func_def_675, type, order_4130057895858720880d_enat: (extended_enat > extended_enat) > $o).
% 62.42/14.85 thf(func_def_676, type, order_mono_nat_nat: (nat > nat) > $o).
% 62.42/14.85 thf(func_def_677, type, order_mono_nat_real: (nat > real) > $o).
% 62.42/14.85 thf(func_def_678, type, order_5726023648592871131at_nat: (nat > nat) > $o).
% 62.42/14.85 thf(func_def_679, type, order_7092887310737990675l_real: (real > real) > $o).
% 62.42/14.85 thf(func_def_680, type, ordering_top_nat: (nat > nat > $o) > (nat > nat > $o) > nat > $o).
% 62.42/14.85 thf(func_def_681, type, top_to3028658606643905974d_enat: extended_enat).
% 62.42/14.85 thf(func_def_682, type, top_top_set_o: set_o).
% 62.42/14.85 thf(func_def_683, type, top_top_set_nat: set_nat).
% 62.42/14.85 thf(func_def_684, type, top_to1996260823553986621t_unit: set_Product_unit).
% 62.42/14.85 thf(func_def_685, type, top_top_set_real: set_real).
% 62.42/14.85 thf(func_def_686, type, top_top_set_char: set_char).
% 62.42/14.85 thf(func_def_687, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 62.42/14.85 thf(func_def_688, type, power_power_complex: complex > nat > complex).
% 62.42/14.85 thf(func_def_689, type, power_power_int: int > nat > int).
% 62.42/14.85 thf(func_def_690, type, power_power_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_691, type, power_power_rat: rat > nat > rat).
% 62.42/14.85 thf(func_def_692, type, power_power_real: real > nat > real).
% 62.42/14.85 thf(func_def_693, type, produc3209952032786966637at_nat: (nat > nat > nat) > produc7248412053542808358at_nat > produc4471711990508489141at_nat).
% 62.42/14.85 thf(func_def_694, type, produc851828971589881931at_num: (nat > num > num) > produc2963631642982155120at_num > produc3368934014287244435at_num).
% 62.42/14.85 thf(func_def_695, type, product_Pair_o_o: $o > $o > product_prod_o_o).
% 62.42/14.85 thf(func_def_696, type, product_Pair_o_int: $o > int > product_prod_o_int).
% 62.42/14.85 thf(func_def_697, type, product_Pair_o_nat: $o > nat > product_prod_o_nat).
% 62.42/14.85 thf(func_def_698, type, produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT).
% 62.42/14.85 thf(func_def_699, type, produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o).
% 62.42/14.85 thf(func_def_700, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 62.42/14.85 thf(func_def_701, type, product_Pair_int_int: int > int > product_prod_int_int).
% 62.42/14.85 thf(func_def_702, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_703, type, product_Pair_nat_num: nat > num > product_prod_nat_num).
% 62.42/14.85 thf(func_def_704, type, produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat).
% 62.42/14.85 thf(func_def_705, type, produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num).
% 62.42/14.85 thf(func_def_706, type, product_Pair_num_num: num > num > product_prod_num_num).
% 62.42/14.85 thf(func_def_707, type, produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat).
% 62.42/14.85 thf(func_def_708, type, produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o).
% 62.42/14.85 thf(func_def_709, type, produc581526299967858633d_enat: vEBT_VEBT > extended_enat > produc7272778201969148633d_enat).
% 62.42/14.85 thf(func_def_710, type, produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int).
% 62.42/14.85 thf(func_def_711, type, produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat).
% 62.42/14.85 thf(func_def_712, type, produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT).
% 62.42/14.85 thf(func_def_713, type, produc457027306803732586at_nat: set_nat > (nat > set_nat) > set_Pr1261947904930325089at_nat).
% 62.42/14.85 thf(func_def_714, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 62.42/14.85 thf(func_def_715, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 62.42/14.85 thf(func_def_716, type, produc1555791787009142072er_nat: (code_integer > code_integer > nat) > produc8923325533196201883nteger > nat).
% 62.42/14.85 thf(func_def_717, type, produc7336495610019696514er_num: (code_integer > code_integer > num) > produc8923325533196201883nteger > num).
% 62.42/14.85 thf(func_def_718, type, produc9125791028180074456eger_o: (code_integer > code_integer > produc6271795597528267376eger_o) > produc8923325533196201883nteger > produc6271795597528267376eger_o).
% 62.42/14.85 thf(func_def_719, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 62.42/14.85 thf(func_def_720, type, produc6771430404735790350plex_o: (complex > complex > $o) > produc4411394909380815293omplex > $o).
% 62.42/14.85 thf(func_def_721, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 62.42/14.85 thf(func_def_722, type, produc8211389475949308722nt_int: (int > int > int) > product_prod_int_int > int).
% 62.42/14.85 thf(func_def_723, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 62.42/14.85 thf(func_def_724, type, produc8739625826339149834_nat_o: (nat > nat > product_prod_nat_nat > $o) > product_prod_nat_nat > product_prod_nat_nat > $o).
% 62.42/14.85 thf(func_def_725, 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).
% 62.42/14.85 thf(func_def_726, type, produc6081775807080527818_nat_o: (nat > nat > $o) > product_prod_nat_nat > $o).
% 62.42/14.85 thf(func_def_727, type, produc2761476792215241774st_nat: (nat > nat > list_nat) > product_prod_nat_nat > list_nat).
% 62.42/14.85 thf(func_def_728, type, produc6842872674320459806at_nat: (nat > nat > nat) > product_prod_nat_nat > nat).
% 62.42/14.85 thf(func_def_729, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_730, type, produc478579273971653890on_num: (nat > num > option_num) > product_prod_nat_num > option_num).
% 62.42/14.85 thf(func_def_731, type, produc5414030515140494994real_o: (real > real > $o) > produc2422161461964618553l_real > $o).
% 62.42/14.85 thf(func_def_732, type, produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer).
% 62.42/14.85 thf(func_def_733, type, product_fst_int_int: product_prod_int_int > int).
% 62.42/14.85 thf(func_def_734, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 62.42/14.85 thf(func_def_735, type, produc6174133586879617921nteger: produc8923325533196201883nteger > code_integer).
% 62.42/14.85 thf(func_def_736, type, product_snd_int_int: product_prod_int_int > int).
% 62.42/14.85 thf(func_def_737, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 62.42/14.85 thf(func_def_738, type, abs_Rat: product_prod_int_int > rat).
% 62.42/14.85 thf(func_def_739, type, fract: int > int > rat).
% 62.42/14.85 thf(func_def_740, type, frct: product_prod_int_int > rat).
% 62.42/14.85 thf(func_def_741, type, rep_Rat: rat > product_prod_int_int).
% 62.42/14.85 thf(func_def_742, type, field_5140801741446780682s_real: set_real).
% 62.42/14.85 thf(func_def_743, type, field_7254667332652039916t_real: rat > real).
% 62.42/14.85 thf(func_def_744, type, normalize: product_prod_int_int > product_prod_int_int).
% 62.42/14.85 thf(func_def_745, type, pcr_rat: product_prod_int_int > rat > $o).
% 62.42/14.85 thf(func_def_746, type, positive: rat > $o).
% 62.42/14.85 thf(func_def_747, type, quotient_of: rat > product_prod_int_int).
% 62.42/14.85 thf(func_def_748, type, ratrel: product_prod_int_int > product_prod_int_int > $o).
% 62.42/14.85 thf(func_def_749, type, real2: (nat > rat) > real).
% 62.42/14.85 thf(func_def_750, type, cauchy: (nat > rat) > $o).
% 62.42/14.85 thf(func_def_751, type, cr_real: (nat > rat) > real > $o).
% 62.42/14.85 thf(func_def_752, type, pcr_real: (nat > rat) > real > $o).
% 62.42/14.85 thf(func_def_753, type, positive2: real > $o).
% 62.42/14.85 thf(func_def_754, type, realrel: (nat > rat) > (nat > rat) > $o).
% 62.42/14.85 thf(func_def_755, type, rep_real: real > nat > rat).
% 62.42/14.85 thf(func_def_756, type, vanishes: (nat > rat) > $o).
% 62.42/14.85 thf(func_def_757, type, real_V2521375963428798218omplex: set_complex).
% 62.42/14.85 thf(func_def_758, type, real_V5970128139526366754l_real: (real > real) > $o).
% 62.42/14.85 thf(func_def_759, type, real_V3694042436643373181omplex: complex > complex > real).
% 62.42/14.85 thf(func_def_760, type, real_V975177566351809787t_real: real > real > real).
% 62.42/14.85 thf(func_def_761, type, real_V1022390504157884413omplex: complex > real).
% 62.42/14.85 thf(func_def_762, type, real_V7735802525324610683m_real: real > real).
% 62.42/14.85 thf(func_def_763, type, real_V4546457046886955230omplex: real > complex).
% 62.42/14.85 thf(func_def_764, type, real_V1803761363581548252l_real: real > real).
% 62.42/14.85 thf(func_def_765, type, real_V2046097035970521341omplex: real > complex > complex).
% 62.42/14.85 thf(func_def_766, type, real_V1485227260804924795R_real: real > real > real).
% 62.42/14.85 thf(func_def_767, type, transp_nat_rat: ((nat > rat) > (nat > rat) > $o) > $o).
% 62.42/14.85 thf(func_def_768, type, algebr932160517623751201me_int: int > int > $o).
% 62.42/14.85 thf(func_def_769, type, algebr934650988132801477me_nat: nat > nat > $o).
% 62.42/14.85 thf(func_def_770, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_771, type, divide1717551699836669952omplex: complex > complex > complex).
% 62.42/14.85 thf(func_def_772, type, divide_divide_int: int > int > int).
% 62.42/14.85 thf(func_def_773, type, divide_divide_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_774, type, divide_divide_rat: rat > rat > rat).
% 62.42/14.85 thf(func_def_775, type, divide_divide_real: real > real > real).
% 62.42/14.85 thf(func_def_776, type, dvd_dvd_Code_integer: code_integer > code_integer > $o).
% 62.42/14.85 thf(func_def_777, type, dvd_dvd_complex: complex > complex > $o).
% 62.42/14.85 thf(func_def_778, type, dvd_dvd_int: int > int > $o).
% 62.42/14.85 thf(func_def_779, type, dvd_dvd_nat: nat > nat > $o).
% 62.42/14.85 thf(func_def_780, type, dvd_dvd_rat: rat > rat > $o).
% 62.42/14.85 thf(func_def_781, type, dvd_dvd_real: real > real > $o).
% 62.42/14.85 thf(func_def_782, type, modulo364778990260209775nteger: code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_783, type, modulo_modulo_int: int > int > int).
% 62.42/14.85 thf(func_def_784, type, modulo_modulo_nat: nat > nat > nat).
% 62.42/14.85 thf(func_def_785, type, zero_n356916108424825756nteger: $o > code_integer).
% 62.42/14.85 thf(func_def_786, type, zero_n1201886186963655149omplex: $o > complex).
% 62.42/14.85 thf(func_def_787, type, zero_n2684676970156552555ol_int: $o > int).
% 62.42/14.85 thf(func_def_788, type, zero_n2687167440665602831ol_nat: $o > nat).
% 62.42/14.85 thf(func_def_789, type, zero_n2052037380579107095ol_rat: $o > rat).
% 62.42/14.85 thf(func_def_790, type, zero_n3304061248610475627l_real: $o > real).
% 62.42/14.85 thf(func_def_791, type, suminf_complex: (nat > complex) > complex).
% 62.42/14.85 thf(func_def_792, type, suminf_real: (nat > real) > real).
% 62.42/14.85 thf(func_def_793, type, summable_complex: (nat > complex) > $o).
% 62.42/14.85 thf(func_def_794, type, summable_real: (nat > real) > $o).
% 62.42/14.85 thf(func_def_795, type, sums_real: (nat > real) > real > $o).
% 62.42/14.85 thf(func_def_796, type, collect_o: ($o > $o) > set_o).
% 62.42/14.85 thf(func_def_797, type, collect_Code_integer: (code_integer > $o) > set_Code_integer).
% 62.42/14.85 thf(func_def_798, type, collect_complex: (complex > $o) > set_complex).
% 62.42/14.85 thf(func_def_799, type, collec4429806609662206161d_enat: (extended_enat > $o) > set_Extended_enat).
% 62.42/14.85 thf(func_def_800, type, collect_int: (int > $o) > set_int).
% 62.42/14.85 thf(func_def_801, type, collect_list_nat: (list_nat > $o) > set_list_nat).
% 62.42/14.85 thf(func_def_802, type, collect_nat: (nat > $o) > set_nat).
% 62.42/14.85 thf(func_def_803, type, collec8663557070575231912omplex: (produc4411394909380815293omplex > $o) > set_Pr5085853215250843933omplex).
% 62.42/14.85 thf(func_def_804, type, collec213857154873943460nt_int: (product_prod_int_int > $o) > set_Pr958786334691620121nt_int).
% 62.42/14.85 thf(func_def_805, type, collec3392354462482085612at_nat: (product_prod_nat_nat > $o) > set_Pr1261947904930325089at_nat).
% 62.42/14.85 thf(func_def_806, type, collec3799799289383736868l_real: (produc2422161461964618553l_real > $o) > set_Pr6218003697084177305l_real).
% 62.42/14.85 thf(func_def_807, type, collect_real: (real > $o) > set_real).
% 62.42/14.85 thf(func_def_808, type, collect_VEBT_VEBT: (vEBT_VEBT > $o) > set_VEBT_VEBT).
% 62.42/14.85 thf(func_def_809, type, image_80655429650038917d_enat: (extended_enat > extended_enat) > set_Extended_enat > set_Extended_enat).
% 62.42/14.85 thf(func_def_810, type, image_int_int: (int > int) > set_int > set_int).
% 62.42/14.85 thf(func_def_811, type, image_nat_int: (nat > int) > set_nat > set_int).
% 62.42/14.85 thf(func_def_812, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 62.42/14.85 thf(func_def_813, type, image_nat_real: (nat > real) > set_nat > set_real).
% 62.42/14.85 thf(func_def_814, type, image_nat_set_nat: (nat > set_nat) > set_nat > set_set_nat).
% 62.42/14.85 thf(func_def_815, type, image_nat_char: (nat > char) > set_nat > set_char).
% 62.42/14.85 thf(func_def_816, type, image_5971271580939081552omplex: (real > filter6041513312241820739omplex) > set_real > set_fi4554929511873752355omplex).
% 62.42/14.85 thf(func_def_817, type, image_2178119161166701260l_real: (real > filter2146258269922977983l_real) > set_real > set_fi7789364187291644575l_real).
% 62.42/14.85 thf(func_def_818, type, image_real_real: (real > real) > set_real > set_real).
% 62.42/14.85 thf(func_def_819, type, image_char_nat: (char > nat) > set_char > set_nat).
% 62.42/14.85 thf(func_def_820, type, insert_int: int > set_int > set_int).
% 62.42/14.85 thf(func_def_821, type, insert_nat: nat > set_nat > set_nat).
% 62.42/14.85 thf(func_def_822, type, insert_real: real > set_real > set_real).
% 62.42/14.85 thf(func_def_823, type, vimage_nat_nat: (nat > nat) > set_nat > set_nat).
% 62.42/14.85 thf(func_def_824, type, set_fo1517530859248394432omplex: (nat > complex > complex) > nat > nat > complex > complex).
% 62.42/14.85 thf(func_def_825, type, set_fo2581907887559384638at_int: (nat > int > int) > nat > nat > int > int).
% 62.42/14.85 thf(func_def_826, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_827, type, set_fo1949268297981939178at_rat: (nat > rat > rat) > nat > nat > rat > rat).
% 62.42/14.85 thf(func_def_828, type, set_fo3111899725591712190t_real: (nat > real > real) > nat > nat > real > real).
% 62.42/14.85 thf(func_def_829, type, set_or5403411693681687835d_enat: extended_enat > extended_enat > set_Extended_enat).
% 62.42/14.85 thf(func_def_830, type, set_or1266510415728281911st_int: int > int > set_int).
% 62.42/14.85 thf(func_def_831, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 62.42/14.85 thf(func_def_832, type, set_or7049704709247886629st_num: num > num > set_num).
% 62.42/14.85 thf(func_def_833, type, set_or633870826150836451st_rat: rat > rat > set_rat).
% 62.42/14.85 thf(func_def_834, type, set_or1222579329274155063t_real: real > real > set_real).
% 62.42/14.85 thf(func_def_835, type, set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat).
% 62.42/14.85 thf(func_def_836, type, set_or4662586982721622107an_int: int > int > set_int).
% 62.42/14.85 thf(func_def_837, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 62.42/14.85 thf(func_def_838, type, set_ord_atLeast_nat: nat > set_nat).
% 62.42/14.85 thf(func_def_839, type, set_ord_atLeast_real: real > set_real).
% 62.42/14.85 thf(func_def_840, type, set_ord_atMost_nat: nat > set_nat).
% 62.42/14.85 thf(func_def_841, type, set_or6656581121297822940st_int: int > int > set_int).
% 62.42/14.85 thf(func_def_842, type, set_or6659071591806873216st_nat: nat > nat > set_nat).
% 62.42/14.85 thf(func_def_843, type, set_or5832277885323065728an_int: int > int > set_int).
% 62.42/14.85 thf(func_def_844, type, set_or5834768355832116004an_nat: nat > nat > set_nat).
% 62.42/14.85 thf(func_def_845, type, set_or1633881224788618240n_real: real > real > set_real).
% 62.42/14.85 thf(func_def_846, type, set_or1210151606488870762an_nat: nat > set_nat).
% 62.42/14.85 thf(func_def_847, type, set_or5849166863359141190n_real: real > set_real).
% 62.42/14.85 thf(func_def_848, type, set_ord_lessThan_nat: nat > set_nat).
% 62.42/14.85 thf(func_def_849, type, set_or5984915006950818249n_real: real > set_real).
% 62.42/14.85 thf(func_def_850, type, abort_real: literal > (product_unit > real) > real).
% 62.42/14.85 thf(func_def_851, type, literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal).
% 62.42/14.85 thf(func_def_852, type, ascii_of: char > char).
% 62.42/14.85 thf(func_def_853, type, char2: $o > $o > $o > $o > $o > $o > $o > $o > char).
% 62.42/14.85 thf(func_def_854, type, comm_s629917340098488124ar_nat: char > nat).
% 62.42/14.85 thf(func_def_855, type, integer_of_char: char > code_integer).
% 62.42/14.85 thf(func_def_856, type, unique3096191561947761185of_nat: nat > char).
% 62.42/14.85 thf(func_def_857, type, topolo4422821103128117721l_real: filter_real > (real > real) > $o).
% 62.42/14.85 thf(func_def_858, type, topolo5044208981011980120l_real: set_real > (real > real) > $o).
% 62.42/14.85 thf(func_def_859, type, topolo6980174941875973593q_real: (nat > real) > $o).
% 62.42/14.85 thf(func_def_860, type, topolo2177554685111907308n_real: real > set_real > filter_real).
% 62.42/14.85 thf(func_def_861, type, topolo7531315842566124627t_real: (nat > real) > $o).
% 62.42/14.85 thf(func_def_862, type, topolo2815343760600316023s_real: real > filter_real).
% 62.42/14.85 thf(func_def_863, type, topolo6517432010174082258omplex: (nat > complex) > $o).
% 62.42/14.85 thf(func_def_864, type, topolo4055970368930404560y_real: (nat > real) > $o).
% 62.42/14.85 thf(func_def_865, type, topolo896644834953643431omplex: filter6041513312241820739omplex).
% 62.42/14.85 thf(func_def_866, type, topolo1511823702728130853y_real: filter2146258269922977983l_real).
% 62.42/14.85 thf(func_def_867, type, arccos: real > real).
% 62.42/14.85 thf(func_def_868, type, arcosh_real: real > real).
% 62.42/14.85 thf(func_def_869, type, arcsin: real > real).
% 62.42/14.85 thf(func_def_870, type, arctan: real > real).
% 62.42/14.85 thf(func_def_871, type, arsinh_real: real > real).
% 62.42/14.85 thf(func_def_872, type, artanh_real: real > real).
% 62.42/14.85 thf(func_def_873, type, cos_complex: complex > complex).
% 62.42/14.85 thf(func_def_874, type, cos_real: real > real).
% 62.42/14.85 thf(func_def_875, type, cos_coeff: nat > real).
% 62.42/14.85 thf(func_def_876, type, cosh_complex: complex > complex).
% 62.42/14.85 thf(func_def_877, type, cosh_real: real > real).
% 62.42/14.85 thf(func_def_878, type, cot_complex: complex > complex).
% 62.42/14.85 thf(func_def_879, type, cot_real: real > real).
% 62.42/14.85 thf(func_def_880, type, exp_complex: complex > complex).
% 62.42/14.85 thf(func_def_881, type, exp_real: real > real).
% 62.42/14.85 thf(func_def_882, type, ln_ln_real: real > real).
% 62.42/14.85 thf(func_def_883, type, log: real > real > real).
% 62.42/14.85 thf(func_def_884, type, pi: real).
% 62.42/14.85 thf(func_def_885, type, powr_real: real > real > real).
% 62.42/14.85 thf(func_def_886, type, powr_real2: real > real > real).
% 62.42/14.85 thf(func_def_887, type, sin_complex: complex > complex).
% 62.42/14.85 thf(func_def_888, type, sin_real: real > real).
% 62.42/14.85 thf(func_def_889, type, sin_coeff: nat > real).
% 62.42/14.85 thf(func_def_890, type, sinh_complex: complex > complex).
% 62.42/14.85 thf(func_def_891, type, sinh_real: real > real).
% 62.42/14.85 thf(func_def_892, type, tan_complex: complex > complex).
% 62.42/14.85 thf(func_def_893, type, tan_real: real > real).
% 62.42/14.85 thf(func_def_894, type, tanh_complex: complex > complex).
% 62.42/14.85 thf(func_def_895, type, tanh_real: real > real).
% 62.42/14.85 thf(func_def_896, type, transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 62.42/14.85 thf(func_def_897, type, transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 62.42/14.85 thf(func_def_898, type, vEBT_Leaf: $o > $o > vEBT_VEBT).
% 62.42/14.85 thf(func_def_899, type, vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_900, type, vEBT_size_VEBT: vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_901, type, vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o).
% 62.42/14.85 thf(func_def_902, type, vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT).
% 62.42/14.85 thf(func_def_903, type, vEBT_V312737461966249ad_rel: produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o).
% 62.42/14.85 thf(func_def_904, type, vEBT_VEBT_high: nat > nat > nat).
% 62.42/14.85 thf(func_def_905, type, vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o).
% 62.42/14.85 thf(func_def_906, type, vEBT_VEBT_low: nat > nat > nat).
% 62.42/14.85 thf(func_def_907, type, vEBT_VEBT_membermima: vEBT_VEBT > nat > $o).
% 62.42/14.85 thf(func_def_908, type, vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.42/14.85 thf(func_def_909, type, vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o).
% 62.42/14.85 thf(func_def_910, type, vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.42/14.85 thf(func_def_911, type, vEBT_VEBT_valid: vEBT_VEBT > nat > $o).
% 62.42/14.85 thf(func_def_912, type, vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.42/14.85 thf(func_def_913, type, vEBT_invar_vebt: vEBT_VEBT > nat > $o).
% 62.42/14.85 thf(func_def_914, type, vEBT_set_vebt: vEBT_VEBT > set_nat).
% 62.42/14.85 thf(func_def_915, type, vEBT_vebt_buildup: nat > vEBT_VEBT).
% 62.42/14.85 thf(func_def_916, type, vEBT_v4011308405150292612up_rel: nat > nat > $o).
% 62.42/14.85 thf(func_def_917, type, vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT).
% 62.42/14.85 thf(func_def_918, type, vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.42/14.85 thf(func_def_919, type, vEBT_VEBT_bit_concat: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_920, type, vEBT_VEBT_minNull: vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_921, type, vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat).
% 62.42/14.85 thf(func_def_922, type, vEBT_vebt_member: vEBT_VEBT > nat > $o).
% 62.42/14.85 thf(func_def_923, type, vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.42/14.85 thf(func_def_924, type, vEBT_VEBT_add: option_nat > option_nat > option_nat).
% 62.42/14.85 thf(func_def_925, type, vEBT_VEBT_greater: option_nat > option_nat > $o).
% 62.42/14.85 thf(func_def_926, type, vEBT_VEBT_less: option_nat > option_nat > $o).
% 62.42/14.85 thf(func_def_927, type, vEBT_VEBT_lesseq: option_nat > option_nat > $o).
% 62.42/14.85 thf(func_def_928, type, vEBT_VEBT_max_in_set: set_nat > nat > $o).
% 62.42/14.85 thf(func_def_929, type, vEBT_VEBT_min_in_set: set_nat > nat > $o).
% 62.42/14.85 thf(func_def_930, type, vEBT_VEBT_mul: option_nat > option_nat > option_nat).
% 62.42/14.85 thf(func_def_931, type, vEBT_V4262088993061758097ft_nat: (nat > nat > nat) > option_nat > option_nat > option_nat).
% 62.42/14.85 thf(func_def_932, type, vEBT_VEBT_power: option_nat > option_nat > option_nat).
% 62.42/14.85 thf(func_def_933, type, vEBT_vebt_maxt: vEBT_VEBT > option_nat).
% 62.42/14.85 thf(func_def_934, type, vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_935, type, vEBT_vebt_mint: vEBT_VEBT > option_nat).
% 62.42/14.85 thf(func_def_936, type, vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_937, type, vEBT_is_succ_in_set: set_nat > nat > nat > $o).
% 62.42/14.85 thf(func_def_938, type, vEBT_vebt_succ: vEBT_VEBT > nat > option_nat).
% 62.42/14.85 thf(func_def_939, type, vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.42/14.85 thf(func_def_940, type, accp_list_nat: (list_nat > list_nat > $o) > list_nat > $o).
% 62.42/14.85 thf(func_def_941, type, accp_nat: (nat > nat > $o) > nat > $o).
% 62.42/14.85 thf(func_def_942, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 62.42/14.85 thf(func_def_943, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 62.42/14.85 thf(func_def_944, type, accp_P3113834385874906142um_num: (product_prod_num_num > product_prod_num_num > $o) > product_prod_num_num > $o).
% 62.42/14.85 thf(func_def_945, type, accp_P6183159247885693666d_enat: (produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o) > produc7272778201969148633d_enat > $o).
% 62.42/14.85 thf(func_def_946, type, accp_P2887432264394892906BT_nat: (produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o) > produc9072475918466114483BT_nat > $o).
% 62.42/14.85 thf(func_def_947, type, accp_VEBT_VEBT: (vEBT_VEBT > vEBT_VEBT > $o) > vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_948, type, less_than: set_Pr1261947904930325089at_nat).
% 62.42/14.85 thf(func_def_949, type, pred_nat: set_Pr1261947904930325089at_nat).
% 62.42/14.85 thf(func_def_950, type, fChoice_real: (real > $o) > real).
% 62.42/14.85 thf(func_def_951, type, member_o: $o > set_o > $o).
% 62.42/14.85 thf(func_def_952, type, member_Code_integer: code_integer > set_Code_integer > $o).
% 62.42/14.85 thf(func_def_953, type, member_complex: complex > set_complex > $o).
% 62.42/14.85 thf(func_def_954, type, member_Extended_enat: extended_enat > set_Extended_enat > $o).
% 62.42/14.85 thf(func_def_955, type, member_int: int > set_int > $o).
% 62.42/14.85 thf(func_def_956, type, member_list_nat: list_nat > set_list_nat > $o).
% 62.42/14.85 thf(func_def_957, type, member_nat: nat > set_nat > $o).
% 62.42/14.85 thf(func_def_958, type, member_num: num > set_num > $o).
% 62.42/14.85 thf(func_def_959, type, member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o).
% 62.42/14.85 thf(func_def_960, type, member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o).
% 62.42/14.85 thf(func_def_961, type, member_rat: rat > set_rat > $o).
% 62.42/14.85 thf(func_def_962, type, member_real: real > set_real > $o).
% 62.42/14.85 thf(func_def_963, type, member_set_nat: set_nat > set_set_nat > $o).
% 62.42/14.85 thf(func_def_964, type, member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_965, type, deg: nat).
% 62.42/14.85 thf(func_def_966, type, m: nat).
% 62.42/14.85 thf(func_def_967, type, ma: nat).
% 62.42/14.85 thf(func_def_968, type, mi: nat).
% 62.42/14.85 thf(func_def_969, type, miny: nat).
% 62.42/14.85 thf(func_def_970, type, na: nat).
% 62.42/14.85 thf(func_def_971, type, res: nat).
% 62.42/14.85 thf(func_def_972, type, sc: nat).
% 62.42/14.85 thf(func_def_973, type, summary: vEBT_VEBT).
% 62.42/14.85 thf(func_def_974, type, treeList: list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_975, type, xa: nat).
% 62.42/14.85 thf(func_def_976, type, za: nat).
% 62.42/14.85 thf(func_def_995, type, sP0: int > (int > $o) > $o).
% 62.42/14.85 thf(func_def_996, type, sP1: nat > nat > nat > $o).
% 62.42/14.85 thf(func_def_997, type, sP2: nat > nat > nat > $o).
% 62.42/14.85 thf(func_def_998, type, sP3: nat > nat > nat > $o).
% 62.42/14.85 thf(func_def_999, type, sP4: int > int > int > $o).
% 62.42/14.85 thf(func_def_1000, type, sP5: int > int > int > $o).
% 62.42/14.85 thf(func_def_1001, type, sP6: int > int > int > $o).
% 62.42/14.85 thf(func_def_1002, type, sP7: vEBT_VEBT > nat > $o).
% 62.42/14.85 thf(func_def_1003, type, sP8: rat > rat > rat > $o).
% 62.42/14.85 thf(func_def_1004, type, sP9: rat > rat > rat > $o).
% 62.42/14.85 thf(func_def_1005, type, sP10: rat > rat > rat > $o).
% 62.42/14.85 thf(func_def_1006, type, sP11: vEBT_VEBT > option_nat > $o).
% 62.42/14.85 thf(func_def_1007, type, sP12: (int > $o) > $o).
% 62.42/14.85 thf(func_def_1008, type, sP13: nat > $o > vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1009, type, sP14: $o > vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1010, type, sP15: vEBT_VEBT > nat > $o).
% 62.42/14.85 thf(func_def_1011, type, sP16: real > (real > real > $o) > real > $o).
% 62.42/14.85 thf(func_def_1012, type, sK17: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_1013, type, sK18: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1014, type, sK19: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1015, type, sK20: (real > real) > real).
% 62.42/14.85 thf(func_def_1016, type, sK21: (real > real) > real).
% 62.42/14.85 thf(func_def_1017, type, sK22: rat > rat).
% 62.42/14.85 thf(func_def_1018, type, sK23: real > int).
% 62.42/14.85 thf(func_def_1019, type, sK24: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1020, type, sK25: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1021, type, sK26: int > int).
% 62.42/14.85 thf(func_def_1022, type, sK27: real > (real > $o) > real).
% 62.42/14.85 thf(func_def_1023, type, sK28: real > (real > $o) > real).
% 62.42/14.85 thf(func_def_1024, type, sK29: real > (real > $o) > real).
% 62.42/14.85 thf(func_def_1025, type, sK30: real > (real > $o) > real).
% 62.42/14.85 thf(func_def_1026, type, sK31: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1027, type, sK32: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1028, type, sK33: real > nat > real).
% 62.42/14.85 thf(func_def_1029, type, sK34: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1030, type, sK35: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1031, type, sK36: nat > nat > nat).
% 62.42/14.85 thf(func_def_1032, type, sK37: real > nat).
% 62.42/14.85 thf(func_def_1033, type, sK38: int > nat).
% 62.42/14.85 thf(func_def_1034, type, sK39: int > nat).
% 62.42/14.85 thf(func_def_1035, type, sK40: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_1036, type, sK41: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1037, type, sK42: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1038, type, sK43: real > real > real > real).
% 62.42/14.85 thf(func_def_1039, type, sK44: (nat > nat) > nat).
% 62.42/14.85 thf(func_def_1040, type, sK45: real > real).
% 62.42/14.85 thf(func_def_1041, type, sK46: nat > nat > nat).
% 62.42/14.85 thf(func_def_1042, type, sK47: rat > rat).
% 62.42/14.85 thf(func_def_1043, type, sK48: real).
% 62.42/14.85 thf(func_def_1044, type, sK49: (real > rat) > real).
% 62.42/14.85 thf(func_def_1045, type, sK50: (real > rat) > real).
% 62.42/14.85 thf(func_def_1046, type, sK51: int > int).
% 62.42/14.85 thf(func_def_1047, type, sK52: nat > nat).
% 62.42/14.85 thf(func_def_1048, type, sK53: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1049, type, sK54: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1050, type, sK55: nat > nat).
% 62.42/14.85 thf(func_def_1051, type, sK56: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1052, type, sK57: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1053, type, sK58: (int > rat) > int).
% 62.42/14.85 thf(func_def_1054, type, sK59: (int > rat) > int).
% 62.42/14.85 thf(func_def_1055, type, sK60: rat > rat).
% 62.42/14.85 thf(func_def_1056, type, sK61: nat > nat > nat).
% 62.42/14.85 thf(func_def_1057, type, sK62: nat > nat > nat).
% 62.42/14.85 thf(func_def_1058, type, sK63: nat > nat > nat).
% 62.42/14.85 thf(func_def_1059, type, sK64: nat > nat > nat).
% 62.42/14.85 thf(func_def_1060, type, sK65: nat > nat > nat).
% 62.42/14.85 thf(func_def_1061, type, sK66: real > real > real).
% 62.42/14.85 thf(func_def_1062, type, sK67: real > real > real).
% 62.42/14.85 thf(func_def_1063, type, sK68: (real > $o) > nat).
% 62.42/14.85 thf(func_def_1064, type, sK69: (real > $o) > real).
% 62.42/14.85 thf(func_def_1065, type, sK70: (real > $o) > real).
% 62.42/14.85 thf(func_def_1066, type, sK71: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1067, type, sK72: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1068, type, sK73: int > (int > $o) > int).
% 62.42/14.85 thf(func_def_1069, type, sK74: int > (int > $o) > int).
% 62.42/14.85 thf(func_def_1070, type, sK75: int > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1071, type, sK76: (int > $o) > int).
% 62.42/14.85 thf(func_def_1072, type, sK77: real > real).
% 62.42/14.85 thf(func_def_1073, type, sK78: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1074, type, sK79: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1075, type, sK80: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1076, type, sK81: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1077, type, sK82: (real > real) > real).
% 62.42/14.85 thf(func_def_1078, type, sK83: (real > real) > real).
% 62.42/14.85 thf(func_def_1079, type, sK84: (int > $o) > int).
% 62.42/14.85 thf(func_def_1080, type, sK85: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1081, type, sK86: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1082, type, sK87: int > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1083, type, sK88: nat > nat > nat).
% 62.42/14.85 thf(func_def_1084, type, sK89: rat > rat > rat).
% 62.42/14.85 thf(func_def_1085, type, sK90: rat > rat > rat > rat).
% 62.42/14.85 thf(func_def_1086, type, sK91: nat > nat > nat).
% 62.42/14.85 thf(func_def_1087, type, sK92: int > int).
% 62.42/14.85 thf(func_def_1088, type, sK93: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1089, type, sK94: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1090, type, sK95: (int > real) > int).
% 62.42/14.85 thf(func_def_1091, type, sK96: (int > real) > int).
% 62.42/14.85 thf(func_def_1092, type, sK97: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1093, type, sK98: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1094, type, sK99: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1095, type, sK100: int > nat).
% 62.42/14.85 thf(func_def_1096, type, sK101: int > nat).
% 62.42/14.85 thf(func_def_1097, type, sK102: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1098, type, sK103: (real > rat) > real).
% 62.42/14.85 thf(func_def_1099, type, sK104: (real > rat) > real).
% 62.42/14.85 thf(func_def_1100, type, sK105: int > nat).
% 62.42/14.85 thf(func_def_1101, type, sK106: rat > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1102, type, sK107: rat > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1103, type, sK108: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1104, type, sK109: int > (int > $o) > int).
% 62.42/14.85 thf(func_def_1105, type, sK110: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1106, type, sK111: real > real).
% 62.42/14.85 thf(func_def_1107, type, sK112: real > real).
% 62.42/14.85 thf(func_def_1108, type, sK113: real > real > real > real).
% 62.42/14.85 thf(func_def_1109, type, sK114: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1110, type, sK115: nat > nat > nat).
% 62.42/14.85 thf(func_def_1111, type, sK116: real > real).
% 62.42/14.85 thf(func_def_1112, type, sK117: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1113, type, sK118: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1114, type, sK119: int > int > int).
% 62.42/14.85 thf(func_def_1115, type, sK120: (nat > real) > nat).
% 62.42/14.85 thf(func_def_1116, type, sK121: (real > real) > real).
% 62.42/14.85 thf(func_def_1117, type, sK122: (real > real) > real).
% 62.42/14.85 thf(func_def_1118, type, sK123: nat > nat).
% 62.42/14.85 thf(func_def_1119, type, sK124: (nat > nat) > nat).
% 62.42/14.85 thf(func_def_1120, type, sK125: nat > nat).
% 62.42/14.85 thf(func_def_1121, type, sK126: real > real > real).
% 62.42/14.85 thf(func_def_1122, type, sK127: (nat > nat) > nat).
% 62.42/14.85 thf(func_def_1123, type, sK128: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1124, type, sK129: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1125, type, sK130: int > int).
% 62.42/14.85 thf(func_def_1126, type, sK131: int > int).
% 62.42/14.85 thf(func_def_1127, type, sK132: (int > $o) > nat).
% 62.42/14.85 thf(func_def_1128, type, sK133: (int > $o) > nat).
% 62.42/14.85 thf(func_def_1129, type, sK134: (real > nat) > real).
% 62.42/14.85 thf(func_def_1130, type, sK135: (real > nat) > real).
% 62.42/14.85 thf(func_def_1131, type, sK136: nat > nat).
% 62.42/14.85 thf(func_def_1132, type, sK137: int > int).
% 62.42/14.85 thf(func_def_1133, type, sK138: real > int).
% 62.42/14.85 thf(func_def_1134, type, sK139: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1135, type, sK140: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1136, type, sK141: real > real).
% 62.42/14.85 thf(func_def_1137, type, sK142: int > int > int).
% 62.42/14.85 thf(func_def_1138, type, sK143: (real > $o) > nat).
% 62.42/14.85 thf(func_def_1139, type, sK144: (real > $o) > real).
% 62.42/14.85 thf(func_def_1140, type, sK145: (real > $o) > real).
% 62.42/14.85 thf(func_def_1141, type, sK146: rat > rat).
% 62.42/14.85 thf(func_def_1142, type, sK147: (int > real) > int).
% 62.42/14.85 thf(func_def_1143, type, sK148: (int > real) > int).
% 62.42/14.85 thf(func_def_1144, type, sK149: rat > rat).
% 62.42/14.85 thf(func_def_1145, type, sK150: nat > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1146, type, sK151: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1147, type, sK152: nat > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1148, type, sK153: int > int > int).
% 62.42/14.85 thf(func_def_1149, type, sK154: (nat > int) > nat).
% 62.42/14.85 thf(func_def_1150, type, sK155: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1151, type, sK156: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1152, type, sK157: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1153, type, sK158: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1154, type, sK159: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_1155, type, sK160: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_1156, type, sK161: nat > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1157, type, sK162: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1158, type, sK163: (int > int > $o) > int).
% 62.42/14.85 thf(func_def_1159, type, sK164: (int > int > $o) > int).
% 62.42/14.85 thf(func_def_1160, type, sK165: (int > int > $o) > int).
% 62.42/14.85 thf(func_def_1161, type, sK166: (int > int > $o) > int).
% 62.42/14.85 thf(func_def_1162, type, sK167: (int > int > $o) > int).
% 62.42/14.85 thf(func_def_1163, type, sK168: real > real > nat).
% 62.42/14.85 thf(func_def_1164, type, sK169: rat > rat).
% 62.42/14.85 thf(func_def_1165, type, sK170: real > real > real).
% 62.42/14.85 thf(func_def_1166, type, sK171: real > int).
% 62.42/14.85 thf(func_def_1167, type, sK172: nat > nat).
% 62.42/14.85 thf(func_def_1168, type, sK173: nat > nat).
% 62.42/14.85 thf(func_def_1169, type, sK174: int > (int > $o) > int).
% 62.42/14.85 thf(func_def_1170, type, sK175: real > real > nat).
% 62.42/14.85 thf(func_def_1171, type, sK176: nat > nat).
% 62.42/14.85 thf(func_def_1172, type, sK177: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1173, type, sK178: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1174, type, sK179: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1175, type, sK180: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1176, type, sK181: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1177, type, sK182: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1178, type, sK183: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1179, type, sK184: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1180, type, sK185: nat > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1181, type, sK186: nat > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1182, type, sK187: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1183, type, sK188: vEBT_VEBT > nat > nat).
% 62.42/14.85 thf(func_def_1184, type, sK189: vEBT_VEBT > nat > nat).
% 62.42/14.85 thf(func_def_1185, type, sK190: vEBT_VEBT > nat > nat).
% 62.42/14.85 thf(func_def_1186, type, sK191: vEBT_VEBT > nat > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1187, type, sK192: vEBT_VEBT > nat > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1188, type, sK193: nat > vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1189, type, sK194: nat > vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1190, type, sK195: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1191, type, sK196: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1192, type, sK197: nat > nat > nat).
% 62.42/14.85 thf(func_def_1193, type, sK198: int > nat).
% 62.42/14.85 thf(func_def_1194, type, sK199: int > nat).
% 62.42/14.85 thf(func_def_1195, type, sK200: rat > rat).
% 62.42/14.85 thf(func_def_1196, type, sK201: int > int).
% 62.42/14.85 thf(func_def_1197, type, sK202: int > int > int).
% 62.42/14.85 thf(func_def_1198, type, sK203: rat > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1199, type, sK204: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1200, type, sK205: rat > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1201, type, sK206: nat > nat > nat).
% 62.42/14.85 thf(func_def_1202, type, sK207: (nat > nat) > nat).
% 62.42/14.85 thf(func_def_1203, type, sK208: (nat > nat) > nat).
% 62.42/14.85 thf(func_def_1204, type, sK209: int > (int > $o) > int).
% 62.42/14.85 thf(func_def_1205, type, sK210: nat > nat).
% 62.42/14.85 thf(func_def_1206, type, sK211: real > real > nat).
% 62.42/14.85 thf(func_def_1207, type, sK212: rat > rat).
% 62.42/14.85 thf(func_def_1208, type, sK213: (real > real) > real).
% 62.42/14.85 thf(func_def_1209, type, sK214: (real > real) > real).
% 62.42/14.85 thf(func_def_1210, type, sK215: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1211, type, sK216: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1212, type, sK217: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1213, type, sK218: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1214, type, sK219: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1215, type, sK220: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1216, type, sK221: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1217, type, sK222: rat > rat).
% 62.42/14.85 thf(func_def_1218, type, sK223: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_1219, type, sK224: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_1220, type, sK225: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1221, type, sK226: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1222, type, sK227: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1223, type, sK228: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1224, type, sK229: rat > rat > rat).
% 62.42/14.85 thf(func_def_1225, type, sK230: (nat > real) > nat).
% 62.42/14.85 thf(func_def_1226, type, sK231: (nat > real) > nat).
% 62.42/14.85 thf(func_def_1227, type, sK232: (real > int) > real).
% 62.42/14.85 thf(func_def_1228, type, sK233: (real > int) > real).
% 62.42/14.85 thf(func_def_1229, type, sK234: int > nat).
% 62.42/14.85 thf(func_def_1230, type, sK235: list_VEBT_VEBT > list_VEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_1231, type, sK236: real > real).
% 62.42/14.85 thf(func_def_1232, type, sK237: real > int).
% 62.42/14.85 thf(func_def_1233, type, sK238: (real > int) > real).
% 62.42/14.85 thf(func_def_1234, type, sK239: (real > int) > real).
% 62.42/14.85 thf(func_def_1235, type, sK240: nat > nat > nat).
% 62.42/14.85 thf(func_def_1236, type, sK241: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1237, type, sK242: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1238, type, sK243: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1239, type, sK244: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1240, type, sK245: nat > nat > nat).
% 62.42/14.85 thf(func_def_1241, type, sK246: (nat > real) > nat).
% 62.42/14.85 thf(func_def_1242, type, sK247: (real > real) > real).
% 62.42/14.85 thf(func_def_1243, type, sK248: (real > real) > real).
% 62.42/14.85 thf(func_def_1244, type, sK249: int > nat).
% 62.42/14.85 thf(func_def_1245, type, sK250: real > real > real).
% 62.42/14.85 thf(func_def_1246, type, sK251: code_integer > code_integer).
% 62.42/14.85 thf(func_def_1247, type, sK252: (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1248, type, sK253: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1249, type, sK254: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_1250, type, sK255: nat > nat > nat).
% 62.42/14.85 thf(func_def_1251, type, sK256: nat > nat > nat).
% 62.42/14.85 thf(func_def_1252, type, sK257: nat > nat > nat).
% 62.42/14.85 thf(func_def_1253, type, sK258: real > nat).
% 62.42/14.85 thf(func_def_1254, type, sK259: real > real).
% 62.42/14.85 thf(func_def_1255, type, sK260: rat > rat > rat).
% 62.42/14.85 thf(func_def_1256, type, sK261: nat > nat).
% 62.42/14.85 thf(func_def_1257, type, sK262: real > nat).
% 62.42/14.85 thf(func_def_1258, type, sK263: real > real).
% 62.42/14.85 thf(func_def_1259, type, sK264: real > int).
% 62.42/14.85 thf(func_def_1260, type, sK265: real > real).
% 62.42/14.85 thf(func_def_1261, type, sK266: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1262, type, sK267: nat > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1263, type, sK268: nat > nat).
% 62.42/14.85 thf(func_def_1264, type, sK269: (int > rat) > int).
% 62.42/14.85 thf(func_def_1265, type, sK270: (int > rat) > int).
% 62.42/14.85 thf(func_def_1266, type, sK271: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1267, type, sK272: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1268, type, sK273: int > (int > $o) > int).
% 62.42/14.85 thf(func_def_1269, type, sK274: real > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1270, type, sK275: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1271, type, sK276: real > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1272, type, sK277: rat > rat > rat > rat).
% 62.42/14.85 thf(func_def_1273, type, sK278: rat > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1274, type, sK279: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1275, type, sK280: rat > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1276, type, sK281: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1277, type, sK282: real > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1278, type, sK283: real > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1279, type, sK284: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1280, type, sK285: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1281, type, sK286: rat > rat > rat).
% 62.42/14.85 thf(func_def_1282, type, sK287: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1283, type, sK288: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1284, type, sK289: int > int > int).
% 62.42/14.85 thf(func_def_1285, type, sK290: real > real).
% 62.42/14.85 thf(func_def_1286, type, sK291: nat > nat).
% 62.42/14.85 thf(func_def_1287, type, sK292: real > real).
% 62.42/14.85 thf(func_def_1288, type, sK293: nat).
% 62.42/14.85 thf(func_def_1289, type, sK294: int > int > int > int).
% 62.42/14.85 thf(func_def_1290, type, sK295: (real > rat) > real).
% 62.42/14.85 thf(func_def_1291, type, sK296: (real > rat) > real).
% 62.42/14.85 thf(func_def_1292, type, sK297: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1293, type, sK298: nat > nat).
% 62.42/14.85 thf(func_def_1294, type, sK299: (real > nat) > real).
% 62.42/14.85 thf(func_def_1295, type, sK300: (real > nat) > real).
% 62.42/14.85 thf(func_def_1296, type, sK301: real > real > real).
% 62.42/14.85 thf(func_def_1297, type, sK302: int > int > int).
% 62.42/14.85 thf(func_def_1298, type, sK303: nat > nat > nat).
% 62.42/14.85 thf(func_def_1299, type, sK304: nat > real > real).
% 62.42/14.85 thf(func_def_1300, type, sK305: real > real).
% 62.42/14.85 thf(func_def_1301, type, sK306: (int > rat) > int).
% 62.42/14.85 thf(func_def_1302, type, sK307: (int > rat) > int).
% 62.42/14.85 thf(func_def_1303, type, sK308: real > real > real).
% 62.42/14.85 thf(func_def_1304, type, sK309: (nat > $o) > nat > nat > nat).
% 62.42/14.85 thf(func_def_1305, type, sK310: rat > rat > rat).
% 62.42/14.85 thf(func_def_1306, type, sK311: int > int).
% 62.42/14.85 thf(func_def_1307, type, sK312: nat > vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_1308, type, sK313: real > real > real > real).
% 62.42/14.85 thf(func_def_1309, type, sK314: vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1310, type, sK315: vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1311, type, sK316: int > int > int).
% 62.42/14.85 thf(func_def_1312, type, sK317: nat > nat).
% 62.42/14.85 thf(func_def_1313, type, sK318: nat > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1314, type, sK319: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1315, type, sK320: nat > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1316, type, sK321: (real > real > $o) > real).
% 62.42/14.85 thf(func_def_1317, type, sK322: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1318, type, sK323: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1319, type, sK324: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1320, type, sK325: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1321, type, sK326: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1322, type, sK327: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1323, type, sK328: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1324, type, sK329: real > nat).
% 62.42/14.85 thf(func_def_1325, type, sK330: real > real).
% 62.42/14.85 thf(func_def_1326, type, sK331: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1327, type, sK332: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1328, type, sK333: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1329, type, sK334: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1330, type, sK335: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1331, type, sK336: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1332, type, sK337: int > int).
% 62.42/14.85 thf(func_def_1333, type, sK338: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1334, type, sK339: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1335, type, sK340: int > int).
% 62.42/14.85 thf(func_def_1336, type, sK341: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1337, type, sK342: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1338, type, sK343: rat > rat).
% 62.42/14.85 thf(func_def_1339, type, sK344: (nat > nat) > nat).
% 62.42/14.85 thf(func_def_1340, type, sK345: (nat > nat) > nat).
% 62.42/14.85 thf(func_def_1341, type, sK346: real > real).
% 62.42/14.85 thf(func_def_1342, type, sK347: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1343, type, sK348: nat > real > real).
% 62.42/14.85 thf(func_def_1344, type, sK349: vEBT_VEBT > option_nat > $o).
% 62.42/14.85 thf(func_def_1345, type, sK350: vEBT_VEBT > option_nat > $o).
% 62.42/14.85 thf(func_def_1346, type, sK351: vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_1347, type, sK352: vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1348, type, sK353: vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1349, type, sK354: option_nat > vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_1350, type, sK355: option_nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1351, type, sK356: option_nat > vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1352, type, sK357: option_nat > vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_1353, type, sK358: option_nat > vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_1354, type, sK359: rat > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1355, type, sK360: rat > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1356, type, sK361: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 62.42/14.85 thf(func_def_1357, type, sK362: nat > nat).
% 62.42/14.85 thf(func_def_1358, type, sK363: rat > rat).
% 62.42/14.85 thf(func_def_1359, type, sK364: (nat > int) > nat).
% 62.42/14.85 thf(func_def_1360, type, sK365: (real > $o) > real > real).
% 62.42/14.85 thf(func_def_1361, type, sK366: (real > $o) > real > real).
% 62.42/14.85 thf(func_def_1362, type, sK367: (real > $o) > real > real).
% 62.42/14.85 thf(func_def_1363, type, sK368: (real > $o) > real > real).
% 62.42/14.85 thf(func_def_1364, type, sK369: real > real > real > real).
% 62.42/14.85 thf(func_def_1365, type, sK370: nat > nat > nat).
% 62.42/14.85 thf(func_def_1366, type, sK371: nat > nat).
% 62.42/14.85 thf(func_def_1367, type, sK372: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1368, type, sK373: nat > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1369, type, sK374: nat > (nat > $o) > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1370, type, sK375: (real > rat) > real).
% 62.42/14.85 thf(func_def_1371, type, sK376: (real > rat) > real).
% 62.42/14.85 thf(func_def_1372, type, sK377: nat > nat > nat).
% 62.42/14.85 thf(func_def_1373, type, sK378: nat > nat > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1374, type, sK379: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1375, type, sK380: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1376, type, sK381: int > int).
% 62.42/14.85 thf(func_def_1377, type, sK382: real > real > nat).
% 62.42/14.85 thf(func_def_1378, type, sK383: int > (int > $o) > int).
% 62.42/14.85 thf(func_def_1379, type, sK384: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1380, type, sK385: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1381, type, sK386: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1382, type, sK387: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1383, type, sK388: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1384, type, sK389: real > nat).
% 62.42/14.85 thf(func_def_1385, type, sK390: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1386, type, sK391: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1387, type, sK392: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1388, type, sK393: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1389, type, sK394: real > real > real).
% 62.42/14.85 thf(func_def_1390, type, sK395: rat > rat).
% 62.42/14.85 thf(func_def_1391, type, sK396: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1392, type, sK397: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1393, type, sK398: nat > nat > nat).
% 62.42/14.85 thf(func_def_1394, type, sK399: code_integer > code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_1395, type, sK400: (nat > rat) > nat).
% 62.42/14.85 thf(func_def_1396, type, sK401: (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1397, type, sK402: nat > nat).
% 62.42/14.85 thf(func_def_1398, type, sK403: num > num).
% 62.42/14.85 thf(func_def_1399, type, sK404: num > num).
% 62.42/14.85 thf(func_def_1400, type, sK405: int > int).
% 62.42/14.85 thf(func_def_1401, type, sK406: (list_VEBT_VEBT > $o) > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1402, type, sK407: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_1403, type, sK408: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_1404, type, sK409: real > real > real).
% 62.42/14.85 thf(func_def_1405, type, sK410: int > int).
% 62.42/14.85 thf(func_def_1406, type, sK411: real > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1407, type, sK412: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1408, type, sK413: real > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1409, type, sK414: int > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1410, type, sK415: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1411, type, sK416: int > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1412, type, sK417: (nat > nat) > nat).
% 62.42/14.85 thf(func_def_1413, type, sK418: int > nat).
% 62.42/14.85 thf(func_def_1414, type, sK419: int > nat).
% 62.42/14.85 thf(func_def_1415, type, sK420: (real > real > $o) > real).
% 62.42/14.85 thf(func_def_1416, type, sK421: (real > real > $o) > real).
% 62.42/14.85 thf(func_def_1417, type, sK422: (real > real > $o) > real).
% 62.42/14.85 thf(func_def_1418, type, sK423: (real > real > $o) > real).
% 62.42/14.85 thf(func_def_1419, type, sK424: (real > real > $o) > real).
% 62.42/14.85 thf(func_def_1420, type, sK425: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1421, type, sK426: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1422, type, sK427: (int > real) > int).
% 62.42/14.85 thf(func_def_1423, type, sK428: (int > real) > int).
% 62.42/14.85 thf(func_def_1424, type, sK429: nat > nat > nat).
% 62.42/14.85 thf(func_def_1425, type, sK430: rat > rat).
% 62.42/14.85 thf(func_def_1426, type, sK431: real > real).
% 62.42/14.85 thf(func_def_1427, type, sK432: nat > nat > nat).
% 62.42/14.85 thf(func_def_1428, type, sK433: nat > nat > nat).
% 62.42/14.85 thf(func_def_1429, type, sK434: nat > nat > nat).
% 62.42/14.85 thf(func_def_1430, type, sK435: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1431, type, sK436: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1432, type, sK437: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1433, type, sK438: real > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1434, type, sK439: real > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1435, type, sK440: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 62.42/14.85 thf(func_def_1436, type, sK441: rat > rat).
% 62.42/14.85 thf(func_def_1437, type, sK442: int > int > int > int).
% 62.42/14.85 thf(func_def_1438, type, sK443: int > int > int > int).
% 62.42/14.85 thf(func_def_1439, type, sK444: int > int).
% 62.42/14.85 thf(func_def_1440, type, sK445: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1441, type, sK446: int > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1442, type, sK447: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1443, type, sK448: int > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1444, type, sK449: (real > real) > real).
% 62.42/14.85 thf(func_def_1445, type, sK450: (real > real) > real).
% 62.42/14.85 thf(func_def_1446, type, sK451: (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1447, type, sK452: real > real > real).
% 62.42/14.85 thf(func_def_1448, type, sK453: int > nat).
% 62.42/14.85 thf(func_def_1449, type, sK454: (rat > rat > $o) > rat).
% 62.42/14.85 thf(func_def_1450, type, sK455: (rat > rat > $o) > rat).
% 62.42/14.85 thf(func_def_1451, type, sK456: (rat > rat > $o) > rat).
% 62.42/14.85 thf(func_def_1452, type, sK457: (rat > rat > $o) > rat).
% 62.42/14.85 thf(func_def_1453, type, sK458: real > real).
% 62.42/14.85 thf(func_def_1454, type, sK459: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1455, type, sK460: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1456, type, sK461: (real > rat) > real).
% 62.42/14.85 thf(func_def_1457, type, sK462: (real > rat) > real).
% 62.42/14.85 thf(func_def_1458, type, sK463: real > real).
% 62.42/14.85 thf(func_def_1459, type, sK464: (int > int > $o) > int).
% 62.42/14.85 thf(func_def_1460, type, sK465: (int > int > $o) > int).
% 62.42/14.85 thf(func_def_1461, type, sK466: (int > int > $o) > int).
% 62.42/14.85 thf(func_def_1462, type, sK467: (int > int > $o) > int).
% 62.42/14.85 thf(func_def_1463, type, sK468: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1464, type, sK469: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1465, type, sK470: (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1466, type, sK471: (nat > real) > nat).
% 62.42/14.85 thf(func_def_1467, type, sK472: (nat > real) > nat).
% 62.42/14.85 thf(func_def_1468, type, sK473: (nat > real) > nat).
% 62.42/14.85 thf(func_def_1469, type, sK474: (nat > real) > nat).
% 62.42/14.85 thf(func_def_1470, type, sK475: real > real).
% 62.42/14.85 thf(func_def_1471, type, sK476: rat > rat > rat).
% 62.42/14.85 thf(func_def_1472, type, sK477: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1473, type, sK478: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1474, type, sK479: real > real > nat).
% 62.42/14.85 thf(func_def_1475, type, sK480: (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1476, type, sK481: (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1477, type, sK482: code_integer > code_integer).
% 62.42/14.85 thf(func_def_1478, type, sK483: rat > rat).
% 62.42/14.85 thf(func_def_1479, type, sK484: rat > rat).
% 62.42/14.85 thf(func_def_1480, type, sK485: code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_1481, type, sK486: real > real > real).
% 62.42/14.85 thf(func_def_1482, type, sK487: (real > nat) > real).
% 62.42/14.85 thf(func_def_1483, type, sK488: (real > nat) > real).
% 62.42/14.85 thf(func_def_1484, type, sK489: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1485, type, sK490: (rat > nat) > rat).
% 62.42/14.85 thf(func_def_1486, type, sK491: (nat > int) > nat).
% 62.42/14.85 thf(func_def_1487, type, sK492: nat > nat).
% 62.42/14.85 thf(func_def_1488, type, sK493: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1489, type, sK494: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1490, type, sK495: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1491, type, sK496: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1492, type, sK497: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1493, type, sK498: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1494, type, sK499: int > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1495, type, sK500: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1496, type, sK501: int > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1497, type, sK502: (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1498, type, sK503: (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1499, type, sK504: nat > vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1500, type, sK505: nat > vEBT_VEBT > option4927543243414619207at_nat).
% 62.42/14.85 thf(func_def_1501, type, sK506: nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1502, type, sK507: nat > nat).
% 62.42/14.85 thf(func_def_1503, type, sK508: (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1504, type, sK509: real > real > real).
% 62.42/14.85 thf(func_def_1505, type, sK510: (nat > int) > nat).
% 62.42/14.85 thf(func_def_1506, type, sK511: code_integer > code_integer > code_integer).
% 62.42/14.85 thf(func_def_1507, type, sK512: int > int).
% 62.42/14.85 thf(func_def_1508, type, sK513: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1509, type, sK514: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1510, type, sK515: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1511, type, sK516: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1512, type, sK517: real > nat).
% 62.42/14.85 thf(func_def_1513, type, sK518: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1514, type, sK519: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1515, type, sK520: vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1516, type, sK521: vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1517, type, sK522: nat > nat).
% 62.42/14.85 thf(func_def_1518, type, sK523: rat > rat).
% 62.42/14.85 thf(func_def_1519, type, sK524: real > nat).
% 62.42/14.85 thf(func_def_1520, type, sK525: nat > nat).
% 62.42/14.85 thf(func_def_1521, type, sK526: nat > nat > nat).
% 62.42/14.85 thf(func_def_1522, type, sK527: rat > rat).
% 62.42/14.85 thf(func_def_1523, type, sK528: int > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1524, type, sK529: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1525, type, sK530: int > (int > $o) > (int > $o) > int).
% 62.42/14.85 thf(func_def_1526, type, sK531: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1527, type, sK532: (rat > int) > rat).
% 62.42/14.85 thf(func_def_1528, type, sK533: real > real).
% 62.42/14.85 thf(func_def_1529, type, sK534: (real > rat) > real).
% 62.42/14.85 thf(func_def_1530, type, sK535: (real > rat) > real).
% 62.42/14.85 thf(func_def_1531, type, sK536: (int > rat) > int).
% 62.42/14.85 thf(func_def_1532, type, sK537: (int > rat) > int).
% 62.42/14.85 thf(func_def_1533, type, sK538: (int > $o) > int).
% 62.42/14.85 thf(func_def_1534, type, sK539: (int > $o) > int).
% 62.42/14.85 thf(func_def_1535, type, sK540: int > int > int).
% 62.42/14.85 thf(func_def_1536, type, sK541: nat > nat).
% 62.42/14.85 thf(func_def_1537, type, sK542: nat > nat).
% 62.42/14.85 thf(func_def_1538, type, sK543: int > int).
% 62.42/14.85 thf(func_def_1539, type, sK544: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1540, type, sK545: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1541, type, sK546: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1542, type, sK547: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1543, type, sK548: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1544, type, sK549: (rat > rat > $o) > rat).
% 62.42/14.85 thf(func_def_1545, type, sK550: (rat > rat > $o) > rat).
% 62.42/14.85 thf(func_def_1546, type, sK551: (rat > rat > $o) > rat).
% 62.42/14.85 thf(func_def_1547, type, sK552: (rat > rat > $o) > rat).
% 62.42/14.85 thf(func_def_1548, type, sK553: (rat > rat > $o) > rat).
% 62.42/14.85 thf(func_def_1549, type, sK554: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1550, type, sK555: (rat > real) > rat).
% 62.42/14.85 thf(func_def_1551, type, sK556: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1552, type, sK557: vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_1553, type, sK558: vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1554, type, sK559: vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1555, type, sK560: nat > $o > vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_1556, type, sK561: nat > $o > vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_1557, type, sK562: nat > $o > vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_1558, type, sK563: nat > $o > vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1559, type, sK564: nat > $o > vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1560, type, sK565: nat > vEBT_VEBT > $o > $o).
% 62.42/14.85 thf(func_def_1561, type, sK566: nat > vEBT_VEBT > $o > $o).
% 62.42/14.85 thf(func_def_1562, type, sK567: vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1563, type, sK568: vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1564, type, sK569: vEBT_VEBT > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_1565, type, sK570: vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1566, type, sK571: vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1567, type, sK572: vEBT_VEBT > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_1568, type, sK573: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1569, type, sK574: (rat > rat) > rat).
% 62.42/14.85 thf(func_def_1570, type, sK575: nat > nat > nat).
% 62.42/14.85 thf(func_def_1571, type, sK576: int > int > int > int).
% 62.42/14.85 thf(func_def_1572, type, sK577: int > int > int > int).
% 62.42/14.85 thf(func_def_1573, type, sK578: vEBT_VEBT > nat > nat).
% 62.42/14.85 thf(func_def_1574, type, sK579: vEBT_VEBT > nat > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1575, type, sK580: vEBT_VEBT > nat > nat).
% 62.42/14.85 thf(func_def_1576, type, sK581: vEBT_VEBT > nat > nat).
% 62.42/14.85 thf(func_def_1577, type, sK582: vEBT_VEBT > nat > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1578, type, sK583: vEBT_VEBT > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_1579, type, sK584: vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1580, type, sK585: vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1581, type, sK586: nat > vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1582, type, sK587: nat > vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1583, type, sK588: vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1584, type, sK589: vEBT_VEBT > product_prod_nat_nat).
% 62.42/14.85 thf(func_def_1585, type, sK590: vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1586, type, sK591: vEBT_VEBT > list_VEBT_VEBT).
% 62.42/14.85 thf(func_def_1587, type, sK592: vEBT_VEBT > vEBT_VEBT).
% 62.42/14.85 thf(func_def_1588, type, sK593: vEBT_VEBT > nat).
% 62.42/14.85 thf(func_def_1589, type, sK594: (real > int) > real).
% 62.42/14.85 thf(func_def_1590, type, sK595: (real > int) > real).
% 62.42/14.85 thf(func_def_1591, type, sK596: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1592, type, sK597: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1593, type, sK598: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1594, type, sK599: (nat > nat > $o) > nat).
% 62.42/14.85 thf(func_def_1595, type, sK600: real > (real > real > $o) > real > real).
% 62.42/14.85 thf(func_def_1596, type, sK601: real > real > (real > real > $o) > real > real).
% 62.42/14.85 thf(func_def_1597, type, sK602: real > real > (real > real > $o) > real > real).
% 62.42/14.85 thf(func_def_1598, type, sK603: (real > real > $o) > real).
% 62.42/14.85 thf(func_def_1599, type, sK604: (real > real > $o) > real).
% 62.42/14.85 thf(func_def_1600, type, sK605: (real > real > $o) > real).
% 62.42/14.85 thf(func_def_1601, type, sK606: nat > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1602, type, sK607: nat > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1603, type, sK608: nat > (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1604, type, sK609: real > nat).
% 62.42/14.85 thf(func_def_1605, type, sK610: real > nat).
% 62.42/14.85 thf(func_def_1606, type, sK611: real > nat).
% 62.42/14.85 thf(func_def_1607, type, sK612: nat > nat).
% 62.42/14.85 thf(func_def_1608, type, sK613: int > int > int).
% 62.42/14.85 thf(func_def_1609, type, sK614: real > nat).
% 62.42/14.85 thf(func_def_1610, type, sK615: real > nat).
% 62.42/14.85 thf(func_def_1611, type, sK616: real > int).
% 62.42/14.85 thf(func_def_1612, type, sK617: code_integer > (code_integer > $o) > code_integer).
% 62.42/14.85 thf(func_def_1613, type, sK618: code_integer > (code_integer > $o) > code_integer).
% 62.42/14.85 thf(func_def_1614, type, sK619: nat > (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1615, type, sK620: nat > (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1616, type, sK621: nat > nat > nat).
% 62.42/14.85 thf(func_def_1617, type, sK622: real > int).
% 62.42/14.85 thf(func_def_1618, type, sK623: real > int).
% 62.42/14.85 thf(func_def_1619, type, sK624: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1620, type, sK625: (int > $o) > int > int).
% 62.42/14.85 thf(func_def_1621, type, sK626: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_1622, type, sK627: nat > nat > nat > nat).
% 62.42/14.85 thf(func_def_1623, type, sK628: (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1624, type, sK629: (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1625, type, sK630: (nat > $o) > nat > nat).
% 62.42/14.85 thf(func_def_1626, type, sK631: real > nat).
% 62.42/14.85 thf(func_def_1627, type, sK632: real > nat).
% 62.42/14.85 thf(func_def_1628, type, sK633: real > nat).
% 62.42/14.85 thf(func_def_1629, type, sK634: nat > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1630, type, sK635: nat > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1631, type, sK636: nat > (nat > $o) > nat).
% 62.42/14.85 thf(func_def_1632, type, sK637: nat > nat > nat).
% 62.42/14.85 thf(func_def_1633, type, sK638: vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1634, type, sK639: vEBT_VEBT > $o).
% 62.42/14.85 thf(func_def_1635, type, sK640: real > int).
% 62.42/14.85 thf(func_def_1636, type, ph641: !>[X0: $tType]:(X0)).
% 62.42/14.85 thf(f46774,plain,(
% 62.42/14.85 $false),
% 62.42/14.85 inference(subsumption_resolution,[status(thm)],[f46773,f43977])).
% 62.42/14.85 thf(f43977,plain,(
% 62.42/14.85 ((vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (suc @ one_one_nat))) != (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (suc @ one_one_nat))))),
% 62.42/14.85 inference(backward_demodulation,[status(thm)],[f30550,f28429])).
% 62.42/14.85 thf(f28429,plain,(
% 62.42/14.85 ((numeral_numeral_nat @ (bit0 @ one)) = (suc @ one_one_nat))),
% 62.42/14.85 inference(cnf_transformation,[status(thm)],[f336])).
% 62.42/14.85 thf(f336,axiom,(
% 62.42/14.85 ((numeral_numeral_nat @ (bit0 @ one)) = (suc @ one_one_nat))),
% 62.42/14.85 file('/export/starexec/sandbox2/tmp/tmp.5KBf3uDrBi/DTF2THF_21372.p',fact_335_Suc__1)).
% 62.42/14.85 thf(f30550,plain,(
% 62.42/14.85 ((vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) != (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 62.42/14.85 inference(cnf_transformation,[status(thm)],[f22876])).
% 62.42/14.85 thf(f22876,plain,(
% 62.42/14.85 ((vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) != (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 62.42/14.85 inference(flattening,[status(thm)],[f10258])).
% 62.42/14.85 thf(f10258,negated_conjecture,(
% 62.42/14.85 ~((vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) = (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 62.42/14.85 inference(negated_conjecture,[status(cth)],[f10257])).
% 62.42/14.85 thf(f10257,conjecture,(
% 62.42/14.85 ((vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) = (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 62.42/14.85 file('/export/starexec/sandbox2/tmp/tmp.5KBf3uDrBi/DTF2THF_21372.p',conj_0)).
% 62.42/14.85 thf(f46773,plain,(
% 62.42/14.85 ((vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (suc @ one_one_nat))) = (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (suc @ one_one_nat))))),
% 62.42/14.85 inference(subsumption_resolution,[status(thm)],[f46765,f43979])).
% 62.42/14.85 thf(f43979,plain,(
% 62.42/14.85 ((ord_less_eq_nat @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (suc @ one_one_nat))) @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (suc @ one_one_nat)))) = $true)),
% 62.42/14.85 inference(backward_demodulation,[status(thm)],[f32345,f28429])).
% 62.42/14.85 thf(f32345,plain,(
% 62.42/14.85 ((ord_less_eq_nat @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) = $true)),
% 62.42/14.85 inference(cnf_transformation,[status(thm)],[f15945])).
% 62.42/14.85 thf(f15945,plain,(
% 62.42/14.85 ((ord_less_eq_nat @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) = $true)),
% 62.42/14.85 inference(fool_elimination,[status(thm)],[f15944])).
% 62.42/14.85 thf(f15944,plain,(
% 62.42/14.85 (ord_less_eq_nat @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 62.42/14.85 inference(rectify,[status(thm)],[f3])).
% 62.42/14.85 thf(f3,axiom,(
% 62.42/14.85 (ord_less_eq_nat @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 62.42/14.85 file('/export/starexec/sandbox2/tmp/tmp.5KBf3uDrBi/DTF2THF_21372.p',fact_2_aaaa)).
% 62.42/14.85 thf(f46765,plain,(
% 62.42/14.85 ((ord_less_eq_nat @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (suc @ one_one_nat))) @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (suc @ one_one_nat)))) != $true) | ((vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (suc @ one_one_nat))) = (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (suc @ one_one_nat))))),
% 62.42/14.85 inference(trivial_inequality_removal,[status(thm)],[f46750])).
% 62.42/14.85 thf(f46750,plain,(
% 62.42/14.85 ((ord_less_eq_nat @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (suc @ one_one_nat))) @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (suc @ one_one_nat)))) != $true) | ($true != $true) | ((vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (suc @ one_one_nat))) = (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (suc @ one_one_nat))))),
% 62.42/14.85 inference(superposition,[status(thm)],[f29645,f43976])).
% 62.42/14.85 thf(f43976,plain,(
% 62.42/14.85 ((ord_less_eq_nat @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (suc @ one_one_nat))) @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (suc @ one_one_nat)))) = $true)),
% 62.42/14.85 inference(backward_demodulation,[status(thm)],[f30397,f28429])).
% 62.42/14.85 thf(f30397,plain,(
% 62.42/14.85 ((ord_less_eq_nat @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) = $true)),
% 62.42/14.85 inference(cnf_transformation,[status(thm)],[f14471])).
% 62.42/14.85 thf(f14471,plain,(
% 62.42/14.85 ((ord_less_eq_nat @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) = $true)),
% 62.42/14.85 inference(fool_elimination,[status(thm)],[f14470])).
% 62.42/14.85 thf(f14470,plain,(
% 62.42/14.85 (ord_less_eq_nat @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 62.42/14.85 inference(rectify,[status(thm)],[f1])).
% 62.42/14.85 thf(f1,axiom,(
% 62.42/14.85 (ord_less_eq_nat @ (vEBT_VEBT_high @ za @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 62.42/14.85 file('/export/starexec/sandbox2/tmp/tmp.5KBf3uDrBi/DTF2THF_21372.p',fact_0_bbbb)).
% 62.42/14.85 thf(f29645,plain,(
% 62.42/14.85 ( ! [X0 : nat,X1 : nat] : (((ord_less_eq_nat @ X1 @ X0) != $true) | ((ord_less_eq_nat @ X0 @ X1) != $true) | (X0 = X1)) )),
% 62.42/14.85 inference(cnf_transformation,[status(thm)],[f22966])).
% 62.42/14.85 thf(f22966,plain,(
% 62.42/14.85 ! [X0 : nat,X1 : nat] : (((ord_less_eq_nat @ X0 @ X1) != $true) | (X0 = X1) | ((ord_less_eq_nat @ X1 @ X0) != $true))),
% 62.42/14.85 inference(flattening,[status(thm)],[f17277])).
% 62.42/14.85 thf(f17277,plain,(
% 62.42/14.85 ! [X0 : nat,X1 : nat] : ((X0 = X1) | ~((ord_less_eq_nat @ X0 @ X1) = $true) | ~((ord_less_eq_nat @ X1 @ X0) = $true))),
% 62.42/14.85 inference(fool_elimination,[status(thm)],[f17276])).
% 62.42/14.85 thf(f17276,plain,(
% 62.42/14.85 ! [X0 : nat,X1 : nat] : ((X0 = X1) | ~(ord_less_eq_nat @ X0 @ X1) | ~(ord_less_eq_nat @ X1 @ X0))),
% 62.42/14.85 inference(rectify,[status(thm)],[f50])).
% 62.42/14.85 thf(f50,axiom,(
% 62.42/14.85 ! [X6 : nat,X11 : nat] : ((X6 = X11) | ~(ord_less_eq_nat @ X6 @ X11) | ~(ord_less_eq_nat @ X11 @ X6))),
% 62.42/14.85 file('/export/starexec/sandbox2/tmp/tmp.5KBf3uDrBi/DTF2THF_21372.p',fact_49_verit__la__disequality)).
% 62.42/14.85 % SZS output end Proof for DTF2THF_21372
% 62.42/14.85 % (21950)------------------------------
% 62.42/14.85 % (21950)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.42/14.85 % (21950)Termination reason: Refutation
% 62.42/14.85
% 62.42/14.85 % (21950)Memory used [KB]: 31726
% 62.42/14.85 % (21950)Time elapsed: 1.263 s
% 62.42/14.85 % (21950)Instructions burned: 4085 (million)
% 62.42/14.85 % (21950)------------------------------
% 62.42/14.85 % (21950)------------------------------
% 62.42/14.85 % (21590)Success in time 7.828 s
%------------------------------------------------------------------------------